Are circadian amplitudes and periods correlated? A new twist in the story

Marta del Olmo; Christoph Schmal; Camillo Mizaikoff; Saskia Grabe; Christian Gabriel; Achim Kramer; Hanspeter Herzel

doi:10.12688/f1000research.135533.2

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Research Article

Revised

Are circadian amplitudes and periods correlated? A new twist in the story

[version 2; peer review: 3 approved]

Marta del Olmo ¹, Christoph Schmal¹, Camillo Mizaikoff¹, [...] Saskia Grabe¹, Christian Gabriel², Achim Kramer², Hanspeter Herzel¹

Marta del Olmo ¹, Christoph Schmal¹, [...] Camillo Mizaikoff¹, Saskia Grabe¹, Christian Gabriel², Achim Kramer², Hanspeter Herzel¹

PUBLISHED 23 Apr 2024

Author details Author details

¹ Institute for Theoretical Biology, Humboldt Universität zu Berlin and Charité Universitätsmedizin Berlin, Philippstr. 13, 10115 Berlin, Germany
² Laboratory of Chronobiology, Institute for Medical Immunology, Charite Universitätsmedizin Berlin, Charitéplatz 1, 10117 Berlin, Germany

Marta del Olmo
Roles: Conceptualization, Data Curation, Investigation, Methodology, Resources, Software, Visualization, Writing – Original Draft Preparation, Writing – Review & Editing

Christoph Schmal
Roles: Data Curation, Investigation, Methodology, Resources, Software, Visualization, Writing – Review & Editing

Camillo Mizaikoff
Roles: Data Curation, Resources, Software, Visualization

Saskia Grabe
Roles: Writing – Review & Editing

Christian Gabriel
Roles: Writing – Review & Editing

Achim Kramer
Roles: Funding Acquisition, Supervision, Writing – Review & Editing

Hanspeter Herzel
Roles: Conceptualization, Funding Acquisition, Investigation, Methodology, Supervision, Writing – Original Draft Preparation, Writing – Review & Editing

OPEN PEER REVIEW

REVIEWER STATUS

This article is included in the Cell & Molecular Biology gateway.

This article is included in the Circadian Clocks in Health and Disease collection.

Abstract

Three parameters are important to characterize a circadian and in general any biological clock: period, phase and amplitude. While circadian periods have been shown to correlate with entrainment phases, and clock amplitude influences the phase response of an oscillator to pulse-like zeitgeber signals, the co-modulations of amplitude and periods, which we term twist, have not been studied in detail. In this paper we define two concepts: parametric twist refers to amplitude-period correlations arising in ensembles of self-sustained, limit cycle clocks in the absence of external inputs, and phase space twist refers to the co-modulation of an individual clock’s amplitude and period in response to external zeitgebers. Our findings show that twist influences the interaction of oscillators with the environment, facilitating entrainment, speeding upfastening recovery to pulse-like perturbations or modifying the response of an individual clock to coupling. This theoretical framework might be applied to understand the emerging properties of other oscillating systems.

Keywords

circadian clocks, mathematical modeling, amplitudes, periods, twist, heterogeneity, entrainment, coupling

Corresponding author: Marta del Olmo

Competing interests: No competing interests were disclosed.

Grant information: This study was supported by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) Project-ID 278001972 – TRR 186 to H.H., A.K., C.G. and M.dO.; SCHM 3362/2-1 as well as SCHM 3362/4-1, project-IDs 414704559 and 511886499 to C.S.; RTG2424 CompCancer to S.G.
The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Copyright: © 2024 del Olmo M et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

How to cite: del Olmo M, Schmal C, Mizaikoff C et al. Are circadian amplitudes and periods correlated? A new twist in the story [version 2; peer review: 3 approved]. F1000Research 2024, 12:1077 (https://doi.org/10.12688/f1000research.135533.2) First published: 31 Aug 2023, 12:1077 (https://doi.org/10.12688/f1000research.135533.1) Latest published: 23 Apr 2024, 12:1077 (https://doi.org/10.12688/f1000research.135533.2)

Revised Amendments from Version 1

We thank the three reviewers for their constructive and insightful comments, which have helped improve our manuscript greatly. In response to their reports, we have implemented several changes:
1. We have clarified the differences between parametric and phase space twist effects, terms that we have introduced for amplitude-period correlations in limit cycles. These correlations must be studied differently from those in non-limit cycle oscillations. We have emphasized that these terms relate to the observable and measurable correlations.
2. Figure 4 has been updated to display the maxima and minima of the x variable in the Gonze model, but after the time series was normalized to its mean. This allows the interpretation of the y axes in panels A and C as relative amplitude, opposite to the previous version’s presentation of absolute values.
3. Figure 8 in the revised manuscript now includes additional panels (D–F), illustrating how the amplitude relaxation rate λ from the Poincare model also governs phase space twist effects, i.e., the correlation between peak-to-peak period and peak-to-trough amplitude.
4. We have added explanatory text in the revised manuscript to justify the selection of each model.
5. Typos and wording suggestions from the reviewers for better clarity have been addressed.
6. We have expanded the discussion with relevant literature, particularly regarding how
parametric twist may impact coupling or the role of specific coupling among subregions in the central circadian pacemaker in mammals, the SCN, which can generate wave-like patterns or encode seasonality.
7. We have included additional references on prior studies analyzing parametric twist across different kinetic oscillator models. We have elaborated on how, although our focus is on models with inhibition based on Hill functions, other models with different repression formulas may exhibit different twist effects, as different terms affect circadian dynamics differently.

See the authors' detailed response to the review by Jae Kim
See the authors' detailed response to the review by Stephanie Taylor
See the authors' detailed response to the review by Jihwan Myung

Introduction

Oscillations are happening all around us, from the vibrating atoms constituting matter to the beating of the animal heart or to circadian clocks present in all kingdoms of life. Circadian clocks are autonomous clocks that tick in the absence of external timing cues with a period of about 24 h and regulate our behavior, physiology and metabolism. A fundamental property of circadian clocks is that their phase and periodicity can be adjusted to external timing signals (zeitgebers) in a process known as entrainment. It is believed that natural selection has acted on the phase relationship between biological rhythms and the environmental cycle, and thus, this phase of entrainment is of central importance for the fitness of the organism, allowing it to anticipate to changes in the external world.¹

Three key properties of a circadian rhythm are its period, amplitude and phase. Chronobiological studies have usually focused on period because there are established tools that allow their direct measurement, including running wheels for mice or race tubes for fungi. Phase, which refers to the position of a point on the oscillation cycle relative to a reference point, can be measured similarly. Circadian rhythms can be entrained to various zeitgeber periods $T$ as reviewed in,² but under the natural conditions of $T = 24$ h, variations of intrinsic periods lead to different phases of entrainment that are the basis of chronotypes: faster running clocks (shorter endogenous periods) lead to early phases (‘morning larks’) and slower clocks (longer periods) correspond to later phases (‘night owls’).³ ^– ⁷

Measuring amplitudes, however, is less straightforward. Some studies have considered activity recordings⁸ ^, ⁹ or conidiation in race tubes,¹⁰ but one might argue that these measures do not represent the clockwork’s amplitude, as they reflect outputs of the circadian system. Other studies have quantified gene expression profiles after careful normalization,¹¹ but from the $\sim$ 20 core clock genes that constitute the mammalian circadian oscillator,¹² it is not immediately evident which gene or protein is best representing the core clock amplitude. Actually, reporter signals monitoring expression of different clock genes and proteins have been used to quantify amplitudes.⁴ ^, ¹³ ^, ¹⁴ Others have approached the amplitude challenge indirectly by measuring the response of an oscillator to zeitgeber pulses.¹⁵ ^– ¹⁸ While small-amplitude clocks exhibit larger pulse-induced phase shifts and are easier to phase-reset, larger amplitude rhythms display smaller phase shifts,¹⁷ ^– ²² with consequences in the size of the phase response curve⁵ or in jet lag duration.²³ Amplitudes, together with periods, also govern entrainment²³ ^, ²⁴ and seasonality.²² ^, ²³ ^, ²⁵ There have been various theoretical and experimental studies showing, for example, how clocks with larger amplitudes display narrower ranges of entrainment than rhythms of lower amplitude,²⁰ and how the phase of entrainment is modulated by oscillator amplitude.²² ^, ²⁴

Taken together, these observations indicate, firstly, that the phase of entrainment is correlated with the intrinsic period, and secondly, that both phase of entrainment and phase changes in response to perturbations also correlate with oscillator amplitude. This leads to the question of whether amplitudes and periods are also co-modulated and what insights these interdependencies provide about the underlying oscillator. These questions are the focus of this paper. Do faster-running clocks have larger or smaller amplitudes than slower clocks? What are the implications? Experimental observations have provided evidence for both: in a human osteosarcoma cell line in culture, clocks with longer periods display larger amplitudes;²⁶ but in cells from the choroid plexus, the major producer of cerebrospinal fluid of the central nervous system, clocks with shorter periods are associated with larger amplitudes¹⁴ (scheme in Figure 1). This dependence between periods and amplitudes is what we here refer to as twist, also known as shear in the literature.²⁷ ^, ²⁸ By convention, negative twist describes oscillators in which amplitude increases are accompanied by a decreasing period (also termed hard oscillators) and vice versa for positive twist (soft oscillators).

Figure 1. Schematic of experimental observations of circadian amplitude-period correlations.

(A) Positive correlations (i.e., positive twist) have been observed in U-2 OS cells kept in culture²⁶; (B) negative correlations have been observed in cells from the choroid plexus in the mouse brain.¹⁴

In this paper we provide definitions for two important concepts in the context of limit cycle oscillations: parametric twist and phase space twist. Parametric twist refers to the amplitude-period correlations observed when analyzing a population of heterogeneous clocks; phase space twist describes the amplitude-period correlations within the time series of an individual oscillator, as influenced by its interaction with the environment.

We show, firstly, that nonlinearities can introduce amplitude-period correlations in simple oscillator models. Moreover, clock models of different complexity can reproduce the experimentally observed positive²⁶ and negative¹⁴ twist effects, with the type of correlation depending on the model, on parameters, as well as on the variable being measured, illustrating the complexity in defining circadian amplitudes. Lastly, we show how twist effects can speed up or slow down zeitgeber-induced amplitude changes in a simple oscillator model. This helps the clock phase adapt and modulate entrainment of oscillators to a periodic signal or their response to coupling. Our results support the use of oscillator theory as a framework to understand emerging properties of circadian clocks with different twist. Moreover, they provide insights into how temporal or spatial phase patterning might arise in coupled networks, as well as how amplitude changes could help in stabilizing the circadian period in the face of temperature changes. Although we focus on circadian clocks, the presented theory can be applied to any other oscillatory system such as cardiac rhythms, flashing fireflies or voice production.

Materials and Methods

Conservative linear and non-linear oscillators: Harmonic vs. Duffing oscillators

In classical mechanics, a mass-spring harmonic oscillator is a system of mass $m$ that, when displaced from its equilibrium position, experiences a restoring force $F$ proportional to the displacement $x$ , namely $F = - kx$ , where $k$ is the spring constant. When $F$ is the only restoring force, the system undergoes harmonic motion (sinusoidal oscillations) around its equilibrium point. In the absence of damping terms, the harmonic motion can be mathematically described by the following linear second order ordinary differential equation (ODE)

(1)

\frac{d^{2} x}{d t^{2}} + \frac{k}{m} x = 0 .

The solution to this differential equation is given by the function $x (t) = A cos (\frac{2 π}{τ} + ϕ)$ ,²⁹ ^, ³⁰ where $A$ represents the amplitude; $ϕ$ , the phase; and $τ$ represents the period of the motion, $τ = 2 π \sqrt{\frac{m}{k}}$ . Thus the oscillatory period is determined only by the mass $m$ and the spring constant $k$ . The amplitude $A$ , on the other hand, is determined solely by the starting conditions (by both initial displacement $x$ and velocity $v = \dot{x}$ ).

Introducing a non-linear term in the restoring force such that $F = - kx - β x^{3}$ allows the conversion of the simple harmonic oscillator into a Duffing oscillator.³¹ Depending on the sign of $β$ , the coefficient that determines the strength of the non-linear term, the spring is termed hard or soft oscillator. The nonlinearity introduces a dependency between the amplitude and period, meaning that varying initial conditions will produce oscillations with different, yet related, periods and amplitudes. The equation of the Duffing oscillator, in the absence of damping terms, reads

(2)

\frac{d^{2} x}{d t^{2}} + \frac{k}{m} x + \frac{β}{m} x^{3} = 0 .

Due to the non-linear term introduced in equation 2, it is helpful to write this system in a form that can be easily treated via numerical integration. Considering the following change of variable $v = \dot{x}$ , the equation can be reformulated as a system of two first order ODEs:

(3)

\begin{array}{l} \frac{d x}{d t} = v \\ \frac{d v}{d t} = - \frac{k}{m} x - \frac{β}{m} x^{3} . \end{array}

Goodwin-like models

The Goodwin model is a minimal model that has been widely used to describe the emergence of oscillations in simple biochemical systems. It is based on a single negative feedback loop, where the final product of a 3-step chain of reactions inhibits the production of the first component. In the context of circadian rhythms, the model can be interpreted as a clock gene mRNA x that gets translated into a clock protein y, that then activates the repressor z, which ultimately inhibits the transcription of x. All synthesis and degradation terms are linear, with the exception of the repression that $z$ exerts on $x$ which is modeled with a Hill curve. The equations that describe the dynamics read

(4)

\begin{array}{l} \frac{d x}{d t} = k_{1} \frac{K_{1}^{n}}{K_{1}^{n} + z^{n}} - k_{2} x \\ \frac{d y}{d t} = k_{3} x - k_{4} y \\ \frac{d z}{d t} = k_{5} y - k_{6} z, \end{array}

where

k_{1}

k_{3}

and

k_{5}

represent the rates of synthesis of

x

y

and

z

, respectively;

k_{2}

k_{4}

and

k_{6}

, the degradation rates; and

n

, the Hill exponent.

The Goodwin model, however, requires a very large Hill exponent ( $n > 8$ ) to produce self-sustained oscillations,³² which biologists and modelers have often considered unrealistic. Gonze³³ Kurosawa³⁴ and others have shown that, by introducing additional nonlinearities in the system, the need for such high value of the Hill exponent can be reduced. In contrast to the linear degradation of variables of the original Goodwin model, the Gonze model³³ describes degradation processes with Michaelis-Menten kinetics as follows

(5)

\begin{array}{l} \frac{d x}{d t} = k_{1} \frac{K_{1}^{n}}{K_{1}^{n} + z^{n}} - k_{2} \frac{x}{K_{2} + x} \\ \frac{d y}{d t} = k_{3} x - k_{4} \frac{y}{K_{4} + y} \\ \frac{d z}{d t} = k_{5} y - k_{6} \frac{z}{K_{6} + z} . \end{array}

To mimic clock heterogeneity and evaluate the amplitude-period correlations among ensembles of Gonze or Goodwin oscillators, degradation rates were varied around $\pm 10 %$ their default value (Table 1).

Table 1. Default parameters of the Goodwin and Gonze models used for our simulations.

Their biological interpretation as well as the default values from publications are shown.

Parameter	Biological interpretation	Value in Goodwin model³⁵ ^, ³⁶	Value in Gonze model³³
$k_{1}$	transcription rate of core clock gene $x$	1 nM	$0.7 nM h^{- 1}$
$k_{2}$	degradation rate of $x$	$0.2 h^{- 1}$	$0.35 nM h^{- 1}$
$k_{3}$	translation rate of core clock protein $y$	$1 h^{- 1}$	$0.7 h^{- 1}$
$k_{4}$	degradation rate of $y$	$0.15 h^{- 1}$ *	$0.35 nM h^{- 1}$
$k_{5}$	nuclear import rate of the repressor $z$	$1 h^{- 1}$	$0.7 h^{- 1}$
$k_{6}$	degradation rate of $z$	$0.1 h^{- 1}$	$0.35 nM h^{- 1}$
$K_{1}$	Michaelis constant of $z$ -mediated repression	1 nM	1 nM
$K_{2}$	Michaelis constant of $x$ degradation	-	1 nM
$K_{4}$	Michaelis constant of $y$ degradation	-	1 nM
$K_{6}$	Michaelis constant of $z$ degradation	-	1 nM
$n$	Hill exponent of cooperative inhibition of $z$ on $x$	9.5 *	4

* Parameters have been adapted to obtain limit cycle oscillations, since the default degradation parameters from Ref. 36 produced weakly damped rhythms.

Almeida model

The Almeida model³⁷ is a protein model of the mammalian clockwork that includes 7 core clock proteins along with the PER:CRY complex. Unlike prior more extensive models,³⁸ ^, ³⁹ it does not consider any post-translational modifications of clock proteins or any nuclear import or export processes, but it takes into account the regulation (activatory or repressive) that these clock proteins exert on DNA binding sites (known as clock-controlled elements) to regulate circadian gene expression. The regulation at these clock-controlled elements, namely E-boxes, D-boxes and RORE elements, is described by the following terms:

\begin{array}{l} E_{box} = V_{E} \frac{BMAL 1}{BMAL 1 + k_{E} + (k_{Er} BMAL 1 CRY)} \\ RORE = V_{R} \frac{ROR}{k_{R} + ROR} \frac{k_{Rr}^{2}}{k_{Rr}^{2} + {REV}^{2}} \\ D_{box} = V_{D} \frac{DBP}{DBP + k_{D}} \frac{k_{Dr}}{k_{Dr} + E 4 BP 4} . \end{array}

The system of ODEs that describes the dynamics of the clock proteins in the Almeida model reads

(6)

\begin{array}{l} \frac{dBMAL 1}{d t} = RORE - γ_{BP} BMAL 1 PERCRY \\ \frac{dROR}{d t} = E_{box} + RORE - γ_{Ror} ROR \\ \frac{dREV}{d t} = 2 E_{box} + D_{box} - γ_{Rev} REV \\ \frac{dDBP}{d t} = E_{box} - γ_{Db} DBP \\ \frac{dE 4 BP 4}{d t} = 2 R_{box} - γ_{E 4} E 4 BP 4 \\ \frac{dCRY}{d t} = E_{box} + 2 R_{box} - γ_{PC} PERCRY + γ_{CP} PERCRY - γ_{C} CRY \\ \frac{dPER}{d t} = E_{box} + D_{box} - γ_{PC} PERCRY + γ_{CP} PERCRY - γ_{P} PER \\ \frac{dPERCRY}{d t} = γ_{PC} PER CRY - γ_{CP} PERCRY - γ_{BP} BMAL 1 PERCRY . \end{array}

All parameter descriptions along with their default values are given in Table 2.

Table 2. List of parameters of the Almeida model.

Time units are given in hours and concentration units as arbitrary units. The default values are taken from the original reference, Almeida et al. ³⁷

Parameter	Biological interpretation	Value	Parameter	Biological interpretation	Value
$V_{R}$	rate of RORE activation	$44.4 h^{- 1}$	$γ_{Ror}$	ROR degradation rate	$2.55 h^{- 1}$
$k_{R}$	strength of RORE activation	3.54	$γ_{Rev}$	REV degradation rate	$0.241 h^{- 1}$
$k_{Rr}$	strength of RORE inhibition	80.1	$γ_{P}$	PER degradation rate	$0.844 h^{- 1}$
$V_{E}$	rate of E box activation	$30.3 h^{- 1}$	$γ_{C}$	CRY degradation rate	$2.34 h^{- 1}$
$k_{E}$	strength of E box activation	214	$γ_{DB}$	DBP degradation rate	$0.156 h^{- 1}$
$k_{Er}$	strength of E box inhibition	1.24	$γ_{E 4}$	E4BP4 degradation rate	$0.295 h^{- 1}$
$V_{D}$	rate of D box activation	$202 h^{- 1}$	$γ_{PC}$	PER:CRY degradation rate	$0.191 h^{- 1}$
$k_{D}$	strength of D box activation	5.32	$γ_{CP}$	PER:CRY formation rate	$0.141 h^{- 1}$
$k_{Dr}$	strength of D box inhibition	94.7	$γ_{BP}$	BMAL1 nuclear export rate	$2.58 h^{- 1}$

To analyze the twist effects that arise from a population of heterogeneous clocks (i.e., parametric twist), all 18 model parameters were randomly varied around $\pm 20 %$ their default value (Table 2) one at a time. Since this model can lead to period-doubling effects upon changes of certain parameters,⁴⁰ those oscillations whose period change resulted in oscillations with period-doubling were removed from the analysis. Moreover, if changing the default parameter in the ensemble resulted in a range of ratio of amplitude variation relative to the default amplitude $< 0.1$ , then we considered that ensemble to have no twist for that particular control parameter. Periods and amplitudes were determined from the peaks and troughs of oscillations using the continuation software XPP-AUTO.

Poincaré model

The intrinsic dynamical properties from single oscillators and their interaction to external stimuli can be very conveniently described by means of a Poincaré model.⁴¹ We here propose a modification of its generic formulation that can take into account phase space twist effects through the twist parameter $ϵ$ , explicitly introduced in the equations. The modified Poincaré model with twist reads

(7)

\begin{array}{l} \frac{d r}{d t} = λr (A - r) \\ \frac{d ϕ}{d t} = ω + ϵ (A - r) . \end{array}

The first equation describes the rate of change of the radial coordinate $r (t)$ (i.e., the time-dependent distance from the origin), whereas the second equation determines the rate of change of the angular coordinate $ϕ (t)$ , where $ω = \frac{2 π}{τ}$ . The parameters $τ$ , $A$ , $λ$ and $ϵ$ denote the free-running period (in units of time), amplitude (arbitrary units), amplitude relaxation rate (in units of time^-1) and twist parameter of the oscillator (in units of time^-1), respectively. In the absence of twist, namely $ϵ = 0$ , the phase changes constantly along the limit cycle at a rate $\frac{2 π}{τ}$ , independently of the radius. In the case of $ϵ \neq 0$ , the phase changes at a constant rate only when $r = A$ ; if any perturbation is to modify $r$ such that $r \neq A$ , then the phase change will be accelerated or decelerated depending on the sign of $ϵ$ and on whether $r > A$ or $r < A$ . This can generate amplitude-period correlations within the time series of the oscillator during its relaxation time, i.e., what we denote as phase space twist. The model parameters, unless otherwise specified in the figures or captions, are the following: $A = 1$ a.u., $λ = 0.05 h^{- 1}$ , $τ = 24$ h and $ϵ$ values of 0 or $\pm 0.1 h^{- 1}$ .

To study phase space twist and the effects of different $ϵ$ values on the oscillator’s response to pulse-like perturbations $pert (t)$ , periodic zeitgeber input $Z (t)$ or mean-field coupling $M$ , the individual Poincaré oscillators $i$ were converted into Cartesian coordinates and the respective terms were added in the equations of the $x_{i}$ variable as follows:

(8)

\begin{array}{l} \frac{d x_{i}}{d t} = λ x_{i} (A - r_{i}) - y_{i} (ω + ϵ (A - r_{i})) + Z (t) + M + pert (t) \\ \frac{d y_{i}}{d t} = λ y_{i} (A - r_{i}) + x_{i} (ω + ϵ (A - r_{i})), \end{array}

where

r_{i} = \sqrt{x_{i}^{2} + y_{i}^{2}}

. The zeitgeber

Z (t)

is given by:

Z (t) = F_{Z} cos (\frac{2 π}{T} t + \frac{π}{2}),

where

T

represents the zeitgeber period and

F_{Z}

the strength (amplitude) of the zeitgeber input. The mean-field

M

coupling is given by:

\begin{array}{l} M & = \frac{K}{N} \sum_{i = 1}^{N} x_{i} (t), \end{array}

where

K

represents the coupling strength and

N

is the number of oscillators in the coupled ensemble. Lastly, the square-like perturbation

pert (t)

is defined as follows:

\begin{array}{l} pert (t) = \{\begin{cases} 0, & if t_{start} \leq t \leq t_{start} + 1 \\ F_{P} & otherwise, \end{cases} \end{array}

where

F_{P}

is the strength of the perturbation and

t_{start}

is the time at which the perturbation starts. The perturbation lasts 1 h and is set to

F_{P} = 0.7

a.u. in all our simulations.

It is important to differentiate between the twist parameter, denoted as $ϵ$ , and the concept of phase space twist, as they are related but not equivalent terms. While $ϵ$ serves as one of the characteristic parameters in the Poincaré model, phase space twist represents the observable outcome of this parameter's influence on how the oscillator's amplitude and period respond to environmental perturbations. Phase space twist is measurable and influenced by $ϵ$ , but also by the amplitude relaxation rate λ. This will become clear in a further section.

Analytical derivation of isochrones

Arthur T. Winfree introduced the concept of isochrones as any set of dynamical states which oscillate with the same phase when they reach the limit cycle at time $t \to \infty$ .⁴² That asymptotic phase at $t \to \infty$ is what Winfree termed latent phase $Φ$ . Since the dynamical flow of the Poincaré model has polar symmetry, the isochrones must also have polar symmetry such that

(9)

Φ = g (ϕ, r) \overset{!}{=} ϕ - f (r) .

By definition, the latent phase velocity $\dot{Φ}$ necessarily increases at units of the angular velocity $ω = \frac{2 π}{τ}$ as the oscillator follows its kinetic equation:

(10)

\frac{dΦ}{d t} \overset{!}{=} ω

Combining the two previous equations and with the use of the chain rule, we can calculate the radius dependency of the latent phase $Φ$ :

\frac{dΦ}{d t} \overset{9}{=} \frac{d ϕ}{d t} - \frac{d f (r)}{d r} \frac{d r}{d t} \overset{10}{=} ω

The terms $\frac{d ϕ}{d t}$ and $\frac{d r}{d t}$ are defined in the Poincaré model (equation 7), so that the equation above can be rewritten as:

ω + ϵ (A - r) - \frac{d f (r)}{d r} λr (A - r) \overset{10}{=} ω

Solving for $\frac{d f (r)}{d r}$ :

\frac{d f (r)}{d r} = \frac{ϵ (A - r)}{λr (A - r)} = \frac{ϵ}{λr}

Next, the solution for $f (r)$ can be found through integration:

f (r) = \frac{ϵ}{λ} \int_{r = A}^{r} \frac{1}{r} d r = \frac{ϵ}{λ} ln r - \frac{ϵ}{λ} ln A

If $A = 1$ (like in all our simulations), the last term can be neglected such that

f (r) = \frac{ϵ}{λ} ln r

Finally, the solution of $f (r)$ can be inserted in equation 9 to end up with the equation for isochrones as a function of the radius:

(11)

Φ = ϕ - \frac{ϵ}{λ} ln r

Isochrones are thus loci of polar coordinates $(ϕ, r)$ in phase space with the same latent phase $Φ$ . Equation 11 shows how the shape of the isochrone (and, consequently, phase space twist effects) for the Poincaré model in equation 7 depends on the ratio of twist $ϵ$ to relaxation rate $λ$ . Specifically, higher $ϵ$ values (in modulus) and lower relaxation rates $λ$ lead to more curved isochrones and greater phase space twist effects. In the Results section, we further elaborate on how to interpret the isochrones graphically.

To plot the isochrones, we simply reorder equation 11 to plot $ϕ$ as a function of $r$ at isochrones with fixed values of latent phases $Φ = \{0, \frac{π}{4}, \frac{π}{2}, \frac{3 π}{4}, π, - \frac{3 π}{4}, - \frac{π}{2}, - \frac{π}{4}\}$ :

ϕ = \frac{ϵ}{λ} ln r - Φ

Computer simulations and data analysis

All numerical simulations were performed and analyzed in Python with the numpy, scipy, pandas and astropy libraries. The function odeint from scipy was used to numerically solve all ordinary differential equations. Bifurcation analyses were computed in XPP-AUTO⁴³ using the parameters $N_{tst} = 150, N_{\max} = 20000, {Ds}_{\min} = 0.0001$ and ${Ds}_{\max} = 0.0002$ .

Throughout our analyses, periods were determined by (i) normalizing the solutions to their mean, (ii) centering them around 0 (by subtracting one unit from the normalized solution), and (iii) by then computing the zeroes of the normalized rhythms. The period was defined as the distance between two consecutive zeros with a negative slope. Amplitudes were determined as the average peak-to-trough distance of the last (normalized) oscillations after removing transients.

Results

Nonlinearities can result in amplitude-period correlations across oscillators

The simple harmonic oscillator (Figure 2A) is a classical model of a system that oscillates with a restoring force proportional to its displacement, namely $F = - kx$ .⁴⁴ The ordinary differential equation that describes the motion of a mass attached on a spring is linear (equation 1 in Materials and Methods) and the solution can be found analytically.²⁹ ^, ³⁰ The period is determined by the size of the mass $m$ and the force constant $k$ (see Materials and Methods), while the amplitude and phase are determined by the starting position and the velocity. Thus, an ensemble of harmonic oscillators with different initial conditions will produce results that differ in amplitudes but whose periods are the same (Figure 2B, C). When plotting amplitudes against periods, no correlation or twist is observed: the period of a simple harmonic oscillator is independent of its amplitude (Figure 2D).

Figure 2. Nonlinearities introduce amplitude-period correlations in oscillator models.

The mass-spring harmonic oscillator (A) is a linear oscillator that, in the absence of friction, produces persistent undamped oscillations (B, in time series and C, in phase space). Different starting conditions produce oscillations that differ in amplitudes but whose period is independent of amplitude (D). The harmonic oscillator is converted to a Duffing oscillator (E, I) by introducing nonlinearities in the restoring force of the spring (the non-linear behavior is represented with deformations of the spring). The new restoring force results in slight changes in the initial conditions producing oscillations whose amplitude and period change (F, G, J, K) and become co-dependent (H, L) and where the correlations depend on the sign of the non-linear term. The terms soft and hard oscillators refer to, by convention, those with positive and negative twist, respectively. Period values in (F–H, J–L) are color-coded.

The simple harmonic oscillator can be converted into a Duffing oscillator³¹ by including a cubic nonlinearity in its equation. In the Duffing oscillator (equations 2, 3 in Materials and Methods), the restoring force is no longer linear (see deformed springs in Figure 2) but instead described by $F = - kx - β x^{3}$ , where $β$ represents the coefficient of non-linear elasticity. These classical conservative oscillators instead are known to have an amplitude-dependent period. A network of Duffing clocks with different starting conditions will produce oscillations of different amplitudes as well as periods. Duffing oscillators with a negative cubic term have been termed soft oscillators and display periods that grow with amplitudes (Figure 2E–H), similar to Kepler’s Third Law of planetary motion,⁴⁵ where planets with larger distances to the Sun run at slower periods than those that are closer. On the other hand, Duffing oscillators with a positive $β$ term are known as hard oscillators and show negative twist (amplitude-period correlations) (Figure 2I–L).⁴⁶ ^, ⁴⁷

In short, nonlinearities in oscillator models can introduce twist effects among ensembles of oscillators with slight differences in their properties (initial conditions, parameters, etc.). Thus, models for the circadian clock, which are based on nonlinearities, are expected to show amplitude-period correlations.

Oscillator heterogeneity produces parametric twist effects in limit cycle clock models

Most circadian clock models generate stable limit cycle oscillations. Limit cycles are isolated closed periodic orbits with a given amplitude and period, where neighboring trajectories (e.g. perturbations applied to the cycle) spiral either towards or out of the limit cycle.⁴¹ ^, ⁴⁸ Stable limit cycles are examples of attractors: they imply self-sustained oscillations. The closed trajectory describes the perfect periodic behavior of the system, and any small perturbation from this trajectory causes the system to return to it, to be attracted back to it.

Limit cycles are inherently non-linear phenomena and they cannot occur in a linear system (i.e., a system in the form of $\dot{\vec{x}} = A \vec{x}$ , like the harmonic oscillator). From the previous section we have learned that non-linear terms can introduce amplitude-period co-dependencies. In this section we show that kinetic limit cycle models of the circadian clock also show twist effects. But twist in limit cycle oscillators has to be studied differently, as different inidial conditions all return to the same cycle. Instead of studying the amplitude-period correlation of an oscillator model with fixed parameters and changing initial conditions as in Figure 2, we study here the correlations that arise among different uncoupled oscillators due to oscillator heterogeneity (i.e., differences in biochemical parameters), as found experimentally.¹⁴ ^, ²⁶ We refer to amplitude-period correlations that become evident in a population of heterogeneous oscillators as parametric twist.

Single negative feedback loop models

The Goodwin model is a simple kinetic oscillator model⁴⁹ that is based on a delayed negative feedback loop, where the final product of a 3-step chain of reactions inhibits the production of the first component (equation 4 in Materials and Methods). In the context of circadian rhythms,³⁵ ^, ³⁶ the model can be interpreted as a clock activator $x$ that produces a clock protein $y$ that, in turn, activates a transcriptional inhibitor $z$ that represses $x$ (Figure 3A). The Goodwin model has been extensively studied and fine-tuned by Gonze,³³ Kurosawa³⁴ and others to study fundamental properties of circadian clocks⁵⁰ ^– ⁵³ or synchronization and entrainment.³³ ^, ⁵⁴ ^, ⁵⁵ The Gonze model³³ (equation 5) includes additional nonlinearities, where the degradation of all 3 variables is modeled with non-linear Michaelis Menten kinetics, to reduce the need of very large Hill exponents ( $n > 8$ ), required in the original Goodwin model to generate self-sustained oscillations,³² that have been questioned to be biologically meaningful. These Michaelian degradation processes can be interpreted as positive feedback loops which aid in the generation of oscillations⁵⁶ (Figure 3B).

Figure 3. Parametric twist in an ensemble of 100 uncoupled Goodwin-like models: correlations are model- and parameter-dependent.

Scheme of the Goodwin (A) and Gonze (B) oscillator models. Changes in degradation parameters result in twist effects when varied randomly around $\pm$ 10% their default parameter value, mimicking oscillator heterogeneity of the Goodwin (C) and Gonze (D) models. Simultaneous co-variation of representative degradation parameters results in parametric twist: changes of $k_{2}$ and $k_{4}$ produce significant positive amplitude-period correlations in both models (E, F); co-variations of $k_{6}$ and $k_{4}$ result in amplitude-period correlations that are not significant in the Goodwin model (G), but significant positive parametric twist in the Gonze model (H). Shown in all panels is Spearman’s $R$ and the significance of the correlation (n. s.: not significant; ***: p value $<$ 0.001). Default model parameters are summarized in Table 1. Relative amplitudes are computed as the peak-to-trough distance of the $x$ variable, normalized to its mean. Oscillators from the ensemble whose amplitude was $< 0.1$ have been removed from the plots; the total number of oscillators ( $n$ ) is indicated in the plots.

To study whether twist effects are present in an ensemble of 100 uncoupled Goodwin and Gonze oscillators with different parameter values, we randomly varied the degradation parameters of $x$ , $y$ or $z$ ( $k_{2}$ , $k_{4}$ and $k_{6}$ , respectively) in each oscillator around $\pm 10 %$ their default parameter value (given in Table 1) and analyzed the resulting amplitude-period correlation. The resulting parametric twist depends on the model and on the parameter being changed: in the Goodwin model, variations in the degradation rate of the transcriptional activator ( $k_{2}$ ) or of the clock protein ( $k_{4}$ ) result in positive twist effects (i.e., soft twist-control), while changes in the transcriptional repressor’s degradation rate ( $k_{6}$ ) result in a hard twist-control, i.e., negative amplitude-period correlation (Figure 3C). In the Gonze model, $k_{2}$ , $k_{4}$ and $k_{6}$ are all soft twist-control parameters, as individual changes of any of them all produce positive parametric twist effects (Figure 3D).

To mimic cell-to-cell variability in a more realistic manner, we introduced heterogeneity by changing combinations of the degradation parameters simultaneously around $\pm 10$ % their default parameter value and analyzing the resulting periods and amplitudes. We observed that the overall twist behavior depends on the particular influence that each parameter has individually. Random co-variations of $k_{2}$ and $k_{4}$ produce significant positive parametric twist effects in ensembles of both Goodwin (Figure 3E) or Gonze clocks (Figure 3F), consistent with the positive correlations when either of the parameters is changed individually (Figure 3C, D). Nevertheless, when $k_{6}$ , which has a negative twist effect in the Goodwin model, is changed at the same time as $k_{4}$ (or $k_{2}$ , data not shown) the correlation is not significant (Figure 3G). Random co-variation of $k_{4}$ and $k_{6}$ in an ensemble of Gonze clocks results in significant positive parametric twist effects (Figure 3H).

Changes in parameter values can result in significant alterations to a system’s long-term behavior, which can include differences in the number of steady-states, limit cycles, or their stability properties. Such qualitative changes in non-linear dynamics are known as bifurcations, with the corresponding parameter values at which they occur being referred to as bifurcation points. In oscillatory systems, Hopf bifurcations are an important type of bifurcation point. They occur when a limit cycle arises from a stable steady-state that loses its stability. A 1-dimensional bifurcation diagram illustrates how changes in a mathematical model’s control parameter affect its final states, for example the period or amplitude of oscillations. The Hopf bubble refers to the region in parameter space where the limit cycle exists, and it is commonly represented with the peaks and troughs of a measured variable in the $y$ axis, with the control parameter plotted on the $x$ axis (Figure 4A). The term “bubble” is used because of the shape of the curve, that resembles a bubble that grows or shrinks as the parameter is changed. Such bifurcation analyses can be used to predict the type of twist that a system shows: the amplitude will increase with parameter changes if the default parameter value is close to the opening of the Hopf bifurcation, or will decrease if the value is near the closing of the bubble.

Figure 4. The type of parametric twist depends on the parameter location within the Hopf bubble.

Bifurcation analysis of Gonze oscillators with changing $k_{4}$ , simulating heterogeneous oscillators: (A) Hopf bubble representing the peaks (maxima) and troughs (minima) of variable $x$ (normalized to its mean) as a function of changes in the degradation rate of $y$ ( $k_{4}$ ); (B) monotonic period decrease for increasing $k_{4}$ . Represented in (C) is the combination of (A) and (B), namely how the difference between the maxima and minima of $x$ changes with the period: a non-monotonic behavior of parametric twist is observed, which depends on the values of $k_{4}$ . Note that, since $k_{4}$ results in a decrease in period length, the $k_{4}$ dimension is included implicitly in (C) and $k_{4}$ increases from the right to the left of the plot. Stars indicate the default parameter value, period or amplitude. All panels display the numerical results from bifurcation analyses of Gonze oscillators obtained with XPP-AUTO.

We observed that Gonze oscillators display self-sustained oscillations for values of $k_{4}$ between 0.2 and 0.43 (Figure 4A) and that, for increasing $k_{4}$ , periods decrease monotonically (Figure 4B). For oscillators with increasing $k_{4}$ , the parametric twist effects from the ensemble are first negative (amplitudes increase and periods decrease); however, when Gonze clocks have $k_{4}$ values that are part of the region of the bubble where amplitudes decrease, positive amplitude-period correlations appear in the ensemble (Figure 4C). In summary, the type of parametric twist depends on both the model and the parameter being studied but also on where the parameter is located within the Hopf bubble. Other more complex kinetic models of the mammalian circadian clockwork have shown that changes in some of the degradation parameters produce non-monotonic period changes,³⁸ and thus the twist picture is expected to become even more complex.

Model with synergistic feedback loops

The Almeida model³⁷ is a more detailed model of the core clock network in mammals that includes seven core clock proteins that exert their regulation at E-boxes, D-boxes and ROR binding elements (RORE) through multiple positive and negative feedback loops (Figure 5A, equation 6 in Materials and Methods). It is a simple model in the sense that, unlike more complex models like the Relógio model,³⁸ it neglects nuclear import/export processes or post-translational modifications of the clock proteins; but complex enough to capture the synergistic positive and negative regulations and feedback processes that the 7 core clock proteins exert on the DNA clock-controlled elements, unlike older models.³⁹ To study parametric twist in an ensemble of uncoupled Almeida oscillators, we randomly changed all 18 parameters individually around $\pm 20 %$ their default value (Table 2 in Materials and Methods) and computed the amplitude-period correlations. In this case, we calculated the ratio of amplitude variation after the parameter change relative to the default amplitude.

Figure 5. The type of parametric twist in models of the circadian clockwork with synergistic feedback loops depends on the parameter being studied and on the variable that is measured.

(A) Scheme of the Almeida model,³⁷ illustrating activation (green arrows) and inhibition (red arrows) of different clock-controlled elements (at E-boxes, D-boxes and ROR elements) by the respective core clock proteins. Parametric twist was studied by modeling a heterogeneous uncoupled ensemble where model parameters were randomly and individually varied around $\pm 20 %$ of their default value (shown in Table 2) and assessing the period-amplitude correlation of the ensemble. (B–D) Almeida oscillators ( $n = 40$ ) with heterogeneous values in the activation rate of D-boxes ( $V_{D}$ ): the twist is negative from the perspective of BMAL1, positive from the perspective of PER:CRY, but the amplitude of PER is not greatly affected by changing $V_{D}$ . (E–G) Almeida oscillators ( $n = 40$ ) with heterogeneous values in the degradation rate of PER ( $γ_{P}$ ) show positive parametric twist effects for BMAL1, PER and PER:CRY. The twist in panels (B, E) is evaluated by comparing the relative amplitude of the respective protein/protein complex (computed as the average peak-to-trough distance normalized to its mean) after parameter change to the default amplitude. Shown in (C, D, F, G) are representative oscillations with long and short periods: dashed lines represent rhythms of smallest amplitudes.

We found, interestingly, that the overall twist effects depend not only on the parameter being studied, but also on the variable which is measured. For example, changes in the rate of D-box activation parameter $V_{D}$ result in negative twist effects for BMAL1 (i.e., lower amplitude BMAL1 rhythms run slower than oscillators with higher amplitude BMAL1 rhythms), positive parametric twist for the PER:CRY complex but almost no parametric twist from the perspective of PER (Figure 5B–D). Changes in PER degradation $γ_{P}$ produce positive parametric twist for BMAL1, PER and PER:CRY but of different magnitudes (Figure 5E–G). Supplementary Table S1 (in⁵⁷) summarizes the parametric twist effects for the additional parameters from the Almeida model, highlighting the complexity that arises with synergies of feedback loops and bringing us again to the question of defining what the relevant amplitude of a complex oscillator is.

Phase space twist in single oscillators: On amplitude-phase models, isochrones and perturbed trajectories

Up until now, our results have focused on studying amplitude-correlations among ensembles of self-sustained oscillators with differences in their intrinsic properties (e.g. biochemical parameters) in the absence of external cues. However, another form of amplitude-period correlations can occur within an oscillator's time series, because when a clock is exposed to external stimuli, its amplitude and period undergo adaptation. We refer to these amplitude-period correlations in individual clocks as they return to their steady-state oscillation after a stimulus as phase space twist.

To further explore the correlation and interdependence between the frequency of an oscillator and its amplitude upon an external stimulus, more generalized models can be of use. The Poincaré oscillator model (equation 7 in Materials and Methods) is a simple conceptual oscillator model with only two variables, amplitude and phase, that has been widely used in chronobiology research.²⁰ ^, ²² ^, ²⁴ ^, ⁴¹ ^, ⁴² ^, ⁵⁷ This amplitude-phase model, regardless of molecular details, can capture the dynamics of an oscillating system and what happens when perturbations push the system away from the limit cycle. When a pulse is applied and an oscillating system is ‘kicked out’ from the limit cycle, the perturbed trajectory is attracted back to it at a rate $λ$ . Within the limit cycle, the dynamics are strictly periodic: if one takes a point in phase space and observes where the system returns to after exactly one period, the answer is trivial: to exactly the same spot (Figure 6, red dots). Nevertheless, outside the limit cycle, during the transient relaxation time (i.e., time between the perturbation and the moment that the trajectory reaches the limit cycle), the time between two consecutive peaks might be shorter or longer than the period of the limit cycle orbit.

Figure 6. Schematic of an isochrone in a Poincaré oscillator with (A) no twist, (B) positive twist or (C) negative twist.

To understand the concept of isochrones, a simple experiment is performed: if one considers a point in phase space (shown in red) within a limit cycle (black) and observes where the system returns to after exactly one period, the answer is trivial: to the same spot. If, however, now a perturbation is applied (red arrow) such that the system starts in a section of the state space which is outside of the stable periodic orbit, and one maps the points that the system ‘leaves as a footprint’ after exactly one period as it relaxes back to the limit cycle (blue trajectories), these points (shown in yellow) will outline an isochrone (grey). The twist parameter $ϵ$ affects the curvature of the isochrone, with implications in the response of the oscillating system to the perturbation. The oscillator with positive twist (B) arrives at a later phase than that with no twist (A), resulting in a phase delay with respect the clock with no twist, whereas the clock with negative twist (C) arrives to the limit cycle at an earlier phase (i.e., advanced with respect to the clock with no twist).

Arthur T. Winfree introduced a practical term, isochrones, to conceptualize timing relations in oscillators perturbed off their attracting cycles,⁴² which becomes important in physiological applications because often, biological oscillators are not on their attracting limit cycles. To understand what isochrones are, Winfree proposes in⁴² a simple experiment where a pulse-like perturbation applied in a system produces a ‘bump’ in the limit cycle. For stable limit cycles, the perturbation will relax and spiral towards the attracting cycle (blue spiral in Figure 6, until eventually the simulation returns to the limit cycle after enough cycles). If during this relaxation, one records the position of the oscillator in time steps which equal the period of the unperturbed limit cycle, the sequence of points left as a footprint (yellow points in Figure 6) will converge to a fixed point on the cycle and will outline an isochrone (Figure 6). All points in the same isochrone have the same latent phase Φ (see equation 9).

Mathematically, isochrones are related to the oscillator's twist parameter $ϵ$ , and whereas in a Poincaré oscillator with no twist, the isochrones are straight and radial (Figure 6A), when twist is present, the isochrones can become bent or skewed (Figure 6B, C) because of the radius dependency that $ϵ$ introduces on the phase dynamics. The phase changes at a constant rate $ω = \frac{2 π}{τ}$ in the limit cycle, but in conditions outside the limit cycle, the phase is made to be modulated by the radius through the twist parameter $ϵ$ until r = A (equation 7 in Materials and Methods). In an individual oscillator with positive twist $ϵ$ , a perturbation that increases the oscillator’s instantaneous amplitude will relax back and intersect the isochrone at an angle $< 360^{\circ}$ (Figure 6B) than an oscillator with no twist in the time of one period, whereas an oscillator with negative twist will cover an angle $> 360 °$ in the time of one period (Figure 6C). This parameter $ϵ$ thus determines how perturbations away from the limit cycle are decelerated or accelerated during the course of relaxation. Ultimately, this influences the amplitude-period correlations within the oscillator's time series (i.e., phase space twist effects) as it returns to its steady-state oscillation following a perturbation.

Phase space twist in single oscillators affects their interaction with the environment

We have seen how, in an individual oscillator, the twist parameter $ϵ$ acts to slow down (in case of positive twist, Figure 6B) or to speed up (in case of negative twist, Figure 6C) trajectories further away from the limit cycle compared to those closer to it. As a result, $ϵ$ has a direct consequence on how the individual oscillator responds to pulse-like perturbations or periodic zeitgebers coming from the environment.

To analyze how the acceleration or deceleration of perturbations induced by $ϵ$ affects phase space twist and the response of an oscillator to a zeitgeber pulse, we applied a pulse-like perturbation to individual Poincaré oscillators with different twist values $ϵ$ . The pulse was applied at CT3 and was made to increase the instantaneous amplitude (such that $r > A$ ). If the oscillator has no twist ( $ϵ = 0$ , Figure 7A), the isochrones are straight and radial, but for a soft or a hard oscillator with positive or negative $ϵ$ values, respectively, isochrones get skewed (Figure 7B, C, also Figure 6). In the case of a soft oscillator with positive $ϵ$ (Figure 7B), the pulse at CT3 gets decelerated during the course of its relaxation, and as a consequence, the oscillator arrives back to the limit cycle at an later phase than the oscillator with $ϵ = 0$ (Figure 7D, E). The same can also be inferred mathematically: for this particular perturbation where $r > A$ , $A - r < 0$ holds and hence the phase velocity outside the limit cycle is smaller ( $\dot{ϕ} < ω$ ) than at the limit cycle for the oscillator with positive twist, since $\dot{ϕ} = ω + ϵ (A - r)$ (equation 7). The opposite holds for the oscillator with negative twist: this clock arrives at an earlier phase to the limit cycle (Figure 7C, F).

Figure 7. Phase space twist in a single oscillator and its effect on responses to pulse-like perturbations and to entrainment to periodic zeitgebers.

(A–C) Poincaré oscillator models for different twist values $ϵ$ , shown in phase space. The limit cycle is shown in black; perturbed trajectories are shown in red; isochrones are depicted in grey. (D–F) Time series corresponding to panels (A–C): the unperturbed limit cycle oscillations are shown with a dashed black line; perturbed trajectories are shown in red. (G–I) Representation of phase space twist effects: peak-to-trough distance as a function of the peak-to-peak distance from the perturbed time series from (D–F) relaxing back to the limit cycle (the implicit time dimension is indicated in the panels). (J–L) Phase response curves: the shape of the PRC and the extent of the phase shifts (in hours) depends on the twist $ϵ$ . (M–O) Arnold tongues illustrating entrainment ranges of an individual oscillator with different phase space twist values to a periodic sinusoidal zeitgeber input of different periods $T$ . The amplitude of the entrained clock is color-coded, with yellow colors corresponding to larger amplitudes. (P–R) The amplitude of an individual oscillator driven by a sinusoidal zeitgeber also depends on the twist of the individual oscillator. Shown are the resonance curves as a response to a zeitgeber input of strength $F = 0.05$ and varying periods $T$ . Amplitudes are defined as the average peak-to-trough distances of the entrained signal.

The positive and negative amplitude-period correlations characteristic of phase space twist effects are evident in how, upon the pulse-like perturbation, the peak-to-peak distance changes (compared to the limit cycle period) as the perturbation returns to the periodic orbit. For a clock with $ϵ = 0$ , the peak-to-peak distance outside the limit cycle after a perturbation coincides with the 24 h peak-to-peak distance within the periodic orbit (Figure 7G). In the case of the Poincaré oscillator with positive twist $ϵ$ , the peak-to-peak distance of the oscillator after the perturbation that increases the instantaneous amplitude is longer than the period of the steady-state rhythm (24 h), since $\dot{ϕ} < ω$ . As the perturbed amplitude decreases and is attracted back to the stable cycle, the peak-to-peak distance approaches 24 hours. Consequently, a positive phase space twist correlation is observed (Figure 7H). Conversely, the clock with negative twist exhibits a shorter peak-to-peak distance than the 24 h rhythm during the relaxation time $(\dot{ϕ} > ω)$ , which subsequently increases as the system returns to the stable periodic orbit, resulting in negative phase space twist effects (Figure 7I). When considering the findings from the last two paragraphs as a whole, it becomes clear how the extent of the phase shift between a perturbed and an unperturbed oscillator (red and black dashed lines in Figure 7D–F, respectively) depends on $ϵ$ and thus the shape of the phase response curve (PRC) and magnitude of the phase shift depend on this twist parameter (Figure 7J–L) and on the amplitude-period correlations (i.e., phase space twist) that it induces.

We then evaluated how twist affects the response of an individual oscillator to a periodic zeitgeber input. We observed how the range of entrainment becomes larger with larger values of absolute value $ϵ$ , as seen by the wider Arnold tongues in Figure 7M–O. It is widely known that, when the frequency of an applied periodic force is equal or close to the natural frequency of the system on which it acts, the amplitude of the oscillator on which the driving force (zeitgeber) acts on increases due to resonance effects. Interestingly, also the resonance curve was affected by twist: for the oscillator with no twist, the maximum amplitude occurred as expected for a zeitgeber period of 24 h, matching the oscillator's intrinsic period (Figure 7P). In oscillators with positive or negative twist, however, the maximum amplitude after zeitgeber forcing increases and decreases with zeitgeber period, respectively (Figure 7Q, R), resulting in skewed resonance curves.⁵⁸ Interestingly, we observed that twist also affects the entrainment of an individual clock, and oscillators with high absolute values of twist cannot entrain to a periodic signal (Supplementary Figure S1 in Ref. 59), a phenomenon that has previously been described as shear-induced chaos²⁷ ^, ⁶⁰ ^, ⁶¹ and that arises because the isochrones become so skewed, that the trajectory of the relaxation gets stretched and folded.

Relaxation rate also modulates phase space twist effects

Not only the twist parameter $ϵ$ , but also the amplitude relaxation rate $λ$ affects the response of oscillators to perturbations and consequently the entrainment range. We have seen how, in the Poincaré model (equation 7), the twist parameter $ϵ$ dictates how much trajectories are ‘slowed down’ or ‘sped up’ dependent on their distance from the limit cycle. The amplitude relaxation rate $λ$ describes the rate of attraction back to the limit cycle, which is independent of $ϵ$ . Together, these parameters both dictate how ‘skewed’ the isochrones are in phase space (see analytical expression in Materials and Methods, equation 11) and modulate phase space twist. For Poincaré oscillators with twist, the isochrones become more radial and straight as the amplitude relaxation rate increases (Figure 8A–C). The correlations between peak-to-peak periods and peak-to-trough amplitudes within the oscillator's time series also change (Figure 8D–F), with the implications that this has on PRCs and entrainment that have been mentioned before. A more ‘plastic’ clock (with a lower $λ$ value) responds to an external pulse with larger phase shifts than a more rigid clock (with higher $λ$ ), resulting in phase response curves of larger amplitude (Figure 8G–I). This is also intuitive, as it is clear that for larger values of $λ$ , the twist $ϵ$ has shorter effective time to act upon perturbed trajectories because the perturbation spends less time outside the limit cycle. As a consequence, phase space twist effects can become less evident, as shown in Figure 8D–F.

Figure 8. Role of amplitude relaxation rate on the skewing of isochrones, phase space twist and consequently on the response of oscillators to perturbations.

(A–C) Poincaré oscillator models with different amplitude relaxation values $λ$ , but otherwise identical (free running period $τ = 24$ h, amplitude $A = 1$ and twist $ϵ = 0.05 h^{- 1}$ ), shown in phase space. The limit cycle is shown in black; perturbed trajectories are shown in red; isochrones are depicted in grey. (D–F) Phase space twist effects: amplitude relaxation rate λ modulates the correlations between period (defined as peak-to-peak distance) and amplitude (peak-to-trough distance). (G–I) Phase response curves: the shape of the PRC and the extent of the phase shifts depends on the relaxation rate $λ$ . See Materials and Methods for details on the analytical derivation of the equation of isochrones and roles of $λ$ and $ϵ$ .

Phase space twist in single oscillators affects coupled networks

Biological oscillators are rarely alone and uncoupled. Coupled oscillators, instead, are at the heart of a wide spectrum of living things: pacemaker cells in the heart,⁴¹ insulin- and glucagon-secreting cells in the pancreas⁶² or neural networks in the brain and spinal cord that control rhythmic behavior as breathing, running and chewing.⁶³ A number of studies have shown that networks of coupled oscillators behave in a fundamentally different way than ‘plain’ uncoupled oscillators.³³ ^, ⁶⁴ ^, ⁶⁵ Moreover, the findings in Figure 7 highlight that individual oscillators respond differently to environmental perturbations depending on the presence or absence of twist. This implies that the impact of single oscillator twist goes beyond the individual behavior, extending to how coupling synchronizes an entire network of oscillators.

To analyze the effect of twist on coupled oscillators, we study systems of Poincaré oscillators coupled through a mean-field (equation 8). We start by coupling two identical oscillators and analyzing the effect that different twist values have on the behavior of the coupled network. The numerical simulations show that increasing coupling strengths affect the period of the coupled system: oscillators with positive twist show longer periods as the coupling strength increases, whereas oscillators with negative twist show period-shortening (Figure 9A and Supplementary Figure S2A–C in Ref. 59). Coupling results in an increase in amplitude due to resonance but, for our default relaxation rate value ( $λ = 0.05 h^{- 1}$ ), no significant differences across oscillators with different twist values were found (Supplementary Figure S2 in Ref. 59). Increasing relaxation rates (oscillators that are attracted faster back to the limit cycle upon a perturbation) nevertheless resulted in less coupling-induced period or amplitude changes (Supplementary Figure S3 in Ref. 59).

Figure 9. Phase space twist $ϵ$ affects the response of a network to coupling.

(A) Mean-field coupling of two identical Poincaré oscillators (sharing the same value of twist $ϵ$ ) and the effect on the period of the coupled system: whereas coupling among oscillators with positive twist results in longer periods of the coupled network, mean-field coupling among oscillators with negative twist results in the network running at a faster pace (period shortening). (B) Mean-field coupling can synchronize a network of oscillators with different twist values. Shown are three oscillators that only differ in their twist parameters (but else identical: amplitude, amplitude relaxation rate and period are the same for the three oscillators). All oscillations overlap during the first part of the time series, since twist has no effect within the limit cycle. At a certain point (indicated by the red arrow), mean-field coupling is turned on and it is observed how the oscillators respond differently to that mean-field coupling, ending up with different individual amplitudes as well as relative phases to the mean-field (depicted in dashed black line).

We then analyzed what happens when oscillators with different twist values are coupled through their mean-field. For this purpose, we took a system of 3 oscillators (with twist values $ϵ = 0$ , $ϵ < 0$ and $ϵ > 0$ ), we let them free run in the absence of coupling, and turned on mean-field coupling after a certain time (red arrow in Figure 9B). We observed that all three oscillators synchronize to the mean-field for low values of the absolute value of twist (Figure 9B). Nevertheless, due to the period differences induced by coupling (Figure 9A), the relative phases of the individual oscillators to the mean-field depend on the specific twist values of the individual clocks: those with negative twist oscillate ‘ahead’ of the mean-field (with a phase advance) but clocks with positive twist cycle with a phase delay (Figure 9B). Very large absolute values of twist, interestingly, produce again complex chaotic dynamics (Supplementary Figure S4 in Ref. 59). The oscillator with a large value of ϵ does not synchronize to the mean-field and, as a result of the inter-oscillator coupling, this desynchronized oscillator ‘pulls’ to desynchronization the other two oscillators which otherwise would have remained in sync.

Discussion

This study aimed at characterizing period-amplitude correlations across circadian clock models of different complexity. Body clocks have to cope with cellular heterogeneity, what results in cellular clocks being heterogeneous across networks and tissues. We addressed whether the amplitude-period correlations that have been observed experimentally can be explained through heterogeneity in the cellular clocks, and what design principles are needed to produce such twist effects that we term parametric twist. Moreover, clocks live in constantly changing environments, that requires them to get adapted in the face of external changes. With the concept of phase space twist we refer to the co-modulations of period and amplitude that an individual oscillator experiences within its time series when it encounters an external stimulus. Ultimately, these twist effects tune the oscillator’s response to coupling, entrainment and pulse-like perturbations. We also retrieve the ‘old’ terminology of hard versus soft oscillators to refer to oscillations with negative and positive amplitude-period correlations, respectively.

The concept twist was historically coined on conservative, non-limit cycle oscillations. These type of oscillations, like the Duffing oscillator,³¹ have an amplitude defined by the initial conditions, and hence any change in initial conditions results in a new cycle. In contrast, limit cycles oscillate with a characteristic period and amplitude regardless of the initial conditions. For this reason, twist cannot be studied in non-limit cycle oscillators in the same way it can be done for limit cycle oscillators. This is the reason why, in the context of the Duffing oscillators, twist is not termed neither parametric nor phase space. We have coined the terms parametric twist and phase space twist as the amplitude-period correlations that one can study in limit cycle oscillators.

Parametric twist effects become evident when analyzing populations of cells and require nonlinearities in oscillator models. The feedback loops needed to generate oscillations, which are commonly modeled with non-linear terms,³³ ^, ³⁷ ^, ³⁸ ^, ⁵⁰ result in variations of parameters producing oscillations of different but correlated amplitudes and periods. We found that the type of parametric twist (positive or negative) depends on a number of factors: Firstly, on the biochemical parameter being affected, since different model parameters control the oscillation properties (amplitude and period) differently (Figures 3 and 5). Second, on the region within the Hopf bubble where the clock’s parameter set is at: as seen in Figure 4, small changes in the default parameter value can lead to either an increase or decrease in amplitude, depending on the specific value of the default parameter. Lastly, the type of parametric twist also depends on the type of model and variable being measured: simple models with single negative feedback loops show the same parametric twist effects for all variables because the parameter of interest has the same effect on the amplitude of all variables. In complex models with synergies of loops, a change in one parameter might increase the amplitude of one variable but decrease the amplitude of a second variable (Figure 5, also Ref. 66 and Supplementary Table S1 in Ref. 59). Additional clock models have found different types of parametric twist: for instance, Goldbeter found that, in a model of the Drosophila clock, the rate of PER degradation led to amplitude increasing and period decreasing⁶⁷ (i.e., negative parametric twist), unlike in the Almeida model (Figure 5E). This highlights the challenge of defining circadian amplitude (and twist effects) to find a metric for the whole oscillating system. Our findings explain why some experimental studies have found positive twist,²⁶ others negative,¹⁴ whereas some other works found very little correlations.⁶⁸

Bokka et al. ⁶⁶ recently conducted a comprehensive analysis of parametric twist in 11 ODE-based oscillator models, including the Goodwin model. In their analysis of the Goodwin model, they found that changes in k ₂ resulted in amplitude increases with a constant period (no twist). Changes in k ₄ led to increased amplitude and decreased period (negative twist), while changes in k ₆ resulted in decreases in both amplitude and period (positive twist).⁶⁶ However, our bifurcation analyses revealed different correlations. We observed positive twist when modifying k ₂ or k ₄, but a negative correlation with changes in k ₆ (Figure 3). Such discrepancies may arise from different variables used for amplitude determination, from varying parameter sets, or from different amplitude definitions (we use relative amplitude, whereas Bokka et al. define it in absolute terms, i.e., as the distance from peak to trough). There have been several studies that have addressed the question of defining amplitude in circadian studies. One approach has been to propose a metric in which amplitude is defined as the geometric mean of all biomolecular species in the model of interest.⁶⁶ ^, ⁶⁹ This metric, however, poses difficulties for experimental application. Defining which are the essential oscillatory species in a system is challenging, and even if agreed upon, measuring all biomolecular species within one experiment is often unfeasible.

The populations of oscillators considered in our and other⁶⁶ analyses to study parametric twist are uncoupled. But coupling across network of oscillators can narrow the distribution of oscillator periods,⁶⁴ thereby also reducing the extent of parametric twist effects. However, the decrease in parametric twist effects upon coupling should be reversible; if inter-oscillator coupling is disrupted, the original amplitude-period correlations are expected to return.

It is important to point out that our study has focused on kinetic oscillator models that use Hill functions to model transcriptional repression.³³ ^, ³⁷ ^, ⁴⁹ However, different repression mechanisms exist across different species, and such mechanisms might be modelled with different terms. For example, in Drosophila ⁷⁰ and in mammals,⁷¹ ^– ⁷⁵ transcriptional inhibition involves the formation of a 1:1 stoichiometric complex between activators and inhibitors – a process known as protein sequestration-based repression.⁷⁶ ^, ⁷⁷ Here, repressors bind tightly to activators, creating an inactive complex and effectively 'sequestering' the activator. If the ratio greatly differs from 1:1, oscillations quickly dampen out.⁷⁶ ^, ⁷⁷ The description of such protein sequestration processes has been described with terms different from Hill functions.⁷⁶ ^, ⁷⁷ In contrast, species like Neurospora show a lower than 1:1 stoichiometry between activators and repressors in nuclei.⁷⁸ In this fungus, self-sustained oscillations require numerous phosphorylations in the activator.⁷⁸ ^, ⁷⁹ As the number of mutated phosphorylation sites increases, the circadian rhythms become weaker and finally arrhythmic.⁷⁸ Such multistep cooperative reactions can be described with Hill curves.⁸⁰ ^, ⁸¹

We also addressed the concept of phase space twist. It is important to clarify that the parameter $ϵ$ that we introduced in the modified Poincaré oscillator is referred to as 'twist' parameter throughout our manuscript (in line with prior studies¹⁴ ^, ⁴¹ ^, ⁴²). But phase space twist, strictly speaking, refers to the observable phenomenon: the amplitude period correlations of a perturbed oscillator as it returns to its steady state rhythm. In the Poincaré oscillator, phase space twist effects are a consequence of the $ϵ$ parameter, but other models might also display such correlations without explicit parameters. Zeitgeber pulses, recurring zeitgeber inputs or coupling can all be regarded as ‘perturbations’, as any of these inputs modifies the natural clock’s limit cycle in phase space.

A feature of phase space twist is that it characterizes the adaptation of an oscillator to a perturbation and thus has significant implications for phase response curves (PRCs), entrainment and coupling. This new concept is therefore particularly interesting because, although it does not give direct insights into what the mechanistic or kinetic details of a system are, it does provide predictions on the dynamical properties of oscillators.

We start by discussing the role of phase space twist in PRCs. The twist parameter ϵ induces amplitude-period correlations within the oscillator's relaxation time (Figure 7G–I) thus affecting the extent of the phase shift after a zeitgeber pulse (Figure 7D–F): increasing the twist parameter in absolute value results in alterations in the phase response curves and consequently in the resetting properties of oscillators. In particular, increasing $| ϵ |$ increases the amplitude of the PRC (Figure 7J–L) until the PRC is converted from a type 1 PRC to a type 0 PRC.¹⁹ ^, ⁸² Consistently, amplitudes¹⁷ ^, ¹⁸ ^, ²⁰ ^, ²¹ ^, ⁸³ and periods¹⁸ of oscillators have also been shown to modulate their resetting properties. In these examples, clocks with short periods¹⁸ and small amplitudes (in mathematical models¹⁷ ^, ¹⁸ ^, ²⁰ or in experiments with reduced coupling²⁰ ^, ²¹ or mutations⁸³ that disrupt the normal rhythmicity) are easier to reset.

One can easily extrapolate these findings to responses to jet lag. If twist modulates the extent of a phase shift in response to a perturbation, it will also be critical in the adaptation to jet lag. This has indeed been found by Ananthasubramaniam et al. ²³ Interestingly, the authors found that the instantaneous amplitude effects induced by jet lag are also modulated by twist. In particular, positive twist aided recovery to jet lag in simple Goodwin-like models: reduced amplitudes were accompanied by faster clocks (i.e., shorter periods) upon a phase advance (e.g. when traveling eastwards), but larger amplitudes coincided with longer periods when the phase had to be delayed (e.g. when traveling westwards). Consistent with these theoretical observations are experiments performed in mice, where compromising coupling in the SCN (and thus decreasing amplitudes) reduce jet lag drastically, since resetting signals are much more efficient.²¹

Transient amplitude-period correlations also affect the entrainment properties of an oscillator. Phase space twist modulates the range of zeitgeber periods to which the Poincaré oscillator can entrain ( $ϵ$ ≠ 0 results in larger entrainment ranges, i.e., wider Arnold tongues, Figure 7M–O) and skews the resonance curves (Figure 7P–R). Some studies have even found coexisting limit cycles for driven oscillators with twist.³¹ ^, ⁸⁴ Of note, however, is that the period of the Poincaré oscillator in all our simulations was set to 24 h. Other theoretical studies have shown that clocks with different intrinsic periods show different amplitude responses to entrainment,²³ ^, ⁸⁵ thus combining phase space and parametric twist effects.

Single cells harbor self-sustained clocks, but they are coupled (see Ref. 86 for a review on coupling) to produce a coherent rhythm at the level of tissues and organisms. Coupled networks show different rhythmic properties than clocks in isolation: phases synchronize and ensemble amplitudes increase for over-critical values of coupling strengths.⁶⁴ Our simulations provide an additional level of regulation, by showing that phase space twist influences period length of individual clocks due to coupling. The presence of coupling-induced changes in the oscillation period or amplitude can provide insights into the underlying oscillator type and the presence of twist. For instance, the longer period that has been observed in dispersed (and presumably uncoupled) U-2 OS cells compared to high-density cultures,⁸⁷ but not in dispersed SCN neurons in culture,⁸⁸ ^, ⁸⁹ could be explained by different implicit oscillator twist and amplitude relaxation rate values in different tissues. However, a study by Myung et al. suggests that the picture may be more intricate: The authors show how pharmacological or physical disruption of SCN coupling does not affect the period of Per2 rhythms, in contrast to Bmal1 oscillations, which display longer periods.⁹⁰

In addition, the twist-induced modulation of period also affects the phase relation with which the clock oscillates respect to the mean-field, consistent with chronotypes. It should be noted, however, that different phase relationships to the mean-field can also be obtained without twist, in networks of coupled oscillators with different intrinsic periods.⁶⁴ But here again, the slower running clocks will tend to be phase-delayed in comparison to the faster-running clocks which will be phase-advanced. This shows that twist plays an important role in how synchronization arises in network of coupled oscillators.

Our simulations assume that the coupling strength is constant across all oscillators. This assumption, however, might be questioned, considering that oscillators may 'communicate' with different strengths to each other, especially because individual clocks are spatially organized within tissues. For example, rhythms in cellular activity across SCN neurons do not occur simultaneously: depending on the developmental stage and the environmental conditions, the SCN can show complex spatio-temporal patterns such as phase waves or phase clustering.⁶⁴ ^, ⁹⁰ ^, ⁹⁷ Such phase organizations have been shown to depend on the previous light schedule that the SCN network was entrained to⁹⁸: while small phase differences of PER2::LUC rhythms are observed in cultured SCN slices after 12:12 light-dark schedules, entrainment to long days with 20 h of light and 4 h of darkness leads to phase clusters and region-specific phase differences of up to 12 h.⁹² ^, ⁹³ Moreover, it has been shown that long summertime days result in a reduced synchronization within the SCN, which might facilitate fast adaptations to a new light-dark cycle.⁹⁹ Experimental evidence suggests that these seasonal changes are regulated by coupling via neurotransmitters such as GABA¹⁰⁰ ^, ¹⁰¹ or VIP.¹⁰² These results imply that adaptation to day length and thus to seasonal variations requires network re-organizations of the SCN. (For complete reviews on coupling in the SCN and seasonality, refer to Refs. 86, 98.) In light of these findings, a more plausible scenario may involve local coupling, where clocks couple to neighbor clocks with a strength proportional to the oscillator distance. Alternatively, differences in coupling may occur, with the same agent acting as a synchronizing or desynchronizing factor based on the network's state, as has been suggested for GABA.⁹³ ^, ¹⁰⁰ Thus, twist might not only be critical in how temporal synchronization but also spatial patterning arises in coupled ensembles.

It is important to remark that also the relaxation rate $λ$ (i.e., how rigid/plastic an oscillator is) modulates phase space twist effects (Figure 8D–F) and affects the response of oscillators to perturbations (Figure 8G–I), consistent with previous computational work.²⁰ ^, ⁵⁷ It is in fact the ratio of $ϵ$ to $λ$ what determines the skewing of isochrones (see the analytical derivation in Materials and Methods) and the oscillator’s response to zeitgebers. Larger values of $λ$ (more ‘rigid’ oscillators) imply that any perturbation is attracted back to the limit cycle at a higher rate and thus, perturbations ‘spend less time’ outside the limit cycle. Consequently, the twist parameter $ϵ$ has a shorter effective time to act on the perturbed trajectory, and the isochrones become more straight and radial (Figure 8A–C). Relaxation rate also has implications on coupling and entrainment. Rigid oscillators with $ϵ \neq 0$ display less coupling-induced amplitude expansions (Supplementary Figure S3B in Ref. 59, also Ref. 20) and less period variations (Supplementary Figure S3A in Ref. 59) in response to coupling than more ‘plastic’ oscillators with lower $λ$ values. Moreover, rigid oscillators have smaller ranges of entrainment and narrower Arnold tongues.²⁰ Again, our results imply that any observation of resonant behavior and/or period changes might provide information on whether the underlying oscillator is a rigid versus plastic and hard versus soft clock.

We finish with an open question. Throughout our work we have claimed that period-amplitude correlations are widespread in both in vivo and in in silico clocks, and how they are critical to define how oscillators function in their environment. But, how does one integrate the circadian clock’s property of temperature compensation within this framework? The circadian clockwork is temperature-compensated,¹⁸ ^, ⁵² ^, ⁶⁹ ^, ¹⁰³ ^, ¹⁰⁴ which means that increasing temperatures do not speed up significantly the clock, which still runs at approximately 24 h. A proposed hypothesis for temperature compensation suggests that temperature-sensitivity of oscillation amplitude could stabilize the period.¹⁸ ^, ⁶⁹ In support of this supposition is evidence from Neurospora ¹⁸ and Gonyaulax ¹⁰⁵ suggesting that amplitude increases in response to temperature, but decreases in Drosophila.¹⁰⁶ What are the twist effects behind this adaptation? Twist might play a role in how fast that temperature variation is sensed in the clockwork and modulate the adaptation to find the new steady-state rhythm of larger or smaller amplitude at the new temperature.

Conclusions

Despite the complexities in quantifying amplitude, our models stress the important role of circadian amplitudes and their correlation with oscillator period. Although amplitudes are known to be regulated by a number of internal and external cellular factors, including cellular biochemical rates,¹⁰⁷ light conditions,¹⁰⁸ coupling,⁶⁴ genetic and epigenetic factors¹⁰⁹ or ageing,¹¹⁰ our findings stress that twist effects (i.e., co-modulations of amplitudes and periods) also ‘feed back’ and affect the interaction of oscillators with the environment, facilitating entrainment, fastening response to pulse-like perturbations or modifying the response of a system to coupling. The theory of our conceptual models can also be applied to other oscillating system such as cardiac rhythms, somite formation, central pattern generators or voice production.

Data and code availability

Extended data

GitHub: Simulated (numerical) data generated for the manuscript. https://github.com/olmom/twist

GitHub: Supplementary Material. https://github.com/olmom/twist

Analysis code

GitHub: Source code to generate and analyze the data and to reproduce all figures. https://github.com/olmom/twist

Data and codes are available under the terms of the Creative Commons Attribution 4.0 International license (CC-BY 4.0).

Acknowledgements

The authors thank Gianmarco Ducci, Bharath Ananthasubramaniam and Adrián Granada for stimulating and helpful comments as well as constructive criticism. The authors would also like to acknowledge the work done by the Bachelor and Master students Pia Rose, Julia K. Schlichting, Barbara Zita Peters Couto and Rebekka Trenkle. We acknowledge financial support from the Open Access Publication Fund of Charité – Universitätsmedizin Berlin and the German Research Foundation (DFG).

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Version 2

VERSION 2 PUBLISHED 31 Aug 2023

Author details Author details

Marta del Olmo
Roles: Conceptualization, Data Curation, Investigation, Methodology, Resources, Software, Visualization, Writing – Original Draft Preparation, Writing – Review & Editing

Christoph Schmal
Roles: Data Curation, Investigation, Methodology, Resources, Software, Visualization, Writing – Review & Editing

Camillo Mizaikoff
Roles: Data Curation, Resources, Software, Visualization

Saskia Grabe
Roles: Writing – Review & Editing

Christian Gabriel
Roles: Writing – Review & Editing

Achim Kramer
Roles: Funding Acquisition, Supervision, Writing – Review & Editing

Hanspeter Herzel
Roles: Conceptualization, Funding Acquisition, Investigation, Methodology, Supervision, Writing – Original Draft Preparation, Writing – Review & Editing

Competing interests

No competing interests were disclosed.

Grant information

This study was supported by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) Project-ID 278001972 – TRR 186 to H.H., A.K., C.G. and M.dO.; SCHM 3362/2-1 as well as SCHM 3362/4-1, project-IDs 414704559 and 511886499 to C.S.; RTG2424 CompCancer to S.G.
The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

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del Olmo M, Schmal C, Mizaikoff C et al. Are circadian amplitudes and periods correlated? A new twist in the story [version 2; peer review: 3 approved]. F1000Research 2024, 12:1077 (https://doi.org/10.12688/f1000research.135533.2)

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Jae Kim, Department of Mathematical Sciences, KAIST, Daejeon, South Korea

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https://doi.org/10.5256/f1000research.148657.r202730

The authors have conducted a study on the relationship between amplitude and period in various circadian clock models. However, it is crucial to address the novelty and distinctions of this study compared to prior work, specifically the

The authors have conducted a study on the relationship between amplitude and period in various circadian clock models. However, it is crucial to address the novelty and distinctions of this study compared to prior work, specifically the study by Bokka et al. in 2018 (https://doi.org/10.1049/iet-syb.2018.0015). Additionally, while the models in this manuscript are based on Hill-type repression (such as the Goodwin and Gonze models), it is important to acknowledge recent research findings that reveal different repression mechanisms in the Drosophila (https://doi.org/10.1073/pnas.2113403119) and mammalian circadian clocks (https://doi.org/10.1098/rsfs.2021.0084). These studies demonstrate that the use of multiple repression mechanisms, including protein sequestration, can substantially alter the oscillator dynamics (https://doi.org/10.1049/iet-syb.2015.0090). Given these discoveries, it would be valuable to explore or discuss how different repression mechanisms impact the relationship between period and amplitude, as demonstrated by Bokka et al. in 2018.
In the Conclusions section on page 19, please provide a reference for 'cellular biochemical rates' as a factor influencing circadian amplitudes. For instance, it was demonstrated that faster production and degradation rates of proteins can generate circadian rhythms with higher amplitudes in the Drosophila circadian clock (https://doi.org/10.1073/pnas.2113403119).

Is the work clearly and accurately presented and does it cite the current literature?

Partly
Is the study design appropriate and is the work technically sound?

Yes
Are sufficient details of methods and analysis provided to allow replication by others?

Yes
If applicable, is the statistical analysis and its interpretation appropriate?

Yes
Are all the source data underlying the results available to ensure full reproducibility?

Yes
Are the conclusions drawn adequately supported by the results?

Yes

References

1. Bokka V, Dey A, Sen S: Period-amplitude co-variation in biomolecular oscillators.IET Syst Biol. 2018; 12 (4): 190-198 PubMed Abstract | Publisher Full Text
2. Jeong EM, Kwon M, Cho E, Lee SH, et al.: Systematic modeling-driven experiments identify distinct molecular clockworks underlying hierarchically organized pacemaker neurons.Proc Natl Acad Sci U S A. 2022; 119 (8). PubMed Abstract | Publisher Full Text
3. Jeong EM, Song YM, Kim JK: Combined multiple transcriptional repression mechanisms generate ultrasensitivity and oscillations.Interface Focus. 2022; 12 (3): 20210084 PubMed Abstract | Publisher Full Text
4. Kim JK: Protein sequestration versus Hill-type repression in circadian clock models.IET Syst Biol. 2016; 10 (4): 125-35 PubMed Abstract | Publisher Full Text

Competing Interests: No competing interests were disclosed.

Reviewer Expertise: Circadian rhythms, mathematical modeling

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.

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Marta del Olmo, Institute for Theoretical Biology, Charité Universitätsmedizin Berlin, Philippstr. 13, 10115, Germany

20 Jun 2024

Author Response
Reviewer 1:
Jae Kim, Department of Mathematical Sciences, KAIST, Daejeon, South Korea

The authors have conducted a study on the relationship between amplitude and period in various circadian clock ... Continue reading
Reviewer 1:
Jae Kim, Department of Mathematical Sciences, KAIST, Daejeon, South Korea

The authors have conducted a study on the relationship between amplitude and period in various circadian clock models. However, it is crucial to address the novelty and distinctions of this study compared to prior work, specifically the study by Bokka et al. in 2018 (https://doi.org/10.1049/iet-syb.2018.0015). Additionally, while the models in this manuscript are based on Hill-type repression (such as the Goodwin and Gonze models), it is important to acknowledge recent research findings that reveal different repression mechanisms in the Drosophila (https://doi.org/10.1073/pnas.2113403119) and mammalian circadian clocks (https://doi.org/10.1098/rsfs.2021.0084). These studies demonstrate that the use of multiple repression mechanisms, including protein sequestration, can substantially alter the oscillator dynamics (https://doi.org/10.1049/iet-syb.2015.0090). Given these discoveries, it would be valuable to explore or discuss how different repression mechanisms impact the relationship between period and amplitude, as demonstrated by Bokka et al. in 2018.

We thank all three reviewers for their constructive comments and literature suggestions, which have helped improve our manuscript greatly.

We have included a paragraph in the Discussion addressing how our manuscript supports and builds upon the findings from Bokka et al. [1]. The authors conduct a comprehensive analysis of parametric twist in different ODE-based models that have been used to model biological oscillatory processes but do not study phase space twist. They classify the models depending on how amplitudes and periods co-vary, where their type 1 corresponds to positive amplitude-period co-variation (what we have termed positive parametric twist in our manuscript) and type 2, to negative amplitude-period correlations (negative twist). The authors consider additional correlations, e.g. constant amplitude but changing period, constant period but changing amplitude (both of which we consider no twist) or more complex co-variations of combinations of positive and negative twist effects.

Among these models, the authors study the Goodwin model. Their analysis of the Goodwin model shows that, for changes in k2, amplitude increases but period remains constant (i.e. no twist). For changes in k4, amplitude increases and period decreases (negative twist). Lastly, for changes in k6, both amplitude and period decrease (positive twist). This is in contrast to our bifurcation analyses, which revealed positive correlations of amplitude and period upon changes in k2 or k4, but a negative correlation upon changes of k6. The differences in our and their results might stem from 3 sources:

A possible explanation might come from the fact that the authors define amplitude as an absolute value of the difference between peak and trough of oscillations, whereas we define amplitude as a relative measure (difference between peak and trough of the oscillation but normalized to the mean).

While we have used the x variable from the Goodwin system to define amplitude, it is unclear from the main text in [1] which variable is used from the whole ODE system to define amplitude. Period is expected to change equally for all the variables upon change of one parameter; amplitude, however, might change differently for the different model variables (as shown in our Figure 5B, Figure 5E). This highlights again the difficulty in defining amplitude in circadian studies. Bokka et al. [1] partly circumvent this issue by defining amplitude as the geometric mean of all biomolecular species in the model. This metric, nevertheless, is difficult to measure in or translate to an experimental setup, as (i) the circadian network does not yet agree on what are the core elements of the clock network for a particular species, and (ii) even if we all agreed on the same core clock elements, it is often not possible to measure all the species at the same time within one experiment.

Moreover, additional differences may stem from the fact that we have used a different parameter set than the one in the original Goodwin model [2] to generate circadian oscillations (taken and adapted from [3] as indicated in Table 1 in the main text), and thus our default parameter set might be at a different region of the Hopf bubble.

Interestingly, the authors describe the presence of a ‘forbidden region’, where oscillations with large amplitude but small periods (or fast frequencies) might be difficult. This is consistent with oscillations emerging from Hopf bifurcations. It would be interesting to study the amplitude-period correlations on oscillations that arise after other types of bifurcation points, including a saddle-node-on-invariant cycle (SNIC), but such analyses are out of the scope of this manuscript. Tsai and colleagues showed in [4] how systems with positive feedback loops can present high amplitude and high frequency rhythms and thus the ‘forbidden region’ might differ. In a more theoretical study by Bose [5], it is also shown how the onset of oscillations through a Hopf bifurcation does not allow for high frequency rhythms, whereas when oscillations emerge through a SNIC bifurcation, periods increase more gradually.

We would also like to acknowledge the reviewer’s on-point comment on the existence of different repression mechanisms in different species, which we also have discussed in the manuscript. For example, in Drosophila and in mammals, inhibition of transcriptional activity is achieved through the formation of a 1:1 stoichiometric complex between activators and inhibitors [6-10]. In this type of repression mechanism, frequently termed protein sequestration-based repression [11, 12], repressors tightly bind to activators to form an inactive 1:1 stoichiometric complex, thus ‘sequestering’ the activator. Such protein sequestration processes can be described with terms that differ from the Hill function [11, 12]. In other species, like Neurospora, the stoichiometry between activators and repressors in nuclei is much less than 1:1 [13]. In this fungus, a large number of phosphorylation sites in the activator is required for self-sustained oscillations. As the number of mutated phosphorylation sites increases, the circadian rhythms become weaker and finally arrhythmic [13]. Such multistep cooperative reactions can be described with Hill curves [14, 15].

It is important to note that there is also evidence supporting an additional mode of repression in the mammalian TTFL mediated by CRY1 alone. After inhibition of the CLOCK:BMAL1 activator complex by the macromolecular repressive complex containing PER and CRY proteins, delayed expression of CRY1 can still independently repress transcription from E-boxes [16-19]. This mode of repression cannot be explained according to a protein sequestration repression, and occurs through direct blocking of the activators [18-20]. Such a blocking-type of inhibition might be accurately modeled with a Hill function, especially provided the observations that CRY1 is gradually phosphorylated until it becomes the late monomeric repressive CRY1 [21]. Such multistep phosphorylations, as mentioned earlier, can be described with the use of Hill functions.

Depending on whether the design of a TTFL contains a Hill-type or a protein sequestration-based repression mechanism, the conditions to achieve self-sustained limit cycle oscillations are different [11]. Models that are based on a Hill-type repression need a large Hill exponent in order to generate self-sustained oscillations, whereas models based on a sequestration repression need a 1:1 stoichiometry between repressor and activator for a successful onset of oscillations. Our manuscript only addresses models that use Hill functions to model transcriptional repression and we have partly extended the discussion to account for additional modes of repression. However, given our and others’ [1] observations that all kinds of parametric twist effects are possible upon changing one parameter in a certain model, we expect that protein sequestration-based models might also result in complex parametric twist effects upon changes of one parameter that likely depend on the default parameter set of the model (and where this is located at within the Hopf bubble) as well as on the variable whose amplitude is being measured.

In the Conclusions section on page 19, please provide a reference for 'cellular biochemical rates' as a factor influencing circadian amplitudes. For instance, it was demonstrated that faster production and degradation rates of proteins can generate circadian rhythms with higher amplitudes in the Drosophila circadian clock (https://doi.org/10.1073/pnas.2113403119).

We have now added the reference [22] to our conclusions.

References

Bokka, V., Dey, A., & Sen, S. (2018). Period–amplitude co‐variation in biomolecular oscillators. IET Systems Biology, 12(4), 190-198.

Goodwin, B. C. (1965). Oscillatory behavior in enzymatic control processes. Advances in Enzyme Regulation, 3, 425-437.

Ruoff, P., Vinsjevik, M., Monnerjahn, C., & Rensing, L. (1999). The Goodwin oscillator: on the importance of degradation reactions in the circadian clock. Journal of Biological Rhythms, 14(6), 469-479.

Tsai, T. Y. C., Choi, Y. S., Ma, W., Pomerening, J. R., Tang, C., & Ferrell Jr, J. E. (2008). Robust, tunable biological oscillations from interlinked positive and negative feedback loops. Science, 321(5885), 126-129.

Bose, A. (2014). Bifurcations Dynamics of Single Neurons and Small Networks. In: Jaeger, D., Jung, R. (eds) Encyclopedia of Computational Neuroscience. Springer, New York, NY.

Nangle, S. N., Rosensweig, C., Koike, N., Tei, H., Takahashi, J. S., Green, C. B., & Zheng, N. (2014). Molecular assembly of the period-cryptochrome circadian transcriptional repressor complex. Elife, 3, e03674.

Aryal, R. P., Kwak, P. B., Tamayo, A. G., Gebert, M., Chiu, P. L., Walz, T., & Weitz, C. J. (2017). Macromolecular assemblies of the mammalian circadian clock. Molecular Cell, 67(5), 770-782.

Schmalen, I., Reischl, S., Wallach, T., Klemz, R., Grudziecki, A., Prabu, J. R., ... & Wolf, E. (2014). Interaction of circadian clock proteins CRY1 and PER2 is modulated by zinc binding and disulfide bond formation. Cell, 157(5), 1203-1215.

Ye, R., Selby, C. P., Chiou, Y. Y., Ozkan-Dagliyan, I., Gaddameedhi, S., & Sancar, A. (2014). Dual modes of CLOCK: BMAL1 inhibition mediated by Cryptochrome and Period proteins in the mammalian circadian clock. Genes & Development, 28(18), 1989-1998.

Menet, J. S., Abruzzi, K. C., Desrochers, J., Rodriguez, J., & Rosbash, M. (2010). Dynamic PER repression mechanisms in the Drosophila circadian clock: from on-DNA to off-DNA. Genes & Development, 24(4), 358-367.

Kim, J. K. (2016). Protein sequestration versus Hill‐type repression in circadian clock models. IET Systems Biology, 10(4), 125-135.

Kim, J. K., & Forger, D. B. (2012). A mechanism for robust circadian timekeeping via stoichiometric balance. Molecular Systems Biology, 8(1), 630.

Schafmeier, T., Haase, A., Káldi, K., Scholz, J., Fuchs, M., & Brunner, M. (2005). Transcriptional feedback of Neurospora circadian clock gene by phosphorylation-dependent inactivation of its transcription factor. Cell, 122(2), 235-246.

Ferrell, J. E., & Ha, S. H. (2014). Ultrasensitivity part II: multisite phosphorylation, stoichiometric inhibitors, and positive feedback. Trends in Biochemical Sciences, 39(11), 556-569.

Gonze, D., & Abou-Jaoudé, W. (2013). The Goodwin model: behind the Hill function. PloS One, 8(8), e69573.

Koike, N., Yoo, S. H., Huang, H. C., Kumar, V., Lee, C., Kim, T. K., & Takahashi, J. S. (2012). Transcriptional architecture and chromatin landscape of the core circadian clock in mammals. Science, 338(6105), 349-354.

Ukai-Tadenuma, M., Yamada, R. G., Xu, H., Ripperger, J. A., Liu, A. C., & Ueda, H. R. (2011). Delay in feedback repression by cryptochrome 1 is required for circadian clock function. Cell, 144(2), 268-281.

Cao, X., Yang, Y., Selby, C. P., Liu, Z., & Sancar, A. (2021). Molecular mechanism of the repressive phase of the mammalian circadian clock. Proceedings of the National Academy of Sciences, 118(2), e2021174118.

Cao, X., Wang, L., Selby, C. P., Lindsey-Boltz, L. A., & Sancar, A. (2023). Analysis of mammalian circadian clock protein complexes over a circadian cycle. Journal of Biological Chemistry, 299(3).

Jeong, E. M., Song, Y. M., & Kim, J. K. (2022). Combined multiple transcriptional repression mechanisms generate ultrasensitivity and oscillations. Interface Focus, 12(3), 20210084.

Liu, N., & Zhang, E. E. (2016). Phosphorylation regulating the ratio of intracellular CRY1 protein determines the circadian period. Frontiers in Neurology, 7, 159.

Jeong, E. M., Kwon, M., Cho, E., Lee, S. H., Kim, H., Kim, E. Y., & Kim, J. K. (2022). Systematic modeling-driven experiments identify distinct molecular clockworks underlying hierarchically organized pacemaker neurons. Proceedings of the National Academy of Sciences, 119(8), e2113403119.
Reviewer 1:
Jae Kim, Department of Mathematical Sciences, KAIST, Daejeon, South Korea

The authors have conducted a study on the relationship between amplitude and period in various circadian clock models. However, it is crucial to address the novelty and distinctions of this study compared to prior work, specifically the study by Bokka et al. in 2018 (https://doi.org/10.1049/iet-syb.2018.0015). Additionally, while the models in this manuscript are based on Hill-type repression (such as the Goodwin and Gonze models), it is important to acknowledge recent research findings that reveal different repression mechanisms in the Drosophila (https://doi.org/10.1073/pnas.2113403119) and mammalian circadian clocks (https://doi.org/10.1098/rsfs.2021.0084). These studies demonstrate that the use of multiple repression mechanisms, including protein sequestration, can substantially alter the oscillator dynamics (https://doi.org/10.1049/iet-syb.2015.0090). Given these discoveries, it would be valuable to explore or discuss how different repression mechanisms impact the relationship between period and amplitude, as demonstrated by Bokka et al. in 2018.

We thank all three reviewers for their constructive comments and literature suggestions, which have helped improve our manuscript greatly.

We have included a paragraph in the Discussion addressing how our manuscript supports and builds upon the findings from Bokka et al. [1]. The authors conduct a comprehensive analysis of parametric twist in different ODE-based models that have been used to model biological oscillatory processes but do not study phase space twist. They classify the models depending on how amplitudes and periods co-vary, where their type 1 corresponds to positive amplitude-period co-variation (what we have termed positive parametric twist in our manuscript) and type 2, to negative amplitude-period correlations (negative twist). The authors consider additional correlations, e.g. constant amplitude but changing period, constant period but changing amplitude (both of which we consider no twist) or more complex co-variations of combinations of positive and negative twist effects.

Among these models, the authors study the Goodwin model. Their analysis of the Goodwin model shows that, for changes in k2, amplitude increases but period remains constant (i.e. no twist). For changes in k4, amplitude increases and period decreases (negative twist). Lastly, for changes in k6, both amplitude and period decrease (positive twist). This is in contrast to our bifurcation analyses, which revealed positive correlations of amplitude and period upon changes in k2 or k4, but a negative correlation upon changes of k6. The differences in our and their results might stem from 3 sources:

A possible explanation might come from the fact that the authors define amplitude as an absolute value of the difference between peak and trough of oscillations, whereas we define amplitude as a relative measure (difference between peak and trough of the oscillation but normalized to the mean).

While we have used the x variable from the Goodwin system to define amplitude, it is unclear from the main text in [1] which variable is used from the whole ODE system to define amplitude. Period is expected to change equally for all the variables upon change of one parameter; amplitude, however, might change differently for the different model variables (as shown in our Figure 5B, Figure 5E). This highlights again the difficulty in defining amplitude in circadian studies. Bokka et al. [1] partly circumvent this issue by defining amplitude as the geometric mean of all biomolecular species in the model. This metric, nevertheless, is difficult to measure in or translate to an experimental setup, as (i) the circadian network does not yet agree on what are the core elements of the clock network for a particular species, and (ii) even if we all agreed on the same core clock elements, it is often not possible to measure all the species at the same time within one experiment.

Moreover, additional differences may stem from the fact that we have used a different parameter set than the one in the original Goodwin model [2] to generate circadian oscillations (taken and adapted from [3] as indicated in Table 1 in the main text), and thus our default parameter set might be at a different region of the Hopf bubble.

Interestingly, the authors describe the presence of a ‘forbidden region’, where oscillations with large amplitude but small periods (or fast frequencies) might be difficult. This is consistent with oscillations emerging from Hopf bifurcations. It would be interesting to study the amplitude-period correlations on oscillations that arise after other types of bifurcation points, including a saddle-node-on-invariant cycle (SNIC), but such analyses are out of the scope of this manuscript. Tsai and colleagues showed in [4] how systems with positive feedback loops can present high amplitude and high frequency rhythms and thus the ‘forbidden region’ might differ. In a more theoretical study by Bose [5], it is also shown how the onset of oscillations through a Hopf bifurcation does not allow for high frequency rhythms, whereas when oscillations emerge through a SNIC bifurcation, periods increase more gradually.

We would also like to acknowledge the reviewer’s on-point comment on the existence of different repression mechanisms in different species, which we also have discussed in the manuscript. For example, in Drosophila and in mammals, inhibition of transcriptional activity is achieved through the formation of a 1:1 stoichiometric complex between activators and inhibitors [6-10]. In this type of repression mechanism, frequently termed protein sequestration-based repression [11, 12], repressors tightly bind to activators to form an inactive 1:1 stoichiometric complex, thus ‘sequestering’ the activator. Such protein sequestration processes can be described with terms that differ from the Hill function [11, 12]. In other species, like Neurospora, the stoichiometry between activators and repressors in nuclei is much less than 1:1 [13]. In this fungus, a large number of phosphorylation sites in the activator is required for self-sustained oscillations. As the number of mutated phosphorylation sites increases, the circadian rhythms become weaker and finally arrhythmic [13]. Such multistep cooperative reactions can be described with Hill curves [14, 15].

It is important to note that there is also evidence supporting an additional mode of repression in the mammalian TTFL mediated by CRY1 alone. After inhibition of the CLOCK:BMAL1 activator complex by the macromolecular repressive complex containing PER and CRY proteins, delayed expression of CRY1 can still independently repress transcription from E-boxes [16-19]. This mode of repression cannot be explained according to a protein sequestration repression, and occurs through direct blocking of the activators [18-20]. Such a blocking-type of inhibition might be accurately modeled with a Hill function, especially provided the observations that CRY1 is gradually phosphorylated until it becomes the late monomeric repressive CRY1 [21]. Such multistep phosphorylations, as mentioned earlier, can be described with the use of Hill functions.

Depending on whether the design of a TTFL contains a Hill-type or a protein sequestration-based repression mechanism, the conditions to achieve self-sustained limit cycle oscillations are different [11]. Models that are based on a Hill-type repression need a large Hill exponent in order to generate self-sustained oscillations, whereas models based on a sequestration repression need a 1:1 stoichiometry between repressor and activator for a successful onset of oscillations. Our manuscript only addresses models that use Hill functions to model transcriptional repression and we have partly extended the discussion to account for additional modes of repression. However, given our and others’ [1] observations that all kinds of parametric twist effects are possible upon changing one parameter in a certain model, we expect that protein sequestration-based models might also result in complex parametric twist effects upon changes of one parameter that likely depend on the default parameter set of the model (and where this is located at within the Hopf bubble) as well as on the variable whose amplitude is being measured.

In the Conclusions section on page 19, please provide a reference for 'cellular biochemical rates' as a factor influencing circadian amplitudes. For instance, it was demonstrated that faster production and degradation rates of proteins can generate circadian rhythms with higher amplitudes in the Drosophila circadian clock (https://doi.org/10.1073/pnas.2113403119).

We have now added the reference [22] to our conclusions.

References

Bokka, V., Dey, A., & Sen, S. (2018). Period–amplitude co‐variation in biomolecular oscillators. IET Systems Biology, 12(4), 190-198.

Goodwin, B. C. (1965). Oscillatory behavior in enzymatic control processes. Advances in Enzyme Regulation, 3, 425-437.

Ruoff, P., Vinsjevik, M., Monnerjahn, C., & Rensing, L. (1999). The Goodwin oscillator: on the importance of degradation reactions in the circadian clock. Journal of Biological Rhythms, 14(6), 469-479.

Tsai, T. Y. C., Choi, Y. S., Ma, W., Pomerening, J. R., Tang, C., & Ferrell Jr, J. E. (2008). Robust, tunable biological oscillations from interlinked positive and negative feedback loops. Science, 321(5885), 126-129.

Bose, A. (2014). Bifurcations Dynamics of Single Neurons and Small Networks. In: Jaeger, D., Jung, R. (eds) Encyclopedia of Computational Neuroscience. Springer, New York, NY.

Nangle, S. N., Rosensweig, C., Koike, N., Tei, H., Takahashi, J. S., Green, C. B., & Zheng, N. (2014). Molecular assembly of the period-cryptochrome circadian transcriptional repressor complex. Elife, 3, e03674.

Aryal, R. P., Kwak, P. B., Tamayo, A. G., Gebert, M., Chiu, P. L., Walz, T., & Weitz, C. J. (2017). Macromolecular assemblies of the mammalian circadian clock. Molecular Cell, 67(5), 770-782.

Schmalen, I., Reischl, S., Wallach, T., Klemz, R., Grudziecki, A., Prabu, J. R., ... & Wolf, E. (2014). Interaction of circadian clock proteins CRY1 and PER2 is modulated by zinc binding and disulfide bond formation. Cell, 157(5), 1203-1215.

Ye, R., Selby, C. P., Chiou, Y. Y., Ozkan-Dagliyan, I., Gaddameedhi, S., & Sancar, A. (2014). Dual modes of CLOCK: BMAL1 inhibition mediated by Cryptochrome and Period proteins in the mammalian circadian clock. Genes & Development, 28(18), 1989-1998.

Menet, J. S., Abruzzi, K. C., Desrochers, J., Rodriguez, J., & Rosbash, M. (2010). Dynamic PER repression mechanisms in the Drosophila circadian clock: from on-DNA to off-DNA. Genes & Development, 24(4), 358-367.

Kim, J. K. (2016). Protein sequestration versus Hill‐type repression in circadian clock models. IET Systems Biology, 10(4), 125-135.

Kim, J. K., & Forger, D. B. (2012). A mechanism for robust circadian timekeeping via stoichiometric balance. Molecular Systems Biology, 8(1), 630.

Schafmeier, T., Haase, A., Káldi, K., Scholz, J., Fuchs, M., & Brunner, M. (2005). Transcriptional feedback of Neurospora circadian clock gene by phosphorylation-dependent inactivation of its transcription factor. Cell, 122(2), 235-246.

Ferrell, J. E., & Ha, S. H. (2014). Ultrasensitivity part II: multisite phosphorylation, stoichiometric inhibitors, and positive feedback. Trends in Biochemical Sciences, 39(11), 556-569.

Gonze, D., & Abou-Jaoudé, W. (2013). The Goodwin model: behind the Hill function. PloS One, 8(8), e69573.

Koike, N., Yoo, S. H., Huang, H. C., Kumar, V., Lee, C., Kim, T. K., & Takahashi, J. S. (2012). Transcriptional architecture and chromatin landscape of the core circadian clock in mammals. Science, 338(6105), 349-354.

Ukai-Tadenuma, M., Yamada, R. G., Xu, H., Ripperger, J. A., Liu, A. C., & Ueda, H. R. (2011). Delay in feedback repression by cryptochrome 1 is required for circadian clock function. Cell, 144(2), 268-281.

Cao, X., Yang, Y., Selby, C. P., Liu, Z., & Sancar, A. (2021). Molecular mechanism of the repressive phase of the mammalian circadian clock. Proceedings of the National Academy of Sciences, 118(2), e2021174118.

Cao, X., Wang, L., Selby, C. P., Lindsey-Boltz, L. A., & Sancar, A. (2023). Analysis of mammalian circadian clock protein complexes over a circadian cycle. Journal of Biological Chemistry, 299(3).

Jeong, E. M., Song, Y. M., & Kim, J. K. (2022). Combined multiple transcriptional repression mechanisms generate ultrasensitivity and oscillations. Interface Focus, 12(3), 20210084.

Liu, N., & Zhang, E. E. (2016). Phosphorylation regulating the ratio of intracellular CRY1 protein determines the circadian period. Frontiers in Neurology, 7, 159.

Jeong, E. M., Kwon, M., Cho, E., Lee, S. H., Kim, H., Kim, E. Y., & Kim, J. K. (2022). Systematic modeling-driven experiments identify distinct molecular clockworks underlying hierarchically organized pacemaker neurons. Proceedings of the National Academy of Sciences, 119(8), e2113403119.
Competing Interests: The authors declare no competing interests. Close
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Author Response 20 Jun 2024

Marta del Olmo, Institute for Theoretical Biology, Charité Universitätsmedizin Berlin, Philippstr. 13, 10115, Germany

20 Jun 2024

Author Response
Reviewer 1:
Jae Kim, Department of Mathematical Sciences, KAIST, Daejeon, South Korea

The authors have conducted a study on the relationship between amplitude and period in various circadian clock ... Continue reading
Reviewer 1:
Jae Kim, Department of Mathematical Sciences, KAIST, Daejeon, South Korea

The authors have conducted a study on the relationship between amplitude and period in various circadian clock models. However, it is crucial to address the novelty and distinctions of this study compared to prior work, specifically the study by Bokka et al. in 2018 (https://doi.org/10.1049/iet-syb.2018.0015). Additionally, while the models in this manuscript are based on Hill-type repression (such as the Goodwin and Gonze models), it is important to acknowledge recent research findings that reveal different repression mechanisms in the Drosophila (https://doi.org/10.1073/pnas.2113403119) and mammalian circadian clocks (https://doi.org/10.1098/rsfs.2021.0084). These studies demonstrate that the use of multiple repression mechanisms, including protein sequestration, can substantially alter the oscillator dynamics (https://doi.org/10.1049/iet-syb.2015.0090). Given these discoveries, it would be valuable to explore or discuss how different repression mechanisms impact the relationship between period and amplitude, as demonstrated by Bokka et al. in 2018.

We thank all three reviewers for their constructive comments and literature suggestions, which have helped improve our manuscript greatly.

We have included a paragraph in the Discussion addressing how our manuscript supports and builds upon the findings from Bokka et al. [1]. The authors conduct a comprehensive analysis of parametric twist in different ODE-based models that have been used to model biological oscillatory processes but do not study phase space twist. They classify the models depending on how amplitudes and periods co-vary, where their type 1 corresponds to positive amplitude-period co-variation (what we have termed positive parametric twist in our manuscript) and type 2, to negative amplitude-period correlations (negative twist). The authors consider additional correlations, e.g. constant amplitude but changing period, constant period but changing amplitude (both of which we consider no twist) or more complex co-variations of combinations of positive and negative twist effects.

Among these models, the authors study the Goodwin model. Their analysis of the Goodwin model shows that, for changes in k2, amplitude increases but period remains constant (i.e. no twist). For changes in k4, amplitude increases and period decreases (negative twist). Lastly, for changes in k6, both amplitude and period decrease (positive twist). This is in contrast to our bifurcation analyses, which revealed positive correlations of amplitude and period upon changes in k2 or k4, but a negative correlation upon changes of k6. The differences in our and their results might stem from 3 sources:

A possible explanation might come from the fact that the authors define amplitude as an absolute value of the difference between peak and trough of oscillations, whereas we define amplitude as a relative measure (difference between peak and trough of the oscillation but normalized to the mean).

While we have used the x variable from the Goodwin system to define amplitude, it is unclear from the main text in [1] which variable is used from the whole ODE system to define amplitude. Period is expected to change equally for all the variables upon change of one parameter; amplitude, however, might change differently for the different model variables (as shown in our Figure 5B, Figure 5E). This highlights again the difficulty in defining amplitude in circadian studies. Bokka et al. [1] partly circumvent this issue by defining amplitude as the geometric mean of all biomolecular species in the model. This metric, nevertheless, is difficult to measure in or translate to an experimental setup, as (i) the circadian network does not yet agree on what are the core elements of the clock network for a particular species, and (ii) even if we all agreed on the same core clock elements, it is often not possible to measure all the species at the same time within one experiment.

Moreover, additional differences may stem from the fact that we have used a different parameter set than the one in the original Goodwin model [2] to generate circadian oscillations (taken and adapted from [3] as indicated in Table 1 in the main text), and thus our default parameter set might be at a different region of the Hopf bubble.

Interestingly, the authors describe the presence of a ‘forbidden region’, where oscillations with large amplitude but small periods (or fast frequencies) might be difficult. This is consistent with oscillations emerging from Hopf bifurcations. It would be interesting to study the amplitude-period correlations on oscillations that arise after other types of bifurcation points, including a saddle-node-on-invariant cycle (SNIC), but such analyses are out of the scope of this manuscript. Tsai and colleagues showed in [4] how systems with positive feedback loops can present high amplitude and high frequency rhythms and thus the ‘forbidden region’ might differ. In a more theoretical study by Bose [5], it is also shown how the onset of oscillations through a Hopf bifurcation does not allow for high frequency rhythms, whereas when oscillations emerge through a SNIC bifurcation, periods increase more gradually.

We would also like to acknowledge the reviewer’s on-point comment on the existence of different repression mechanisms in different species, which we also have discussed in the manuscript. For example, in Drosophila and in mammals, inhibition of transcriptional activity is achieved through the formation of a 1:1 stoichiometric complex between activators and inhibitors [6-10]. In this type of repression mechanism, frequently termed protein sequestration-based repression [11, 12], repressors tightly bind to activators to form an inactive 1:1 stoichiometric complex, thus ‘sequestering’ the activator. Such protein sequestration processes can be described with terms that differ from the Hill function [11, 12]. In other species, like Neurospora, the stoichiometry between activators and repressors in nuclei is much less than 1:1 [13]. In this fungus, a large number of phosphorylation sites in the activator is required for self-sustained oscillations. As the number of mutated phosphorylation sites increases, the circadian rhythms become weaker and finally arrhythmic [13]. Such multistep cooperative reactions can be described with Hill curves [14, 15].

It is important to note that there is also evidence supporting an additional mode of repression in the mammalian TTFL mediated by CRY1 alone. After inhibition of the CLOCK:BMAL1 activator complex by the macromolecular repressive complex containing PER and CRY proteins, delayed expression of CRY1 can still independently repress transcription from E-boxes [16-19]. This mode of repression cannot be explained according to a protein sequestration repression, and occurs through direct blocking of the activators [18-20]. Such a blocking-type of inhibition might be accurately modeled with a Hill function, especially provided the observations that CRY1 is gradually phosphorylated until it becomes the late monomeric repressive CRY1 [21]. Such multistep phosphorylations, as mentioned earlier, can be described with the use of Hill functions.

Depending on whether the design of a TTFL contains a Hill-type or a protein sequestration-based repression mechanism, the conditions to achieve self-sustained limit cycle oscillations are different [11]. Models that are based on a Hill-type repression need a large Hill exponent in order to generate self-sustained oscillations, whereas models based on a sequestration repression need a 1:1 stoichiometry between repressor and activator for a successful onset of oscillations. Our manuscript only addresses models that use Hill functions to model transcriptional repression and we have partly extended the discussion to account for additional modes of repression. However, given our and others’ [1] observations that all kinds of parametric twist effects are possible upon changing one parameter in a certain model, we expect that protein sequestration-based models might also result in complex parametric twist effects upon changes of one parameter that likely depend on the default parameter set of the model (and where this is located at within the Hopf bubble) as well as on the variable whose amplitude is being measured.

In the Conclusions section on page 19, please provide a reference for 'cellular biochemical rates' as a factor influencing circadian amplitudes. For instance, it was demonstrated that faster production and degradation rates of proteins can generate circadian rhythms with higher amplitudes in the Drosophila circadian clock (https://doi.org/10.1073/pnas.2113403119).

We have now added the reference [22] to our conclusions.

References

Bokka, V., Dey, A., & Sen, S. (2018). Period–amplitude co‐variation in biomolecular oscillators. IET Systems Biology, 12(4), 190-198.

Goodwin, B. C. (1965). Oscillatory behavior in enzymatic control processes. Advances in Enzyme Regulation, 3, 425-437.

Ruoff, P., Vinsjevik, M., Monnerjahn, C., & Rensing, L. (1999). The Goodwin oscillator: on the importance of degradation reactions in the circadian clock. Journal of Biological Rhythms, 14(6), 469-479.

Tsai, T. Y. C., Choi, Y. S., Ma, W., Pomerening, J. R., Tang, C., & Ferrell Jr, J. E. (2008). Robust, tunable biological oscillations from interlinked positive and negative feedback loops. Science, 321(5885), 126-129.

Bose, A. (2014). Bifurcations Dynamics of Single Neurons and Small Networks. In: Jaeger, D., Jung, R. (eds) Encyclopedia of Computational Neuroscience. Springer, New York, NY.

Nangle, S. N., Rosensweig, C., Koike, N., Tei, H., Takahashi, J. S., Green, C. B., & Zheng, N. (2014). Molecular assembly of the period-cryptochrome circadian transcriptional repressor complex. Elife, 3, e03674.

Aryal, R. P., Kwak, P. B., Tamayo, A. G., Gebert, M., Chiu, P. L., Walz, T., & Weitz, C. J. (2017). Macromolecular assemblies of the mammalian circadian clock. Molecular Cell, 67(5), 770-782.

Schmalen, I., Reischl, S., Wallach, T., Klemz, R., Grudziecki, A., Prabu, J. R., ... & Wolf, E. (2014). Interaction of circadian clock proteins CRY1 and PER2 is modulated by zinc binding and disulfide bond formation. Cell, 157(5), 1203-1215.

Ye, R., Selby, C. P., Chiou, Y. Y., Ozkan-Dagliyan, I., Gaddameedhi, S., & Sancar, A. (2014). Dual modes of CLOCK: BMAL1 inhibition mediated by Cryptochrome and Period proteins in the mammalian circadian clock. Genes & Development, 28(18), 1989-1998.

Menet, J. S., Abruzzi, K. C., Desrochers, J., Rodriguez, J., & Rosbash, M. (2010). Dynamic PER repression mechanisms in the Drosophila circadian clock: from on-DNA to off-DNA. Genes & Development, 24(4), 358-367.

Kim, J. K. (2016). Protein sequestration versus Hill‐type repression in circadian clock models. IET Systems Biology, 10(4), 125-135.

Kim, J. K., & Forger, D. B. (2012). A mechanism for robust circadian timekeeping via stoichiometric balance. Molecular Systems Biology, 8(1), 630.

Schafmeier, T., Haase, A., Káldi, K., Scholz, J., Fuchs, M., & Brunner, M. (2005). Transcriptional feedback of Neurospora circadian clock gene by phosphorylation-dependent inactivation of its transcription factor. Cell, 122(2), 235-246.

Ferrell, J. E., & Ha, S. H. (2014). Ultrasensitivity part II: multisite phosphorylation, stoichiometric inhibitors, and positive feedback. Trends in Biochemical Sciences, 39(11), 556-569.

Gonze, D., & Abou-Jaoudé, W. (2013). The Goodwin model: behind the Hill function. PloS One, 8(8), e69573.

Koike, N., Yoo, S. H., Huang, H. C., Kumar, V., Lee, C., Kim, T. K., & Takahashi, J. S. (2012). Transcriptional architecture and chromatin landscape of the core circadian clock in mammals. Science, 338(6105), 349-354.

Ukai-Tadenuma, M., Yamada, R. G., Xu, H., Ripperger, J. A., Liu, A. C., & Ueda, H. R. (2011). Delay in feedback repression by cryptochrome 1 is required for circadian clock function. Cell, 144(2), 268-281.

Cao, X., Yang, Y., Selby, C. P., Liu, Z., & Sancar, A. (2021). Molecular mechanism of the repressive phase of the mammalian circadian clock. Proceedings of the National Academy of Sciences, 118(2), e2021174118.

Cao, X., Wang, L., Selby, C. P., Lindsey-Boltz, L. A., & Sancar, A. (2023). Analysis of mammalian circadian clock protein complexes over a circadian cycle. Journal of Biological Chemistry, 299(3).

Jeong, E. M., Song, Y. M., & Kim, J. K. (2022). Combined multiple transcriptional repression mechanisms generate ultrasensitivity and oscillations. Interface Focus, 12(3), 20210084.

Liu, N., & Zhang, E. E. (2016). Phosphorylation regulating the ratio of intracellular CRY1 protein determines the circadian period. Frontiers in Neurology, 7, 159.

Jeong, E. M., Kwon, M., Cho, E., Lee, S. H., Kim, H., Kim, E. Y., & Kim, J. K. (2022). Systematic modeling-driven experiments identify distinct molecular clockworks underlying hierarchically organized pacemaker neurons. Proceedings of the National Academy of Sciences, 119(8), e2113403119.
Reviewer 1:
Jae Kim, Department of Mathematical Sciences, KAIST, Daejeon, South Korea

The authors have conducted a study on the relationship between amplitude and period in various circadian clock models. However, it is crucial to address the novelty and distinctions of this study compared to prior work, specifically the study by Bokka et al. in 2018 (https://doi.org/10.1049/iet-syb.2018.0015). Additionally, while the models in this manuscript are based on Hill-type repression (such as the Goodwin and Gonze models), it is important to acknowledge recent research findings that reveal different repression mechanisms in the Drosophila (https://doi.org/10.1073/pnas.2113403119) and mammalian circadian clocks (https://doi.org/10.1098/rsfs.2021.0084). These studies demonstrate that the use of multiple repression mechanisms, including protein sequestration, can substantially alter the oscillator dynamics (https://doi.org/10.1049/iet-syb.2015.0090). Given these discoveries, it would be valuable to explore or discuss how different repression mechanisms impact the relationship between period and amplitude, as demonstrated by Bokka et al. in 2018.

We thank all three reviewers for their constructive comments and literature suggestions, which have helped improve our manuscript greatly.

We have included a paragraph in the Discussion addressing how our manuscript supports and builds upon the findings from Bokka et al. [1]. The authors conduct a comprehensive analysis of parametric twist in different ODE-based models that have been used to model biological oscillatory processes but do not study phase space twist. They classify the models depending on how amplitudes and periods co-vary, where their type 1 corresponds to positive amplitude-period co-variation (what we have termed positive parametric twist in our manuscript) and type 2, to negative amplitude-period correlations (negative twist). The authors consider additional correlations, e.g. constant amplitude but changing period, constant period but changing amplitude (both of which we consider no twist) or more complex co-variations of combinations of positive and negative twist effects.

Among these models, the authors study the Goodwin model. Their analysis of the Goodwin model shows that, for changes in k2, amplitude increases but period remains constant (i.e. no twist). For changes in k4, amplitude increases and period decreases (negative twist). Lastly, for changes in k6, both amplitude and period decrease (positive twist). This is in contrast to our bifurcation analyses, which revealed positive correlations of amplitude and period upon changes in k2 or k4, but a negative correlation upon changes of k6. The differences in our and their results might stem from 3 sources:

A possible explanation might come from the fact that the authors define amplitude as an absolute value of the difference between peak and trough of oscillations, whereas we define amplitude as a relative measure (difference between peak and trough of the oscillation but normalized to the mean).

While we have used the x variable from the Goodwin system to define amplitude, it is unclear from the main text in [1] which variable is used from the whole ODE system to define amplitude. Period is expected to change equally for all the variables upon change of one parameter; amplitude, however, might change differently for the different model variables (as shown in our Figure 5B, Figure 5E). This highlights again the difficulty in defining amplitude in circadian studies. Bokka et al. [1] partly circumvent this issue by defining amplitude as the geometric mean of all biomolecular species in the model. This metric, nevertheless, is difficult to measure in or translate to an experimental setup, as (i) the circadian network does not yet agree on what are the core elements of the clock network for a particular species, and (ii) even if we all agreed on the same core clock elements, it is often not possible to measure all the species at the same time within one experiment.

Moreover, additional differences may stem from the fact that we have used a different parameter set than the one in the original Goodwin model [2] to generate circadian oscillations (taken and adapted from [3] as indicated in Table 1 in the main text), and thus our default parameter set might be at a different region of the Hopf bubble.

Interestingly, the authors describe the presence of a ‘forbidden region’, where oscillations with large amplitude but small periods (or fast frequencies) might be difficult. This is consistent with oscillations emerging from Hopf bifurcations. It would be interesting to study the amplitude-period correlations on oscillations that arise after other types of bifurcation points, including a saddle-node-on-invariant cycle (SNIC), but such analyses are out of the scope of this manuscript. Tsai and colleagues showed in [4] how systems with positive feedback loops can present high amplitude and high frequency rhythms and thus the ‘forbidden region’ might differ. In a more theoretical study by Bose [5], it is also shown how the onset of oscillations through a Hopf bifurcation does not allow for high frequency rhythms, whereas when oscillations emerge through a SNIC bifurcation, periods increase more gradually.

We would also like to acknowledge the reviewer’s on-point comment on the existence of different repression mechanisms in different species, which we also have discussed in the manuscript. For example, in Drosophila and in mammals, inhibition of transcriptional activity is achieved through the formation of a 1:1 stoichiometric complex between activators and inhibitors [6-10]. In this type of repression mechanism, frequently termed protein sequestration-based repression [11, 12], repressors tightly bind to activators to form an inactive 1:1 stoichiometric complex, thus ‘sequestering’ the activator. Such protein sequestration processes can be described with terms that differ from the Hill function [11, 12]. In other species, like Neurospora, the stoichiometry between activators and repressors in nuclei is much less than 1:1 [13]. In this fungus, a large number of phosphorylation sites in the activator is required for self-sustained oscillations. As the number of mutated phosphorylation sites increases, the circadian rhythms become weaker and finally arrhythmic [13]. Such multistep cooperative reactions can be described with Hill curves [14, 15].

It is important to note that there is also evidence supporting an additional mode of repression in the mammalian TTFL mediated by CRY1 alone. After inhibition of the CLOCK:BMAL1 activator complex by the macromolecular repressive complex containing PER and CRY proteins, delayed expression of CRY1 can still independently repress transcription from E-boxes [16-19]. This mode of repression cannot be explained according to a protein sequestration repression, and occurs through direct blocking of the activators [18-20]. Such a blocking-type of inhibition might be accurately modeled with a Hill function, especially provided the observations that CRY1 is gradually phosphorylated until it becomes the late monomeric repressive CRY1 [21]. Such multistep phosphorylations, as mentioned earlier, can be described with the use of Hill functions.

Depending on whether the design of a TTFL contains a Hill-type or a protein sequestration-based repression mechanism, the conditions to achieve self-sustained limit cycle oscillations are different [11]. Models that are based on a Hill-type repression need a large Hill exponent in order to generate self-sustained oscillations, whereas models based on a sequestration repression need a 1:1 stoichiometry between repressor and activator for a successful onset of oscillations. Our manuscript only addresses models that use Hill functions to model transcriptional repression and we have partly extended the discussion to account for additional modes of repression. However, given our and others’ [1] observations that all kinds of parametric twist effects are possible upon changing one parameter in a certain model, we expect that protein sequestration-based models might also result in complex parametric twist effects upon changes of one parameter that likely depend on the default parameter set of the model (and where this is located at within the Hopf bubble) as well as on the variable whose amplitude is being measured.

In the Conclusions section on page 19, please provide a reference for 'cellular biochemical rates' as a factor influencing circadian amplitudes. For instance, it was demonstrated that faster production and degradation rates of proteins can generate circadian rhythms with higher amplitudes in the Drosophila circadian clock (https://doi.org/10.1073/pnas.2113403119).

We have now added the reference [22] to our conclusions.

References

Bokka, V., Dey, A., & Sen, S. (2018). Period–amplitude co‐variation in biomolecular oscillators. IET Systems Biology, 12(4), 190-198.

Goodwin, B. C. (1965). Oscillatory behavior in enzymatic control processes. Advances in Enzyme Regulation, 3, 425-437.

Ruoff, P., Vinsjevik, M., Monnerjahn, C., & Rensing, L. (1999). The Goodwin oscillator: on the importance of degradation reactions in the circadian clock. Journal of Biological Rhythms, 14(6), 469-479.

Tsai, T. Y. C., Choi, Y. S., Ma, W., Pomerening, J. R., Tang, C., & Ferrell Jr, J. E. (2008). Robust, tunable biological oscillations from interlinked positive and negative feedback loops. Science, 321(5885), 126-129.

Bose, A. (2014). Bifurcations Dynamics of Single Neurons and Small Networks. In: Jaeger, D., Jung, R. (eds) Encyclopedia of Computational Neuroscience. Springer, New York, NY.

Nangle, S. N., Rosensweig, C., Koike, N., Tei, H., Takahashi, J. S., Green, C. B., & Zheng, N. (2014). Molecular assembly of the period-cryptochrome circadian transcriptional repressor complex. Elife, 3, e03674.

Aryal, R. P., Kwak, P. B., Tamayo, A. G., Gebert, M., Chiu, P. L., Walz, T., & Weitz, C. J. (2017). Macromolecular assemblies of the mammalian circadian clock. Molecular Cell, 67(5), 770-782.

Schmalen, I., Reischl, S., Wallach, T., Klemz, R., Grudziecki, A., Prabu, J. R., ... & Wolf, E. (2014). Interaction of circadian clock proteins CRY1 and PER2 is modulated by zinc binding and disulfide bond formation. Cell, 157(5), 1203-1215.

Ye, R., Selby, C. P., Chiou, Y. Y., Ozkan-Dagliyan, I., Gaddameedhi, S., & Sancar, A. (2014). Dual modes of CLOCK: BMAL1 inhibition mediated by Cryptochrome and Period proteins in the mammalian circadian clock. Genes & Development, 28(18), 1989-1998.

Menet, J. S., Abruzzi, K. C., Desrochers, J., Rodriguez, J., & Rosbash, M. (2010). Dynamic PER repression mechanisms in the Drosophila circadian clock: from on-DNA to off-DNA. Genes & Development, 24(4), 358-367.

Kim, J. K. (2016). Protein sequestration versus Hill‐type repression in circadian clock models. IET Systems Biology, 10(4), 125-135.

Kim, J. K., & Forger, D. B. (2012). A mechanism for robust circadian timekeeping via stoichiometric balance. Molecular Systems Biology, 8(1), 630.

Schafmeier, T., Haase, A., Káldi, K., Scholz, J., Fuchs, M., & Brunner, M. (2005). Transcriptional feedback of Neurospora circadian clock gene by phosphorylation-dependent inactivation of its transcription factor. Cell, 122(2), 235-246.

Ferrell, J. E., & Ha, S. H. (2014). Ultrasensitivity part II: multisite phosphorylation, stoichiometric inhibitors, and positive feedback. Trends in Biochemical Sciences, 39(11), 556-569.

Gonze, D., & Abou-Jaoudé, W. (2013). The Goodwin model: behind the Hill function. PloS One, 8(8), e69573.

Koike, N., Yoo, S. H., Huang, H. C., Kumar, V., Lee, C., Kim, T. K., & Takahashi, J. S. (2012). Transcriptional architecture and chromatin landscape of the core circadian clock in mammals. Science, 338(6105), 349-354.

Ukai-Tadenuma, M., Yamada, R. G., Xu, H., Ripperger, J. A., Liu, A. C., & Ueda, H. R. (2011). Delay in feedback repression by cryptochrome 1 is required for circadian clock function. Cell, 144(2), 268-281.

Cao, X., Yang, Y., Selby, C. P., Liu, Z., & Sancar, A. (2021). Molecular mechanism of the repressive phase of the mammalian circadian clock. Proceedings of the National Academy of Sciences, 118(2), e2021174118.

Cao, X., Wang, L., Selby, C. P., Lindsey-Boltz, L. A., & Sancar, A. (2023). Analysis of mammalian circadian clock protein complexes over a circadian cycle. Journal of Biological Chemistry, 299(3).

Jeong, E. M., Song, Y. M., & Kim, J. K. (2022). Combined multiple transcriptional repression mechanisms generate ultrasensitivity and oscillations. Interface Focus, 12(3), 20210084.

Liu, N., & Zhang, E. E. (2016). Phosphorylation regulating the ratio of intracellular CRY1 protein determines the circadian period. Frontiers in Neurology, 7, 159.

Jeong, E. M., Kwon, M., Cho, E., Lee, S. H., Kim, H., Kim, E. Y., & Kim, J. K. (2022). Systematic modeling-driven experiments identify distinct molecular clockworks underlying hierarchically organized pacemaker neurons. Proceedings of the National Academy of Sciences, 119(8), e2113403119.
Competing Interests: The authors declare no competing interests. Close
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Reviewer Report 18 Sep 2023

Stephanie Taylor, Department of Computer Science, Colby College, Waterville, Maine, USA

Approved

https://doi.org/10.5256/f1000research.148657.r202731

In this article, the authors describe the phenomenon of twist (period-amplitude inter-dependence) using a range of differential equation oscillator models, ultimately with a view to understanding circadian clocks. The simplest models (harmonic and Duffing) are used to demonstrate the non-linearity introduces a relationship between amplitude and period. The authors then switch to limit-cycle oscillators, because limit-cycle oscillators are more appropriate for studying circadian clocks. They use two approaches to studying twist. The first, referred to as "parametric twist", computes the periods and amplitudes of oscillators with varied parameter values, creating graphs that relates period to amplitude (and thus demonstrate twist - if the correlation is positive, there is positive twist, and negative correlation indicates negative twist). This parametric twist analysis is performed using two limit cycle models, one with one feedback loop and one with more (synergistic) loops. The second, referred to as "phase space twist", examines the response of a single oscillator to a temporary change in one parameter, mimicking signaling. A separate limit-cycle model (a Poincare oscillator) is used for this analysis. Important to the discussion of phase space twist is the concept of isochrons, which are useful for understanding the effects of perturbations on the phase and speed of the clock (does it speed up or slow down the clock?) and it is easy to see how amplitude and phase response are related, when we look at how curved the isochrons are. The authors relate these analyses to phase response curves, recovery from jetlag, the range of entrainment, and temperature compensation and conclude, that although measuring amplitude and understanding relationship between amplitude and period are highly complicated, it is nonetheless important to be mindful of the role of amplitude and its correlation to period when studying clocks.

On the one hand, I really appreciated that the authors chose models that best illustrated the points and that the paper is organized to go from simple ideas to more complex ideas. It was helpful to have the initial explanation of twist use duffing oscillator (which has non-linearity) as a contrast to a harmonic oscillator (which doesn't). It provided a nice introduction to the cause of twist and helped me to develop an intuition. Then, it made sense to use a very simple (3-ODE Goodwin oscillator) as the first example of a limit cycle oscillator. Since it has so few parameters, it was possible to perturb them one at a time and in pairs to illustrate a good portion of the possibilities for how one might observe heterogeneity in the system. Then, moving to Almeida's oscillator demonstrated how quickly the analysis can become very complex. All of this provides a nice progression of complexity. Then, we pivot a bit to the Poincare oscillator, to study twist in a more dynamical way - instead of permanently choosing different parameters, the authors model light signaling (at least, initially) and study the effects of the model returning to its limit cycle after the perturbation is over. This model is particularly appropriate for studying the dynamics of recovery because it is a 2-ode model (so we can examine it in "phase space", which is helpful for visualizing the clock) and because there is a parameter that controls how related period and amplitude are. It is also straight-forward to adjust how curved the isochrons are, which makes illustrating twist very clear. Then, after studying the response of a single oscillator to light, they use the same model to study the effects of intercellular signaling and the effects of different twist on the properties of the ensemble oscillation (e.g. how long the ensemble period is in contrast to that of individual oscillators).

On the other hand, I would have appreciated a more explicit connection between the discussion of "parametric twist" and "phase space twist". They are both due to changes in parameter values causing changes in period and amplitude. In the one case, we measure the period and amplitude of oscillators with different (constant) parameter values. In the second case, we measure instantaneous changes in period and amplitude due to a temporary changes in parameter values (I am referring to Z(t) as a parameter here). In both cases, the shape of the isochrons in the unperturbed condition should help us to predict what will happen in the perturbed condition. There are two challenges with making that point clear with the Goodwin and Almeida models - 1) the isochrons aren't lines, they are multi-dimensional manifolds, and 2) the isochrons in the unperturbed and perturbed systems are different. So I am asking for too much to relate the two discussions by using images of isochrons, but a more explicit comparison would help the reader understand how the analyses are related to each other.

In the analysis of the modified Poincare oscillator, the term "twist" is used differently than it is in the "parametric twist" analysis. In the parametric analysis, the twist captures the full effects of the perturbations (we plot the measured periods and amplitudes across the heterogeneous set of oscillators). In the phase space analysis, "twist" refers to a particular parameter \epsilon that, when non-zero, introduces a dependence of the speed and amplitude on each other. However, that twist parameter (\epsilon) is not the only parameter that affects the skewness of the isochrons. The relaxation rate \lambda also affects the skewness of the isochrons. And the skewness of the isochrons affects the changes in speed and amplitude when the oscillator is signaled (i.e. a parameter is temporarily perturbed). So, if we were to measure the effects on period and amplitude (as is done with figures 7G, H, and I), that reflects that values of both \lambda and \epsilon. When I was thinking about how to relate the "parametric twist" and "phase twist" analysis, I was looking for aspects of figures 3 (Gonze model), 5 (Almeida model), and 7 (poincare model) to relate and it seemed to me that that the subfigures that related the amplitude of the oscillator to its period (3C, 3D, 3E, 3F, 3G, 3H, 5B, 5B, 5E, 7G, 7H, 7I) should be the figures that I can connect. They are labeled as twist in figures 3 and 5. In Figure 7, they show an approximation of instantaneous amplitude and period (rather than steady-state amplitude and period), so it isn't a perfect 1-1 mapping, but it is still the closest that makes sense to me. And in all of these cases, the output could be labeled twist if \epsilon didn't get to claim that label. I think it would make the concept of twist clearer to the reader if it were consistent across both analyses that twist is the "effect" we can measure. My reading of the text as it is currently written is that twist is a phenomenon we can measure in "parametric twist" analyses and that twist is a parameter that causes one change in the isochrons (but not all of the changes that can happen) that ultimately leads to something we can measure. Also, labeling the parameter as twist in the Poincare oscillator makes it seem like we need to use the Poincare oscillator to perform a "phase space twist" analysis. But we aren't limited to the Poincare oscillator for such analyses.

Specific Points

It would be helpful to have a little more framing text that indicates the different models are used to make different points.
I think of the "phase space twist" analysis as analysis of recovery from temporary changes to parameters and "parametric twist" as analysis of permanent differences in parameters. If that is the way you intend readers to think about it, then I would suggest changing the names to reflect those ideas more directly. Right now, the term "phase space twist" stands out as a highly specific analysis - the model most be 2D so you can draw the isochrons. But I think that is limiting. I totally embrace idea that there is both heterogeneity among oscillators and temporary changes due to intercellular signaling, so I would love to see it characterized that way more explicitly. What about "twist due to heterogeneity" and "twist due to signaling"? They aren't quite zippy enough and rely on my characterization of twist as a measurable phenomenon, so I don't push those terms exactly.
On page 8, when you are introducing isochrons, it would be helpful to have a figure to refer to. The most natural figure is Figure 7, but that would force you to make an unnatural re-ordering of the figures. So maybe introduce a new figure?
On Page 11, (analyses shown in Figure 3), how did you choose which parameters to pair together?
On Page 11, there is a discussion of the Hopf bifurcation that is interesting (and highlights how sensitivity twist is to our choice of parameters). It occurs to me that this has implications for modeling SCN oscillators as they recover from TTX treatment. The individual oscillator amplitude crashes and the oscillators lose synchrony during TTX and both recover after TTX is washed out. Is there anything you could add to the discussion about this?
On Page 12, in the paragraph about how overall twist effects depend not only on the parameter being studies, but also on the variable that is measured, I was reminded of early modeling studies that showed not-immediately intuitive results. In his 1995 fly model paper (A model of circadian oscillations in the Drosophila period protein, Proc. R. Soc. Lond. B, 1995), Goldbeter showed that the rate of (doubly-phosphorylated) PER degradation affected both period and amplitude and that increasing this rate led to PER amplitude increasing (rather than decreasing, as one might expect). This can be reasoned through, but it is another illustration of the complexity of the non-linear feedback and the relationship between period and amplitude. Is this worth bringing up here or on the discussion?
Figure 6. Why are there no yellow dots on the limit cycle for A and B? Also, maybe make the dot color on the limit cycle different from yellow (e.g. to blue), so that you can describe what is happening over time with even more clarity (i.e. that the simulation eventually reaches the "blue" dot after enough cycles).
Figure 8 and text referring to it. I appreciate that you relate twist to entrainment and PRCs. I think this is interesting and shows why twist is important to understand.
Page 16 first paragraph. "This suggests that single oscillator twist might also affect how coupling synchronizes a network of oscillators" is too weak. Given that coupling (M) and the perturbation used in Figure 7 (Z(t)) both affect the model the same way (they both are added to the rate equation for the x variable), Figure 7 shows that the PRCs are affected by twist. Not only are they affected, they are affected dramatically - Figures 7K and 7L look like they are opposite each other (i.e. flipped sign). This means we are guaranteed that the oscillators will respond differently to intercellular signaling and that that the ensemble oscillation will be different.
Page 16, in the discussion of the ensemble period, we see that for this model that positive twist leads to longer periods (and negative twist to shorter). What is missing from this discussion is that the relative phase of the signal and of the PRC of the receiver determine the period of the ensemble. Positive twist leads to longer periods because the mean field is determined by the value of the x variable. If it were determined by a variable with the opposite phase, then we would have the opposite effect. This affects the related paragraph in the Discussion as well.
Page 17. The meaning of "For this reason we have not termed twist, in the context of conservative oscillations, parametric or phase space" wasn't clear to me. I think you mean you can't study twist in non-limit-cycle oscillators in the same way you can for limit cycle oscillators. Is that correct?

Is the work clearly and accurately presented and does it cite the current literature?

Yes
Is the study design appropriate and is the work technically sound?

Yes
Are sufficient details of methods and analysis provided to allow replication by others?

Yes
If applicable, is the statistical analysis and its interpretation appropriate?

Yes
Are all the source data underlying the results available to ensure full reproducibility?

Yes
Are the conclusions drawn adequately supported by the results?

Yes

Competing Interests: No competing interests were disclosed.

Reviewer Expertise: I study the response of limit cycle oscillators to signals and sensitivity analysis involving isochrons.

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.

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Author Response 20 Jun 2024

Marta del Olmo, Institute for Theoretical Biology, Charité Universitätsmedizin Berlin, Philippstr. 13, 10115, Germany

20 Jun 2024

Author Response
Reviewer 2:
Stephanie Taylor, Department of Computer Science, Colby College, Waterville, Maine, USA

We thank all three reviewers for their constructive comments and literature suggestions, which have helped improve ... Continue reading
Reviewer 2:
Stephanie Taylor, Department of Computer Science, Colby College, Waterville, Maine, USA

We thank all three reviewers for their constructive comments and literature suggestions, which have helped improve our manuscript greatly. Below we give a detailed response to the individual comments.

In this article, the authors describe the phenomenon of twist (period-amplitude inter-dependence) using a range of differential equation oscillator models, ultimately with a view to understanding circadian clocks. The simplest models (harmonic and Duffing) are used to demonstrate that nonlinearity introduces a relationship between amplitude and period. The authors then switch to limit-cycle oscillators, because limit-cycle oscillators are more appropriate for studying circadian clocks. They use two approaches to studying twist. The first, referred to as "parametric twist", computes the periods and amplitudes of oscillators with varied parameter values, creating graphs that relates period to amplitude (and thus demonstrate twist - if the correlation is positive, there is positive twist, and negative correlation indicates negative twist). This parametric twist analysis is performed using two limit cycle models, one with one feedback loop and one with more (synergistic) loops. The second, referred to as "phase space twist", examines the response of a single oscillator to a temporary change in one parameter, mimicking signaling. A separate limit-cycle model (a Poincare oscillator) is used for this analysis. Important to the discussion of phase space twist is the concept of isochrons, which are useful for understanding the effects of perturbations on the phase and speed of the clock (does it speed up or slow down the clock?) and it is easy to see how amplitude and phase response are related, when we look at how curved the isochrons are. The authors relate these analyses to phase response curves, recovery from jetlag, the range of entrainment, and temperature compensation and conclude, that although measuring amplitude and understanding relationship between amplitude and period are highly complicated, it is nonetheless important to be mindful of the role of amplitude and its correlation to period when studying clocks.

On the one hand, I really appreciated that the authors chose models that best illustrated the points and that the paper is organized to go from simple ideas to more complex ideas. It was helpful to have the initial explanation of twist using a duffing oscillator (which has non-linearity) as a contrast to a harmonic oscillator (which doesn't). It provided a nice introduction to the cause of twist and helped me to develop an intuition. Then, it made sense to use a very simple (3-ODE Goodwin oscillator) as the first example of a limit cycle oscillator. Since it has so few parameters, it was possible to perturb them one at a time and in pairs to illustrate a good portion of the possibilities for how one might observe heterogeneity in the system. Then, moving to Almeida's oscillator demonstrated how quickly the analysis can become very complex. All of this provides a nice progression of complexity. Then, we pivot a bit to the Poincare oscillator, to study twist in a more dynamical way - instead of permanently choosing different parameters, the authors model light signaling (at least, initially) and study the effects of the model returning to its limit cycle after the perturbation is over. This model is particularly appropriate for studying the dynamics of recovery because it is a 2-ode model (so we can examine it in "phase space", which is helpful for visualizing the clock) and because there is a parameter that controls how related period and amplitude are. It is also straight-forward to adjust how curved the isochrons are, which makes illustrating twist very clear. Then, after studying the response of a single oscillator to light, they use the same model to study the effects of intercellular signaling and the effects of different twist on the properties of the ensemble oscillation (e.g. how long the ensemble period is in contrast to that of individual oscillators).

On the other hand, I would have appreciated a more explicit connection between the discussion of "parametric twist" and "phase space twist". They are both due to changes in parameter values causing changes in period and amplitude. In the one case, we measure the period and amplitude of oscillators with different (constant) parameter values. In the second case, we measure instantaneous changes in period and amplitude due to temporary changes in parameter values (I am referring to Z(t) as a parameter here). In both cases, the shape of the isochrons in the unperturbed condition should help us to predict what will happen in the perturbed condition. There are two challenges with making that point clear with the Goodwin and Almeida models - 1) the isochrons aren't lines, they are multi-dimensional manifolds, and 2) the isochrons in the unperturbed and perturbed systems are different. So I am asking for too much to relate the two discussions by using images of isochrons, but a more explicit comparison would help the reader understand how the analyses are related to each other.

We thank the reviewer for this insightful observation that we have also spent a lot of time thinking about, and for which we lack a clear answer. However, we aim to shed light on some thoughts that may help clarify the distinctions (and similarities) between parametric and phase space twist.

The concept “twist” was historically coined on conservative, non-limit cycle oscillations. It was first described in the Duffing oscillator [1] –an adaptation of the harmonic oscillator with an additional nonlinear term. This nonlinear term introduces unique properties to the system, as depicted in our Figure 1 from the manuscript: whereas harmonic, linear oscillators have no amplitude-period co-dependencies, the introduction of nonlinearities in the harmonic oscillator result in amplitude-period correlations, i.e. twist. Unlike limit cycle oscillators, non-limit cycle oscillators are conservative and do not dissipate energy. In more practical terms, this means that the initial conditions persist indefinitely, as no energy is dissipated, and thus the oscillator’s amplitude is defined by initial conditions (or by the energy of the system).

Most biological oscillations, however, are of limit cycle type. Limit cycles are dissipative, and different initial conditions relax (i.e., lose energy) and converge to the stable cycle, characterized by a given period and amplitude. In limit cycle oscillations, twist has to be studied in a different way than it is studied for non-limit cycle oscillators. Because different initial conditions do not produce different cycles, if we want to study how amplitudes and periods correlate in different cycles, one has to study this by altering parameters in the model. This is what we have termed parametric twist: the amplitude-period correlations arising from changes in model parameters, thus altering the original limit cycle slightly.

Another form of amplitude-period correlations can occur within an oscillator’s time series: what we term phase space twist. In a perfect stable limit cycle, period and amplitude do not change from cycle to cycle. But upon its interaction with the environment, the oscillator encounters perturbations that modify its amplitude and period transiently. The transient correlations between period and amplitude that appear as the oscillator relaxes and “adapts” to the perturbation is what we have termed “phase space twist” throughout our analyses. This is related to Winfree’s “invention” of isochrones in phase space [2]. Such “transient” correlations (i.e., phase space twist), have implications for PRCs, entrainment and coupling, as shown in Figure 7 and Figure 9.

In the realm of conservative systems, one does not need the distinction between phase space twist and parametric twist, as isochrones lack the relevance that they hold in limit cycles. It is for this reason that we have exclusively introduced these terms in the “limit cycle world”. While in phase space twist, isochrones are well defined manifolds, in the “world of parametric twist”, each new limit cycle has its own properties and isochrone manifolds, which makes the manifolds of parametric twist very difficult to visualize graphically.

In the analysis of the modified Poincare oscillator, the term "twist" is used differently than it is in the "parametric twist" analysis. In the parametric analysis, the twist captures the full effects of the perturbations (we plot the measured periods and amplitudes across the heterogeneous set of oscillators). In the phase space analysis, "twist" refers to a particular parameter \epsilon that, when non-zero, introduces a dependence of the speed and amplitude on each other. However, that twist parameter (\epsilon) is not the only parameter that affects the skewness of the isochrons. The relaxation rate \lambda also affects the skewness of the isochrons. And the skewness of the isochrons affects the changes in speed and amplitude when the oscillator is signaled (i.e. a parameter is temporarily perturbed). So, if we were to measure the effects on period and amplitude (as is done with figures 7G, H, and I), that reflects that values of both \lambda and \epsilon. When I was thinking about how to relate the "parametric twist" and "phase twist" analysis, I was looking for aspects of figures 3 (Gonze model), 5 (Almeida model), and 7 (poincare model) to relate and it seemed to me that that the subfigures that related the amplitude of the oscillator to its period (3C, 3D, 3E, 3F, 3G, 3H, 5B, 5B, 5E, 7G, 7H, 7I) should be the figures that I can connect. They are labeled as twist in figures 3 and 5. In Figure 7, they show an approximation of instantaneous amplitude and period (rather than steady-state amplitude and period), so it isn't a perfect 1-1 mapping, but it is still the closest that makes sense to me. And in all of these cases, the output could be labeled twist if \epsilon didn't get to claim that label. I think it would make the concept of twist clearer to the reader if it were consistent across both analyses that twist is the "effect" we can measure. My reading of the text as it is currently written is that twist is a phenomenon we can measure in "parametric twist" analyses and that twist is a parameter that causes one change in the isochrons (but not all of the changes that can happen) that ultimately leads to something we can measure. Also, labeling the parameter as twist in the Poincare oscillator makes it seem like we need to use the Poincare oscillator to perform a "phase space twist" analysis. But we aren't limited to the Poincare oscillator for such analyses.

We acknowledge the reviewer's valid point regarding the unclear distinction between the \epsilon parameter and phase space twist effects, which sometimes led to confusion. To address this, we now explicitly define phase space twist as the observable amplitude-period correlations in an oscillator's response to a perturbation—the measurable 'phenotype'. The reviewer is right that in the Poincaré oscillator, phase space twist effects are a consequence of the \epsilon parameter. However, other models might also display such correlations without the explicit \epsilon parameter. We have clarified this in the Methods and in the Discussion.

Moreover, as the reviewer correctly points out, \epsilon is not the only parameter influencing the skewness of isochrones, also \lambda plays a role. Thus, both \epsilon and \lambda modulate the extent of phase space twist effects that become apparent in an oscillator as it encounters and responds to a perturbation. We have revised Figure 8 in the manuscript and added panels to show the phase space twist effects for oscillators with the same \epsilon but changing \lambda values. We now show how the correlation between both variables decreases for oscillators with higher amplitude relaxation rates \lambda.

We have revised the manuscript to consistently refer to the \epsilon parameter as the "twist parameter \epsilon" (like in prior studies [3]), clarifying that phase space twist denotes the amplitude-period correlations emerging from varying \epsilon and \lambda values.

Specific Points:

It would be helpful to have a little more framing text that indicates the different models are used to make different points.

We have justified the use of the different models.
I think of the "phase space twist" analysis as analysis of recovery from temporary changes to parameters and "parametric twist" as analysis of permanent differences in parameters. If that is the way you intend readers to think about it, then I would suggest changing the names to reflect those ideas more directly. Right now, the term "phase space twist" stands out as a highly specific analysis - the model must be 2D so you can draw the isochrons. But I think that is limiting. I totally embrace the idea that there is both heterogeneity among oscillators and temporary changes due to intercellular signaling, so I would love to see it characterized that way more explicitly. What about "twist due to heterogeneity" and "twist due to signaling"? They aren't quite zippy enough and rely on my characterization of twist as a measurable phenomenon, so I don't push those terms exactly.

In response to the reviewer’s on-point comment, we have clarified what are the “practical” implications of parametric and phase space twist throughout the revised text, but without changing the terminology. We wish to highlight here again, that whereas phase space twist refers to the transient amplitude-period changes that a limit cycle oscillator suffers when it encounters a perturbation (with consequences for coupling and entrainment), the concept of parametric twist “lives” in the domain of oscillator heterogeneity. Although parametric twist might be an intrinsic property of an oscillator (i.e. cells of the choroid plexus show negative twist; whereas U-2 OS cells are positive twist oscillators), it only becomes evident in heterogeneous systems, when different parameter schemes (i.e. oscillator heterogeneity) come into play.

On page 8, when you are introducing isochrons, it would be helpful to have a figure to refer to. The most natural figure is Figure 7, but that would force you to make an unnatural re-ordering of the figures. So maybe introduce a new figure?

We refer to the section from the Results already in the Materials and Methods, when we introduce the definition of isochrones. We opted against adding a new figure in the Methods section because, in our eyes, it would be redundant with the current Figure 7. We believe that this figure is more needed in the Results section, where we emphasize in the text how isochrones and the parameters \epsilon and \lambda can affect the acceleration at which a perturbation relaxes back to the limit cycle.

On Page 11, (analyses shown in Figure 3), how did you choose which parameters to pair together?

We paired all degradation parameters with each other, namely k2 with k4, k4 with k6 and k6 with k2, but we only show two representative pairs. We have clarified that in the revised text and in the caption of Figure 3.

On Page 11, there is a discussion of the Hopf bifurcation that is interesting (and highlights how sensitivity twist is to our choice of parameters). It occurs to me that this has implications for modeling SCN oscillators as they recover from TTX treatment. The individual oscillator amplitude crashes and the oscillators lose synchrony during TTX and both recover after TTX is washed out. Is there anything you could add to the discussion about this?

We thank the reviewer for this observation. One of the messages of Figure 4 is indeed that parametric twist in an ensemble of oscillators is sensitive to the default parameter set of an individual oscillator. However, it is important to highlight that the analysis in that figure refers to uncoupled oscillators. The results shown in the Figure could be regarded as a population of oscillators with heterogeneous k4 values. Depending on how ‘spreaded’ such heterogeneity is, the expected parametric twist will be different. While a population of oscillators with small deviations around the default value k4=0.35, let’s say ±5%, would likely result in a population where one would almost have no (or partly positive) parametric twist (see Figure 4C, the variation in k4 would result in very minor differences in periods and peak-trough distances), a population of oscillators with larger heterogeneity in their k4 values would span a larger region of the bubble, resulting in complex parametric twist effects (both positive and negative, see Figure 4C).

Coupling in the SCN (or in any other oscillatory network) is expected to reduce the phenotypic differences in their periods and amplitude distributions, reducing the period heterogeneity (at least to some extent) and increasing amplitude due to resonance effects [4]. For this reason, we would expect that, upon overcritical coupling, the population of uncoupled oscillators with heterogeneous k4 values oscillates at a common period and amplitude. In other words, coupling would reduce, at least to some extent, the degree of parametric twist in a population of heterogeneous oscillators. Upon disruption of inter-oscillator coupling, such as TTX treatment in the SCN, one would expect the oscillators to return to their ‘original’ degree of parametric twist. We have added a short paragraph on the Discussion on how coupling is expected to affect parametric twist.

On Page 12, in the paragraph about how overall twist effects depend not only on the parameter being studied, but also on the variable that is measured, I was reminded of early modeling studies that showed not-immediately intuitive results. In his 1995 fly model paper (A model of circadian oscillations in the Drosophila period protein, Proc. R. Soc. Lond. B, 1995), Goldbeter showed that the rate of (doubly-phosphorylated) PER degradation affected both period and amplitude and that increasing this rate led to PER amplitude increasing (rather than decreasing, as one might expect). This can be reasoned through, but it is another illustration of the complexity of the non-linear feedback and the relationship between period and amplitude. Is this worth bringing up here or on the discussion?

We have discussed this point (and added the respective reference [5]) in the discussion and highlighted that different oscillator models (which have different nonlinear dependencies) result in different parametric twist effects. For this reason, it is important to know what is the type of twist in a population of oscillators (experimentally), to be able to devise a model that captures such properties.

Figure 6. Why are there no yellow dots on the limit cycle for A and B? Also, maybe make the dot color on the limit cycle different from yellow (e.g. to blue), so that you can describe what is happening over time with even more clarity (i.e. that the simulation eventually reaches the "blue" dot after enough cycles).

We have added yellow dots on the limit cycles for A and B. We have revised the text to more clearly describe what happens over time.

Figure 8 and text referring to it. I appreciate that you relate the twist to entrainment and PRCs. I think this is interesting and shows why twist is important to understand.

We would like to thank the reviewer for the positive comment.

Page 16 first paragraph. "This suggests that single oscillator twist might also affect how coupling synchronizes a network of oscillators" is too weak. Given that coupling (M) and the perturbation used in Figure 7 (Z(t)) both affect the model the same way (they both are added to the rate equation for the x variable), Figure 7 shows that the PRCs are affected by twist. Not only are they affected, they are affected dramatically - Figures 7K and 7L look like they are opposite each other (i.e. flipped sign). This means we are guaranteed that the oscillators will respond differently to intercellular signaling and that the ensemble oscillation will be different.

We thank the reviewer for the suggestion. We have now strengthened that point in the revised manuscript.

Page 16, in the discussion of the ensemble period, we see that for this model that positive twist leads to longer periods (and negative twist to shorter). What is missing from this discussion is that the relative phase of the signal and of the PRC of the receiver determine the period of the ensemble. Positive twist leads to longer periods because the mean field is determined by the value of the x variable. If it were determined by a variable with the opposite phase, then we would have the opposite effect. This affects the related paragraph in the Discussion as well.

We appreciate the reviewer's clarification regarding the dependence of phase on the direction of pulse-like perturbations, and we agree with the observation that pulses of opposite directions induce different phase shifts.

However, in Figure 9 we present analyses that offer a comparable scenario to the dynamics observed in chronotypes, albeit without an external zeitgeber. Instead, we focus on examining the long-term effects on the rhythmic properties of coupled oscillators. First, we show that mean-field coupling between two identical oscillators leads to amplitude expansions (Supplementary Figure S2) and period alterations dependent on the twist parameter \epsilon (Figure 9A).

Subsequently, we couple three Poincaré oscillators, which differ only in their \epsilon values, via their mean field (the arrow in Figure 9B indicates when mean-field coupling is ‘turned on’). We observe an increase in the amplitude of individual oscillators (consistent with Supplementary Figure S2). Although the periods of all three oscillators (as well as that of the mean-field) synchronize due to coupling, their phases relative to the mean field rhythm align with our predictions from Figure 9A and the theory from chronotypes: the oscillator with negative \epsilon exhibits an advanced rhythm compared to the mean field, while the oscillator with positive \epsilon displays a delayed rhythm relative to the mean field. These results remain consistent regardless of whether mean-field coupling is introduced in the x or y coordinate of the Poincaré oscillator.

The observation from Figure 9B is consistent with our findings in Figure 9A, where the phase delay of the oscillator with positive \epsilon aligns with this clock displaying longer periods upon coupling. This phenomenon resonates with chronotypes, where individuals with longer free-running periods have later phases of entrainment.

Page 17. The meaning of "For this reason we have not termed twist, in the context of conservative oscillations, parametric or phase space" wasn't clear to me. I think you mean you can't study twist in non-limit-cycle oscillators in the same way you can for limit cycle oscillators. Is that correct?

That is correct (also related to the first point of the report). We have clarified it throughout the revised manuscript.

References

Duffing, G. (1918). Erzwungene Schwingungen bei veränderlicher Eigenfrequenz und ihre technische Bedeutung (No. 41-42). Vieweg.

Winfree, A. T. (1980). The geometry of biological time (Vol. 2). New York: Springer.

Myung, J., Schmal, C., Hong, S., Tsukizawa, Y., Rose, P., Zhang, Y., ... & Takumi, T. (2018). The choroid plexus is an important circadian clock component. Nature Communications, 9(1), 1062.

Schmal, C., Herzog, E. D., & Herzel, H. (2018). Measuring relative coupling strength in circadian systems. Journal of Biological Rhythms, 33(1), 84-98.

Goldbeter, A. (1995). A model for circadian oscillations in the Drosophila period protein (PER). Proceedings of the Royal Society of London. Series B: Biological Sciences, 261(1362), 319-324.
Reviewer 2:
Stephanie Taylor, Department of Computer Science, Colby College, Waterville, Maine, USA

We thank all three reviewers for their constructive comments and literature suggestions, which have helped improve our manuscript greatly. Below we give a detailed response to the individual comments.

In this article, the authors describe the phenomenon of twist (period-amplitude inter-dependence) using a range of differential equation oscillator models, ultimately with a view to understanding circadian clocks. The simplest models (harmonic and Duffing) are used to demonstrate that nonlinearity introduces a relationship between amplitude and period. The authors then switch to limit-cycle oscillators, because limit-cycle oscillators are more appropriate for studying circadian clocks. They use two approaches to studying twist. The first, referred to as "parametric twist", computes the periods and amplitudes of oscillators with varied parameter values, creating graphs that relates period to amplitude (and thus demonstrate twist - if the correlation is positive, there is positive twist, and negative correlation indicates negative twist). This parametric twist analysis is performed using two limit cycle models, one with one feedback loop and one with more (synergistic) loops. The second, referred to as "phase space twist", examines the response of a single oscillator to a temporary change in one parameter, mimicking signaling. A separate limit-cycle model (a Poincare oscillator) is used for this analysis. Important to the discussion of phase space twist is the concept of isochrons, which are useful for understanding the effects of perturbations on the phase and speed of the clock (does it speed up or slow down the clock?) and it is easy to see how amplitude and phase response are related, when we look at how curved the isochrons are. The authors relate these analyses to phase response curves, recovery from jetlag, the range of entrainment, and temperature compensation and conclude, that although measuring amplitude and understanding relationship between amplitude and period are highly complicated, it is nonetheless important to be mindful of the role of amplitude and its correlation to period when studying clocks.

On the one hand, I really appreciated that the authors chose models that best illustrated the points and that the paper is organized to go from simple ideas to more complex ideas. It was helpful to have the initial explanation of twist using a duffing oscillator (which has non-linearity) as a contrast to a harmonic oscillator (which doesn't). It provided a nice introduction to the cause of twist and helped me to develop an intuition. Then, it made sense to use a very simple (3-ODE Goodwin oscillator) as the first example of a limit cycle oscillator. Since it has so few parameters, it was possible to perturb them one at a time and in pairs to illustrate a good portion of the possibilities for how one might observe heterogeneity in the system. Then, moving to Almeida's oscillator demonstrated how quickly the analysis can become very complex. All of this provides a nice progression of complexity. Then, we pivot a bit to the Poincare oscillator, to study twist in a more dynamical way - instead of permanently choosing different parameters, the authors model light signaling (at least, initially) and study the effects of the model returning to its limit cycle after the perturbation is over. This model is particularly appropriate for studying the dynamics of recovery because it is a 2-ode model (so we can examine it in "phase space", which is helpful for visualizing the clock) and because there is a parameter that controls how related period and amplitude are. It is also straight-forward to adjust how curved the isochrons are, which makes illustrating twist very clear. Then, after studying the response of a single oscillator to light, they use the same model to study the effects of intercellular signaling and the effects of different twist on the properties of the ensemble oscillation (e.g. how long the ensemble period is in contrast to that of individual oscillators).

On the other hand, I would have appreciated a more explicit connection between the discussion of "parametric twist" and "phase space twist". They are both due to changes in parameter values causing changes in period and amplitude. In the one case, we measure the period and amplitude of oscillators with different (constant) parameter values. In the second case, we measure instantaneous changes in period and amplitude due to temporary changes in parameter values (I am referring to Z(t) as a parameter here). In both cases, the shape of the isochrons in the unperturbed condition should help us to predict what will happen in the perturbed condition. There are two challenges with making that point clear with the Goodwin and Almeida models - 1) the isochrons aren't lines, they are multi-dimensional manifolds, and 2) the isochrons in the unperturbed and perturbed systems are different. So I am asking for too much to relate the two discussions by using images of isochrons, but a more explicit comparison would help the reader understand how the analyses are related to each other.

We thank the reviewer for this insightful observation that we have also spent a lot of time thinking about, and for which we lack a clear answer. However, we aim to shed light on some thoughts that may help clarify the distinctions (and similarities) between parametric and phase space twist.

The concept “twist” was historically coined on conservative, non-limit cycle oscillations. It was first described in the Duffing oscillator [1] –an adaptation of the harmonic oscillator with an additional nonlinear term. This nonlinear term introduces unique properties to the system, as depicted in our Figure 1 from the manuscript: whereas harmonic, linear oscillators have no amplitude-period co-dependencies, the introduction of nonlinearities in the harmonic oscillator result in amplitude-period correlations, i.e. twist. Unlike limit cycle oscillators, non-limit cycle oscillators are conservative and do not dissipate energy. In more practical terms, this means that the initial conditions persist indefinitely, as no energy is dissipated, and thus the oscillator’s amplitude is defined by initial conditions (or by the energy of the system).

Most biological oscillations, however, are of limit cycle type. Limit cycles are dissipative, and different initial conditions relax (i.e., lose energy) and converge to the stable cycle, characterized by a given period and amplitude. In limit cycle oscillations, twist has to be studied in a different way than it is studied for non-limit cycle oscillators. Because different initial conditions do not produce different cycles, if we want to study how amplitudes and periods correlate in different cycles, one has to study this by altering parameters in the model. This is what we have termed parametric twist: the amplitude-period correlations arising from changes in model parameters, thus altering the original limit cycle slightly.

Another form of amplitude-period correlations can occur within an oscillator’s time series: what we term phase space twist. In a perfect stable limit cycle, period and amplitude do not change from cycle to cycle. But upon its interaction with the environment, the oscillator encounters perturbations that modify its amplitude and period transiently. The transient correlations between period and amplitude that appear as the oscillator relaxes and “adapts” to the perturbation is what we have termed “phase space twist” throughout our analyses. This is related to Winfree’s “invention” of isochrones in phase space [2]. Such “transient” correlations (i.e., phase space twist), have implications for PRCs, entrainment and coupling, as shown in Figure 7 and Figure 9.

In the realm of conservative systems, one does not need the distinction between phase space twist and parametric twist, as isochrones lack the relevance that they hold in limit cycles. It is for this reason that we have exclusively introduced these terms in the “limit cycle world”. While in phase space twist, isochrones are well defined manifolds, in the “world of parametric twist”, each new limit cycle has its own properties and isochrone manifolds, which makes the manifolds of parametric twist very difficult to visualize graphically.

In the analysis of the modified Poincare oscillator, the term "twist" is used differently than it is in the "parametric twist" analysis. In the parametric analysis, the twist captures the full effects of the perturbations (we plot the measured periods and amplitudes across the heterogeneous set of oscillators). In the phase space analysis, "twist" refers to a particular parameter \epsilon that, when non-zero, introduces a dependence of the speed and amplitude on each other. However, that twist parameter (\epsilon) is not the only parameter that affects the skewness of the isochrons. The relaxation rate \lambda also affects the skewness of the isochrons. And the skewness of the isochrons affects the changes in speed and amplitude when the oscillator is signaled (i.e. a parameter is temporarily perturbed). So, if we were to measure the effects on period and amplitude (as is done with figures 7G, H, and I), that reflects that values of both \lambda and \epsilon. When I was thinking about how to relate the "parametric twist" and "phase twist" analysis, I was looking for aspects of figures 3 (Gonze model), 5 (Almeida model), and 7 (poincare model) to relate and it seemed to me that that the subfigures that related the amplitude of the oscillator to its period (3C, 3D, 3E, 3F, 3G, 3H, 5B, 5B, 5E, 7G, 7H, 7I) should be the figures that I can connect. They are labeled as twist in figures 3 and 5. In Figure 7, they show an approximation of instantaneous amplitude and period (rather than steady-state amplitude and period), so it isn't a perfect 1-1 mapping, but it is still the closest that makes sense to me. And in all of these cases, the output could be labeled twist if \epsilon didn't get to claim that label. I think it would make the concept of twist clearer to the reader if it were consistent across both analyses that twist is the "effect" we can measure. My reading of the text as it is currently written is that twist is a phenomenon we can measure in "parametric twist" analyses and that twist is a parameter that causes one change in the isochrons (but not all of the changes that can happen) that ultimately leads to something we can measure. Also, labeling the parameter as twist in the Poincare oscillator makes it seem like we need to use the Poincare oscillator to perform a "phase space twist" analysis. But we aren't limited to the Poincare oscillator for such analyses.

We acknowledge the reviewer's valid point regarding the unclear distinction between the \epsilon parameter and phase space twist effects, which sometimes led to confusion. To address this, we now explicitly define phase space twist as the observable amplitude-period correlations in an oscillator's response to a perturbation—the measurable 'phenotype'. The reviewer is right that in the Poincaré oscillator, phase space twist effects are a consequence of the \epsilon parameter. However, other models might also display such correlations without the explicit \epsilon parameter. We have clarified this in the Methods and in the Discussion.

Moreover, as the reviewer correctly points out, \epsilon is not the only parameter influencing the skewness of isochrones, also \lambda plays a role. Thus, both \epsilon and \lambda modulate the extent of phase space twist effects that become apparent in an oscillator as it encounters and responds to a perturbation. We have revised Figure 8 in the manuscript and added panels to show the phase space twist effects for oscillators with the same \epsilon but changing \lambda values. We now show how the correlation between both variables decreases for oscillators with higher amplitude relaxation rates \lambda.

We have revised the manuscript to consistently refer to the \epsilon parameter as the "twist parameter \epsilon" (like in prior studies [3]), clarifying that phase space twist denotes the amplitude-period correlations emerging from varying \epsilon and \lambda values.

Specific Points:

It would be helpful to have a little more framing text that indicates the different models are used to make different points.

We have justified the use of the different models.
I think of the "phase space twist" analysis as analysis of recovery from temporary changes to parameters and "parametric twist" as analysis of permanent differences in parameters. If that is the way you intend readers to think about it, then I would suggest changing the names to reflect those ideas more directly. Right now, the term "phase space twist" stands out as a highly specific analysis - the model must be 2D so you can draw the isochrons. But I think that is limiting. I totally embrace the idea that there is both heterogeneity among oscillators and temporary changes due to intercellular signaling, so I would love to see it characterized that way more explicitly. What about "twist due to heterogeneity" and "twist due to signaling"? They aren't quite zippy enough and rely on my characterization of twist as a measurable phenomenon, so I don't push those terms exactly.

In response to the reviewer’s on-point comment, we have clarified what are the “practical” implications of parametric and phase space twist throughout the revised text, but without changing the terminology. We wish to highlight here again, that whereas phase space twist refers to the transient amplitude-period changes that a limit cycle oscillator suffers when it encounters a perturbation (with consequences for coupling and entrainment), the concept of parametric twist “lives” in the domain of oscillator heterogeneity. Although parametric twist might be an intrinsic property of an oscillator (i.e. cells of the choroid plexus show negative twist; whereas U-2 OS cells are positive twist oscillators), it only becomes evident in heterogeneous systems, when different parameter schemes (i.e. oscillator heterogeneity) come into play.

On page 8, when you are introducing isochrons, it would be helpful to have a figure to refer to. The most natural figure is Figure 7, but that would force you to make an unnatural re-ordering of the figures. So maybe introduce a new figure?

We refer to the section from the Results already in the Materials and Methods, when we introduce the definition of isochrones. We opted against adding a new figure in the Methods section because, in our eyes, it would be redundant with the current Figure 7. We believe that this figure is more needed in the Results section, where we emphasize in the text how isochrones and the parameters \epsilon and \lambda can affect the acceleration at which a perturbation relaxes back to the limit cycle.

On Page 11, (analyses shown in Figure 3), how did you choose which parameters to pair together?

We paired all degradation parameters with each other, namely k2 with k4, k4 with k6 and k6 with k2, but we only show two representative pairs. We have clarified that in the revised text and in the caption of Figure 3.

On Page 11, there is a discussion of the Hopf bifurcation that is interesting (and highlights how sensitivity twist is to our choice of parameters). It occurs to me that this has implications for modeling SCN oscillators as they recover from TTX treatment. The individual oscillator amplitude crashes and the oscillators lose synchrony during TTX and both recover after TTX is washed out. Is there anything you could add to the discussion about this?

We thank the reviewer for this observation. One of the messages of Figure 4 is indeed that parametric twist in an ensemble of oscillators is sensitive to the default parameter set of an individual oscillator. However, it is important to highlight that the analysis in that figure refers to uncoupled oscillators. The results shown in the Figure could be regarded as a population of oscillators with heterogeneous k4 values. Depending on how ‘spreaded’ such heterogeneity is, the expected parametric twist will be different. While a population of oscillators with small deviations around the default value k4=0.35, let’s say ±5%, would likely result in a population where one would almost have no (or partly positive) parametric twist (see Figure 4C, the variation in k4 would result in very minor differences in periods and peak-trough distances), a population of oscillators with larger heterogeneity in their k4 values would span a larger region of the bubble, resulting in complex parametric twist effects (both positive and negative, see Figure 4C).

Coupling in the SCN (or in any other oscillatory network) is expected to reduce the phenotypic differences in their periods and amplitude distributions, reducing the period heterogeneity (at least to some extent) and increasing amplitude due to resonance effects [4]. For this reason, we would expect that, upon overcritical coupling, the population of uncoupled oscillators with heterogeneous k4 values oscillates at a common period and amplitude. In other words, coupling would reduce, at least to some extent, the degree of parametric twist in a population of heterogeneous oscillators. Upon disruption of inter-oscillator coupling, such as TTX treatment in the SCN, one would expect the oscillators to return to their ‘original’ degree of parametric twist. We have added a short paragraph on the Discussion on how coupling is expected to affect parametric twist.

On Page 12, in the paragraph about how overall twist effects depend not only on the parameter being studied, but also on the variable that is measured, I was reminded of early modeling studies that showed not-immediately intuitive results. In his 1995 fly model paper (A model of circadian oscillations in the Drosophila period protein, Proc. R. Soc. Lond. B, 1995), Goldbeter showed that the rate of (doubly-phosphorylated) PER degradation affected both period and amplitude and that increasing this rate led to PER amplitude increasing (rather than decreasing, as one might expect). This can be reasoned through, but it is another illustration of the complexity of the non-linear feedback and the relationship between period and amplitude. Is this worth bringing up here or on the discussion?

We have discussed this point (and added the respective reference [5]) in the discussion and highlighted that different oscillator models (which have different nonlinear dependencies) result in different parametric twist effects. For this reason, it is important to know what is the type of twist in a population of oscillators (experimentally), to be able to devise a model that captures such properties.

Figure 6. Why are there no yellow dots on the limit cycle for A and B? Also, maybe make the dot color on the limit cycle different from yellow (e.g. to blue), so that you can describe what is happening over time with even more clarity (i.e. that the simulation eventually reaches the "blue" dot after enough cycles).

We have added yellow dots on the limit cycles for A and B. We have revised the text to more clearly describe what happens over time.

Figure 8 and text referring to it. I appreciate that you relate the twist to entrainment and PRCs. I think this is interesting and shows why twist is important to understand.

We would like to thank the reviewer for the positive comment.

Page 16 first paragraph. "This suggests that single oscillator twist might also affect how coupling synchronizes a network of oscillators" is too weak. Given that coupling (M) and the perturbation used in Figure 7 (Z(t)) both affect the model the same way (they both are added to the rate equation for the x variable), Figure 7 shows that the PRCs are affected by twist. Not only are they affected, they are affected dramatically - Figures 7K and 7L look like they are opposite each other (i.e. flipped sign). This means we are guaranteed that the oscillators will respond differently to intercellular signaling and that the ensemble oscillation will be different.

We thank the reviewer for the suggestion. We have now strengthened that point in the revised manuscript.

Page 16, in the discussion of the ensemble period, we see that for this model that positive twist leads to longer periods (and negative twist to shorter). What is missing from this discussion is that the relative phase of the signal and of the PRC of the receiver determine the period of the ensemble. Positive twist leads to longer periods because the mean field is determined by the value of the x variable. If it were determined by a variable with the opposite phase, then we would have the opposite effect. This affects the related paragraph in the Discussion as well.

We appreciate the reviewer's clarification regarding the dependence of phase on the direction of pulse-like perturbations, and we agree with the observation that pulses of opposite directions induce different phase shifts.

However, in Figure 9 we present analyses that offer a comparable scenario to the dynamics observed in chronotypes, albeit without an external zeitgeber. Instead, we focus on examining the long-term effects on the rhythmic properties of coupled oscillators. First, we show that mean-field coupling between two identical oscillators leads to amplitude expansions (Supplementary Figure S2) and period alterations dependent on the twist parameter \epsilon (Figure 9A).

Subsequently, we couple three Poincaré oscillators, which differ only in their \epsilon values, via their mean field (the arrow in Figure 9B indicates when mean-field coupling is ‘turned on’). We observe an increase in the amplitude of individual oscillators (consistent with Supplementary Figure S2). Although the periods of all three oscillators (as well as that of the mean-field) synchronize due to coupling, their phases relative to the mean field rhythm align with our predictions from Figure 9A and the theory from chronotypes: the oscillator with negative \epsilon exhibits an advanced rhythm compared to the mean field, while the oscillator with positive \epsilon displays a delayed rhythm relative to the mean field. These results remain consistent regardless of whether mean-field coupling is introduced in the x or y coordinate of the Poincaré oscillator.

The observation from Figure 9B is consistent with our findings in Figure 9A, where the phase delay of the oscillator with positive \epsilon aligns with this clock displaying longer periods upon coupling. This phenomenon resonates with chronotypes, where individuals with longer free-running periods have later phases of entrainment.

Page 17. The meaning of "For this reason we have not termed twist, in the context of conservative oscillations, parametric or phase space" wasn't clear to me. I think you mean you can't study twist in non-limit-cycle oscillators in the same way you can for limit cycle oscillators. Is that correct?

That is correct (also related to the first point of the report). We have clarified it throughout the revised manuscript.

References

Duffing, G. (1918). Erzwungene Schwingungen bei veränderlicher Eigenfrequenz und ihre technische Bedeutung (No. 41-42). Vieweg.

Winfree, A. T. (1980). The geometry of biological time (Vol. 2). New York: Springer.

Myung, J., Schmal, C., Hong, S., Tsukizawa, Y., Rose, P., Zhang, Y., ... & Takumi, T. (2018). The choroid plexus is an important circadian clock component. Nature Communications, 9(1), 1062.

Schmal, C., Herzog, E. D., & Herzel, H. (2018). Measuring relative coupling strength in circadian systems. Journal of Biological Rhythms, 33(1), 84-98.

Goldbeter, A. (1995). A model for circadian oscillations in the Drosophila period protein (PER). Proceedings of the Royal Society of London. Series B: Biological Sciences, 261(1362), 319-324.
Competing Interests: The authors declare no conflict of interest. Close
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COMMENTS ON THIS REPORT

Author Response 20 Jun 2024

Marta del Olmo, Institute for Theoretical Biology, Charité Universitätsmedizin Berlin, Philippstr. 13, 10115, Germany

20 Jun 2024

Author Response
Reviewer 2:
Stephanie Taylor, Department of Computer Science, Colby College, Waterville, Maine, USA

We thank all three reviewers for their constructive comments and literature suggestions, which have helped improve ... Continue reading
Reviewer 2:
Stephanie Taylor, Department of Computer Science, Colby College, Waterville, Maine, USA

We thank all three reviewers for their constructive comments and literature suggestions, which have helped improve our manuscript greatly. Below we give a detailed response to the individual comments.

In this article, the authors describe the phenomenon of twist (period-amplitude inter-dependence) using a range of differential equation oscillator models, ultimately with a view to understanding circadian clocks. The simplest models (harmonic and Duffing) are used to demonstrate that nonlinearity introduces a relationship between amplitude and period. The authors then switch to limit-cycle oscillators, because limit-cycle oscillators are more appropriate for studying circadian clocks. They use two approaches to studying twist. The first, referred to as "parametric twist", computes the periods and amplitudes of oscillators with varied parameter values, creating graphs that relates period to amplitude (and thus demonstrate twist - if the correlation is positive, there is positive twist, and negative correlation indicates negative twist). This parametric twist analysis is performed using two limit cycle models, one with one feedback loop and one with more (synergistic) loops. The second, referred to as "phase space twist", examines the response of a single oscillator to a temporary change in one parameter, mimicking signaling. A separate limit-cycle model (a Poincare oscillator) is used for this analysis. Important to the discussion of phase space twist is the concept of isochrons, which are useful for understanding the effects of perturbations on the phase and speed of the clock (does it speed up or slow down the clock?) and it is easy to see how amplitude and phase response are related, when we look at how curved the isochrons are. The authors relate these analyses to phase response curves, recovery from jetlag, the range of entrainment, and temperature compensation and conclude, that although measuring amplitude and understanding relationship between amplitude and period are highly complicated, it is nonetheless important to be mindful of the role of amplitude and its correlation to period when studying clocks.

On the one hand, I really appreciated that the authors chose models that best illustrated the points and that the paper is organized to go from simple ideas to more complex ideas. It was helpful to have the initial explanation of twist using a duffing oscillator (which has non-linearity) as a contrast to a harmonic oscillator (which doesn't). It provided a nice introduction to the cause of twist and helped me to develop an intuition. Then, it made sense to use a very simple (3-ODE Goodwin oscillator) as the first example of a limit cycle oscillator. Since it has so few parameters, it was possible to perturb them one at a time and in pairs to illustrate a good portion of the possibilities for how one might observe heterogeneity in the system. Then, moving to Almeida's oscillator demonstrated how quickly the analysis can become very complex. All of this provides a nice progression of complexity. Then, we pivot a bit to the Poincare oscillator, to study twist in a more dynamical way - instead of permanently choosing different parameters, the authors model light signaling (at least, initially) and study the effects of the model returning to its limit cycle after the perturbation is over. This model is particularly appropriate for studying the dynamics of recovery because it is a 2-ode model (so we can examine it in "phase space", which is helpful for visualizing the clock) and because there is a parameter that controls how related period and amplitude are. It is also straight-forward to adjust how curved the isochrons are, which makes illustrating twist very clear. Then, after studying the response of a single oscillator to light, they use the same model to study the effects of intercellular signaling and the effects of different twist on the properties of the ensemble oscillation (e.g. how long the ensemble period is in contrast to that of individual oscillators).

On the other hand, I would have appreciated a more explicit connection between the discussion of "parametric twist" and "phase space twist". They are both due to changes in parameter values causing changes in period and amplitude. In the one case, we measure the period and amplitude of oscillators with different (constant) parameter values. In the second case, we measure instantaneous changes in period and amplitude due to temporary changes in parameter values (I am referring to Z(t) as a parameter here). In both cases, the shape of the isochrons in the unperturbed condition should help us to predict what will happen in the perturbed condition. There are two challenges with making that point clear with the Goodwin and Almeida models - 1) the isochrons aren't lines, they are multi-dimensional manifolds, and 2) the isochrons in the unperturbed and perturbed systems are different. So I am asking for too much to relate the two discussions by using images of isochrons, but a more explicit comparison would help the reader understand how the analyses are related to each other.

We thank the reviewer for this insightful observation that we have also spent a lot of time thinking about, and for which we lack a clear answer. However, we aim to shed light on some thoughts that may help clarify the distinctions (and similarities) between parametric and phase space twist.

The concept “twist” was historically coined on conservative, non-limit cycle oscillations. It was first described in the Duffing oscillator [1] –an adaptation of the harmonic oscillator with an additional nonlinear term. This nonlinear term introduces unique properties to the system, as depicted in our Figure 1 from the manuscript: whereas harmonic, linear oscillators have no amplitude-period co-dependencies, the introduction of nonlinearities in the harmonic oscillator result in amplitude-period correlations, i.e. twist. Unlike limit cycle oscillators, non-limit cycle oscillators are conservative and do not dissipate energy. In more practical terms, this means that the initial conditions persist indefinitely, as no energy is dissipated, and thus the oscillator’s amplitude is defined by initial conditions (or by the energy of the system).

Most biological oscillations, however, are of limit cycle type. Limit cycles are dissipative, and different initial conditions relax (i.e., lose energy) and converge to the stable cycle, characterized by a given period and amplitude. In limit cycle oscillations, twist has to be studied in a different way than it is studied for non-limit cycle oscillators. Because different initial conditions do not produce different cycles, if we want to study how amplitudes and periods correlate in different cycles, one has to study this by altering parameters in the model. This is what we have termed parametric twist: the amplitude-period correlations arising from changes in model parameters, thus altering the original limit cycle slightly.

Another form of amplitude-period correlations can occur within an oscillator’s time series: what we term phase space twist. In a perfect stable limit cycle, period and amplitude do not change from cycle to cycle. But upon its interaction with the environment, the oscillator encounters perturbations that modify its amplitude and period transiently. The transient correlations between period and amplitude that appear as the oscillator relaxes and “adapts” to the perturbation is what we have termed “phase space twist” throughout our analyses. This is related to Winfree’s “invention” of isochrones in phase space [2]. Such “transient” correlations (i.e., phase space twist), have implications for PRCs, entrainment and coupling, as shown in Figure 7 and Figure 9.

In the realm of conservative systems, one does not need the distinction between phase space twist and parametric twist, as isochrones lack the relevance that they hold in limit cycles. It is for this reason that we have exclusively introduced these terms in the “limit cycle world”. While in phase space twist, isochrones are well defined manifolds, in the “world of parametric twist”, each new limit cycle has its own properties and isochrone manifolds, which makes the manifolds of parametric twist very difficult to visualize graphically.

In the analysis of the modified Poincare oscillator, the term "twist" is used differently than it is in the "parametric twist" analysis. In the parametric analysis, the twist captures the full effects of the perturbations (we plot the measured periods and amplitudes across the heterogeneous set of oscillators). In the phase space analysis, "twist" refers to a particular parameter \epsilon that, when non-zero, introduces a dependence of the speed and amplitude on each other. However, that twist parameter (\epsilon) is not the only parameter that affects the skewness of the isochrons. The relaxation rate \lambda also affects the skewness of the isochrons. And the skewness of the isochrons affects the changes in speed and amplitude when the oscillator is signaled (i.e. a parameter is temporarily perturbed). So, if we were to measure the effects on period and amplitude (as is done with figures 7G, H, and I), that reflects that values of both \lambda and \epsilon. When I was thinking about how to relate the "parametric twist" and "phase twist" analysis, I was looking for aspects of figures 3 (Gonze model), 5 (Almeida model), and 7 (poincare model) to relate and it seemed to me that that the subfigures that related the amplitude of the oscillator to its period (3C, 3D, 3E, 3F, 3G, 3H, 5B, 5B, 5E, 7G, 7H, 7I) should be the figures that I can connect. They are labeled as twist in figures 3 and 5. In Figure 7, they show an approximation of instantaneous amplitude and period (rather than steady-state amplitude and period), so it isn't a perfect 1-1 mapping, but it is still the closest that makes sense to me. And in all of these cases, the output could be labeled twist if \epsilon didn't get to claim that label. I think it would make the concept of twist clearer to the reader if it were consistent across both analyses that twist is the "effect" we can measure. My reading of the text as it is currently written is that twist is a phenomenon we can measure in "parametric twist" analyses and that twist is a parameter that causes one change in the isochrons (but not all of the changes that can happen) that ultimately leads to something we can measure. Also, labeling the parameter as twist in the Poincare oscillator makes it seem like we need to use the Poincare oscillator to perform a "phase space twist" analysis. But we aren't limited to the Poincare oscillator for such analyses.

We acknowledge the reviewer's valid point regarding the unclear distinction between the \epsilon parameter and phase space twist effects, which sometimes led to confusion. To address this, we now explicitly define phase space twist as the observable amplitude-period correlations in an oscillator's response to a perturbation—the measurable 'phenotype'. The reviewer is right that in the Poincaré oscillator, phase space twist effects are a consequence of the \epsilon parameter. However, other models might also display such correlations without the explicit \epsilon parameter. We have clarified this in the Methods and in the Discussion.

Moreover, as the reviewer correctly points out, \epsilon is not the only parameter influencing the skewness of isochrones, also \lambda plays a role. Thus, both \epsilon and \lambda modulate the extent of phase space twist effects that become apparent in an oscillator as it encounters and responds to a perturbation. We have revised Figure 8 in the manuscript and added panels to show the phase space twist effects for oscillators with the same \epsilon but changing \lambda values. We now show how the correlation between both variables decreases for oscillators with higher amplitude relaxation rates \lambda.

We have revised the manuscript to consistently refer to the \epsilon parameter as the "twist parameter \epsilon" (like in prior studies [3]), clarifying that phase space twist denotes the amplitude-period correlations emerging from varying \epsilon and \lambda values.

Specific Points:

It would be helpful to have a little more framing text that indicates the different models are used to make different points.

We have justified the use of the different models.
I think of the "phase space twist" analysis as analysis of recovery from temporary changes to parameters and "parametric twist" as analysis of permanent differences in parameters. If that is the way you intend readers to think about it, then I would suggest changing the names to reflect those ideas more directly. Right now, the term "phase space twist" stands out as a highly specific analysis - the model must be 2D so you can draw the isochrons. But I think that is limiting. I totally embrace the idea that there is both heterogeneity among oscillators and temporary changes due to intercellular signaling, so I would love to see it characterized that way more explicitly. What about "twist due to heterogeneity" and "twist due to signaling"? They aren't quite zippy enough and rely on my characterization of twist as a measurable phenomenon, so I don't push those terms exactly.

In response to the reviewer’s on-point comment, we have clarified what are the “practical” implications of parametric and phase space twist throughout the revised text, but without changing the terminology. We wish to highlight here again, that whereas phase space twist refers to the transient amplitude-period changes that a limit cycle oscillator suffers when it encounters a perturbation (with consequences for coupling and entrainment), the concept of parametric twist “lives” in the domain of oscillator heterogeneity. Although parametric twist might be an intrinsic property of an oscillator (i.e. cells of the choroid plexus show negative twist; whereas U-2 OS cells are positive twist oscillators), it only becomes evident in heterogeneous systems, when different parameter schemes (i.e. oscillator heterogeneity) come into play.

On page 8, when you are introducing isochrons, it would be helpful to have a figure to refer to. The most natural figure is Figure 7, but that would force you to make an unnatural re-ordering of the figures. So maybe introduce a new figure?

We refer to the section from the Results already in the Materials and Methods, when we introduce the definition of isochrones. We opted against adding a new figure in the Methods section because, in our eyes, it would be redundant with the current Figure 7. We believe that this figure is more needed in the Results section, where we emphasize in the text how isochrones and the parameters \epsilon and \lambda can affect the acceleration at which a perturbation relaxes back to the limit cycle.

On Page 11, (analyses shown in Figure 3), how did you choose which parameters to pair together?

We paired all degradation parameters with each other, namely k2 with k4, k4 with k6 and k6 with k2, but we only show two representative pairs. We have clarified that in the revised text and in the caption of Figure 3.

On Page 11, there is a discussion of the Hopf bifurcation that is interesting (and highlights how sensitivity twist is to our choice of parameters). It occurs to me that this has implications for modeling SCN oscillators as they recover from TTX treatment. The individual oscillator amplitude crashes and the oscillators lose synchrony during TTX and both recover after TTX is washed out. Is there anything you could add to the discussion about this?

We thank the reviewer for this observation. One of the messages of Figure 4 is indeed that parametric twist in an ensemble of oscillators is sensitive to the default parameter set of an individual oscillator. However, it is important to highlight that the analysis in that figure refers to uncoupled oscillators. The results shown in the Figure could be regarded as a population of oscillators with heterogeneous k4 values. Depending on how ‘spreaded’ such heterogeneity is, the expected parametric twist will be different. While a population of oscillators with small deviations around the default value k4=0.35, let’s say ±5%, would likely result in a population where one would almost have no (or partly positive) parametric twist (see Figure 4C, the variation in k4 would result in very minor differences in periods and peak-trough distances), a population of oscillators with larger heterogeneity in their k4 values would span a larger region of the bubble, resulting in complex parametric twist effects (both positive and negative, see Figure 4C).

Coupling in the SCN (or in any other oscillatory network) is expected to reduce the phenotypic differences in their periods and amplitude distributions, reducing the period heterogeneity (at least to some extent) and increasing amplitude due to resonance effects [4]. For this reason, we would expect that, upon overcritical coupling, the population of uncoupled oscillators with heterogeneous k4 values oscillates at a common period and amplitude. In other words, coupling would reduce, at least to some extent, the degree of parametric twist in a population of heterogeneous oscillators. Upon disruption of inter-oscillator coupling, such as TTX treatment in the SCN, one would expect the oscillators to return to their ‘original’ degree of parametric twist. We have added a short paragraph on the Discussion on how coupling is expected to affect parametric twist.

On Page 12, in the paragraph about how overall twist effects depend not only on the parameter being studied, but also on the variable that is measured, I was reminded of early modeling studies that showed not-immediately intuitive results. In his 1995 fly model paper (A model of circadian oscillations in the Drosophila period protein, Proc. R. Soc. Lond. B, 1995), Goldbeter showed that the rate of (doubly-phosphorylated) PER degradation affected both period and amplitude and that increasing this rate led to PER amplitude increasing (rather than decreasing, as one might expect). This can be reasoned through, but it is another illustration of the complexity of the non-linear feedback and the relationship between period and amplitude. Is this worth bringing up here or on the discussion?

We have discussed this point (and added the respective reference [5]) in the discussion and highlighted that different oscillator models (which have different nonlinear dependencies) result in different parametric twist effects. For this reason, it is important to know what is the type of twist in a population of oscillators (experimentally), to be able to devise a model that captures such properties.

Figure 6. Why are there no yellow dots on the limit cycle for A and B? Also, maybe make the dot color on the limit cycle different from yellow (e.g. to blue), so that you can describe what is happening over time with even more clarity (i.e. that the simulation eventually reaches the "blue" dot after enough cycles).

We have added yellow dots on the limit cycles for A and B. We have revised the text to more clearly describe what happens over time.

Figure 8 and text referring to it. I appreciate that you relate the twist to entrainment and PRCs. I think this is interesting and shows why twist is important to understand.

We would like to thank the reviewer for the positive comment.

Page 16 first paragraph. "This suggests that single oscillator twist might also affect how coupling synchronizes a network of oscillators" is too weak. Given that coupling (M) and the perturbation used in Figure 7 (Z(t)) both affect the model the same way (they both are added to the rate equation for the x variable), Figure 7 shows that the PRCs are affected by twist. Not only are they affected, they are affected dramatically - Figures 7K and 7L look like they are opposite each other (i.e. flipped sign). This means we are guaranteed that the oscillators will respond differently to intercellular signaling and that the ensemble oscillation will be different.

We thank the reviewer for the suggestion. We have now strengthened that point in the revised manuscript.

Page 16, in the discussion of the ensemble period, we see that for this model that positive twist leads to longer periods (and negative twist to shorter). What is missing from this discussion is that the relative phase of the signal and of the PRC of the receiver determine the period of the ensemble. Positive twist leads to longer periods because the mean field is determined by the value of the x variable. If it were determined by a variable with the opposite phase, then we would have the opposite effect. This affects the related paragraph in the Discussion as well.

We appreciate the reviewer's clarification regarding the dependence of phase on the direction of pulse-like perturbations, and we agree with the observation that pulses of opposite directions induce different phase shifts.

However, in Figure 9 we present analyses that offer a comparable scenario to the dynamics observed in chronotypes, albeit without an external zeitgeber. Instead, we focus on examining the long-term effects on the rhythmic properties of coupled oscillators. First, we show that mean-field coupling between two identical oscillators leads to amplitude expansions (Supplementary Figure S2) and period alterations dependent on the twist parameter \epsilon (Figure 9A).

Subsequently, we couple three Poincaré oscillators, which differ only in their \epsilon values, via their mean field (the arrow in Figure 9B indicates when mean-field coupling is ‘turned on’). We observe an increase in the amplitude of individual oscillators (consistent with Supplementary Figure S2). Although the periods of all three oscillators (as well as that of the mean-field) synchronize due to coupling, their phases relative to the mean field rhythm align with our predictions from Figure 9A and the theory from chronotypes: the oscillator with negative \epsilon exhibits an advanced rhythm compared to the mean field, while the oscillator with positive \epsilon displays a delayed rhythm relative to the mean field. These results remain consistent regardless of whether mean-field coupling is introduced in the x or y coordinate of the Poincaré oscillator.

The observation from Figure 9B is consistent with our findings in Figure 9A, where the phase delay of the oscillator with positive \epsilon aligns with this clock displaying longer periods upon coupling. This phenomenon resonates with chronotypes, where individuals with longer free-running periods have later phases of entrainment.

Page 17. The meaning of "For this reason we have not termed twist, in the context of conservative oscillations, parametric or phase space" wasn't clear to me. I think you mean you can't study twist in non-limit-cycle oscillators in the same way you can for limit cycle oscillators. Is that correct?

That is correct (also related to the first point of the report). We have clarified it throughout the revised manuscript.

References

Duffing, G. (1918). Erzwungene Schwingungen bei veränderlicher Eigenfrequenz und ihre technische Bedeutung (No. 41-42). Vieweg.

Winfree, A. T. (1980). The geometry of biological time (Vol. 2). New York: Springer.

Myung, J., Schmal, C., Hong, S., Tsukizawa, Y., Rose, P., Zhang, Y., ... & Takumi, T. (2018). The choroid plexus is an important circadian clock component. Nature Communications, 9(1), 1062.

Schmal, C., Herzog, E. D., & Herzel, H. (2018). Measuring relative coupling strength in circadian systems. Journal of Biological Rhythms, 33(1), 84-98.

Goldbeter, A. (1995). A model for circadian oscillations in the Drosophila period protein (PER). Proceedings of the Royal Society of London. Series B: Biological Sciences, 261(1362), 319-324.
Reviewer 2:
Stephanie Taylor, Department of Computer Science, Colby College, Waterville, Maine, USA

We thank all three reviewers for their constructive comments and literature suggestions, which have helped improve our manuscript greatly. Below we give a detailed response to the individual comments.

In this article, the authors describe the phenomenon of twist (period-amplitude inter-dependence) using a range of differential equation oscillator models, ultimately with a view to understanding circadian clocks. The simplest models (harmonic and Duffing) are used to demonstrate that nonlinearity introduces a relationship between amplitude and period. The authors then switch to limit-cycle oscillators, because limit-cycle oscillators are more appropriate for studying circadian clocks. They use two approaches to studying twist. The first, referred to as "parametric twist", computes the periods and amplitudes of oscillators with varied parameter values, creating graphs that relates period to amplitude (and thus demonstrate twist - if the correlation is positive, there is positive twist, and negative correlation indicates negative twist). This parametric twist analysis is performed using two limit cycle models, one with one feedback loop and one with more (synergistic) loops. The second, referred to as "phase space twist", examines the response of a single oscillator to a temporary change in one parameter, mimicking signaling. A separate limit-cycle model (a Poincare oscillator) is used for this analysis. Important to the discussion of phase space twist is the concept of isochrons, which are useful for understanding the effects of perturbations on the phase and speed of the clock (does it speed up or slow down the clock?) and it is easy to see how amplitude and phase response are related, when we look at how curved the isochrons are. The authors relate these analyses to phase response curves, recovery from jetlag, the range of entrainment, and temperature compensation and conclude, that although measuring amplitude and understanding relationship between amplitude and period are highly complicated, it is nonetheless important to be mindful of the role of amplitude and its correlation to period when studying clocks.

On the one hand, I really appreciated that the authors chose models that best illustrated the points and that the paper is organized to go from simple ideas to more complex ideas. It was helpful to have the initial explanation of twist using a duffing oscillator (which has non-linearity) as a contrast to a harmonic oscillator (which doesn't). It provided a nice introduction to the cause of twist and helped me to develop an intuition. Then, it made sense to use a very simple (3-ODE Goodwin oscillator) as the first example of a limit cycle oscillator. Since it has so few parameters, it was possible to perturb them one at a time and in pairs to illustrate a good portion of the possibilities for how one might observe heterogeneity in the system. Then, moving to Almeida's oscillator demonstrated how quickly the analysis can become very complex. All of this provides a nice progression of complexity. Then, we pivot a bit to the Poincare oscillator, to study twist in a more dynamical way - instead of permanently choosing different parameters, the authors model light signaling (at least, initially) and study the effects of the model returning to its limit cycle after the perturbation is over. This model is particularly appropriate for studying the dynamics of recovery because it is a 2-ode model (so we can examine it in "phase space", which is helpful for visualizing the clock) and because there is a parameter that controls how related period and amplitude are. It is also straight-forward to adjust how curved the isochrons are, which makes illustrating twist very clear. Then, after studying the response of a single oscillator to light, they use the same model to study the effects of intercellular signaling and the effects of different twist on the properties of the ensemble oscillation (e.g. how long the ensemble period is in contrast to that of individual oscillators).

On the other hand, I would have appreciated a more explicit connection between the discussion of "parametric twist" and "phase space twist". They are both due to changes in parameter values causing changes in period and amplitude. In the one case, we measure the period and amplitude of oscillators with different (constant) parameter values. In the second case, we measure instantaneous changes in period and amplitude due to temporary changes in parameter values (I am referring to Z(t) as a parameter here). In both cases, the shape of the isochrons in the unperturbed condition should help us to predict what will happen in the perturbed condition. There are two challenges with making that point clear with the Goodwin and Almeida models - 1) the isochrons aren't lines, they are multi-dimensional manifolds, and 2) the isochrons in the unperturbed and perturbed systems are different. So I am asking for too much to relate the two discussions by using images of isochrons, but a more explicit comparison would help the reader understand how the analyses are related to each other.

We thank the reviewer for this insightful observation that we have also spent a lot of time thinking about, and for which we lack a clear answer. However, we aim to shed light on some thoughts that may help clarify the distinctions (and similarities) between parametric and phase space twist.

The concept “twist” was historically coined on conservative, non-limit cycle oscillations. It was first described in the Duffing oscillator [1] –an adaptation of the harmonic oscillator with an additional nonlinear term. This nonlinear term introduces unique properties to the system, as depicted in our Figure 1 from the manuscript: whereas harmonic, linear oscillators have no amplitude-period co-dependencies, the introduction of nonlinearities in the harmonic oscillator result in amplitude-period correlations, i.e. twist. Unlike limit cycle oscillators, non-limit cycle oscillators are conservative and do not dissipate energy. In more practical terms, this means that the initial conditions persist indefinitely, as no energy is dissipated, and thus the oscillator’s amplitude is defined by initial conditions (or by the energy of the system).

Most biological oscillations, however, are of limit cycle type. Limit cycles are dissipative, and different initial conditions relax (i.e., lose energy) and converge to the stable cycle, characterized by a given period and amplitude. In limit cycle oscillations, twist has to be studied in a different way than it is studied for non-limit cycle oscillators. Because different initial conditions do not produce different cycles, if we want to study how amplitudes and periods correlate in different cycles, one has to study this by altering parameters in the model. This is what we have termed parametric twist: the amplitude-period correlations arising from changes in model parameters, thus altering the original limit cycle slightly.

Another form of amplitude-period correlations can occur within an oscillator’s time series: what we term phase space twist. In a perfect stable limit cycle, period and amplitude do not change from cycle to cycle. But upon its interaction with the environment, the oscillator encounters perturbations that modify its amplitude and period transiently. The transient correlations between period and amplitude that appear as the oscillator relaxes and “adapts” to the perturbation is what we have termed “phase space twist” throughout our analyses. This is related to Winfree’s “invention” of isochrones in phase space [2]. Such “transient” correlations (i.e., phase space twist), have implications for PRCs, entrainment and coupling, as shown in Figure 7 and Figure 9.

In the realm of conservative systems, one does not need the distinction between phase space twist and parametric twist, as isochrones lack the relevance that they hold in limit cycles. It is for this reason that we have exclusively introduced these terms in the “limit cycle world”. While in phase space twist, isochrones are well defined manifolds, in the “world of parametric twist”, each new limit cycle has its own properties and isochrone manifolds, which makes the manifolds of parametric twist very difficult to visualize graphically.

In the analysis of the modified Poincare oscillator, the term "twist" is used differently than it is in the "parametric twist" analysis. In the parametric analysis, the twist captures the full effects of the perturbations (we plot the measured periods and amplitudes across the heterogeneous set of oscillators). In the phase space analysis, "twist" refers to a particular parameter \epsilon that, when non-zero, introduces a dependence of the speed and amplitude on each other. However, that twist parameter (\epsilon) is not the only parameter that affects the skewness of the isochrons. The relaxation rate \lambda also affects the skewness of the isochrons. And the skewness of the isochrons affects the changes in speed and amplitude when the oscillator is signaled (i.e. a parameter is temporarily perturbed). So, if we were to measure the effects on period and amplitude (as is done with figures 7G, H, and I), that reflects that values of both \lambda and \epsilon. When I was thinking about how to relate the "parametric twist" and "phase twist" analysis, I was looking for aspects of figures 3 (Gonze model), 5 (Almeida model), and 7 (poincare model) to relate and it seemed to me that that the subfigures that related the amplitude of the oscillator to its period (3C, 3D, 3E, 3F, 3G, 3H, 5B, 5B, 5E, 7G, 7H, 7I) should be the figures that I can connect. They are labeled as twist in figures 3 and 5. In Figure 7, they show an approximation of instantaneous amplitude and period (rather than steady-state amplitude and period), so it isn't a perfect 1-1 mapping, but it is still the closest that makes sense to me. And in all of these cases, the output could be labeled twist if \epsilon didn't get to claim that label. I think it would make the concept of twist clearer to the reader if it were consistent across both analyses that twist is the "effect" we can measure. My reading of the text as it is currently written is that twist is a phenomenon we can measure in "parametric twist" analyses and that twist is a parameter that causes one change in the isochrons (but not all of the changes that can happen) that ultimately leads to something we can measure. Also, labeling the parameter as twist in the Poincare oscillator makes it seem like we need to use the Poincare oscillator to perform a "phase space twist" analysis. But we aren't limited to the Poincare oscillator for such analyses.

We acknowledge the reviewer's valid point regarding the unclear distinction between the \epsilon parameter and phase space twist effects, which sometimes led to confusion. To address this, we now explicitly define phase space twist as the observable amplitude-period correlations in an oscillator's response to a perturbation—the measurable 'phenotype'. The reviewer is right that in the Poincaré oscillator, phase space twist effects are a consequence of the \epsilon parameter. However, other models might also display such correlations without the explicit \epsilon parameter. We have clarified this in the Methods and in the Discussion.

Moreover, as the reviewer correctly points out, \epsilon is not the only parameter influencing the skewness of isochrones, also \lambda plays a role. Thus, both \epsilon and \lambda modulate the extent of phase space twist effects that become apparent in an oscillator as it encounters and responds to a perturbation. We have revised Figure 8 in the manuscript and added panels to show the phase space twist effects for oscillators with the same \epsilon but changing \lambda values. We now show how the correlation between both variables decreases for oscillators with higher amplitude relaxation rates \lambda.

We have revised the manuscript to consistently refer to the \epsilon parameter as the "twist parameter \epsilon" (like in prior studies [3]), clarifying that phase space twist denotes the amplitude-period correlations emerging from varying \epsilon and \lambda values.

Specific Points:

It would be helpful to have a little more framing text that indicates the different models are used to make different points.

We have justified the use of the different models.
I think of the "phase space twist" analysis as analysis of recovery from temporary changes to parameters and "parametric twist" as analysis of permanent differences in parameters. If that is the way you intend readers to think about it, then I would suggest changing the names to reflect those ideas more directly. Right now, the term "phase space twist" stands out as a highly specific analysis - the model must be 2D so you can draw the isochrons. But I think that is limiting. I totally embrace the idea that there is both heterogeneity among oscillators and temporary changes due to intercellular signaling, so I would love to see it characterized that way more explicitly. What about "twist due to heterogeneity" and "twist due to signaling"? They aren't quite zippy enough and rely on my characterization of twist as a measurable phenomenon, so I don't push those terms exactly.

In response to the reviewer’s on-point comment, we have clarified what are the “practical” implications of parametric and phase space twist throughout the revised text, but without changing the terminology. We wish to highlight here again, that whereas phase space twist refers to the transient amplitude-period changes that a limit cycle oscillator suffers when it encounters a perturbation (with consequences for coupling and entrainment), the concept of parametric twist “lives” in the domain of oscillator heterogeneity. Although parametric twist might be an intrinsic property of an oscillator (i.e. cells of the choroid plexus show negative twist; whereas U-2 OS cells are positive twist oscillators), it only becomes evident in heterogeneous systems, when different parameter schemes (i.e. oscillator heterogeneity) come into play.

On page 8, when you are introducing isochrons, it would be helpful to have a figure to refer to. The most natural figure is Figure 7, but that would force you to make an unnatural re-ordering of the figures. So maybe introduce a new figure?

We refer to the section from the Results already in the Materials and Methods, when we introduce the definition of isochrones. We opted against adding a new figure in the Methods section because, in our eyes, it would be redundant with the current Figure 7. We believe that this figure is more needed in the Results section, where we emphasize in the text how isochrones and the parameters \epsilon and \lambda can affect the acceleration at which a perturbation relaxes back to the limit cycle.

On Page 11, (analyses shown in Figure 3), how did you choose which parameters to pair together?

We paired all degradation parameters with each other, namely k2 with k4, k4 with k6 and k6 with k2, but we only show two representative pairs. We have clarified that in the revised text and in the caption of Figure 3.

On Page 11, there is a discussion of the Hopf bifurcation that is interesting (and highlights how sensitivity twist is to our choice of parameters). It occurs to me that this has implications for modeling SCN oscillators as they recover from TTX treatment. The individual oscillator amplitude crashes and the oscillators lose synchrony during TTX and both recover after TTX is washed out. Is there anything you could add to the discussion about this?

We thank the reviewer for this observation. One of the messages of Figure 4 is indeed that parametric twist in an ensemble of oscillators is sensitive to the default parameter set of an individual oscillator. However, it is important to highlight that the analysis in that figure refers to uncoupled oscillators. The results shown in the Figure could be regarded as a population of oscillators with heterogeneous k4 values. Depending on how ‘spreaded’ such heterogeneity is, the expected parametric twist will be different. While a population of oscillators with small deviations around the default value k4=0.35, let’s say ±5%, would likely result in a population where one would almost have no (or partly positive) parametric twist (see Figure 4C, the variation in k4 would result in very minor differences in periods and peak-trough distances), a population of oscillators with larger heterogeneity in their k4 values would span a larger region of the bubble, resulting in complex parametric twist effects (both positive and negative, see Figure 4C).

Coupling in the SCN (or in any other oscillatory network) is expected to reduce the phenotypic differences in their periods and amplitude distributions, reducing the period heterogeneity (at least to some extent) and increasing amplitude due to resonance effects [4]. For this reason, we would expect that, upon overcritical coupling, the population of uncoupled oscillators with heterogeneous k4 values oscillates at a common period and amplitude. In other words, coupling would reduce, at least to some extent, the degree of parametric twist in a population of heterogeneous oscillators. Upon disruption of inter-oscillator coupling, such as TTX treatment in the SCN, one would expect the oscillators to return to their ‘original’ degree of parametric twist. We have added a short paragraph on the Discussion on how coupling is expected to affect parametric twist.

On Page 12, in the paragraph about how overall twist effects depend not only on the parameter being studied, but also on the variable that is measured, I was reminded of early modeling studies that showed not-immediately intuitive results. In his 1995 fly model paper (A model of circadian oscillations in the Drosophila period protein, Proc. R. Soc. Lond. B, 1995), Goldbeter showed that the rate of (doubly-phosphorylated) PER degradation affected both period and amplitude and that increasing this rate led to PER amplitude increasing (rather than decreasing, as one might expect). This can be reasoned through, but it is another illustration of the complexity of the non-linear feedback and the relationship between period and amplitude. Is this worth bringing up here or on the discussion?

We have discussed this point (and added the respective reference [5]) in the discussion and highlighted that different oscillator models (which have different nonlinear dependencies) result in different parametric twist effects. For this reason, it is important to know what is the type of twist in a population of oscillators (experimentally), to be able to devise a model that captures such properties.

Figure 6. Why are there no yellow dots on the limit cycle for A and B? Also, maybe make the dot color on the limit cycle different from yellow (e.g. to blue), so that you can describe what is happening over time with even more clarity (i.e. that the simulation eventually reaches the "blue" dot after enough cycles).

We have added yellow dots on the limit cycles for A and B. We have revised the text to more clearly describe what happens over time.

Figure 8 and text referring to it. I appreciate that you relate the twist to entrainment and PRCs. I think this is interesting and shows why twist is important to understand.

We would like to thank the reviewer for the positive comment.

Page 16 first paragraph. "This suggests that single oscillator twist might also affect how coupling synchronizes a network of oscillators" is too weak. Given that coupling (M) and the perturbation used in Figure 7 (Z(t)) both affect the model the same way (they both are added to the rate equation for the x variable), Figure 7 shows that the PRCs are affected by twist. Not only are they affected, they are affected dramatically - Figures 7K and 7L look like they are opposite each other (i.e. flipped sign). This means we are guaranteed that the oscillators will respond differently to intercellular signaling and that the ensemble oscillation will be different.

We thank the reviewer for the suggestion. We have now strengthened that point in the revised manuscript.

Page 16, in the discussion of the ensemble period, we see that for this model that positive twist leads to longer periods (and negative twist to shorter). What is missing from this discussion is that the relative phase of the signal and of the PRC of the receiver determine the period of the ensemble. Positive twist leads to longer periods because the mean field is determined by the value of the x variable. If it were determined by a variable with the opposite phase, then we would have the opposite effect. This affects the related paragraph in the Discussion as well.

We appreciate the reviewer's clarification regarding the dependence of phase on the direction of pulse-like perturbations, and we agree with the observation that pulses of opposite directions induce different phase shifts.

However, in Figure 9 we present analyses that offer a comparable scenario to the dynamics observed in chronotypes, albeit without an external zeitgeber. Instead, we focus on examining the long-term effects on the rhythmic properties of coupled oscillators. First, we show that mean-field coupling between two identical oscillators leads to amplitude expansions (Supplementary Figure S2) and period alterations dependent on the twist parameter \epsilon (Figure 9A).

Subsequently, we couple three Poincaré oscillators, which differ only in their \epsilon values, via their mean field (the arrow in Figure 9B indicates when mean-field coupling is ‘turned on’). We observe an increase in the amplitude of individual oscillators (consistent with Supplementary Figure S2). Although the periods of all three oscillators (as well as that of the mean-field) synchronize due to coupling, their phases relative to the mean field rhythm align with our predictions from Figure 9A and the theory from chronotypes: the oscillator with negative \epsilon exhibits an advanced rhythm compared to the mean field, while the oscillator with positive \epsilon displays a delayed rhythm relative to the mean field. These results remain consistent regardless of whether mean-field coupling is introduced in the x or y coordinate of the Poincaré oscillator.

The observation from Figure 9B is consistent with our findings in Figure 9A, where the phase delay of the oscillator with positive \epsilon aligns with this clock displaying longer periods upon coupling. This phenomenon resonates with chronotypes, where individuals with longer free-running periods have later phases of entrainment.

Page 17. The meaning of "For this reason we have not termed twist, in the context of conservative oscillations, parametric or phase space" wasn't clear to me. I think you mean you can't study twist in non-limit-cycle oscillators in the same way you can for limit cycle oscillators. Is that correct?

That is correct (also related to the first point of the report). We have clarified it throughout the revised manuscript.

References

Duffing, G. (1918). Erzwungene Schwingungen bei veränderlicher Eigenfrequenz und ihre technische Bedeutung (No. 41-42). Vieweg.

Winfree, A. T. (1980). The geometry of biological time (Vol. 2). New York: Springer.

Myung, J., Schmal, C., Hong, S., Tsukizawa, Y., Rose, P., Zhang, Y., ... & Takumi, T. (2018). The choroid plexus is an important circadian clock component. Nature Communications, 9(1), 1062.

Schmal, C., Herzog, E. D., & Herzel, H. (2018). Measuring relative coupling strength in circadian systems. Journal of Biological Rhythms, 33(1), 84-98.

Goldbeter, A. (1995). A model for circadian oscillations in the Drosophila period protein (PER). Proceedings of the Royal Society of London. Series B: Biological Sciences, 261(1362), 319-324.
Competing Interests: The authors declare no conflict of interest. Close
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Reviewer Report 18 Sep 2023

Jihwan Myung, Graduate Institute of Mind, Brain and Consciousness (GIMBC), Taipei Medical University, Taipei, Taiwan

Approved

https://doi.org/10.5256/f1000research.148657.r202732

In this manuscript, the authors make a comprehensive exploration of the "twist" (amplitude-period correlation) in nonlinear oscillators, especially in the context of circadian rhythms. The Introduction insightfully summarizes the background and offers a clear explanation of the topic. While some sections exhibit a degree of coarseness in their explanations, the central theme remains consistently and clearly highlighted. Overall, this manuscript is a noteworthy addition to the field, and I recommend its acceptance for indexing with minor revisions.

Minor comments:

1. Abstract: I suggest changing "..., fastening recovery..." to either "speeding up recovery" or "accelerating recovery".

2. p.4, subsection "Goodwin-like models": I recommend changing "sigmoidal Hill curve" to "Hill curve", as "sigmoidal" often refers to forms involving exponential functions.

3. p.4, subsection "Goodwin-like models": Add biological interpretations of the variables x, y, and z (mRNA, ...).

4. p.5. ll.4-5: While the Gonze model[33] describes degradation processes with Michaelis-Menten dynamics, it may not provide sufficient historical context. I recommend citing Kurosawa & Iwasa (2002)¹ for a more comprehensive historical perspective.

5. p.6, Table 2: The reference [36] should be spelled out in "The default values are taken from [36].

6. p.11. Figure 4: It is unclear if the graphs are figurative cartoons or based on actual numerical data from Gonze oscillators.

7. pp.11-12: The justification for the use of the Almeida model can be described. For example, why not use the detailed Gonze model or Becker-Weimann model?

8. p.13, ll.1-2: It is challenging to assert that the "parametric twist" emerges solely due to a heterogeneous population. While it describes the average oscillator behavior in a population (often assumed to be normally distributed), it essentially characterizes the behavior of a single representative oscillator.

9. p.14, "... 24 h, matching the intrinsic oscillator's frequency (Figure 7P).": I suggest to rephrase it to "... 24 h, matching the oscillator's intrinsic period (Figure 7P)." The change from "frequency" to "intrinsic period" would provide more specificity and clarity.

10. p.18 "the longer period that has been observed in dispersed... , but not in dispersed SCN neurons in culture": In light of the findings in Figure 5, it's noteworthy that when Bmal1 oscillation is observed (as opposed to Per2), the SCN period lengthens under pharmacological decoupling or physical sub-sectioning of the SCN (Myung et al., 2012²),.

11. p.18 "Thus, a more plausible scenario may involve local coupling...": Especially in the SCN, there are papers that highlight the role of specific coupling among subregions, which can reproduce the wave-like spatial patterns and seasonal encoding of circadian time. These can be discussed.

Is the work clearly and accurately presented and does it cite the current literature?

Partly
Is the study design appropriate and is the work technically sound?

Yes
Are sufficient details of methods and analysis provided to allow replication by others?

Yes
If applicable, is the statistical analysis and its interpretation appropriate?

Yes
Are all the source data underlying the results available to ensure full reproducibility?

Yes
Are the conclusions drawn adequately supported by the results?

Yes

References

1. Kurosawa G, Iwasa Y: Saturation of enzyme kinetics in circadian clock models.J Biol Rhythms. 2002; 17 (6): 568-77 PubMed Abstract | Publisher Full Text
2. Myung J, Hong S, Hatanaka F, Nakajima Y, et al.: Period coding of Bmal1 oscillators in the suprachiasmatic nucleus.J Neurosci. 2012; 32 (26): 8900-18 PubMed Abstract | Publisher Full Text

Competing Interests: No competing interests were disclosed.

Reviewer Expertise: Circadian rhythms, computational modeling of circadian clocks, suprachiasmatic nucleus (SCN), bioluminescence reporter imaging

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.

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Author Response 20 Jun 2024

Marta del Olmo, Institute for Theoretical Biology, Charité Universitätsmedizin Berlin, Philippstr. 13, 10115, Germany

20 Jun 2024

Author Response
Reviewer 3:
Jihwan Myung, Graduate Institute of Mind, Brain and Consciousness (GIMBC), Taipei Medical University, Taipei, Taiwan

In this manuscript, the authors make a comprehensive exploration of the "twist" ... Continue reading
Reviewer 3:
Jihwan Myung, Graduate Institute of Mind, Brain and Consciousness (GIMBC), Taipei Medical University, Taipei, Taiwan

In this manuscript, the authors make a comprehensive exploration of the "twist" (amplitude-period correlation) in nonlinear oscillators, especially in the context of circadian rhythms. The Introduction insightfully summarizes the background and offers a clear explanation of the topic. While some sections exhibit a degree of coarseness in their explanations, the central theme remains consistently and clearly highlighted. Overall, this manuscript is a noteworthy addition to the field, and I recommend its acceptance for indexing with minor revisions.

Minor comments:
1. Abstract: I suggest changing "..., fastening recovery..." to either "speeding up recovery" or "accelerating recovery".
2. p.4, subsection "Goodwin-like models": I recommend changing "sigmoidal Hill curve" to "Hill curve", as "sigmoidal" often refers to forms involving exponential functions.
3. p.4, subsection "Goodwin-like models": Add biological interpretations of the variables x, y, and z (mRNA, ...).
4. p.5. ll.4-5: While the Gonze model[33] describes degradation processes with Michaelis-Menten dynamics, it may not provide sufficient historical context. I recommend citing Kurosawa & Iwasa (2002) for a more comprehensive historical perspective.
5. p.6, Table 2: The reference [36] should be spelled out in "The default values are taken from [36].
6. p.11. Figure 4: It is unclear if the graphs are figurative cartoons or based on actual numerical data from Gonze oscillators.
7. pp.11-12: The justification for the use of the Almeida model can be described. For example, why not use the detailed Gonze model or Becker-Weimann model?
8. p.13, ll.1-2: It is challenging to assert that the "parametric twist" emerges solely due to a heterogeneous population. While it describes the average oscillator behavior in a population (often assumed to be normally distributed), it essentially characterizes the behavior of a single representative oscillator.
9. p.14, "... 24 h, matching the intrinsic oscillator's frequency (Figure 7P).": I suggest rephrasing it to "... 24 h, matching the oscillator's intrinsic period (Figure 7P)." The change from "frequency" to "intrinsic period" would provide more specificity and clarity.
10. p.18 "the longer period that has been observed in dispersed... , but not in dispersed SCN neurons in culture": In light of the findings in Figure 5, it's noteworthy that when Bmal1 oscillation is observed (as opposed to Per2), the SCN period lengthens under pharmacological decoupling or physical sub-sectioning of the SCN (Myung et al., 2012).
11. p.18 "Thus, a more plausible scenario may involve local coupling...": Especially in the SCN, there are papers that highlight the role of specific coupling among subregions, which can reproduce the wave-like spatial patterns and seasonal encoding of circadian time. These can be discussed.

We thank all three reviewers for their constructive comments and literature suggestions, which have helped improve our manuscript greatly. Below we summarize the changes that we have implemented in response to the reviewer’s suggestions:

We have revised the manuscript and rephrased points 1, 2, 5 and 9.

We have included the biological interpretation of the 3 variables in the Goodwin model in the Methods section.

We have added the more historical reference on saturation kinetics for the degradation processes (Kurosawa and Iwasa, [1]).

We have clarified in the caption of Figure 4, that the results are based on actual numerical data from Gonze oscillators.

We have justified the use of the Almeida model. It is a simple enough model in the sense that it does not take into account post-translational modifications of the core clock proteins, nor nuclear import/export processes (which other more complex models of the mammalian clockwork do consider [2]), and complex enough in the sense that it contains synergies of feedback loops (the Per–Cry–Bmal1 loop, the Bmal1–Rev-Erb loop or Bmal1–Ror loop, which other older models do not take into account [3]).

We have revised the text to clarify that while the term ‘parametric twist’ characterizes the average behavior of a population, it also provides a means to describe the behavior of a representative oscillator within that population.

We have revised the Discussion to comment on the findings of Myung et al. [4], namely that Bmal1 rhythms in the SCN, in contrast to Per2 oscillations, exhibit longer periods when the SCN is pharmacologically decoupled or physically subdivided.

In the discussion, we have further elaborated on coupling within the SCN and how it has been shown to regulate complex phenomena like wave-like spatial patterns and even encode seasonality.

References

Kurosawa, G., & Iwasa, Y. (2002). Saturation of enzyme kinetics in circadian clock models. Journal of Biological Rhythms, 17(6), 568-577.

Relógio, A., Westermark, P. O., Wallach, T., Schellenberg, K., Kramer, A., & Herzel, H. (2011). Tuning the mammalian circadian clock: robust synergy of two loops. PLoS Computational Biology, 7(12), e1002309.

Becker-Weimann, S., Wolf, J., Herzel, H., & Kramer, A. (2004). Modeling feedback loops of the mammalian circadian oscillator. Biophysical Journal, 87(5), 3023-3034.

Myung, J., Hong, S., Hatanaka, F., Nakajima, Y., De Schutter, E., & Takumi, T. (2012). Period coding of Bmal1 oscillators in the suprachiasmatic nucleus. Journal of Neuroscience, 32(26), 8900-8918.
Reviewer 3:
Jihwan Myung, Graduate Institute of Mind, Brain and Consciousness (GIMBC), Taipei Medical University, Taipei, Taiwan

In this manuscript, the authors make a comprehensive exploration of the "twist" (amplitude-period correlation) in nonlinear oscillators, especially in the context of circadian rhythms. The Introduction insightfully summarizes the background and offers a clear explanation of the topic. While some sections exhibit a degree of coarseness in their explanations, the central theme remains consistently and clearly highlighted. Overall, this manuscript is a noteworthy addition to the field, and I recommend its acceptance for indexing with minor revisions.

Minor comments:
1. Abstract: I suggest changing "..., fastening recovery..." to either "speeding up recovery" or "accelerating recovery".
2. p.4, subsection "Goodwin-like models": I recommend changing "sigmoidal Hill curve" to "Hill curve", as "sigmoidal" often refers to forms involving exponential functions.
3. p.4, subsection "Goodwin-like models": Add biological interpretations of the variables x, y, and z (mRNA, ...).
4. p.5. ll.4-5: While the Gonze model[33] describes degradation processes with Michaelis-Menten dynamics, it may not provide sufficient historical context. I recommend citing Kurosawa & Iwasa (2002) for a more comprehensive historical perspective.
5. p.6, Table 2: The reference [36] should be spelled out in "The default values are taken from [36].
6. p.11. Figure 4: It is unclear if the graphs are figurative cartoons or based on actual numerical data from Gonze oscillators.
7. pp.11-12: The justification for the use of the Almeida model can be described. For example, why not use the detailed Gonze model or Becker-Weimann model?
8. p.13, ll.1-2: It is challenging to assert that the "parametric twist" emerges solely due to a heterogeneous population. While it describes the average oscillator behavior in a population (often assumed to be normally distributed), it essentially characterizes the behavior of a single representative oscillator.
9. p.14, "... 24 h, matching the intrinsic oscillator's frequency (Figure 7P).": I suggest rephrasing it to "... 24 h, matching the oscillator's intrinsic period (Figure 7P)." The change from "frequency" to "intrinsic period" would provide more specificity and clarity.
10. p.18 "the longer period that has been observed in dispersed... , but not in dispersed SCN neurons in culture": In light of the findings in Figure 5, it's noteworthy that when Bmal1 oscillation is observed (as opposed to Per2), the SCN period lengthens under pharmacological decoupling or physical sub-sectioning of the SCN (Myung et al., 2012).
11. p.18 "Thus, a more plausible scenario may involve local coupling...": Especially in the SCN, there are papers that highlight the role of specific coupling among subregions, which can reproduce the wave-like spatial patterns and seasonal encoding of circadian time. These can be discussed.

We thank all three reviewers for their constructive comments and literature suggestions, which have helped improve our manuscript greatly. Below we summarize the changes that we have implemented in response to the reviewer’s suggestions:

We have revised the manuscript and rephrased points 1, 2, 5 and 9.

We have included the biological interpretation of the 3 variables in the Goodwin model in the Methods section.

We have added the more historical reference on saturation kinetics for the degradation processes (Kurosawa and Iwasa, [1]).

We have clarified in the caption of Figure 4, that the results are based on actual numerical data from Gonze oscillators.

We have justified the use of the Almeida model. It is a simple enough model in the sense that it does not take into account post-translational modifications of the core clock proteins, nor nuclear import/export processes (which other more complex models of the mammalian clockwork do consider [2]), and complex enough in the sense that it contains synergies of feedback loops (the Per–Cry–Bmal1 loop, the Bmal1–Rev-Erb loop or Bmal1–Ror loop, which other older models do not take into account [3]).

We have revised the text to clarify that while the term ‘parametric twist’ characterizes the average behavior of a population, it also provides a means to describe the behavior of a representative oscillator within that population.

We have revised the Discussion to comment on the findings of Myung et al. [4], namely that Bmal1 rhythms in the SCN, in contrast to Per2 oscillations, exhibit longer periods when the SCN is pharmacologically decoupled or physically subdivided.

In the discussion, we have further elaborated on coupling within the SCN and how it has been shown to regulate complex phenomena like wave-like spatial patterns and even encode seasonality.

References

Kurosawa, G., & Iwasa, Y. (2002). Saturation of enzyme kinetics in circadian clock models. Journal of Biological Rhythms, 17(6), 568-577.

Relógio, A., Westermark, P. O., Wallach, T., Schellenberg, K., Kramer, A., & Herzel, H. (2011). Tuning the mammalian circadian clock: robust synergy of two loops. PLoS Computational Biology, 7(12), e1002309.

Becker-Weimann, S., Wolf, J., Herzel, H., & Kramer, A. (2004). Modeling feedback loops of the mammalian circadian oscillator. Biophysical Journal, 87(5), 3023-3034.

Myung, J., Hong, S., Hatanaka, F., Nakajima, Y., De Schutter, E., & Takumi, T. (2012). Period coding of Bmal1 oscillators in the suprachiasmatic nucleus. Journal of Neuroscience, 32(26), 8900-8918.
Competing Interests: The authors declare no conflict of interest. Close
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Author Response 20 Jun 2024

Marta del Olmo, Institute for Theoretical Biology, Charité Universitätsmedizin Berlin, Philippstr. 13, 10115, Germany

20 Jun 2024

Author Response
Reviewer 3:
Jihwan Myung, Graduate Institute of Mind, Brain and Consciousness (GIMBC), Taipei Medical University, Taipei, Taiwan

In this manuscript, the authors make a comprehensive exploration of the "twist" ... Continue reading
Reviewer 3:
Jihwan Myung, Graduate Institute of Mind, Brain and Consciousness (GIMBC), Taipei Medical University, Taipei, Taiwan

In this manuscript, the authors make a comprehensive exploration of the "twist" (amplitude-period correlation) in nonlinear oscillators, especially in the context of circadian rhythms. The Introduction insightfully summarizes the background and offers a clear explanation of the topic. While some sections exhibit a degree of coarseness in their explanations, the central theme remains consistently and clearly highlighted. Overall, this manuscript is a noteworthy addition to the field, and I recommend its acceptance for indexing with minor revisions.

Minor comments:
1. Abstract: I suggest changing "..., fastening recovery..." to either "speeding up recovery" or "accelerating recovery".
2. p.4, subsection "Goodwin-like models": I recommend changing "sigmoidal Hill curve" to "Hill curve", as "sigmoidal" often refers to forms involving exponential functions.
3. p.4, subsection "Goodwin-like models": Add biological interpretations of the variables x, y, and z (mRNA, ...).
4. p.5. ll.4-5: While the Gonze model[33] describes degradation processes with Michaelis-Menten dynamics, it may not provide sufficient historical context. I recommend citing Kurosawa & Iwasa (2002) for a more comprehensive historical perspective.
5. p.6, Table 2: The reference [36] should be spelled out in "The default values are taken from [36].
6. p.11. Figure 4: It is unclear if the graphs are figurative cartoons or based on actual numerical data from Gonze oscillators.
7. pp.11-12: The justification for the use of the Almeida model can be described. For example, why not use the detailed Gonze model or Becker-Weimann model?
8. p.13, ll.1-2: It is challenging to assert that the "parametric twist" emerges solely due to a heterogeneous population. While it describes the average oscillator behavior in a population (often assumed to be normally distributed), it essentially characterizes the behavior of a single representative oscillator.
9. p.14, "... 24 h, matching the intrinsic oscillator's frequency (Figure 7P).": I suggest rephrasing it to "... 24 h, matching the oscillator's intrinsic period (Figure 7P)." The change from "frequency" to "intrinsic period" would provide more specificity and clarity.
10. p.18 "the longer period that has been observed in dispersed... , but not in dispersed SCN neurons in culture": In light of the findings in Figure 5, it's noteworthy that when Bmal1 oscillation is observed (as opposed to Per2), the SCN period lengthens under pharmacological decoupling or physical sub-sectioning of the SCN (Myung et al., 2012).
11. p.18 "Thus, a more plausible scenario may involve local coupling...": Especially in the SCN, there are papers that highlight the role of specific coupling among subregions, which can reproduce the wave-like spatial patterns and seasonal encoding of circadian time. These can be discussed.

We thank all three reviewers for their constructive comments and literature suggestions, which have helped improve our manuscript greatly. Below we summarize the changes that we have implemented in response to the reviewer’s suggestions:

We have revised the manuscript and rephrased points 1, 2, 5 and 9.

We have included the biological interpretation of the 3 variables in the Goodwin model in the Methods section.

We have added the more historical reference on saturation kinetics for the degradation processes (Kurosawa and Iwasa, [1]).

We have clarified in the caption of Figure 4, that the results are based on actual numerical data from Gonze oscillators.

We have justified the use of the Almeida model. It is a simple enough model in the sense that it does not take into account post-translational modifications of the core clock proteins, nor nuclear import/export processes (which other more complex models of the mammalian clockwork do consider [2]), and complex enough in the sense that it contains synergies of feedback loops (the Per–Cry–Bmal1 loop, the Bmal1–Rev-Erb loop or Bmal1–Ror loop, which other older models do not take into account [3]).

We have revised the text to clarify that while the term ‘parametric twist’ characterizes the average behavior of a population, it also provides a means to describe the behavior of a representative oscillator within that population.

We have revised the Discussion to comment on the findings of Myung et al. [4], namely that Bmal1 rhythms in the SCN, in contrast to Per2 oscillations, exhibit longer periods when the SCN is pharmacologically decoupled or physically subdivided.

In the discussion, we have further elaborated on coupling within the SCN and how it has been shown to regulate complex phenomena like wave-like spatial patterns and even encode seasonality.

References

Kurosawa, G., & Iwasa, Y. (2002). Saturation of enzyme kinetics in circadian clock models. Journal of Biological Rhythms, 17(6), 568-577.

Relógio, A., Westermark, P. O., Wallach, T., Schellenberg, K., Kramer, A., & Herzel, H. (2011). Tuning the mammalian circadian clock: robust synergy of two loops. PLoS Computational Biology, 7(12), e1002309.

Becker-Weimann, S., Wolf, J., Herzel, H., & Kramer, A. (2004). Modeling feedback loops of the mammalian circadian oscillator. Biophysical Journal, 87(5), 3023-3034.

Myung, J., Hong, S., Hatanaka, F., Nakajima, Y., De Schutter, E., & Takumi, T. (2012). Period coding of Bmal1 oscillators in the suprachiasmatic nucleus. Journal of Neuroscience, 32(26), 8900-8918.
Reviewer 3:
Jihwan Myung, Graduate Institute of Mind, Brain and Consciousness (GIMBC), Taipei Medical University, Taipei, Taiwan

In this manuscript, the authors make a comprehensive exploration of the "twist" (amplitude-period correlation) in nonlinear oscillators, especially in the context of circadian rhythms. The Introduction insightfully summarizes the background and offers a clear explanation of the topic. While some sections exhibit a degree of coarseness in their explanations, the central theme remains consistently and clearly highlighted. Overall, this manuscript is a noteworthy addition to the field, and I recommend its acceptance for indexing with minor revisions.

Minor comments:
1. Abstract: I suggest changing "..., fastening recovery..." to either "speeding up recovery" or "accelerating recovery".
2. p.4, subsection "Goodwin-like models": I recommend changing "sigmoidal Hill curve" to "Hill curve", as "sigmoidal" often refers to forms involving exponential functions.
3. p.4, subsection "Goodwin-like models": Add biological interpretations of the variables x, y, and z (mRNA, ...).
4. p.5. ll.4-5: While the Gonze model[33] describes degradation processes with Michaelis-Menten dynamics, it may not provide sufficient historical context. I recommend citing Kurosawa & Iwasa (2002) for a more comprehensive historical perspective.
5. p.6, Table 2: The reference [36] should be spelled out in "The default values are taken from [36].
6. p.11. Figure 4: It is unclear if the graphs are figurative cartoons or based on actual numerical data from Gonze oscillators.
7. pp.11-12: The justification for the use of the Almeida model can be described. For example, why not use the detailed Gonze model or Becker-Weimann model?
8. p.13, ll.1-2: It is challenging to assert that the "parametric twist" emerges solely due to a heterogeneous population. While it describes the average oscillator behavior in a population (often assumed to be normally distributed), it essentially characterizes the behavior of a single representative oscillator.
9. p.14, "... 24 h, matching the intrinsic oscillator's frequency (Figure 7P).": I suggest rephrasing it to "... 24 h, matching the oscillator's intrinsic period (Figure 7P)." The change from "frequency" to "intrinsic period" would provide more specificity and clarity.
10. p.18 "the longer period that has been observed in dispersed... , but not in dispersed SCN neurons in culture": In light of the findings in Figure 5, it's noteworthy that when Bmal1 oscillation is observed (as opposed to Per2), the SCN period lengthens under pharmacological decoupling or physical sub-sectioning of the SCN (Myung et al., 2012).
11. p.18 "Thus, a more plausible scenario may involve local coupling...": Especially in the SCN, there are papers that highlight the role of specific coupling among subregions, which can reproduce the wave-like spatial patterns and seasonal encoding of circadian time. These can be discussed.

We thank all three reviewers for their constructive comments and literature suggestions, which have helped improve our manuscript greatly. Below we summarize the changes that we have implemented in response to the reviewer’s suggestions:

We have revised the manuscript and rephrased points 1, 2, 5 and 9.

We have included the biological interpretation of the 3 variables in the Goodwin model in the Methods section.

We have added the more historical reference on saturation kinetics for the degradation processes (Kurosawa and Iwasa, [1]).

We have clarified in the caption of Figure 4, that the results are based on actual numerical data from Gonze oscillators.

We have justified the use of the Almeida model. It is a simple enough model in the sense that it does not take into account post-translational modifications of the core clock proteins, nor nuclear import/export processes (which other more complex models of the mammalian clockwork do consider [2]), and complex enough in the sense that it contains synergies of feedback loops (the Per–Cry–Bmal1 loop, the Bmal1–Rev-Erb loop or Bmal1–Ror loop, which other older models do not take into account [3]).

We have revised the text to clarify that while the term ‘parametric twist’ characterizes the average behavior of a population, it also provides a means to describe the behavior of a representative oscillator within that population.

We have revised the Discussion to comment on the findings of Myung et al. [4], namely that Bmal1 rhythms in the SCN, in contrast to Per2 oscillations, exhibit longer periods when the SCN is pharmacologically decoupled or physically subdivided.

In the discussion, we have further elaborated on coupling within the SCN and how it has been shown to regulate complex phenomena like wave-like spatial patterns and even encode seasonality.

References

Kurosawa, G., & Iwasa, Y. (2002). Saturation of enzyme kinetics in circadian clock models. Journal of Biological Rhythms, 17(6), 568-577.

Relógio, A., Westermark, P. O., Wallach, T., Schellenberg, K., Kramer, A., & Herzel, H. (2011). Tuning the mammalian circadian clock: robust synergy of two loops. PLoS Computational Biology, 7(12), e1002309.

Becker-Weimann, S., Wolf, J., Herzel, H., & Kramer, A. (2004). Modeling feedback loops of the mammalian circadian oscillator. Biophysical Journal, 87(5), 3023-3034.

Myung, J., Hong, S., Hatanaka, F., Nakajima, Y., De Schutter, E., & Takumi, T. (2012). Period coding of Bmal1 oscillators in the suprachiasmatic nucleus. Journal of Neuroscience, 32(26), 8900-8918.
Competing Interests: The authors declare no conflict of interest. Close
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Jihwan Myung, Taipei Medical University, Taipei, Taiwan
Stephanie Taylor, Colby College, Waterville, USA
Jae Kim, KAIST, Daejeon, South Korea

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29 Sep 2023 | for Version 1

Jae Kim, Department of Mathematical Sciences, KAIST, Daejeon, South Korea

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Approved

The authors have conducted a study on the relationship between amplitude and period in various circadian clock models. However, it is crucial to address the novelty and distinctions of this study compared to prior work, specifically the study by Bokka et al. in 2018 (https://doi.org/10.1049/iet-syb.2018.0015). Additionally, while the models in this manuscript are based on Hill-type repression (such as the Goodwin and Gonze models), it is important to acknowledge recent research findings that reveal different repression mechanisms in the Drosophila (https://doi.org/10.1073/pnas.2113403119) and mammalian circadian clocks (https://doi.org/10.1098/rsfs.2021.0084). These studies demonstrate that the use of multiple repression mechanisms, including protein sequestration, can substantially alter the oscillator dynamics (https://doi.org/10.1049/iet-syb.2015.0090). Given these discoveries, it would be valuable to explore or discuss how different repression mechanisms impact the relationship between period and amplitude, as demonstrated by Bokka et al. in 2018.
In the Conclusions section on page 19, please provide a reference for 'cellular biochemical rates' as a factor influencing circadian amplitudes. For instance, it was demonstrated that faster production and degradation rates of proteins can generate circadian rhythms with higher amplitudes in the Drosophila circadian clock (https://doi.org/10.1073/pnas.2113403119).

Is the work clearly and accurately presented and does it cite the current literature?

Partly
Is the study design appropriate and is the work technically sound?

Yes
Are sufficient details of methods and analysis provided to allow replication by others?

Yes
If applicable, is the statistical analysis and its interpretation appropriate?

Yes
Are all the source data underlying the results available to ensure full reproducibility?

Yes
Are the conclusions drawn adequately supported by the results?

Yes

References

Competing Interests

No competing interests were disclosed.

Reviewer Expertise

Circadian rhythms, mathematical modeling

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.

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Author Response

20 Jun 2024

Marta del Olmo, Institute for Theoretical Biology, Charité Universitätsmedizin Berlin, Philippstr. 13, 10115, Germany

Reviewer 1:
Jae Kim, Department of Mathematical Sciences, KAIST, Daejeon, South Korea

The authors have conducted a study on the relationship between amplitude and period in various circadian clock models. However, it is crucial to address the novelty and distinctions of this study compared to prior work, specifically the study by Bokka et al. in 2018 (https://doi.org/10.1049/iet-syb.2018.0015). Additionally, while the models in this manuscript are based on Hill-type repression (such as the Goodwin and Gonze models), it is important to acknowledge recent research findings that reveal different repression mechanisms in the Drosophila (https://doi.org/10.1073/pnas.2113403119) and mammalian circadian clocks (https://doi.org/10.1098/rsfs.2021.0084). These studies demonstrate that the use of multiple repression mechanisms, including protein sequestration, can substantially alter the oscillator dynamics (https://doi.org/10.1049/iet-syb.2015.0090). Given these discoveries, it would be valuable to explore or discuss how different repression mechanisms impact the relationship between period and amplitude, as demonstrated by Bokka et al. in 2018.

We thank all three reviewers for their constructive comments and literature suggestions, which have helped improve our manuscript greatly.

We have included a paragraph in the Discussion addressing how our manuscript supports and builds upon the findings from Bokka et al. [1]. The authors conduct a comprehensive analysis of parametric twist in different ODE-based models that have been used to model biological oscillatory processes but do not study phase space twist. They classify the models depending on how amplitudes and periods co-vary, where their type 1 corresponds to positive amplitude-period co-variation (what we have termed positive parametric twist in our manuscript) and type 2, to negative amplitude-period correlations (negative twist). The authors consider additional correlations, e.g. constant amplitude but changing period, constant period but changing amplitude (both of which we consider no twist) or more complex co-variations of combinations of positive and negative twist effects.

Among these models, the authors study the Goodwin model. Their analysis of the Goodwin model shows that, for changes in k2, amplitude increases but period remains constant (i.e. no twist). For changes in k4, amplitude increases and period decreases (negative twist). Lastly, for changes in k6, both amplitude and period decrease (positive twist). This is in contrast to our bifurcation analyses, which revealed positive correlations of amplitude and period upon changes in k2 or k4, but a negative correlation upon changes of k6. The differences in our and their results might stem from 3 sources:

A possible explanation might come from the fact that the authors define amplitude as an absolute value of the difference between peak and trough of oscillations, whereas we define amplitude as a relative measure (difference between peak and trough of the oscillation but normalized to the mean).
While we have used the x variable from the Goodwin system to define amplitude, it is unclear from the main text in [1] which variable is used from the whole ODE system to define amplitude. Period is expected to change equally for all the variables upon change of one parameter; amplitude, however, might change differently for the different model variables (as shown in our Figure 5B, Figure 5E). This highlights again the difficulty in defining amplitude in circadian studies. Bokka et al. [1] partly circumvent this issue by defining amplitude as the geometric mean of all biomolecular species in the model. This metric, nevertheless, is difficult to measure in or translate to an experimental setup, as (i) the circadian network does not yet agree on what are the core elements of the clock network for a particular species, and (ii) even if we all agreed on the same core clock elements, it is often not possible to measure all the species at the same time within one experiment.
Moreover, additional differences may stem from the fact that we have used a different parameter set than the one in the original Goodwin model [2] to generate circadian oscillations (taken and adapted from [3] as indicated in Table 1 in the main text), and thus our default parameter set might be at a different region of the Hopf bubble.

Interestingly, the authors describe the presence of a ‘forbidden region’, where oscillations with large amplitude but small periods (or fast frequencies) might be difficult. This is consistent with oscillations emerging from Hopf bifurcations. It would be interesting to study the amplitude-period correlations on oscillations that arise after other types of bifurcation points, including a saddle-node-on-invariant cycle (SNIC), but such analyses are out of the scope of this manuscript. Tsai and colleagues showed in [4] how systems with positive feedback loops can present high amplitude and high frequency rhythms and thus the ‘forbidden region’ might differ. In a more theoretical study by Bose [5], it is also shown how the onset of oscillations through a Hopf bifurcation does not allow for high frequency rhythms, whereas when oscillations emerge through a SNIC bifurcation, periods increase more gradually.

We would also like to acknowledge the reviewer’s on-point comment on the existence of different repression mechanisms in different species, which we also have discussed in the manuscript. For example, in Drosophila and in mammals, inhibition of transcriptional activity is achieved through the formation of a 1:1 stoichiometric complex between activators and inhibitors [6-10]. In this type of repression mechanism, frequently termed protein sequestration-based repression [11, 12], repressors tightly bind to activators to form an inactive 1:1 stoichiometric complex, thus ‘sequestering’ the activator. Such protein sequestration processes can be described with terms that differ from the Hill function [11, 12]. In other species, like Neurospora, the stoichiometry between activators and repressors in nuclei is much less than 1:1 [13]. In this fungus, a large number of phosphorylation sites in the activator is required for self-sustained oscillations. As the number of mutated phosphorylation sites increases, the circadian rhythms become weaker and finally arrhythmic [13]. Such multistep cooperative reactions can be described with Hill curves [14, 15].

It is important to note that there is also evidence supporting an additional mode of repression in the mammalian TTFL mediated by CRY1 alone. After inhibition of the CLOCK:BMAL1 activator complex by the macromolecular repressive complex containing PER and CRY proteins, delayed expression of CRY1 can still independently repress transcription from E-boxes [16-19]. This mode of repression cannot be explained according to a protein sequestration repression, and occurs through direct blocking of the activators [18-20]. Such a blocking-type of inhibition might be accurately modeled with a Hill function, especially provided the observations that CRY1 is gradually phosphorylated until it becomes the late monomeric repressive CRY1 [21]. Such multistep phosphorylations, as mentioned earlier, can be described with the use of Hill functions.

Depending on whether the design of a TTFL contains a Hill-type or a protein sequestration-based repression mechanism, the conditions to achieve self-sustained limit cycle oscillations are different [11]. Models that are based on a Hill-type repression need a large Hill exponent in order to generate self-sustained oscillations, whereas models based on a sequestration repression need a 1:1 stoichiometry between repressor and activator for a successful onset of oscillations. Our manuscript only addresses models that use Hill functions to model transcriptional repression and we have partly extended the discussion to account for additional modes of repression. However, given our and others’ [1] observations that all kinds of parametric twist effects are possible upon changing one parameter in a certain model, we expect that protein sequestration-based models might also result in complex parametric twist effects upon changes of one parameter that likely depend on the default parameter set of the model (and where this is located at within the Hopf bubble) as well as on the variable whose amplitude is being measured.

In the Conclusions section on page 19, please provide a reference for 'cellular biochemical rates' as a factor influencing circadian amplitudes. For instance, it was demonstrated that faster production and degradation rates of proteins can generate circadian rhythms with higher amplitudes in the Drosophila circadian clock (https://doi.org/10.1073/pnas.2113403119).

We have now added the reference [22] to our conclusions.

References

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Goodwin, B. C. (1965). Oscillatory behavior in enzymatic control processes. Advances in Enzyme Regulation, 3, 425-437.
Ruoff, P., Vinsjevik, M., Monnerjahn, C., & Rensing, L. (1999). The Goodwin oscillator: on the importance of degradation reactions in the circadian clock. Journal of Biological Rhythms, 14(6), 469-479.
Tsai, T. Y. C., Choi, Y. S., Ma, W., Pomerening, J. R., Tang, C., & Ferrell Jr, J. E. (2008). Robust, tunable biological oscillations from interlinked positive and negative feedback loops. Science, 321(5885), 126-129.
Bose, A. (2014). Bifurcations Dynamics of Single Neurons and Small Networks. In: Jaeger, D., Jung, R. (eds) Encyclopedia of Computational Neuroscience. Springer, New York, NY.
Nangle, S. N., Rosensweig, C., Koike, N., Tei, H., Takahashi, J. S., Green, C. B., & Zheng, N. (2014). Molecular assembly of the period-cryptochrome circadian transcriptional repressor complex. Elife, 3, e03674.
Aryal, R. P., Kwak, P. B., Tamayo, A. G., Gebert, M., Chiu, P. L., Walz, T., & Weitz, C. J. (2017). Macromolecular assemblies of the mammalian circadian clock. Molecular Cell, 67(5), 770-782.
Schmalen, I., Reischl, S., Wallach, T., Klemz, R., Grudziecki, A., Prabu, J. R., ... & Wolf, E. (2014). Interaction of circadian clock proteins CRY1 and PER2 is modulated by zinc binding and disulfide bond formation. Cell, 157(5), 1203-1215.
Ye, R., Selby, C. P., Chiou, Y. Y., Ozkan-Dagliyan, I., Gaddameedhi, S., & Sancar, A. (2014). Dual modes of CLOCK: BMAL1 inhibition mediated by Cryptochrome and Period proteins in the mammalian circadian clock. Genes & Development, 28(18), 1989-1998.
Menet, J. S., Abruzzi, K. C., Desrochers, J., Rodriguez, J., & Rosbash, M. (2010). Dynamic PER repression mechanisms in the Drosophila circadian clock: from on-DNA to off-DNA. Genes & Development, 24(4), 358-367.
Kim, J. K. (2016). Protein sequestration versus Hill‐type repression in circadian clock models. IET Systems Biology, 10(4), 125-135.
Kim, J. K., & Forger, D. B. (2012). A mechanism for robust circadian timekeeping via stoichiometric balance. Molecular Systems Biology, 8(1), 630.
Schafmeier, T., Haase, A., Káldi, K., Scholz, J., Fuchs, M., & Brunner, M. (2005). Transcriptional feedback of Neurospora circadian clock gene by phosphorylation-dependent inactivation of its transcription factor. Cell, 122(2), 235-246.
Ferrell, J. E., & Ha, S. H. (2014). Ultrasensitivity part II: multisite phosphorylation, stoichiometric inhibitors, and positive feedback. Trends in Biochemical Sciences, 39(11), 556-569.
Gonze, D., & Abou-Jaoudé, W. (2013). The Goodwin model: behind the Hill function. PloS One, 8(8), e69573.
Koike, N., Yoo, S. H., Huang, H. C., Kumar, V., Lee, C., Kim, T. K., & Takahashi, J. S. (2012). Transcriptional architecture and chromatin landscape of the core circadian clock in mammals. Science, 338(6105), 349-354.
Ukai-Tadenuma, M., Yamada, R. G., Xu, H., Ripperger, J. A., Liu, A. C., & Ueda, H. R. (2011). Delay in feedback repression by cryptochrome 1 is required for circadian clock function. Cell, 144(2), 268-281.
Cao, X., Yang, Y., Selby, C. P., Liu, Z., & Sancar, A. (2021). Molecular mechanism of the repressive phase of the mammalian circadian clock. Proceedings of the National Academy of Sciences, 118(2), e2021174118.
Cao, X., Wang, L., Selby, C. P., Lindsey-Boltz, L. A., & Sancar, A. (2023). Analysis of mammalian circadian clock protein complexes over a circadian cycle. Journal of Biological Chemistry, 299(3).
Jeong, E. M., Song, Y. M., & Kim, J. K. (2022). Combined multiple transcriptional repression mechanisms generate ultrasensitivity and oscillations. Interface Focus, 12(3), 20210084.
Liu, N., & Zhang, E. E. (2016). Phosphorylation regulating the ratio of intracellular CRY1 protein determines the circadian period. Frontiers in Neurology, 7, 159.
Jeong, E. M., Kwon, M., Cho, E., Lee, S. H., Kim, H., Kim, E. Y., & Kim, J. K. (2022). Systematic modeling-driven experiments identify distinct molecular clockworks underlying hierarchically organized pacemaker neurons. Proceedings of the National Academy of Sciences, 119(8), e2113403119.

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The authors declare no competing interests.

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Reviewer Report

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18 Sep 2023 | for Version 1

Stephanie Taylor, Department of Computer Science, Colby College, Waterville, Maine, USA

17 Views Cite this report Responses(1)

Approved

It would be helpful to have a little more framing text that indicates the different models are used to make different points.
I think of the "phase space twist" analysis as analysis of recovery from temporary changes to parameters and "parametric twist" as analysis of permanent differences in parameters. If that is the way you intend readers to think about it, then I would suggest changing the names to reflect those ideas more directly. Right now, the term "phase space twist" stands out as a highly specific analysis - the model most be 2D so you can draw the isochrons. But I think that is limiting. I totally embrace idea that there is both heterogeneity among oscillators and temporary changes due to intercellular signaling, so I would love to see it characterized that way more explicitly. What about "twist due to heterogeneity" and "twist due to signaling"? They aren't quite zippy enough and rely on my characterization of twist as a measurable phenomenon, so I don't push those terms exactly.
On page 8, when you are introducing isochrons, it would be helpful to have a figure to refer to. The most natural figure is Figure 7, but that would force you to make an unnatural re-ordering of the figures. So maybe introduce a new figure?
On Page 11, (analyses shown in Figure 3), how did you choose which parameters to pair together?
On Page 11, there is a discussion of the Hopf bifurcation that is interesting (and highlights how sensitivity twist is to our choice of parameters). It occurs to me that this has implications for modeling SCN oscillators as they recover from TTX treatment. The individual oscillator amplitude crashes and the oscillators lose synchrony during TTX and both recover after TTX is washed out. Is there anything you could add to the discussion about this?
On Page 12, in the paragraph about how overall twist effects depend not only on the parameter being studies, but also on the variable that is measured, I was reminded of early modeling studies that showed not-immediately intuitive results. In his 1995 fly model paper (A model of circadian oscillations in the Drosophila period protein, Proc. R. Soc. Lond. B, 1995), Goldbeter showed that the rate of (doubly-phosphorylated) PER degradation affected both period and amplitude and that increasing this rate led to PER amplitude increasing (rather than decreasing, as one might expect). This can be reasoned through, but it is another illustration of the complexity of the non-linear feedback and the relationship between period and amplitude. Is this worth bringing up here or on the discussion?
Figure 6. Why are there no yellow dots on the limit cycle for A and B? Also, maybe make the dot color on the limit cycle different from yellow (e.g. to blue), so that you can describe what is happening over time with even more clarity (i.e. that the simulation eventually reaches the "blue" dot after enough cycles).
Figure 8 and text referring to it. I appreciate that you relate twist to entrainment and PRCs. I think this is interesting and shows why twist is important to understand.
Page 16 first paragraph. "This suggests that single oscillator twist might also affect how coupling synchronizes a network of oscillators" is too weak. Given that coupling (M) and the perturbation used in Figure 7 (Z(t)) both affect the model the same way (they both are added to the rate equation for the x variable), Figure 7 shows that the PRCs are affected by twist. Not only are they affected, they are affected dramatically - Figures 7K and 7L look like they are opposite each other (i.e. flipped sign). This means we are guaranteed that the oscillators will respond differently to intercellular signaling and that that the ensemble oscillation will be different.
Page 16, in the discussion of the ensemble period, we see that for this model that positive twist leads to longer periods (and negative twist to shorter). What is missing from this discussion is that the relative phase of the signal and of the PRC of the receiver determine the period of the ensemble. Positive twist leads to longer periods because the mean field is determined by the value of the x variable. If it were determined by a variable with the opposite phase, then we would have the opposite effect. This affects the related paragraph in the Discussion as well.
Page 17. The meaning of "For this reason we have not termed twist, in the context of conservative oscillations, parametric or phase space" wasn't clear to me. I think you mean you can't study twist in non-limit-cycle oscillators in the same way you can for limit cycle oscillators. Is that correct?

Is the work clearly and accurately presented and does it cite the current literature?

Yes
Is the study design appropriate and is the work technically sound?

Yes
Are sufficient details of methods and analysis provided to allow replication by others?

Yes
If applicable, is the statistical analysis and its interpretation appropriate?

Yes
Are all the source data underlying the results available to ensure full reproducibility?

Yes
Are the conclusions drawn adequately supported by the results?

Yes

Competing Interests

No competing interests were disclosed.

Reviewer Expertise

I study the response of limit cycle oscillators to signals and sensitivity analysis involving isochrons.

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.

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Author Response

20 Jun 2024

Marta del Olmo, Institute for Theoretical Biology, Charité Universitätsmedizin Berlin, Philippstr. 13, 10115, Germany

Reviewer 2:
Stephanie Taylor, Department of Computer Science, Colby College, Waterville, Maine, USA

We thank all three reviewers for their constructive comments and literature suggestions, which have helped improve our manuscript greatly. Below we give a detailed response to the individual comments.

In this article, the authors describe the phenomenon of twist (period-amplitude inter-dependence) using a range of differential equation oscillator models, ultimately with a view to understanding circadian clocks. The simplest models (harmonic and Duffing) are used to demonstrate that nonlinearity introduces a relationship between amplitude and period. The authors then switch to limit-cycle oscillators, because limit-cycle oscillators are more appropriate for studying circadian clocks. They use two approaches to studying twist. The first, referred to as "parametric twist", computes the periods and amplitudes of oscillators with varied parameter values, creating graphs that relates period to amplitude (and thus demonstrate twist - if the correlation is positive, there is positive twist, and negative correlation indicates negative twist). This parametric twist analysis is performed using two limit cycle models, one with one feedback loop and one with more (synergistic) loops. The second, referred to as "phase space twist", examines the response of a single oscillator to a temporary change in one parameter, mimicking signaling. A separate limit-cycle model (a Poincare oscillator) is used for this analysis. Important to the discussion of phase space twist is the concept of isochrons, which are useful for understanding the effects of perturbations on the phase and speed of the clock (does it speed up or slow down the clock?) and it is easy to see how amplitude and phase response are related, when we look at how curved the isochrons are. The authors relate these analyses to phase response curves, recovery from jetlag, the range of entrainment, and temperature compensation and conclude, that although measuring amplitude and understanding relationship between amplitude and period are highly complicated, it is nonetheless important to be mindful of the role of amplitude and its correlation to period when studying clocks.

On the one hand, I really appreciated that the authors chose models that best illustrated the points and that the paper is organized to go from simple ideas to more complex ideas. It was helpful to have the initial explanation of twist using a duffing oscillator (which has non-linearity) as a contrast to a harmonic oscillator (which doesn't). It provided a nice introduction to the cause of twist and helped me to develop an intuition. Then, it made sense to use a very simple (3-ODE Goodwin oscillator) as the first example of a limit cycle oscillator. Since it has so few parameters, it was possible to perturb them one at a time and in pairs to illustrate a good portion of the possibilities for how one might observe heterogeneity in the system. Then, moving to Almeida's oscillator demonstrated how quickly the analysis can become very complex. All of this provides a nice progression of complexity. Then, we pivot a bit to the Poincare oscillator, to study twist in a more dynamical way - instead of permanently choosing different parameters, the authors model light signaling (at least, initially) and study the effects of the model returning to its limit cycle after the perturbation is over. This model is particularly appropriate for studying the dynamics of recovery because it is a 2-ode model (so we can examine it in "phase space", which is helpful for visualizing the clock) and because there is a parameter that controls how related period and amplitude are. It is also straight-forward to adjust how curved the isochrons are, which makes illustrating twist very clear. Then, after studying the response of a single oscillator to light, they use the same model to study the effects of intercellular signaling and the effects of different twist on the properties of the ensemble oscillation (e.g. how long the ensemble period is in contrast to that of individual oscillators).

On the other hand, I would have appreciated a more explicit connection between the discussion of "parametric twist" and "phase space twist". They are both due to changes in parameter values causing changes in period and amplitude. In the one case, we measure the period and amplitude of oscillators with different (constant) parameter values. In the second case, we measure instantaneous changes in period and amplitude due to temporary changes in parameter values (I am referring to Z(t) as a parameter here). In both cases, the shape of the isochrons in the unperturbed condition should help us to predict what will happen in the perturbed condition. There are two challenges with making that point clear with the Goodwin and Almeida models - 1) the isochrons aren't lines, they are multi-dimensional manifolds, and 2) the isochrons in the unperturbed and perturbed systems are different. So I am asking for too much to relate the two discussions by using images of isochrons, but a more explicit comparison would help the reader understand how the analyses are related to each other.

We thank the reviewer for this insightful observation that we have also spent a lot of time thinking about, and for which we lack a clear answer. However, we aim to shed light on some thoughts that may help clarify the distinctions (and similarities) between parametric and phase space twist.

The concept “twist” was historically coined on conservative, non-limit cycle oscillations. It was first described in the Duffing oscillator [1] –an adaptation of the harmonic oscillator with an additional nonlinear term. This nonlinear term introduces unique properties to the system, as depicted in our Figure 1 from the manuscript: whereas harmonic, linear oscillators have no amplitude-period co-dependencies, the introduction of nonlinearities in the harmonic oscillator result in amplitude-period correlations, i.e. twist. Unlike limit cycle oscillators, non-limit cycle oscillators are conservative and do not dissipate energy. In more practical terms, this means that the initial conditions persist indefinitely, as no energy is dissipated, and thus the oscillator’s amplitude is defined by initial conditions (or by the energy of the system).

Most biological oscillations, however, are of limit cycle type. Limit cycles are dissipative, and different initial conditions relax (i.e., lose energy) and converge to the stable cycle, characterized by a given period and amplitude. In limit cycle oscillations, twist has to be studied in a different way than it is studied for non-limit cycle oscillators. Because different initial conditions do not produce different cycles, if we want to study how amplitudes and periods correlate in different cycles, one has to study this by altering parameters in the model. This is what we have termed parametric twist: the amplitude-period correlations arising from changes in model parameters, thus altering the original limit cycle slightly.

Another form of amplitude-period correlations can occur within an oscillator’s time series: what we term phase space twist. In a perfect stable limit cycle, period and amplitude do not change from cycle to cycle. But upon its interaction with the environment, the oscillator encounters perturbations that modify its amplitude and period transiently. The transient correlations between period and amplitude that appear as the oscillator relaxes and “adapts” to the perturbation is what we have termed “phase space twist” throughout our analyses. This is related to Winfree’s “invention” of isochrones in phase space [2]. Such “transient” correlations (i.e., phase space twist), have implications for PRCs, entrainment and coupling, as shown in Figure 7 and Figure 9.

In the realm of conservative systems, one does not need the distinction between phase space twist and parametric twist, as isochrones lack the relevance that they hold in limit cycles. It is for this reason that we have exclusively introduced these terms in the “limit cycle world”. While in phase space twist, isochrones are well defined manifolds, in the “world of parametric twist”, each new limit cycle has its own properties and isochrone manifolds, which makes the manifolds of parametric twist very difficult to visualize graphically.

In the analysis of the modified Poincare oscillator, the term "twist" is used differently than it is in the "parametric twist" analysis. In the parametric analysis, the twist captures the full effects of the perturbations (we plot the measured periods and amplitudes across the heterogeneous set of oscillators). In the phase space analysis, "twist" refers to a particular parameter \epsilon that, when non-zero, introduces a dependence of the speed and amplitude on each other. However, that twist parameter (\epsilon) is not the only parameter that affects the skewness of the isochrons. The relaxation rate \lambda also affects the skewness of the isochrons. And the skewness of the isochrons affects the changes in speed and amplitude when the oscillator is signaled (i.e. a parameter is temporarily perturbed). So, if we were to measure the effects on period and amplitude (as is done with figures 7G, H, and I), that reflects that values of both \lambda and \epsilon. When I was thinking about how to relate the "parametric twist" and "phase twist" analysis, I was looking for aspects of figures 3 (Gonze model), 5 (Almeida model), and 7 (poincare model) to relate and it seemed to me that that the subfigures that related the amplitude of the oscillator to its period (3C, 3D, 3E, 3F, 3G, 3H, 5B, 5B, 5E, 7G, 7H, 7I) should be the figures that I can connect. They are labeled as twist in figures 3 and 5. In Figure 7, they show an approximation of instantaneous amplitude and period (rather than steady-state amplitude and period), so it isn't a perfect 1-1 mapping, but it is still the closest that makes sense to me. And in all of these cases, the output could be labeled twist if \epsilon didn't get to claim that label. I think it would make the concept of twist clearer to the reader if it were consistent across both analyses that twist is the "effect" we can measure. My reading of the text as it is currently written is that twist is a phenomenon we can measure in "parametric twist" analyses and that twist is a parameter that causes one change in the isochrons (but not all of the changes that can happen) that ultimately leads to something we can measure. Also, labeling the parameter as twist in the Poincare oscillator makes it seem like we need to use the Poincare oscillator to perform a "phase space twist" analysis. But we aren't limited to the Poincare oscillator for such analyses.

We acknowledge the reviewer's valid point regarding the unclear distinction between the \epsilon parameter and phase space twist effects, which sometimes led to confusion. To address this, we now explicitly define phase space twist as the observable amplitude-period correlations in an oscillator's response to a perturbation—the measurable 'phenotype'. The reviewer is right that in the Poincaré oscillator, phase space twist effects are a consequence of the \epsilon parameter. However, other models might also display such correlations without the explicit \epsilon parameter. We have clarified this in the Methods and in the Discussion.

Moreover, as the reviewer correctly points out, \epsilon is not the only parameter influencing the skewness of isochrones, also \lambda plays a role. Thus, both \epsilon and \lambda modulate the extent of phase space twist effects that become apparent in an oscillator as it encounters and responds to a perturbation. We have revised Figure 8 in the manuscript and added panels to show the phase space twist effects for oscillators with the same \epsilon but changing \lambda values. We now show how the correlation between both variables decreases for oscillators with higher amplitude relaxation rates \lambda.

We have revised the manuscript to consistently refer to the \epsilon parameter as the "twist parameter \epsilon" (like in prior studies [3]), clarifying that phase space twist denotes the amplitude-period correlations emerging from varying \epsilon and \lambda values.

Specific Points:

It would be helpful to have a little more framing text that indicates the different models are used to make different points.

We have justified the use of the different models.
I think of the "phase space twist" analysis as analysis of recovery from temporary changes to parameters and "parametric twist" as analysis of permanent differences in parameters. If that is the way you intend readers to think about it, then I would suggest changing the names to reflect those ideas more directly. Right now, the term "phase space twist" stands out as a highly specific analysis - the model must be 2D so you can draw the isochrons. But I think that is limiting. I totally embrace the idea that there is both heterogeneity among oscillators and temporary changes due to intercellular signaling, so I would love to see it characterized that way more explicitly. What about "twist due to heterogeneity" and "twist due to signaling"? They aren't quite zippy enough and rely on my characterization of twist as a measurable phenomenon, so I don't push those terms exactly.

In response to the reviewer’s on-point comment, we have clarified what are the “practical” implications of parametric and phase space twist throughout the revised text, but without changing the terminology. We wish to highlight here again, that whereas phase space twist refers to the transient amplitude-period changes that a limit cycle oscillator suffers when it encounters a perturbation (with consequences for coupling and entrainment), the concept of parametric twist “lives” in the domain of oscillator heterogeneity. Although parametric twist might be an intrinsic property of an oscillator (i.e. cells of the choroid plexus show negative twist; whereas U-2 OS cells are positive twist oscillators), it only becomes evident in heterogeneous systems, when different parameter schemes (i.e. oscillator heterogeneity) come into play.

On page 8, when you are introducing isochrons, it would be helpful to have a figure to refer to. The most natural figure is Figure 7, but that would force you to make an unnatural re-ordering of the figures. So maybe introduce a new figure?

We refer to the section from the Results already in the Materials and Methods, when we introduce the definition of isochrones. We opted against adding a new figure in the Methods section because, in our eyes, it would be redundant with the current Figure 7. We believe that this figure is more needed in the Results section, where we emphasize in the text how isochrones and the parameters \epsilon and \lambda can affect the acceleration at which a perturbation relaxes back to the limit cycle.

On Page 11, (analyses shown in Figure 3), how did you choose which parameters to pair together?

We paired all degradation parameters with each other, namely k2 with k4, k4 with k6 and k6 with k2, but we only show two representative pairs. We have clarified that in the revised text and in the caption of Figure 3.

On Page 11, there is a discussion of the Hopf bifurcation that is interesting (and highlights how sensitivity twist is to our choice of parameters). It occurs to me that this has implications for modeling SCN oscillators as they recover from TTX treatment. The individual oscillator amplitude crashes and the oscillators lose synchrony during TTX and both recover after TTX is washed out. Is there anything you could add to the discussion about this?

We thank the reviewer for this observation. One of the messages of Figure 4 is indeed that parametric twist in an ensemble of oscillators is sensitive to the default parameter set of an individual oscillator. However, it is important to highlight that the analysis in that figure refers to uncoupled oscillators. The results shown in the Figure could be regarded as a population of oscillators with heterogeneous k4 values. Depending on how ‘spreaded’ such heterogeneity is, the expected parametric twist will be different. While a population of oscillators with small deviations around the default value k4=0.35, let’s say ±5%, would likely result in a population where one would almost have no (or partly positive) parametric twist (see Figure 4C, the variation in k4 would result in very minor differences in periods and peak-trough distances), a population of oscillators with larger heterogeneity in their k4 values would span a larger region of the bubble, resulting in complex parametric twist effects (both positive and negative, see Figure 4C).

Coupling in the SCN (or in any other oscillatory network) is expected to reduce the phenotypic differences in their periods and amplitude distributions, reducing the period heterogeneity (at least to some extent) and increasing amplitude due to resonance effects [4]. For this reason, we would expect that, upon overcritical coupling, the population of uncoupled oscillators with heterogeneous k4 values oscillates at a common period and amplitude. In other words, coupling would reduce, at least to some extent, the degree of parametric twist in a population of heterogeneous oscillators. Upon disruption of inter-oscillator coupling, such as TTX treatment in the SCN, one would expect the oscillators to return to their ‘original’ degree of parametric twist. We have added a short paragraph on the Discussion on how coupling is expected to affect parametric twist.

On Page 12, in the paragraph about how overall twist effects depend not only on the parameter being studied, but also on the variable that is measured, I was reminded of early modeling studies that showed not-immediately intuitive results. In his 1995 fly model paper (A model of circadian oscillations in the Drosophila period protein, Proc. R. Soc. Lond. B, 1995), Goldbeter showed that the rate of (doubly-phosphorylated) PER degradation affected both period and amplitude and that increasing this rate led to PER amplitude increasing (rather than decreasing, as one might expect). This can be reasoned through, but it is another illustration of the complexity of the non-linear feedback and the relationship between period and amplitude. Is this worth bringing up here or on the discussion?

We have discussed this point (and added the respective reference [5]) in the discussion and highlighted that different oscillator models (which have different nonlinear dependencies) result in different parametric twist effects. For this reason, it is important to know what is the type of twist in a population of oscillators (experimentally), to be able to devise a model that captures such properties.

Figure 6. Why are there no yellow dots on the limit cycle for A and B? Also, maybe make the dot color on the limit cycle different from yellow (e.g. to blue), so that you can describe what is happening over time with even more clarity (i.e. that the simulation eventually reaches the "blue" dot after enough cycles).

We have added yellow dots on the limit cycles for A and B. We have revised the text to more clearly describe what happens over time.

Figure 8 and text referring to it. I appreciate that you relate the twist to entrainment and PRCs. I think this is interesting and shows why twist is important to understand.

We would like to thank the reviewer for the positive comment.

Page 16 first paragraph. "This suggests that single oscillator twist might also affect how coupling synchronizes a network of oscillators" is too weak. Given that coupling (M) and the perturbation used in Figure 7 (Z(t)) both affect the model the same way (they both are added to the rate equation for the x variable), Figure 7 shows that the PRCs are affected by twist. Not only are they affected, they are affected dramatically - Figures 7K and 7L look like they are opposite each other (i.e. flipped sign). This means we are guaranteed that the oscillators will respond differently to intercellular signaling and that the ensemble oscillation will be different.

We thank the reviewer for the suggestion. We have now strengthened that point in the revised manuscript.

Page 16, in the discussion of the ensemble period, we see that for this model that positive twist leads to longer periods (and negative twist to shorter). What is missing from this discussion is that the relative phase of the signal and of the PRC of the receiver determine the period of the ensemble. Positive twist leads to longer periods because the mean field is determined by the value of the x variable. If it were determined by a variable with the opposite phase, then we would have the opposite effect. This affects the related paragraph in the Discussion as well.

We appreciate the reviewer's clarification regarding the dependence of phase on the direction of pulse-like perturbations, and we agree with the observation that pulses of opposite directions induce different phase shifts.

However, in Figure 9 we present analyses that offer a comparable scenario to the dynamics observed in chronotypes, albeit without an external zeitgeber. Instead, we focus on examining the long-term effects on the rhythmic properties of coupled oscillators. First, we show that mean-field coupling between two identical oscillators leads to amplitude expansions (Supplementary Figure S2) and period alterations dependent on the twist parameter \epsilon (Figure 9A).

Subsequently, we couple three Poincaré oscillators, which differ only in their \epsilon values, via their mean field (the arrow in Figure 9B indicates when mean-field coupling is ‘turned on’). We observe an increase in the amplitude of individual oscillators (consistent with Supplementary Figure S2). Although the periods of all three oscillators (as well as that of the mean-field) synchronize due to coupling, their phases relative to the mean field rhythm align with our predictions from Figure 9A and the theory from chronotypes: the oscillator with negative \epsilon exhibits an advanced rhythm compared to the mean field, while the oscillator with positive \epsilon displays a delayed rhythm relative to the mean field. These results remain consistent regardless of whether mean-field coupling is introduced in the x or y coordinate of the Poincaré oscillator.

The observation from Figure 9B is consistent with our findings in Figure 9A, where the phase delay of the oscillator with positive \epsilon aligns with this clock displaying longer periods upon coupling. This phenomenon resonates with chronotypes, where individuals with longer free-running periods have later phases of entrainment.

Page 17. The meaning of "For this reason we have not termed twist, in the context of conservative oscillations, parametric or phase space" wasn't clear to me. I think you mean you can't study twist in non-limit-cycle oscillators in the same way you can for limit cycle oscillators. Is that correct?

That is correct (also related to the first point of the report). We have clarified it throughout the revised manuscript.

References

Duffing, G. (1918). Erzwungene Schwingungen bei veränderlicher Eigenfrequenz und ihre technische Bedeutung (No. 41-42). Vieweg.
Winfree, A. T. (1980). The geometry of biological time (Vol. 2). New York: Springer.
Myung, J., Schmal, C., Hong, S., Tsukizawa, Y., Rose, P., Zhang, Y., ... & Takumi, T. (2018). The choroid plexus is an important circadian clock component. Nature Communications, 9(1), 1062.
Schmal, C., Herzog, E. D., & Herzel, H. (2018). Measuring relative coupling strength in circadian systems. Journal of Biological Rhythms, 33(1), 84-98.
Goldbeter, A. (1995). A model for circadian oscillations in the Drosophila period protein (PER). Proceedings of the Royal Society of London. Series B: Biological Sciences, 261(1362), 319-324.

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Competing Interests

The authors declare no conflict of interest.

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Reviewer Report

20 Views

18 Sep 2023 | for Version 1

Jihwan Myung, Graduate Institute of Mind, Brain and Consciousness (GIMBC), Taipei Medical University, Taipei, Taiwan

20 Views Cite this report Responses(1)

Approved

Is the work clearly and accurately presented and does it cite the current literature?

Partly
Is the study design appropriate and is the work technically sound?

Yes
Are sufficient details of methods and analysis provided to allow replication by others?

Yes
If applicable, is the statistical analysis and its interpretation appropriate?

Yes
Are all the source data underlying the results available to ensure full reproducibility?

Yes
Are the conclusions drawn adequately supported by the results?

Yes

References

Competing Interests

No competing interests were disclosed.

Reviewer Expertise

Circadian rhythms, computational modeling of circadian clocks, suprachiasmatic nucleus (SCN), bioluminescence reporter imaging

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.

Respond to this report

Responses (1)

Author Response

20 Jun 2024

Marta del Olmo, Institute for Theoretical Biology, Charité Universitätsmedizin Berlin, Philippstr. 13, 10115, Germany

Reviewer 3:
Jihwan Myung, Graduate Institute of Mind, Brain and Consciousness (GIMBC), Taipei Medical University, Taipei, Taiwan

In this manuscript, the authors make a comprehensive exploration of the "twist" (amplitude-period correlation) in nonlinear oscillators, especially in the context of circadian rhythms. The Introduction insightfully summarizes the background and offers a clear explanation of the topic. While some sections exhibit a degree of coarseness in their explanations, the central theme remains consistently and clearly highlighted. Overall, this manuscript is a noteworthy addition to the field, and I recommend its acceptance for indexing with minor revisions.

Minor comments:
1. Abstract: I suggest changing "..., fastening recovery..." to either "speeding up recovery" or "accelerating recovery".
2. p.4, subsection "Goodwin-like models": I recommend changing "sigmoidal Hill curve" to "Hill curve", as "sigmoidal" often refers to forms involving exponential functions.
3. p.4, subsection "Goodwin-like models": Add biological interpretations of the variables x, y, and z (mRNA, ...).
4. p.5. ll.4-5: While the Gonze model[33] describes degradation processes with Michaelis-Menten dynamics, it may not provide sufficient historical context. I recommend citing Kurosawa & Iwasa (2002) for a more comprehensive historical perspective.
5. p.6, Table 2: The reference [36] should be spelled out in "The default values are taken from [36].
6. p.11. Figure 4: It is unclear if the graphs are figurative cartoons or based on actual numerical data from Gonze oscillators.
7. pp.11-12: The justification for the use of the Almeida model can be described. For example, why not use the detailed Gonze model or Becker-Weimann model?
8. p.13, ll.1-2: It is challenging to assert that the "parametric twist" emerges solely due to a heterogeneous population. While it describes the average oscillator behavior in a population (often assumed to be normally distributed), it essentially characterizes the behavior of a single representative oscillator.
9. p.14, "... 24 h, matching the intrinsic oscillator's frequency (Figure 7P).": I suggest rephrasing it to "... 24 h, matching the oscillator's intrinsic period (Figure 7P)." The change from "frequency" to "intrinsic period" would provide more specificity and clarity.
10. p.18 "the longer period that has been observed in dispersed... , but not in dispersed SCN neurons in culture": In light of the findings in Figure 5, it's noteworthy that when Bmal1 oscillation is observed (as opposed to Per2), the SCN period lengthens under pharmacological decoupling or physical sub-sectioning of the SCN (Myung et al., 2012).
11. p.18 "Thus, a more plausible scenario may involve local coupling...": Especially in the SCN, there are papers that highlight the role of specific coupling among subregions, which can reproduce the wave-like spatial patterns and seasonal encoding of circadian time. These can be discussed.

We thank all three reviewers for their constructive comments and literature suggestions, which have helped improve our manuscript greatly. Below we summarize the changes that we have implemented in response to the reviewer’s suggestions:

We have revised the manuscript and rephrased points 1, 2, 5 and 9.
We have included the biological interpretation of the 3 variables in the Goodwin model in the Methods section.
We have added the more historical reference on saturation kinetics for the degradation processes (Kurosawa and Iwasa, [1]).
We have clarified in the caption of Figure 4, that the results are based on actual numerical data from Gonze oscillators.
We have justified the use of the Almeida model. It is a simple enough model in the sense that it does not take into account post-translational modifications of the core clock proteins, nor nuclear import/export processes (which other more complex models of the mammalian clockwork do consider [2]), and complex enough in the sense that it contains synergies of feedback loops (the Per–Cry–Bmal1 loop, the Bmal1–Rev-Erb loop or Bmal1–Ror loop, which other older models do not take into account [3]).
We have revised the text to clarify that while the term ‘parametric twist’ characterizes the average behavior of a population, it also provides a means to describe the behavior of a representative oscillator within that population.
We have revised the Discussion to comment on the findings of Myung et al. [4], namely that Bmal1 rhythms in the SCN, in contrast to Per2 oscillations, exhibit longer periods when the SCN is pharmacologically decoupled or physically subdivided.
In the discussion, we have further elaborated on coupling within the SCN and how it has been shown to regulate complex phenomena like wave-like spatial patterns and even encode seasonality.

References

Kurosawa, G., & Iwasa, Y. (2002). Saturation of enzyme kinetics in circadian clock models. Journal of Biological Rhythms, 17(6), 568-577.
Relógio, A., Westermark, P. O., Wallach, T., Schellenberg, K., Kramer, A., & Herzel, H. (2011). Tuning the mammalian circadian clock: robust synergy of two loops. PLoS Computational Biology, 7(12), e1002309.
Becker-Weimann, S., Wolf, J., Herzel, H., & Kramer, A. (2004). Modeling feedback loops of the mammalian circadian oscillator. Biophysical Journal, 87(5), 3023-3034.
Myung, J., Hong, S., Hatanaka, F., Nakajima, Y., De Schutter, E., & Takumi, T. (2012). Period coding of Bmal1 oscillators in the suprachiasmatic nucleus. Journal of Neuroscience, 32(26), 8900-8918.

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Competing Interests

The authors declare no conflict of interest.

Alongside their report, reviewers assign a status to the article:

Approved - the paper is scientifically sound in its current form and only minor, if any, improvements are suggested

Approved with reservations - A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit.

Not approved - fundamental flaws in the paper seriously undermine the findings and conclusions

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Are circadian amplitudes and periods correlated? A new twist in the story

Abstract

Keywords

Revised Amendments from Version 1

Introduction

Figure 1. Schematic of experimental observations of circadian amplitude-period correlations.

Materials and Methods

Conservative linear and non-linear oscillators: Harmonic vs. Duffing oscillators

(1)

(2)

(3)

Goodwin-like models

(4)

(5)

Table 1. Default parameters of the Goodwin and Gonze models used for our simulations.

Almeida model

(6)

Table 2. List of parameters of the Almeida model.

Poincaré model

(7)

(8)

(9)

(10)

(11)

Computer simulations and data analysis

Results

Nonlinearities can result in amplitude-period correlations across oscillators

Figure 2. Nonlinearities introduce amplitude-period correlations in oscillator models.

Oscillator heterogeneity produces parametric twist effects in limit cycle clock models

Figure 3. Parametric twist in an ensemble of 100 uncoupled Goodwin-like models: correlations are model- and parameter-dependent.

Figure 4. The type of parametric twist depends on the parameter location within the Hopf bubble.

Figure 5. The type of parametric twist in models of the circadian clockwork with synergistic feedback loops depends on the parameter being studied and on the variable that is measured.

Phase space twist in single oscillators: On amplitude-phase models, isochrones and perturbed trajectories

Figure 6. Schematic of an isochrone in a Poincaré oscillator with (A) no twist, (B) positive twist or (C) negative twist.

Phase space twist in single oscillators affects their interaction with the environment

Figure 7. Phase space twist in a single oscillator and its effect on responses to pulse-like perturbations and to entrainment to periodic zeitgebers.

Relaxation rate also modulates phase space twist effects

Figure 8. Role of amplitude relaxation rate on the skewing of isochrones, phase space twist and consequently on the response of oscillators to perturbations.

Phase space twist in single oscillators affects coupled networks

Figure 9. Phase space twist ϵ affects the response of a network to coupling.

Discussion

Conclusions

Data and code availability

Extended data

Analysis code

Acknowledgements

References

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Figure 9. Phase space twist $ϵ$ affects the response of a network to coupling.