An equation for the biological transmembrane potential from basic biophysical principles [version 1; peer review: 1 not approved]

Biological membranes mediate different physiological processes necessary for life, many of which depend on ion movement. In turn, the difference between the electrical potentials around a biological membrane, called transmembrane potential, or membrane potential for short, is one of the key biophysical variables affecting ion movement. Most of the existing equations that describe the change in membrane potential are based on analogies with resistive-capacitive electrical circuits. These equivalent circuit models assume resistance and capacitance as measures of the permeable and the impermeable properties of the membrane, respectively. These models have increased our understanding of bioelectricity, and were particularly useful at times when the basic structure, biochemistry, and biophysics of biological membrane systems were not well known. However, the parts in the ohmic circuits from which equations are derived, are not quite like the biological elements present in the spaces around and within biological membranes. Using current, basic knowledge about the structure, biophysics, and biochemical properties of biological membrane systems, it is shown here that it is possible to derive a simple equation for the transmembrane potential. Of note, the resulting equation is not based on electrical circuit analogies. Nevertheless, the classical model for the membrane potential based on an equivalent RC-circuit is recovered as a particular case, thus providing a mathematical justification for the classical models. Examples are presented showing the effects of the voltage dependence of charge aggregation around the membrane, on the timing and shape of neuronal action potentials.


Introduction
Biological membranes mediate communication between cellular compartments and their surrounding environments (Blaustein et al., 2004;Boron & Boulpaep, 2016;Helman & Thompson, 1982;Sten-Knudsen, 2002). One of the most important biophysical variables affecting, and affected by such communication, is the difference between the electrical potentials inside and outside the cellular membrane, called transmembrane potential, or membrane potential, for short (Johnston et al., 1995;Sperelakis, 2012;Weiss, 1996). Paraphrasing Cole (Cole, 1933), the existence of the membrane potential is, by itself, an indication of the fact that a cell is alive. The ongoing changes in the membrane potential are physiological correlates of cellular activity in different levels of organization, within the cell itself and its surroundings. Therefore, it is important to have a basic understanding of the biophysical principles underlying its behaviour.
Modeling approaches have been useful to describe electrical and biochemical interactions involving ions, and how they affect the membrane potential (Fricke, 1925;Osterhout, 1933;Teorell, 1935). Many models of biological membranes are based on constructing equivalent electrical circuits (Hermann, 1899, Hermann, 1905 using conservative paradigms like Kirchoff's law of electrical current for RC-circuits (Cole, 1933;Fricke, 1931;Hermann, 1872;Johnson et al., 1989). Of seminal importance, the Hodgkin & Huxley (1952) model describes the changes in membrane potential with a system of differential equations derived from an RC-circuit. The equations in the model are derived by considering an electrical circuit with resistors, a capacitor, and a battery arranged in parallel, respectively representing what is known today as membrane channels, the neighborhood around the membrane, and the membrane potential. The total current in the model is the sum of a "capacitative" current representing flow of electrical charge around the membrane, and different resistive currents representing different transmembrane ion fluxes, now known to be mediated by channels selective to ions of specific types (Hodgkin et al., 1952). Ionic transmembrane currents are, however, not resistive, as ions cross the membrane either by diffusion (Eisenberg, 1998;Neher & Sakmann, 1976), or via mechanical translocation by protein pumps (Gadsby, 2009). However, ions moving across a membrane are not like electrons through a wire and the lipid bilayer of the membrane is not quite like the air-filled space between the plates in a capacitor.
In partial agreement with the basic postulates of models based on equivalent circuits, present knowledge about the molecular structure and biochemistry of biological membranes supports the idea that charged molecules accumulate next to the hydrophilic heads of the lipid bilayer that forms the outer portion of the membrane, and that the density of charge around the membrane is increasing as a function of the membrane potential (Everitt & Haydon, 1968;Wobschall, 1972). One important assumption for the equivalent circuit, is that the charge aggregation around the membrane is proportional to voltage, which implies that the time-dependent rate of change of such a flow is assumed to be constant. Several studies report that the specific membrane capacitance for membranes of multiple cell types is approximately 1 µF/cm 2 (Cole & Curtis, 1938a;Cole & Curtis, 1938b;Cole & Hodgkin, 1939;Galligan et al., 2000;Gentet et al., 2000). The charge density around the membrane is, however, not necessarily proportional to the membrane potential (Kilic & Lindau, 2001;Weinberg, 2015). One reason for the potential divergence is that there are multiple problems associated to measuring the membrane capacitance experimentally (Golowasch et al., 2009;Weinberg, 2015). Perhaps the most important issue in this regard is the non-isopotentiality of the electric fields around membranes. Shown herein, it is possible to derive a simple differential equation that describes the change in the transmembrane potential in a small patch of membrane from basic biophysical considerations, without using electrical circuit analogies. Importantly, a particular case of this generalised formulation explains the phenomenological formulation based on an equivalent circuit originally proposed by Hermann, Hodgkin and Huxley, and others (Hille, 1992). The generalisation and the possibility of explaining an already existing model are in line with the thinking behind the thermodynamic model for transmembrane transport (Herrera-Valdez, 2018), which proposes a general formulation for the molecular fluxes mediated by different physiological transmembrane transport mechanisms, both passive and active, and explains conductance based models as linear approximations around the reversal potential in the case of electrogenic transport.
Once the derivation is explained, it is illustrated with two simple examples of membrane excitability.

Biophysical derivation to describe the transmembrane potential
Consider a small volume Ω containing a small portion of a cellular membrane, together with the extracellular and intracellular compartments in the immediate neighborhood around the membrane. Assume, each of these compartments is filled with physiological solution containing ions of different species, and assume also that the number of ions for each of the different ion species is constant. Let Q a represent the density of charge around the membrane (Coulombs/µm 2 ). Assume also that each of the ion types has some permeability across the membrane, and let Q m represent the charge density in the space within the lipid bilayer of the membrane (Coulombs/µm 2 ). By extension, the total number of ions in the system and the total charge density Q a + Q m (in Coulombs) are constant with respect to time.
Let U e (t) and U i (t) (in Volts) represent the electrical potential in the extra-and intracellular compartments, respectively. The difference v(t) = U i (t) − U e (t) is the transmembrane potential, or by simplicity, membrane potential. For conceptual simplification, if the extracellular potential is equal to zero, then the transmembrane potential could be thought of as the intracellular potential. Since ions tend to accumulate around the membrane as v increases in size, whether positive or negative, the net charge density (Coulombs/µm 2 ) around the membrane can be assumed to be some smooth, monotonic, increasing function Q a (v), equal to zero for v = 0 ( Figure 1A).
It follows from the assumption of constant total charge that the time-dependent change in the total charge density is zero. That is, Here ∂ t represents the instantaneous rate of change with respect to time. Therefore, the term ∂ t Q m (t) represents the current density (Amperes/µm 2 ) carried by different ions across the membrane and ∂ t Q a (v(t)) represents the current density around the membrane.
Equation (1) yields a differential equation for v in terms of the ionic flux across the membrane, normalized by rate of change in charge density around the membrane with respect to v. To see this, note first that applying the chain rule to the term representing the charge density around the membrane yields with ∂ t v in V/s and ∂ v Q a in Coulombs/µm 2 per Volt. Also, the current density across the membrane can be thought of as proportional to the total transmembrane flux of charge. Substitution of Equation (2) into Equation (1), and rewriting after solving for ∂ t v yields where ∂ t Q m (t) represents the total transmembrane current density carried by the different ion types around the membrane (Amperes/µm 2 ). To build specific models, the transmembrane flux ∂ t Q m can be substituted by a sum of different transmembrane ionic fluxes associated to different physiological transport mechanisms (Herrera-Valdez, 2018).
The multiplicative inverse of ∂ v Q a can be thought of as a scaling factor for the change in the membrane potential. Note Q a is cannot be a constant. If Q a is a linear function of v, then the scaling is a constant. If Q a is nonlinear as a function of v, then the scaling changes dynamically with v, possibly amplifying or dampening, depending on the way v changes in time. These possibilities are explored by analysing the effects of the profile of charge accumulation around the membrane on the generation of neuronal action potentials.
An important particular case that gives theoretical justification to the equivalent circuit analogy for membranes as a linear approximation is analysed in the following section. Having done that, the behaviour of the dynamics for v are explored using two nonlinear profiles for Q a ( Figure 1A) that behave similarly to the linear case near v = 0, but differ in that one saturates, and the other grows exponentially, as the membrane is polarised.
The membrane capacitance from equivalent circuit models Assume that charge aggregation around the membrane is linear as a function of voltage. That is, where C m is a constant that describes the voltage-dependent rate of charge density around the membrane ( Figure 1A, black line). In this case, ∂ v Q a = C m ( Figure 1B, black line), which means that Equation ( as proposed by Hodgkin & Huxley (1952), and others.
The change in the density of charge around the membrane, ∂ v Q a , is in Coulombs per volt per µm 2 , which is equivalent to farads/µm 2 , the units used for electrical capacitance. The plates of a capacitor in an electrical circuit are made of conducting media (e.g. metal), and separated by air, or other insulating material. Therefore, this particular case supports the idea that the intra and extracellular media can be thought of as similar to the metal plates in the capacitor, and the membrane can be regarded as analogue to the insulating air layer between the plates.
Two nonlinear voltage dependent profiles for the density of charge around the membrane Q a with saturation as a function of v It is arguable that there is a voltage-dependent upper bound for the accumulation of charges around the membrane ( Figure 1A, blue curve). That is, if the membrane is polarised to very large voltages (either negative or positive) the charge density around the membrane could be assumed to reach a limit. One possible description for the voltage-dependence of Q a could be where v T = k B T/q e in mV is a thermal potential, with k B , T, and q e are Boltzmann's constant, the absolute temperature, and the elementary charge, respectively. The rate of change of Q a with respect to v is then, Figure 1B, blue curve). Q a (v) in this case an odd and increasing function, bounded by the asymptotic values of ±2v T C m , respectively ( Figure 1A, blue curve). The graph of ∂ v Q a in Equation (6) has a symmetrical shape around a local maximum at v = 0, always taking positive values ( Figure 1B, blue curve). This means that the density of charge around the membrane tends to change less as the membrane is polarised. The amplification factor 1/∂ v Q a for the rate of change in v increases, and its maximum change occurs when the membrane is not polarised. As a consequence, 1/∂ v Q a exerts an amplification effect on ∂ t v that is larger for polarised membranes, and has a minimum at v = 0.
Exponentially increasing charge density around the membrane Another possibility similar to the current densities from the thermodynamic model (Herrera-Valdez, 2018), is that Q a (v) is a hyperbolic sine. In this case and ( ) The graph of Q a (v) is odd and increasing with exponential growth in both directions of the v axis ( Figure 1A, orange curve), qualitatively opposite to that described earlier for the saturating Q a with hyperbolic tangent shape. In this case ∂ v Q a has a "U" shape with a local minimum value of C m at v = 0, which means that ∂ v Q a exerts an attenuation effect on ∂ t v that increases as v is further from 0 ( Figure 1B, orange curve).
Note the three profiles for the charge density around the membrane presented above can be thought of as approximations of one another around v = 0.
Neuronal dynamics assuming voltage dependent density of charge around the membrane How does the voltage-dependence in the charge-density around the membrane influence the changes in the transmembrane potential? Specifically, what are the effects of the nonlinearities in the voltage-dependence of ∂ v Q a on the dynamics of the transmembrane potential? To compare the effects caused by the three profiles (tanh, linear, and sinh) for Q a , the dynamics for the transmembrane potential in a neuron were modelled (Figure 2, blue, black, and orange lines, respectively).
Consider a neuronal membrane and assume that it has three transmembrane transport mechanisms, say, voltage-dependent K + and Na + channels, and a Na + -K + ATPase. If w represents the proportion of open K + channels and the proportion of inactivated Na + channels, then the dynamics of the membrane potential can then be written as with ∂ v Q a as in equations Equation (4), Equation (6), and Equation (8). The transmembrane current densities (pA) given by and auxiliary functions for the transmembrane fluxes, the activation rates and the steady state activation of the K + channels given by for ι ∈ {NaK, K, Na}, and A description of all the parameters and their values can be found in Table 1.
Note that the ranges for the three different profiles for ∂ v Q a ( Figure 1B) can be ordered by magnitude. The hyperbolic profile for Q a yields the lowest values for ∂ v Q a , in comparison to the constant ∂ v Q a = C m , and the hyperbolic sine profile, which yields the largest values for ∂ v Q a . Therefore, the scaling effects of 1/∂ t Q a on the change in transmembrane potential can be thought of in the opposite order, with larger effects caused by saturating Q a , a smaller scaling for the linear profile, and the smallest scaling effect caused by the exponential profile. It is possible to appreciate this by comparing the trajectories of three neuronal action potentials from the same initial conditions (v 0 = −48 mV and w 0 =0.001), obtained from the model in equations Equation (9)-Equation (16) using the same parameters (Herrera-Valdez et al., 2013), and the same initial conditions, except for their Q a profile ( Figure 2).
As anticipated, the smallest scaling effect on the change in v occurs for the hyperbolic sine profile (Figure 2, orange curves, spike time at 6.62 milliseconds approx), as shown by the longer delay in the action potential in comparison with the linear and saturating profiles (Figure 2A-B, black and blue curves, with spike times at 13.63 and 20.82 milliseconds, respectively). The delays to the action potential can be increased if the initial condition is lower, or decreased if higher, without altering the hierarchy in the scaling effect.
The scaling effects on ∂ t v caused by the three Q a profiles can also be appreciated in the trajectories of the (∂ t v,v)-plane, which shows a smaller area contained by the curve for the exponential-profile, in comparison to the areas bounded by trajectories produced by the linear and saturating profiles for Q a , respectively.
The dynamical ranges for Q a for the different profiles show smaller departures from 0 in the saturating profile with respect to the other two profiles ( Figure 2C-D). This suggests that that measuring charge aggregation around the membrane during action potentials would not necessarily show large differences indicative of the Q a profile, at least with the assumptions made here.
In contrast, the scaling induced by the inverse of ∂ v Q a is notorious in both its time-dependence of the scaling factor, and also in its trajectory as a function of ∂ t v ( Figure 2E-F). The larger scaling effect of the saturating profile on ∂ t v can be clearly appreciated in Figure 2F, which shows the values of the scaling factors of the saturating profile during the action potential above any of the values from the linear or exponential profiles.
One indirect way of measuring the effects of the scaling factor 1/∂ t Q a on the change in membrane potential, ∂ t v, is to calculate the efficiency of the Na + current during the upstroke of the action potential (Carter & Bean, 2009). This is done by calculating the total Na + charge during the upstroke, and dividing it by the total charge during the same time interval. Explicitly, if starting from an initial condition v = v 0 , the time interval between the start of the trajectory and the peak time of an action potential is [0, T] .
Scaling factors with more dampening effects would cause the Na + current to be less efficient.
To further illustrate the scaling effects of the Q a profiles on ∂ t v, the maximum upstroke velocities, the time delays to produce action potentials, and the efficiency of the Na + current during the upstroke were calculated from sample trajectories obtained by increasing the values of the initial condition for v (Table (2)).
In agreement with the observations made above, the decreasing amplification caused by the saturating and linear profiles (in that order) with respect to the exponential profile were negatively correlated with the efficiency, and the delay to spiking. Nevertheless, it is worth noticing that the differences decrease for larger values of v 0 .

Discussion
A simple equation describing the time evolution of the transmembrane potential has been derived from basic biophysical principles (Equation (3)). The new equation is in line with the derivation of the thermodynamic model (Herrera-Valdez, 2018), in that it only considers basic biophysical principles to describe the elements in a system that include a membrane, and ions in solution. Importantly, the derivation is not based on an equivalent circuit analogy, but it is general enough to allow the possibility of obtaining the traditional equation for the equivalent RC-circuit. The derivation draws from basic, current knowledge about the membrane and its surroundings and separately taking into account its ion-impermeable and ion-permeable properties, as referred by early works by Cole and Curtis and other authors.
As it is desirable in any generalization, it explains a classical model proposed in seminal work previously done. Explicitly, the expression for the change in voltage-dependent in the charge density around the membrane can be simplified to obtain the so-called "membrane capacitance" in classical models. This is done by assuming that the charge density around the membrane is linearly related to the membrane potential. Interestingly, the term emerges from applying the chain-rule to calculate the time-dependent change in the total charge density around the membrane, which depends on voltage (Coombs et al., 1959;Everitt & Haydon, 1968;Lu et al., 1995).
From a physical perspective, changes in the sign and magnitude of the transmembrane potential should cause the charges around the membrane to redistribute (Huang, 1981;Santos-Sacchi & Navarrete, 2002). Such changes occur within very short times relative to the usual time scales taken into account for studies of membrane excitability, or electrical signalling in general. One important result from the derivation is that the change in the charge density around the membrane is not necessarily a constant, as evidenced by different reports (Amzica & Neckelmann, 1999;Neher & Marty, 1982;Proks & Ashcroft, 1995;Smith et al., 1997).
For historical interest, prior to 1938 approximately, the hypothesis the membrane would disintegrate at the pass of the excitation wave during an action potential was a source of debate and reason for experimentation (Cole & Curtis, 1938a). The model presented here provides a theoretical argument that explains how changes in ∂ v Q a can take place without membrane disintegration, and enables the possibility of studying membrane systems without assuming that the amount of charge aggregation around the membrane is proportional to voltage (Everitt & Haydon, 1968;Kilic & Lindau, 2001). Cole and Curtis dismissed that hypothesis on the basis of finding very small changes in the membrane capacitance using a Wheatstone bridge to record currents from squid axons and other preparations. Of note, the change in accumulation of charge around the membrane, commonly called "membrane capacitance" is typically regarded as a constant since around the time of the seminal work of Cole in the decade of 1930's (Cole & Curtis, 1938b;Cole & Curtis, 1939), and others (Everitt & Haydon, 1968). Further, more recent studies have also found non significant changes in membrane capacitance (Gentet et al., 2000). One problem is that such measurements have been made for membrane potentials around rest. From a theoretical perspective, one reason that experimental measures of membrane capacity have proved to be difficult, is the assumption of isopotentiality, which does not hold almost surely (Chen & Gillis, 2000;Golowasch et al., 2009). Nevertheless, from a physical perspective, the change in the charge density with respect to voltage, ∂ v Q a , can be a non-constant function (Santos-Sacchi & Navarrete, 2002). Indeed, it has been shown in experimental work involving exo and endocytosis (Neher & Marty, 1982;Proks & Ashcroft, 1995;Smith et al., 1997), and in relation to voltage-gated currents (Kilic & Lindau, 2001). The profiles for saturation and exponential aggregation of charge around the membrane defined above suggest that experimental measures of the impermeable aspect of the membrane as originally called by Cole and Curtis and others, may be even more difficult than previously thought (but see (Kilic & Lindau, 2001)).
Of note, the runs with the saturating Q a profile suggest a reasonable explanation for the rapid changes during the initiation of an action potential (Naundorf et al., 2006), as they suggest a way to accelerate the initial depolarisation during the action potential upstroke that cannot be obtained from the Hodgkin and Huxley model.
One issue of possible importance is the timing of action potentials in a network context, which, on top of being influenced by local field potential oscillations (time-dependent forcing on Equation (3), could be subject of non-negligible rescaling effects due to voltage-dependent changes in ∂ v Q a (Amzica & Neckelmann, 1999). Future experimental measurements may shed light on this issue, as it could have large repercussions in our interpretations of network dynamics.

Data availability
All data underlying the results are available as part of the article and no additional source data are required.
Therefore, the term ∂ t Qm(t) represents the current density (Amperes/μm2) carried by different ions across the membrane and ∂ t Qa(v(t)) represents the current density around the membrane" need to better be explained and justified. For example, does the ∂ t Qa(v(t)) represents the current density in the bulk liquid phase around the membrane or not?
Author's Eq. 3 seems missing a negative sign.
The manuscript says: "if the membrane is polarised to very large voltages (either negative or positive) the charge density around the membrane could be assumed to reach a limit". Is there any evidence for such a case? In reality, there is no evidence for such a case in any biological systems. But, the author seems arbitrarily use a hyperbolic tangent "tanh" function to describe the voltage-dependence of Qa (Eq. 5) without justification.
The manuscript also says: "Another possibility similar to the current densities from the thermodynamic model (Herrera-Valdez, 2018), is that Qa(v) is a hyperbolic sine". Is there any experimental evidence or justification for this case?
So, the manuscript leaves with three cases: linear, hyperbolic tangent "tanh", and hyperbolic sine "sinh"; Which one you want people to believe?
It is questionable whether a paper like that would cause confusion in the field.
Therefore, if the author could objectively address these important issues, this reviewer would consider support the indexing of his revised manuscript.