Density artefacts at interfaces caused by multiple time-step effects in molecular dynamics simulations

Introduction

To describe the dynamic processes of biomolecules, such as proteins, DNA, carbohydrates and membranes, molecular dynamics (MD) simulations have been proven to be a powerful technique to provide insights at the atomic level. In MD simulations, physical processes that involve a wide range of time scales have to be considered appropriately. However, the accurate and efficient treatment of various time scales presents a notoriously difficult problem due to resonance artefacts arising from separation into distinct numerical subproblems^1,2. Over the past decades, various methods have been proposed to mediate the issue and to speed up the simulation substantially. For example, fast vibrational modes (e.g. covalent bonds) can be removed completely by constraining the motion, e.g. using SHAKE³, SETTLE⁴, M-SHAKE⁵, LINCS⁶, or FLEXSHAKE⁷. Alternatively, multiple time-step (MTS) integration is an efficient approach to accommodate motions on different time scales. In the twin-range (TR) scheme^8–10, the pairwise nonbonded forces ${\mathbf{\text{f}}}_i^{nb}$ acting on particle i are split into a short-range contribution from particles j within the cutoff R_s , and a mid-range contribution from particles j between R_s and the long-range cutoff R_l (see Figure 1),

Figure 1. Simulated Systems.

Left: Schematic illustration of the twin-range (TR) setup with the inner cutoff R_s and the outer cutoff R_l. Interactions beyond R_l are not calculated explicitly. Middle: Schematic illustration of the planar layer systems. Right: Schematic illustration of the systems with a droplet in a water box.

with

The short-range forces ${\mathbf{\text{f}}}_i^{nb,s}$ are updated every time step ∆t (inner time step), while the mid-range forces ${\mathbf{\text{f}}}_i^{nb,m}$ are updated every n_m-th time step, n_m∆t with n_m ≥ 1 (outer time step). Long-range electrostatic interactions beyond the cutoff radius R_l can be included using the (an)isotropic reaction-field methods (RF or ARF)^11,12, or lattice-sum methods^13,14. Although theoretically possible, the TR scheme is generally not used together with lattice-sum methods. Simulations with (A)RF typically use larger cutoffs R_l than simulations with PME (e.g. 1.4 nm for the GROMOS force field¹⁵ compared to 0.9 nm for the AMBER force field¹⁶), which in the past encouraged the use of a TR scheme to reduce the computational cost.

In practice, for each particle i two pairlists are generated, which track the neighbouring short-range and mid-range particles j. To be consistent, the period of updating both pairlists n_p∆t should be equal to the period of updating the mid-range forces n_m∆t. In the following, the TR scheme is defined by R_s < R_l and n_m = n_p > 1, and the single-range (SR) scheme by R_s = R_l , n_m = 1 and n_p ≥ 1, i.e. there is only one pairlist for SR.

Splitting the forces into different contributions leaves some freedom of how to consider them in the integration. There are two main strategies for the inclusion of the outer contributions: (i) constant mid-range forces application (CFA) ${\mathbf{\text{f}}}_i^{nb,m(CFA)}(t)$ at every inner time step ∆t, or (ii) using a reversible reference system propagator algorithm (RESPA)^8–10, which is based on the Trotter factorisation of the Liouville propagator¹⁷. In the case of RESPA, scaled mid-range forces ${\mathbf{\text{f}}}_i^{nb,m(RESPA)}(t)$ are typically applied impulse-wise, i.e. only at every outer time step n_m∆t. Note that in this work the leap-frog scheme¹⁸ is used for the integration of the equations of motion, which is similar¹⁹ to velocity-Verlet²⁰ as typically employed with RESPA. The resulting discrete integration scheme can be written as follows,

Note that for simplicity, the notation is restricted to the microcanonical case and it is implicitly assumed that t mod ∆t: = ∆t(n mod 1) = 0, i.e. the time t = n∆t is discretized with respect to ∆t, where mod denotes the modulo operation.

Although computationally efficient, MTS algorithms have the disadvantage of introducing numerical artefacts. Due to nonlinearity, it is notoriously difficult to formulate general physical or chemical conditions, which favour or suppress the build-up of these undesired resonances. Instead, specialised thermostats have been proposed as a remedy, which suppress resonance artefacts from MTS integration, for example by imposing isokinetic constraints on every degree of freedom^21–23, or by selectively targeting the high-frequency modes using a colored noise thermostat²⁴. In practice, however, standard thermostats like weak-coupling²⁵, Nosé-Hoover (chain)^26–28, or stochastic thermostats^29–31 remain widely used in combination with a moderate integration time-step difference between fast and slow motions, as this should limit resonance artefacts to an acceptable value.

In this work, artefacts due to MTS integration are investigated with the main focus on a water-chloroform layer system using different thermostats (Figure 1). This system represents a simplified version of a biological membrane, for which an accurate simulation is of high interest in biology. The impact of splitting the nonbonded interactions into a short-range and mid-range contribution on the density profile normal to the interface and the kinetic energy distribution is investigated. In addition, a planar system consisting of cyclohexane and water as well as corresponding droplet systems are investigated. These additional setups allow to assess the impact of different solvents as well as different interface geometries on the formation of MTS artefacts. Overall, the artefacts are larger than those observed for more well-behaved, homogeneous systems^32,33.

Methods

For all simulations, the GROMOS software package³⁴ version 1.4.0 was used with the GROMOS 53A6 force field¹⁵. Periodic boundary conditions were applied. Newton’s equations of motion were integrated using the leap-frog scheme¹⁸ with an inner time step of 2 fs. Different outer time steps for the pairlist update or mid-range force evaluation were used (i.e. n_m , n_p ∈ [1, 20]). If not indicated otherwise, the SR scheme corresponds to a midrange force evaluation every 2 fs (n_m = 1) and a pairlist update every 10 fs (n_p = 5). The TR scheme typically corresponds to a mid-range force evaluation in combination with a pairlist update every 10 fs (n_m = n_p = 5). If not stated otherwise, the CFA method was used for the mid-range force contributions. The inner cutoff radius R_s was set to 0.8 nm and the outer cutoff radius R_l to 1.4 nm. The center of mass motion was stopped every 1 ps. The coordinates were saved every 0.2 ps and the energies every 0.1 ps. Bond lengths were constrained using the SHAKE algorithm with a geometric tolerance of 10⁻⁴. Four different thermostats were tested, i.e. weak coupling²⁵ (WC), Nosé-Hoover^26,27 (NH), Nosé-Hoover chain²⁸ (NHchain), and stochastic dynamics (SD) i.e. a Langevin thermostat³⁵. Different temperature baths were used for different solvents. The coupling time τ was set to 0.1 ps for all non-stochastic thermostatting methods. A chain length of three was employed for the NHchain thermostat. For SD, a friction coefficient γ = 10 ps^-1 was used. A standard homogeneous RF approach¹¹ with a charge-group based pairlist algorithm was used (each solvent molecule is a single charge group) to mimic long-range electrostatic interactions beyond the cutoff R_l. The dielectric permittivity was set to 78.5, which corresponds to the experimentally measured value of water³⁶.

Results & discussion

The density profile of the water-chloroform system was determined along the z-axis normal to the layer. As can be seen in Figure 2, the splitting of the forces into a short-range and a less frequently updated mid-range contribution (given in Equation (1)) using the TR CFA scheme with n_m = n_p = 5 led to a significant density increase in chloroform close to the interface for WC, NH and NHchain thermostats. A corresponding graph with the separate densities of the two phases is given in Figure S2, Supplementary File 1. The small density fluctuations in Figure 2 are due to the relative orientation of the molecules at the interface in combination with the granularity of the molecules and the binning. On the contrary, no significant density increase was found for SD for pairlist-update periods up to 40 fs as shown in Figure 3. However, SD appears to cause slightly more pronounced local density fluctuations in the chloroform phase compared to the non-stochastic thermostats for the chosen simulation time of 1 ns, independent of the pairlist-update period.

Figure 2. Profile of the density ρ in the water-chloroform layer system with respect to the layer normal z.

For the five setups with the twin-range (TR) scheme, the mid-range forces were updated with the same frequency as the pairlist (i.e. n_m = n_p = 5). For the single-range (SR) scheme, the mid-range forces were evaluated every time step ∆t (i.e. n_m = 1), whereas the pairlist was updated every 5th step (i.e. n_p = 5). A weak-coupling (WC), Nosé-Hoover (NH), Nosé-Hoover chain (NHchain), or a stochastic (SD) thermostat were used.

Figure 3. Density profile with respect to the layer normal z and the pairlist update period n_p∆t.

For the single-range (SR) scheme, the mid-range forces were evaluated every time step ∆t (i.e. n_m = 1). For the four setups with the twin-range (TR) scheme, the mid-range forces were updated with the same period as the pairlist (i.e. n_p = n_m). The TR scheme was applied in a constant (CFA) or impulse-wise (RESPA) manner. A weak-coupling (WC), Nosé-Hoover (NH), or a stochastic dynamics (SD) thermostat were used. Note that the density range is chosen to visualize the major artefacts occurring for n_m∆t > 15 fs. A visualisation of the minor artefacts around n_m∆t ≈ 10 fs can be seen in Figure 4.

For the non-stochastic thermostats, the density increase of the TR setup was accompanied by a density decrease further away from the interface due to the constant volume conditions. An analysis of the density resonance pattern with respect to the outer time step n_m∆t is shown in Figure 3. Note that the density artefacts are not perfectly symmetric with respect to the layer normal due to NVT conditions and finite layer thickness. The asymmetry for large outer time steps is due to the small thickness of the layers, which enables aggregation at one interface and depletion at the other interface. It can be seen that the TR CFA scheme leads to large resonance artefacts beyond n_m∆t ≈ 20 fs. On the other hand, evaluating the mid-range forces every step while keeping the pairlist update every fifth step (i.e. SR scheme with n_m = 1 and n_p = 5) resolved the issue almost entirely for all investigated thermostats (Figure 2 and Figure 3, data not shown for NHchain). The minor differences between SR schemes with different update frequencies can only be seen in density difference plots (Figure 4) with respect to the SR reference run (n_m = n_p = 1).

Figure 4. Spacial density difference profiles with respect to a SR reference simulation (n_m = n_p = 1) under NVT conditions for layer systems (left) and droplet systems (right) with water-chloroform composition (top) or water-cyclohexane composition (bottom).

The Nosé-Hoover (NH) thermostat was used in all setups. The single-range (SR) scheme (red and orange lines) and the twin-range scheme with constant-force application (TR CFA, light and dark blue lines) and two different update frequencies (n_p = 5, 10) are compared. Running averages with a window of seven data points are shown.

These findings indicate that the observed density increase among non-stochastic thermostats is mainly caused by resonance artefacts due to the CFA MTS integration. This argumentation is supported by the pairlist error analysis shown in Figure S3, Supplementary File 1. The change in the pairlist after 10 fs is only on the order of 1%, and, even more importantly, it scales sublinearly with n_m. Thus, the error from the pairlist cannot explain the substantial density increases seen in Figure 3. Instead, resonance effects lead to the build-up of the observed density artefacts over time, until a local equilibrium configuration is reached at the interface. This was found to be accompanied by a local change in the diffusion of the molecules at the interface. The mean residence time with respect to the layer normal z shows that regions of high density imply a locally reduced diffusivity of the molecules and vice versa (Figure S4, Supplementary File 1). In addition, it could be shown that the erroneous density at the interface leads to a slight deviation of the molecular orientations (Figure S5, Supplementary File 1). Note that the build-up of the density artefacts for realistic MTS setups evolves on a time scale much shorter than the duration of performed simulations (Figure S6, Supplementary File 1).

The numerical artefacts stem likely from an increasing mismatch between the forces and the position of the atoms the forces are applied to. In line with this interpretation, the density artefacts were suppressed for a typical value of the outer time step n_m∆t = 10 fs when the mid-range forces were applied impulse-wise (TR RESPA scheme, blue line in Figure 2), i.e. the direction of the forces and the atom positions matched at the time point when the forces were applied. However, the TR RESPA scheme does not eliminate the resonance artefacts of non-stochastic thermostats, but only shifts them to longer outer time steps in this context (Figure 3). Thus, the partitioning of the system into different time scales together with the choice of the time steps in the MTS integration remain delicate aspects of current MD implementations⁴⁰, also for impulse-wise MTS algorithms^1,2,41.

On the contrary, only small effects on the density profile could be observed for the SR scheme with pairlist update periods up to n_p∆t = 40 fs (Figure 3 and Figure 4). Due to the large cutoff R_l used here (i.e. 1.4 nm), the changes in the pairlist are small between updates and scale sub-linearly with n_m (Figure S3, Supplementary File 1). This observation is in line with the study of Krieger et al.⁴², where they found a negligible energy drift when using a similarly low pair-list update frequency in a homogeneous system. When insisting on a non-stochastic thermostat, a rough performance analysis of the run time showed that for a realistic setup approximately half of the computational performance loss due to evaluating the mid-range forces at every time step could be partially compensated by a less frequent update of the pairlist (Table 2).

Chemical composition and constant pressure

In order to exclude that the observed artefacts are specific to the water-chloroform system, the densities at a water-cyclohexane interface in a layer system were investigated. The corresponding density difference plots with respect to a SR reference are given in Figure 4. They show a similar behaviour of the TR scheme compared to the one observed for the water-chloroform systems, with slightly less pronounced artefacts. In addition, imposing constant pressure (NPT) conditions in the simulation setup does not reduce the observed density artefacts (Figure 5). The average pressure in the NVT simulations and the average volume in the NPT simulations are listed in Table S1 (Supplementary File 1). Note that under NVT conditions, the observed density artefacts for the TR scheme lead to a reduction of the pressure as a function of the pairlist update frequency. As mentioned the box volume is slightly too small due to the chosen setup. Under NPT conditions and using the SR scheme, the box volume remains relatively constant. However, the semi-anisotropic pressure coupling in the x,y-direction with fixed z-dimension did not allow to preserve the system pressure at 1 atm for the elongated bilayer system. Separate scaling of the z-dimension did not improve the pressure control of the given system (data not shown). Using the TR scheme, the corresponding box volumes decrease as a function of the pairlist update frequency, in line with the NVT results.

Figure 5. Density profile of the water-cyclohexane layer system under NVT conditions (left) or NPT conditions (right).

A semi-aniostropic pressure coupling was applied in x/y-direction using a Berendsen barostat, while the box length in z-direction remained fixed. A Nosé-Hoover (NH) thermostat was used in all setups. The single-range (SR) scheme (purple, red, and orange lines) and the twin-range scheme with constant-force application (TR CFA, light and dark blue lines) and three different update frequencies (n_p = 1, 5, 10) are compared. Running averages with a window of seven data points are shown.

Thermostatting method

The impact of the chosen non-stochastic thermostatting algorithm as well as the MTS integrator, and the energy distributions was investigated for the water-chloroform layer system. The WC thermostat does not give a canonical distribution of the kinetic energy by construction⁴³, whereas the NH and NHchain thermostats do. In Figure 6, the kinetic energy distributions within water and chloroform are shown for the different setups. For chloroform, the difference in the width of the distribution between WC and NH(chain) thermostats is clearly visible, whereas the average kinetic energy remains the same, despite the significant density gradient in this solvent. In water, on the other hand, a clear shift in the kinetic energy distribution towards a higher average value was observed for the TR CFA scheme compared to the SR scheme with the WC thermostat. This shift was less pronounced with the NHchain thermostat, and absent with the NH thermostat. These observations indicate that a temperature shift may occur due to MTS resonances. However, as the density increase occurred with all three thermostatting methods, but the temperature shift only with two of them, the latter cannot be the cause for the former. Therefore, one can conclude that the resonance artefacts caused by the TR scheme cannot be removed with thermostats that control the velocities globally. For this, one would have to control or perturb the velocities selectively as done in SD or as suggested in the literature by imposing isokinetic constraints or restraints on each degree of freedom^21–23.

Figure 6. Histogram of the kinetic energy for chloroform (left) and water (right) using a weak coupling (WC, black), Nosé-Hoover (NH, red) and Nosé-Hoover chain (NHchain, green) thermostat (bin size 1 K).

Note that a different number of molecules is simulated for chloroform (left) and water (right), which leads to a different width of the canonical distribution functions. For the twin-range (TR) scheme (solid lines), the mid-range forces were updated with the same period as the pairlist (i.e. n_m = n_p = 5). For the single-range (SR) scheme (dashed lines), the mid-range forces were evaluated every time step Δt (i.e. n_m = 1), whereas the pairlist was updated every 5th step (i.e. n_p = 5).

Dataset 1.Zip containing simulation input files.

Conclusions

The effect of using the TR scheme with a constant force application (CFA) at solvent-solvent interfaces was investigated with layer and droplet systems of different composition (water-chloroform and water-cyclohexane). These systems represent simplified versions of a biological membrane and a protein in solution. The force splitting was found to lead to substantial MTS resonance artefacts at the interfaces for WC, NH and NHchain thermostats. It could be shown that the presence of the artificial density increase at the interfaces is independent of the interface geometry as well as the chemical composition of the system although the magnitude could vary. In addition, the diffusivity of the solvent molecules at the interface changed due to the density artefacts.

The resonance effects likely stem from a mismatch between the forces and the position of the atoms the forces are applied to. This hypothesis is supported by the fact that the pairlist changes only in the order of 1 % between updates and that the TR scheme with RESPA (where the forces are applied impulse-wise) does not eliminate the numerical artefacts but shifts them to longer time steps. The randomly applied stochastic forces of a SD thermostat seem to disrupt the MTS resonance build-up and thus successfully prevent the occurrence of density artefacts at the interfaces. Based on these findings, the use of TR CFA is not recommended for heterogeneous systems with WC, NH or NHchain thermostats. Instead, the SR scheme should be used, which did not show any significant density errors in the entire range of pairlist updates (analysed up to n_p∆t = 40 fs). Thus, the increase in computational cost for SR can be partially compensated by updating the pairlist less frequently (i.e. n_p∆t > 10 fs). Of course, there is a limit to the latter, as other errors are introduced by less frequent pairlist updates. By using only a single pairlist, the complexity of the implementation can be reduced, as the additional difficulties arising from efficient memory management of two (or more) pairlists can be avoided.

Data availability

We are unable to provide the output files from the simulations directly due to file size. We instead provide the input files that when used as input with the GROMOS software package will generate the same results -

Dataset 1: Zip containing simulation input files 10.5256/f1000research.16715.d222949⁴⁴