Deep learning of position-dependent diffusivity from umbrella sampling molecular dynamics simulations

Abstract

A widely used computational framework to calculate the membrane permeability coefficient of small molecules is the inhomogeneous solubility-diffusion (ISD) model. It requires two ingredients that can be calculated using molecular dynamics simulations: the potential of mean force, which is well-defined, and the position-dependent diffusivity, which is often problematic and challenging. Two methods (Woolf-Roux and Hummer) have been proposed to determine the position-dependent diffusivity profiles using biased umbrella sampling simulations. While both are constructed from similar time-correlation functions, they can disagree quantitatively. Here, a reconciliation of these methods is achieved through deep learning memory functions in the time domain. The diffusivity extracted through this analysis is shown to be in best agreement with the equilibrium counting permeability for the same membrane system compared to both the Woolf-Roux and Hummer diffusivity. The effect of memory on the rate of barrier crossing is assessed through numerical simulations of the generalized Langevin equation (GLE). The GLE is efficiently simulated via Markovian embedding, which relies on the positive, decaying exponential form of the memory functions extracted by deep learning. Our results based on the ISD permeability, the known permeability from equilibrium MD, and the numerical GLE simulations indicate that memory effects most likely do not have a significant effect on the permeation of water.

I. INTRODUCTION

A widely used computational approach to calculate the permeability coefficient of a small molecule across a membrane using molecular dynamics (MD) simulations is the inhomogeneous solubility-diffusion (ISD) model, first derived by Diamond and Katz,¹

$$P={\left[{\int }_{-L/2}^{L/2}\mathrm{d}z\frac{e^{W\left(z\right)/k_{\mathrm{B}}T}}{D\left(z\right)}\right]}^{-1}$$ (1)

where $L$ is the width of the membrane, $z$ is the $z$-component of the distance between the permeant and the center of the membrane, $W\left(z\right)$ is the potential of mean force (PMF), and $D\left(z\right)$ is the position-dependent diffusivity. Part of the appeal of the ISD approximation is its synergy with MD simulations, which can be used to determine both the PMF and position-dependent diffusivity. Indeed, one of the earliest studies of water permeation through lipid membranes based on MD simulations relied upon this model.² It is not without limitations, however. While the PMF can be determined using enhanced sampling techniques such as umbrella sampling,³ estimating the position-dependent diffusivity can be more challenging.^4–7 For a molecule in a membrane (permeant), the diffusivity is expected to vary along $z$ due to the inhomogeneous membrane environment. Therefore, estimating the diffusivity from the classic mean-square deviation (MSD) relation, $D=\underset{t\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}⟨(z(t)-z(0){)}^2⟩/2t$, while valid for homogeneous bulk solution, is not strictly applicable in this case.

The friction coefficient, $\gamma$, is related to the diffusivity $D$ via the Einstein relation, $D=k_{\mathrm{B}}T/\gamma$. The friction can be estimated from the integral of the force-force time-correlation function exerted on a permeant fixed at position $z$.⁸ However, this approach is not always reliable, as it has been shown to significantly overestimate the diffusivity in the case of an ion in water.⁹ Alternatively, the permeant may be harmonically restrained around a position $z$ to determine a local $D(z)$, although special considerations are necessary to extract the diffusivity from the biased dynamics of the restrained molecule. For instance, the integral of the velocity-velocity time-correlation function, which is related to the diffusivity in homogeneous bulk solution according to Green-Kubo linear response, does not provide a useful route when there is a restraining potential.

The generalized Langevin equation (GLE) provides the most rigorous framework to pursue this type of analysis.¹⁰ From this mathematical construct, one may estimate the friction from the integral of the memory function, and the diffusion coefficient from the Einstein relation. In fact, the treatments based on a fixed or restrained permeant are closely related. Berne et al. showed that the memory function of a particle restrained by a harmonic potential in the limit of an increasing spring constant asymptotically approaches the force-force time-correlation function of a fixed particle.¹¹ Ultimately, it is important to keep in mind that the physical picture of overdamped diffusive motion is valid only when certain fundamental conditions are met.^12,13 In particular, the mass of the solute must be larger than the mass of the surrounding solvent, a condition that may or may not be satisfied as the permeant diffuses along $z$.

Despite these fundamental limitations, a practical strategy for extracting diffusivity from umbrella sampling simulations with a harmonic restraining potential can be formulated within the GLE framework. An approach based on the velocity-velocity time-correlation function was first derived by Woolf and Roux¹⁴ following an analysis by Straub et al.¹⁵ This result was revisited by Hummer to derive a mathematically equivalent form in terms of the position-position time-correlation function.¹⁶ This latter method is perhaps more widely used, but it is not necessarily more accurate in practical applications.^4,17,18 It was shown by Rowley and coworkers that the Woolf-Roux and Hummer diffusivity expressions can yield significantly different results using the same set of trajectory data, and both can be unreliable.⁴ Nevertheless, the origin of these unresolved problems has not been fully elucidated.

To circumvent these issues, we implemented a deep learning approach to directly extract the memory function of the GLE from umbrella sampling MD simulations. The deep learning approach, developed by Russo et al.,¹⁹ employs a multilayer perceptron (MLP) to parameterize a sum of positive, decaying exponential functions that correspond to the memory function of a GLE in a harmonic potential in the time domain. The method is applied to the case of water permeation across a DLPC membrane. The memory function extracted by the MLP independently in each umbrella window was built into an effective position-dependent memory function across the membrane. After parameterizing the PMF and the memory function, the position-dependent diffusivity across the membrane was determined and inserted into Eq. (1) to obtain the permeability. It is shown that the MLP is able to reliably extract the memory function of a water molecule across the highly inhomogeneous environment of a lipid bilayer. Using these results, the issues with the Woolf-Roux and Hummer treatment of the GLE are clarified. Lastly, the Grote-Hynes theory of barrier crossing²⁰ is used to show that the physical picture of overdamped diffusive motion is valid for water permeation.

II. THEORETICAL DEVELOPMENTS

The physical model underlying the present analysis begins by representing the dynamics in terms of the generalized Langevin equation (GLE),¹⁰

$$m\ddot{z}(t)=-\frac{\partial W}{\partial z}-{\int }_0^{t}M\left(t-t^{\prime }\right)\dot{z}\left(t^{\prime }\right)\mathrm{d}{t}^{\prime }+f(t)$$ (2)

where $m$ is the mass of the permeant, $W(z)$ is the potential of mean force (PMF), $M(t)$ is the memory function, and $f(t)$ is a Gaussian random noise obeying the second fluctuationdissipation theorem, $⟨f(0)f(t)⟩=k_{\mathrm{B}}TM(t)$. By construction, the total friction given by

$$\gamma ={\int }_0^{\mathrm{\infty }}M(t)dt$$ (3)

and the diffusivity $D$ is recovered via the Einstein relation,

$$D=\frac{k_{\mathrm{B}}T}{\gamma }$$ (4)

In the case of umbrella sampling simulations, the system is restrained by a harmonic potential, $k{z}^2/2$, with force constant $k$. In practice, the technical challenge consists in accurately extracting the memory function $M(t)$ from the biased dynamics of a permeant that is restrained around a specific position $z$.

A. Multilayer perceptron (MLP) position-dependent diffusivity

To treat the data from the umbrella sampling simulations, we reduce the GLE to time-correlation functions by multiplying both sides of Eq. (2) by the initial velocity, $\dot{z}(0)$, which results in,

$$C_{Fv}(t)={\int }_0^{t}M\left(t-t^{\prime }\right)C_{vv}\left(t^{\prime }\right)\mathrm{d}{t}^{\prime }$$ (5)

where $⟨f(t)\dot{z}(0)⟩=0,C_{vv}(t)=⟨\dot{z}(t)\dot{z}(0)⟩$ is the velocity autocorrelation function, and,

$$C_{Fv}(t)=-m⟨\ddot{z}(t)\dot{z}(0)⟩-k⟨z(t)\dot{z}(0)⟩$$ (6)

is the force-velocity time-correlation function.

To solve for the memory function of the permeant in a given umbrella window, we employ the multilayer perceptron (MLP) method developed by Russo et al.¹⁹ The MLP method assumes that the memory function can be expressed as a sum of positive, decaying exponential functions, where each exponential term is linked to a unique neuron during optimization. In the MLP method, the memory function is given by,

$$M(t)=\sum _{i=1}^N{K}_i{e}^{-t/{\tau }_i}$$ (7)

where $N$ is the total number of neurons, and $K_i$ and ${\tau }_i$ are the parameters to be optimized. The coupling constants, $K_i$, are related to the mean-square fluctuation of the force acting on the permeant molecule, which can give rise to “caging” effects in the diffusive dynamics.²¹ In the Markov limit, each exponential component contributes to the total friction according to Eq. (3), yielding

$$\gamma =\sum _{i=1}^N{\gamma }_i$$ (8)

with ${\gamma }_i=K_i{\tau }_i$. Following the Einstein-Smoluchowski relation (Eq. (4)), the effective diffusivity for the memory function of Eq. (7) may be expressed as,

$$D^{MLP}={\left[\sum _{i=1}^N\frac{{\gamma }_i}{k_{\mathrm{B}}T}\right]}^{-1}$$ (9)

The optimization of the MLP proceeds by iteratively minimizing a loss function based on the mean-square error between the current estimate of and the known value of $C_{Fv}(t)$. Here, we expand the original definition of the MLP loss function of Russo and coworkers¹⁹ for the memory function optimization. In our approach, we implement a loss function that incorporates additional information about the memory function, in particular an estimate of the value of its integral, via the Einstein-Smoluchowski relationship to the diffusivity (Eq. (4)). The Woolf-Roux (Eq. (13)) and Hummer (Eq. (15)) diffusivity serve as guidelines to establish a reasonable range for the integral of the predicted memory function. When the the modified loss function is applied, the resulting value of Eq. (9) is effectively constrained by a flat-bottomed harmonic function based on the Woolf-Roux and Hummer diffusivity estimates for each umbrella window, such that

$$\begin{gathered} ℒ=\frac{1}{𝒯}\sum _{n=0}^{𝒯-1}{\left(C_{Fv}(n\mathrm{\Delta }t)-C_{Fv}^{MLP}(n\mathrm{\Delta }t)\right)}^2 \\ +\frac{1}{2}{\left(\frac{1}{D^H}\left(D^{MLP}-D^H\right)\right)}^2\theta \left(D^H-D^{MLP}\right) \\ +\frac{1}{2}{\left(\frac{1}{D^{WR}}\left(D^{MLP}-D^{WR}\right)\right)}^2\theta \left(D^{MLP}-D^{WR}\right) \end{gathered}$$ (10)

where $C_{Fv}^{MLP}(t)$ is the convolution integral of the known $C_{vv}(t)$ with the current estimate of $M(t)$, $\theta (x)=1$ if $x>0$ and 0 otherwise, and $𝒯\mathrm{\Delta }t$ is the maximum lag time to be calculated, after which is it is assumed $C_{Fv}(t)$ and $C_{vv}(t)$ have fully decayed. $D^{WR}$ is the Woolf-Roux diffusivity and $D^H$ is the Hummer diffusivity, which effectively act as upper and lower bounds on Eq. (9), respectively. As an initial justification for the choice of $D^{WR}$ as the upper bound and $D^H$ as the lower bound in Eq. (10), we draw on the findings of Rowley and coworkers⁴ which indicate that the diffusivity estimated by Woolf-Roux is generally larger than that estimated by Hummer.

A comparison of the MLP’s convergence for the original (simple) and modified loss functions is given in Fig. S1 of the supporting information, including the simple loss function’s apparent failure to obtain a reasonable estimate of the position-dependent diffusivity. The convergence of each window is assessed by the mean-square error (MSE) between the known $C_{Fv}(t)$ and the $C_{Fv}^{MLP}(t)$ predicted by the convolution integral of the memory function and the velocity autocorrelation function. A crucial observation is that for the center region of the membrane $(|z|<3\AA )$, the MSE is at a minimum compared to the rest of the membrane, indicating successful optimization. In addition, the resulting diffusivity estimates for this region are in very good agreement between the two loss functions. This finding indicates that the flat-bottomed harmonic loss function does not affect MLP estimates that naturally converge to values in between the Woolf-Roux and Hummer boundaries.

B. Position-dependent diffusivity

Traditionally, the position-dependent diffusivity has been extracted from biased umbrella sampling MD simulations by solving for the Laplace transform of the memory function. It is assumed that, in the presence of the biasing potential, the system undergoes small fluctuations $\delta z(t)$ around the mean value position $⟨z⟩$. To treat the data from the biased simulation, we reduce the GLE to time-correlation functions associated with the local behavior around the mean position $⟨z⟩$ by multiplying both sides of Eq. (2) by the initial velocity, $\delta \dot{z}(0)$ (or equivalently $\dot{z}(0)$), which results in,

$$m{\dot{C}}_{vv}(t)=-\frac{k_{\mathrm{B}}T}{⟨\delta z^2⟩}⟨\delta z(t)\delta \dot{z}(0)⟩-{\int }_0^{t}M\left(t-t^{\prime }\right)C_{vv}\left(t^{\prime }\right)\mathrm{d}{t}^{\prime }$$ (11)

where $k_{\mathrm{B}}T/⟨\delta z^2⟩=\left(W^{\prime \prime }+k\right)$ is the second derivative of the total potential, $C_{vv}(t)=⟨\delta \dot{z}(t)\delta \dot{z}(0)⟩,{\dot{C}}_{vv}(t)=⟨\delta \ddot{z}(t)\delta \dot{z}(0)⟩$, and $⟨\delta z(t)\delta \dot{z}(t)⟩={\int }_0^t{C}_{vv}(t)\mathrm{d}t$. The Laplace transform of the memory function, $\tilde{M}(s)$, is related to the diffusivity via,

$$D=\underset{s\to 0}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{k_{\mathrm{B}}T}{\tilde{M}(s)}$$ (12)

Correspondingly, we can also define $D=\underset{s\to 0}{\mathrm{l}\mathrm{i}\mathrm{m}}\tilde{D}(s)$. The so-called Woolf-Roux diffusivity ascribed to the position $⟨z⟩$ is given by,¹⁴

$$D^{WR}=\underset{s\to 0}{\mathrm{l}\mathrm{i}\mathrm{m}}\left[\frac{-{\tilde{C}}_{vv}(s)⟨\delta z^2⟩⟨\delta {\dot{z}}^2⟩}{{\tilde{C}}_{vv}(s)\left[s⟨\delta z^2⟩+⟨\delta {\dot{z}}^2⟩/s\right]-⟨\delta z^2⟩⟨\delta {\dot{z}}^2⟩}\right]$$ (13)

where ${\tilde{C}}_{vv}(s)$ is the Laplace transform of the velocity autocorrelation function. Because of the singularity located at $s=0$, it is necessary to extrapolate from a well-behaved region of $\tilde{D}(s)$ back to zero to obtain the limit in Eq. (13). The Hummer diffusivity approach begins with a substitution of ${\tilde{C}}_{vv}(s)$ by the Laplace transform of the position autocorrelation function in Eq. (13).¹⁶ In Laplace space, the relationship between these two time-correlation functions is given by,

$${\tilde{C}}_{vv}(s)=-s^2{\tilde{C}}_{zz}(s)+s⟨\delta z^2⟩$$ (14)

where $C_{zz}(t)=⟨\delta z(t)\delta z(0)⟩$ is the position autocorrelation function and ${\tilde{C}}_{zz}(s)$ is its Laplace transform. The Hummer diffusivity is obtained by substituting Eq. (14) in Eq. (13) in the limit of $s\to 0$,¹⁶

$$\begin{gathered} D^H=\underset{s\to 0}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{⟨\delta {\dot{z}}^2⟩⟨\delta z^2⟩\left(⟨\delta z^2⟩-s{\tilde{C}}_{zz}(s)\right)}{{\tilde{C}}_{zz}(s)\left[s^2⟨\delta z^2⟩+⟨\delta {\dot{z}}^2⟩\right]-s{⟨\delta z^2⟩}^2} \\ =\underset{s\to 0}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{{⟨\delta z^2⟩}^2}{{\tilde{C}}_{zz}(s)} \end{gathered}$$ (15)

The final expression can also be written as $\underset{t\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}{\int }_0^t{C}_{zz}\left(t^{\prime }\right)\mathrm{d}{t}^{\prime }$ in the time domain.¹⁶

In either approach, a reliable numerical evaluation of the expression requires a careful examination of its convergence in relation to the limiting process. Complete derivations of the memory functions in Laplace space for the Woolf-Roux and Hummer methods are provided in the appendix, and also in the appendix of the work by Rowley and coworkers.⁴ Eq. (14) is of central importance for understanding the differences and similarities of the two approaches. This expression relates the Laplace transform of the velocity-velocity time-correlation function used in the Woolf-Roux method and the position-position time-correlation function used in the Hummer method.

III. COMPUTATIONAL DETAILS

A. Molecular dynamics simulations

The details of the set of umbrella sampling MD simulations of water in a DLPC membrane that were employed for the extraction of the position-dependent diffusivity with the Woolf-Roux, Hummer, and MLP approaches are as follows. The membrane and water systems each consisted of 72 monomers of 1,2-dilauroyl-sn-glycero-3-phosphocholine (DLPC) lipids and 2880 water molecules, represented by the Drude2023 lipid force field²² and SWM4,²³ respectively. We note that the choice of the Drude polarizable force field for this work ensures good agreement with the experimental viscosity of water. Drude SWM4 water exhibits very good experimental agreement for self-diffusion, while the non-polarizable TIP3P overestimates self-diffusion by approximately a factor of 2 at 298 K.²³ It has also been shown that the Drude2023 lipid model with LJ-PME has improved experimental agreement for membrane water permeability compared to CHARMM36, especially for saturated lipids.²²

The umbrella sampling simulations were conducted in triplicate, with a unique initial configuration for each replicate of each window. The initial configurations were generated from the trajectory of a spontaneous water crossing event from three independent equilibrium simulations, which were conducted in our previous work.¹⁸ The harmonic restraints were spaced by 1 Å from $z=–30$ to +30 Å for a total of 61 windows per replicate set of simulations. The force constant ($k$) of each harmonic potential was set to 10 kcal/mol Ų. We note here that one of the umbrella sampling replicates was conducted for our previous work,²⁴ and two additional replicates were generated for the present work. Each window for each set of umbrella sampling simulations was run for 30 ns, for a total of 5490 ns of sampling. When computing the time-correlation functions for all of the following analysis, the first 5 ns of each trajectory were excluded to allow for equilibration. The value of the collective variable $z$ was recorded every time step (1 fs). The $C_{Fv}(t)$ and $C_{vv}(t)$ time-correlation functions were computed for each replicate of each window along $z$ separately. Each was computed up to a lag time of 15 ps, with increments of 1 fs. The instantaneous velocity and acceleration were evaluated by finite-difference. The key MD protocol is as follows. The MD engine OpenMM version 7.7²⁵ on GPU with the Nose-Hoover integrator was used with a timestep of 1 fs, temperature of 303 K, friction of 10 ps⁻¹, chain length, number of steps, and Yoshida-Suzuki all set to 3. The Drude particles were set to a temperature of 1 K and a friction of 1 ps⁻¹. LJ-PME was used for non-bonded interactions with a cutoff of 10 Å. The simulations were run at a constant volume of 47.3 × 47.3 × 70.8 Å (x × y × z). For convenience, the OpenMM front-end rickflow was used.²⁶

B. Numerical simulation of the GLE

Due to spatially varying friction across the membrane environment, it is expected that the memory function obtained within each umbrella sampling window will have position-dependent parameters, $K_i(z)$ and ${\tau }_i(z)$. The overall position-dependent memory function is modeled as,

$$M(t;z)=\sum _{i=1}^N{K}_i(z)e^{-t/{\tau }_i(z)}$$ (16)

where $N$ was chosen to be 2. The GLE with position-dependent memory is obtained by substituting Eq. (16) into Eq. (2),^27,28

$$\begin{gathered} m\ddot{z}(t)=-\frac{\partial W(z)}{\partial z} \\ -{\int }_0^t\sum _{i=1}^N{K}_i\left(z\left(t^{\prime }\right)\right)e^{-\left(t-t^{\prime }\right)/{\tau }_i\left(z\left(t^{\prime }\right)\right)}\dot{z}\left(t^{\prime }\right)\mathrm{d}{t}^{\prime } \\ +f(t) \end{gathered}$$ (17)

This GLE with position-dependent memory is an approximation,^27–29 although it will be shown to be a good representation for water permeability through a lipid membrane.

We adapted a numerical integration scheme provided by Bond and coworkers³⁰ for the simulation of the GLE with the position-dependent memory function composed of a sum of positive, decaying exponential terms (Prony series) via Markovian embedding. The form of the memory function assumed by the MLP of Russo et al.¹⁹ is well-suited for the numerical integration schemes developed by Bond and coworkers.³⁰ The Markovian embedding scheme adapted to accommodate the position-dependent parameters of the memory function yields,³⁰

$$\dot{z}(t+\mathrm{\Delta }t)=\dot{z}(t)-\frac{\mathrm{\Delta }t}{m}\frac{\partial W(z)}{\partial z}+\frac{\mathrm{\Delta }t}{m}\sum _{i=1}^N{S}_i(t)$$ (18)

$$z(t+\mathrm{\Delta }t)=z(t)+\mathrm{\Delta }t\dot{z}(t+\mathrm{\Delta }t)$$ (19)

$$S_i(t+\mathrm{\Delta }t)=S_i(t)-\frac{\mathrm{\Delta }t}{{\tau }_i(z)}{S}_i(t)-\mathrm{\Delta }t\frac{{\gamma }_i(z)}{{\tau }_i(z)}\dot{z}(t+\mathrm{\Delta }t)+\frac{1}{{\tau }_i(z)}\sqrt{2k_{\mathrm{B}}T{\gamma }_i(z)\mathrm{\Delta }t}{𝒩}_t$$ (20)

where $\mathrm{\Delta }t$ is the timestep, $z(t)$ is the main reaction coordinate, $S_i(t)$ is an embedded Ornstein-Uhlenbeck process that depends on ${\tau }_i(z)$ and the friction ${\gamma }_i(z)=K_i(z){\tau }_i(z)$ in the $i$th term of the memory function, and ${𝒩}_t$ is a normally distributed random number.

We tested and compared two Markovian embedding integration schemes developed by Bond and coworkers,³⁰ each based on the Euler-Maruyama integrator. In the original study, the Euler-Maruyama integrator that demonstrated the best stability for a GLE with a position-independent memory function was called “method 3.” We will refer to this integrator as EM-3. They also developed a multi-stage splitting variation of EM-3 with twice as many velocity (4) and position (2) updates per step, inspired by Leimkuhler and Matthews.³¹ We will refer to this multi-stage splitting integrator as EM-3m. These two schemes were assessed for their stability and accuracy with respect to the mitigation of spurious drift due to the position-dependence of the parameters of the memory function.

The position-dependent memory functions were constructed based on the output of the MLP optimization for each window of the umbrella sampling MD simulations along $z$. Because the MLP was implemented with two neurons, each umbrella window yielded two optimized values each for $K_i$ and ${\tau }_i$. The assignment of $i=1$ or $i=2$ for a given window by the MLP is arbitrary with respect to the other windows. To ensure the stability of the numerical integration and to better visualize the physical significance of each term in the memory function, the larger ${\tau }_i$ were assigned to $i=1$, and the smaller ${\tau }_i$ were assigned to $i=2$. To simplify the implementation and maximize computational efficiency, the each term $(i=1,2)$ in the position-dependent memory function was expressed as a set of $K_i(z)$ and $D_i(z)=k_{\mathrm{B}}T/K_i(z){\tau }_i(z)$, and fit to analytical functions by visual inspection and least-squares.

Test simulations of the GLE were conducted first to assess the importance of numerical issues arising from the implementation of the position-dependent memory function, which corresponds to nonlinear friction in the framework of the standard GLE.^27,32,33 The position-dependent memory function leads to a spurious drift that is comparable to that of Brownian dynamics integrated with a position-dependent diffusivity.³⁴ This effect is also present in Langevin dynamics simulations with position-dependent friction due to the Itô vs. Stratonovich dilemma.^35,36 The validity of the integration scheme was assessed by evaluating the error in the free energy landscape sampled by the simulations with respect to the forces used in the underlying GLE. We did not seek to implement an exact GLE with nonlinear friction, such as was done by Ayaz et al.²⁸

The set of equilibrium GLE simulations run for this study is summarized in Table I. The set of equilibrium simulations cover three different free energy landscapes. To test the stability and accuracy of the EM-3 and EM-3m integration schemes, a zero mean force system was simulated. The zero mean force simulations were also used to calculate the position-dependent diffusivity via mean-square deviations (MSD) in displacements. To ensure efficient sampling of the entire $z$-coordinate, each trajectory was initialized from a random $z$. As an additional test case to compare the two integrators, a system with a mean force corresponding to a barrier of 3 kcal/mol was run. In the presence of the barrier, the trajectories were initialized in the bulk. The superior accuracy of EM-3m over EM-3 is established in Fig. S2 of the supporting information. The main set of equilibrium GLE simulations used the EM-3m integrator and the full PMF of water across the DLPC membrane based on the 1-D probability distribution from equilibrium MD simulations. For computational efficiency, an analytical function was fit to the PMF that was extracted from the MD simulations. The analytical derivative of the fitted PMF was used for calculating forces in the numerical simulations. For all of the equilibrium GLE simulations, the temperature was set to 303 K, the timestep was 1 fs, and the $z$ coordinate was saved every 1 ps. To confine the system, upper and lower half harmonic walls with a force constant of 5 kcal/mol Ų were applied to the particle if it diffused outside of ± 50 Å.

TABLE I.

Equilibrium GLE simulations

integrator	landscape	# trajectories	time/trajectory (μs)	total time (μs)
EM-3	zero mean force (0 kcal/mol)	500	0.05	25
EM-3	half PMF (3 kcal/mol)	500	1	500
EM-3m	zero mean force (0 kcal/mol)	500	0.05	25
EM-3m	half PMF (3 kcal/mol)	500	1	500
EM-3m	full PMF (6 kcal/mol)	1000	10	10000

In addition, GLE umbrella sampling simulations with the full PMF of water were carried out to directly examine the accuracy of the Woolf-Roux and Hummer estimates of the diffusivity. As in the all-atom MD, the harmonic restraints were spaced by 1 Å from $z=–30$ to +30 Å for a total of 61 windows, and the force constant of each harmonic potential was set to 10 kcal/mol Ų. Each window for each of the three sets of umbrella sampling simulations was run for 30 ns, for a total of 5490 ns of sampling. The calculation of the time-correlation functions used the same protocol as the MD simulations.

C. Determination of the diffusivity

MLP method.

The MLP was implemented following the details described by Russo et al.¹⁹ The optimization is conducted for a single umbrella window at a time. The MLP consisted of three layers: input, hidden, and output. Two neurons were chosen for the hidden layer. The input layer contains a time-axis up to 15 ps with 1 fs spacing, corresponding to the same shape as the time-correlation functions from the MD simulations. The input layer is connected to each neuron by edges with weights and biases. For the activator function of each neuron of the hidden layer, the input from the previous layer is exponentiated and then time-convolved with the known $C_{vv}(t)$ using the convolution function of TorchAudio.³⁷ Each neuron in the hidden layer is connected to the output layer by edges with weights (but no biases). The current estimate of $C_{Fv}(t)$ is then returned via the weighted sum of the output of each neuron, which was used to evaluate the loss at each iteration. To build the neural network, PyTorch was used.³⁸ The optimizer used was Rprop, which is the same one used by Russo et al.¹⁹ The optimization of the memory function proceeded for 500 epochs.

To ensure that the memory function is strictly a sum of positive, decaying exponential functions, the weights of the input layer are constrained to be negative, while the weights of the output layer are constrained to be positive. The weights of input layer are initialized to be the negatives of exponentiated Gaussian random numbers. The biases of the input layer are initialized to be Gaussian random numbers. The weights of the output layer are initialized to be exponentiated Gaussian random numbers. At each iteration, it was necessary to calculate the current estimate of the diffusivity for evaluating the modified loss function with respect to the upper and lower bounds imposed by Woolf-Roux and Hummer. The parameter $1/{\tau }_i$ is the negative of the $i$th weight of the input layer. The parameter $K_i$ is the $i$th weight of the output layer times the exponentiated bias of the input layer. The effective diffusivity can then be computed with Eq. (9), and the relationship ${\gamma }_i=K_i{\tau }_i$. The structure of the MLP is shown in Fig. S3 in the supporting information. Eq. (S2) is an expression for the diffusivity in terms of the weights and biases of the model at a given iteration of optimization.

Woolf-Roux method.

The Laplace transform of $C_{vv}(t)$ was calculated by numerical integration with the trapezoid rule with $ds=0.0001$ ps⁻¹ up to s = 1.5 ps⁻¹. At each $s$, the integration was performed over the entire 15 ps that was computed for $C_{vv}(t)$ with dt = 0.001 ps. After calculating $\tilde{D}(s)$ according to Eq. (13) up to s = 1.5 ps⁻¹, the $s=0$ value was determined by the y-intercept of a straight line fit by least-squares to an automatically determined set of points. The set of points for fitting depended on the presence of a singularity (besides s = 0) in the range of $s$ being considered. If the singularity was detected, the fitting was inspired by the method of Rowley and coworkers.⁴ The line was fit to a set of 20 points (20 ds) centered on the point of least curvature, determined by where the finite-difference second derivative of $\tilde{D}(s)$ is closest to zero. In cases where there was no singularity detected, the function generally passes through an inflection point and then increases approximately linearly. The line for this case was fit to all of the points past the point of greatest negative curvature (the minimum of the finite-difference second derivative).

Hummer method.

The position autocorrelation function was computed up to a maximum time of 500 ps for each replicate of each window with an increment 1 fs. The time integral of each position autocorrelation function was computed with the trapezoid rule with dt = 0.001 ps. In order to obtain the best estimate of the diffusivity (without including unnecessary noise in the integral of the position autocorrelation function), the upper limit of integration ($t_{\mathrm{\infty }}$) was optimized for each window and replicate independently. The optimization of $t_{\mathrm{\infty }}$ was inspired by a method employed by Dickson et al.³⁹ The upper limit of $t$ for a window centered at $z$ is set by the point where $0.01\times ⟨\delta z^2⟩>C_{zz}(t)$, but not greater than 500 ps.

IV. RESULTS

A. Position-dependent memory function from MLP

The parameters $K_i$ and ${\tau }_i$ of Eq. (7) were optimized by the MLP for each window of the umbrella sampling simulations of water in a DLPC membrane. It is assumed that the memory function can be accurately modeled with two positive, decaying exponential terms. Several memory functions that were optimized by the MLP in distinct regions of the membrane are shown in Fig. 1. The position-dependence of the parameters of the memory function is apparent in the visually distinct behavior of each region. For instance, the optimized memory function at $z=15\AA$ is shown to decay relatively slowly compared to those of the $z=0\AA$ center of the membrane and the $z=30\AA$ bulk region. The position-dependence of the MLP optimization is further illustrated by the lag time dependent functions that compose the GLE of Eq. (5) in different regions of the membrane. The velocity autocorrelation functions for every umbrella window are shown in Fig. S4, the force-velocity autocorrelation functions are shown in Fig. S5, and the optimized memory functions are shown in Fig. S6 in the supporting information. From the bulk solution to the center of the membrane, the timescale of decay and shape of each time-correlation function gradually changes.

FIG. 1.

Optimized memory functions, $M(t)$, extracted from three distinct regions of the umbrella sampling MD simulations of water in the membrane: $z=30$ (blue), $z=15$ (green), and $z=0$ Å (red). The uncertainty intervals are given by plus and minus the standard deviation of the three replicates.

The optimized parameters of the position-dependent memory function across the membrane are shown in Fig. 2. The distinct timescales of decay for each term ($i=1,2$) are revealed by the plots of ${\tau }_i(z)$ in Fig. 2B. From the set of ${\tau }_i(z)$ and $K_i(z)$, the effective friction coefficients associated with each term $i$ can be resolved, which are shown in Fig. 2C. A physical picture of the GLE emerged after the MLP distinguished two regimes for the terms in the memory function. The first is the existence of a slowly decaying term that dominates the effective diffusivity ($i=1$) and a second contribution that is quickly decaying ($i=2$). Notably, the friction associated with the slower $i=1$ mode, when expressed as $k_{\mathrm{B}}T/{\gamma }_1(z)$, accounts almost entirely for the effective position-dependent diffusivity. These sets of position-dependent parameters, extracted from MD umbrella sampling simulations, can be used to construct a memory function for a GLE that can be efficiently simulated with numerical integration.

FIG. 2.

The position-dependent parameters of the optimized memory function (Eq. (7)) extracted by the MLP with $N=2$ with the modified loss function from umbrella sampling MD simulations of water in a DLPC membrane. A. The coupling factor, $K_i(z)$, B. the timescale of decay, ${\tau }_i(z)$, and C. ${\gamma }_i(z)/k_{\mathrm{B}}T$ for each term the the $i=1$ (blue) and $i=2$ (red) terms of the optimized memory function. The dots correspond to the mean of three independent simulations with error bars given by plus and minus the standard deviation of each quantity. The lines for $K_i(z)$ and ${\gamma }_i(z)/k_{\mathrm{B}}T$ correspond to analytical functions which were fit to the points by visual inspection and least-squares, and subsequently used for the numerical integration of the GLE.

It is interesting to note from Fig. 2 that the coupling factors $K_i(z)$ display a significant dependence on the position $z$, whereas the relaxation times ${\tau }_i(z)$ are nearly constant over $z$. This observation suggests that there are two dominant natural relaxation modes within the bilayer, with relaxation times ${\tau }_1$ and ${\tau }_2$. These relaxation times are inherent features of the membrane fluctuations. On the other hand, the strength of the coupling of the permeant molecule with these two modes is strongly dependent on position.

A constructive explanation of this observation may be drawn from a spectral decomposition of the effective dynamical propagator of the entire system^40,41 under the assumption that the natural fluctuations of the membrane are not perturbed significantly by the presence of the permeant molecule at position $z$. For a fixed molecule, the memory function is related to equilibrium time-correlation function of the force exerted by the membrane in the $z$ direction, $⟨\delta F_z(t)\delta F_z(0)⟩$, with $\delta F_z=F_z-⟨F_z⟩$. The time-correlation function can be expressed as,^40,41

$$⟨\delta F_z(t)\delta F_z(0)⟩=\sum _{i>0}{\left(\delta F_z\cdot {\varphi }_i^{\mathrm{R}}\right)}^2{e}^{-t/{\tau }_i}$$ (21)

with

$$\left(\delta F_z\cdot {\varphi }_i^{\mathrm{R}}\right)=\int d\mathbf{x}\delta F_z(\mathbf{x}){\varphi }_i^{\mathrm{R}}(\mathbf{x})$$ (22)

where $\mathbf{x}$ represents the coordinates of the system, ${\varphi }_i^{\mathrm{R}}(\mathbf{x})$ is the $i$th right-eigenmode of the effective dynamical propagator of the system with its associated eigenvalue $e^{-t/{\tau }_i}$, the equilibrium distribution corresponds to the first right-eigenvector ${\varphi }_0^{\mathrm{R}}(\mathbf{x})$, and $⟨F_z⟩=\left(F_z\cdot {\varphi }_0^{\mathrm{R}}\right)$. According to the MLP analysis, the sum over $i$ is dominated by the contribution from two eigenmodes. The friction is then,

$$\begin{gathered} \gamma (z)=\frac{1}{k_{\mathrm{B}}T}{\int }_0^{\mathrm{\infty }}⟨\delta F_z(t)\delta F_z(0)⟩dt \\ =\frac{1}{k_{\mathrm{B}}T}\sum _{i>0}{\left(\delta F_z\cdot {\varphi }_i^{\mathrm{R}}\right)}^2{\tau }_i \\ =\frac{1}{k_{\mathrm{B}}T}{\left(\delta F_z\cdot {\varphi }_1^{\mathrm{R}}\right)}^2{\tau }_1+\frac{1}{k_{\mathrm{B}}T}{\left(\delta F_z\cdot {\varphi }_2^{\mathrm{R}}\right)}^2{\tau }_2+\dots \\ \approx K_1(z){\tau }_1+K_2(z){\tau }_2 \end{gathered}$$ (23)

The coupling factors $K_i(z)$ are related to the mean-square fluctuations of the force exerted on the permeant projected onto the $i$th right-eigenmode ${\varphi }_i^{\mathrm{R}}$ of the effective dynamical propagator, $⟨\delta F_z\delta F_z⟩=\sum _{i>0}{\left(\delta F_z\cdot {\varphi }_i^{\mathrm{R}}\right)}^2$. Based on the MLP analysis, it appears that the relaxation time of the first two eigenmodes, ${\tau }_1$ and ${\tau }_2$, do not depend strongly on the position $z$. The prefactors ${\left(\delta F_z\cdot {\varphi }_i^{\mathrm{R}}\right)}^2$, however, vary with $z$. This outcome is conceivable if the fixed permeant molecule can be treated as a perturbation and the dominant contribution to $\gamma (z)$ arises from the natural eigenmodes of the unperturbed membrane.

B. Validation of the numerical integration of the GLE

Having extracted the position-dependent memory functions from the umbrella sampling MD simulations, it is possible to proceed with the numerical simulations of the GLE. Comparison of two integration schemes for modeling the GLE with a position-dependent memory function confirms the superiority of the multi-step approach of EM-3m. We suspect that the strength of EM-3m is related to the extra position update within a timestep, which allows for smoother integration of the position-dependent friction, ${\gamma }_i(z)$, and the coupling factors, $K_i(z)$. A comparison of the EM-3 and EM-3m simulations is shown in Fig. S2 of the supporting information. Our first assessment of integrator accuracy is based on simulations with no PMF. The absence of a PMF allows for the evaluation of the mean-square deviation (MSD) diffusivity and the detection of spurious drift due to integration errors. The MSD diffusivity extracted from this test shows virtually no difference between the EM-3 and EM-3m integrators (shown for EM-3m in Fig. 3A). It has the expected behaviors of both position and lag time dependence. For the short lag time of 5 ps, the diffusivity profile overestimates the effective diffusivity of the underlying GLE. By 15 ps, however, there is good agreement, indicating that approximately this lag time is required for the full friction to accumulate. The spurious drift was computed based on the 1-dimensional probability distribution, $p(z)$. The “drift” PMF for these simulations can be determined directly from the marginal distribution as, $-k_{\mathrm{B}}T\mathrm{l}\mathrm{n}[p(z)]$. Comparison shows that deviations from the expected flat free energy are smaller for the EM-3m integrator than the EM-integrator. As an additional assessment, each integrator was tested on a 3 kcal/mol barrier (referred to as the “half PMF” system). The drift PMF in this case is the difference between the $-k_{\mathrm{B}}T\mathrm{l}\mathrm{n}[p(z)]$ extracted from each simulation and the analytical PMF input. In the presence of the half PMF, the EM-3m again has a smaller magnitude of spurious drift, demonstrating its superiority in modeling the position-dependent memory function.

FIG. 3.

Assessment of the EM-3m numerical integration scheme for the GLE with the position-dependent memory function and the PMF of water extracted from MD simulations of water across a DLPC membrane. A. MSD $D(z)$ on flat free-energy surface (zero mean force) for several lag times. The exact diffusivity of the underlying GLE (dashed black line). B. The PMF of water in a DLPC membrane from the marginal probability distribution of unbiased MD simulations (green line). The PMF modeled by an analytical function used to compute the underlying mean force of the GLE (dotted black line). The PMF computed from the 1-D probability distribution of the equilibrium GLE simulations with the EM-3m integrator (blue line). C. The PMF of the zero mean force GLE simulation, or drift PMF (red line). The difference between the 1-D probability distribution PMF and the analytical input from the equilibrium GLE simulations (blue line). The PMFs are symmetrized about $z=0$.

Finally, the accuracy of the EM-3m integrator was assessed based on the GLE simulations used to model the permeation of water. The PMF of water extracted from the equilibrium MD simulations was fit to an analytical function for efficiency. The two PMFs, shown in Fig. 3B, are essentially equivalent. For comparison, the PMF calculated from $p(z)$ from the long equilibrium GLE simulations is also shown. The deviations are, again, very small. The drift PMF for quantifying the integration error of EM-3m with the full PMF of water is shown in Fig. 3C. This difference has a magnitude up to 0.065 kcal/mol at the very center for the full PMF, compared to fluctuations of around several hundredths of a kcal/mol for the zero mean force simulation. The integration error is sufficiently small for the purposes of estimating rates of barrier crossing with the numerical GLE simulations. On the basis of these results it can be concluded that an accurate coarse-grained representation of the the permeation of water molecules through a DLPC membrane is provided by the GLE.⁴²

C. Determination of the position-dependent diffusivity

The position-dependent diffusivity extracted with the Woolf-Roux (Eq. (13)), Hummer (Eq. (15)), and MLP (Eq. (5)) approaches from the biased umbrella sampling simulations of the all-atom DLPC membrane are displayed in Fig. 4. The Woolf-Roux diffusivity is consistently larger than the Hummer diffusivity for the entire range of $z$ across the membrane. The effect of the modified loss function (Eq. (10)) on the MLP diffusivity for a given window is highly dependent on the window’s location in the membrane. Near the bulk solution (|z| > 20 Å), the Woolf-Roux and Hummer diffusivity are in good agreement, and the MLP tends towards the slightly lower boundary. In the lipid tails region (3 < |z| < 20 Å), the MLP converges to the upper limit, as the difference between Woolf-Roux and Hummer widens. At the center of the membrane (|z| < 3 Å), the MLP converges to an intermediate value. This region also exhibits the largest difference between Woolf-Roux and Hummer. The three z = 0 values of the diffusivity are 0.235±0.035, 0.0841±0.0052, and 0.1583±0.0069 Ų/ps for Woolf-Roux, Hummer, and MLP, respectively. The center of the membrane is the peak of the PMF, and therefore has the most critical impact on the resulting ISD permeability estimate.

FIG. 4.

Position-dependent diffusivity ($D(z)$) from umbrella sampling MD simulations of water across a DLPC membrane computed with Woolf-Roux (blue), Hummer (green), and MLP (red) approaches. The plotted values are the mean of the three replicates for each window, which were first symmetrized about $z=0$. The localized error bars are plus and minus the standard deviation between the three replicates.

Because the three approaches attempt to solve the GLE for the diffusivity based on the same set of trajectory data, an explanation of the deviations between each is warranted. First of all, the diffusivity from the Woolf-Roux method is sensitive to the procedure used for extrapolating the Laplace transform of the diffusivity to zero. The difficulty with extracting the diffusivity in terms of the Laplace transform of the velocity autocorrelation function is apparent upon inspection of Eq. (13), where there is a singularity when s= 0. Detailed plots of these extrapolations are available in Fig. S7, Fig. S8, and Fig. S9 in the supporting information. Hummer has a relatively well-behaved diffusivity in Laplace space at s = 0, which is apparent in Eq. (15), where the diffusivity can be computed via the integral of the position autocorrelation function in the time domain. However, the Hummer diffusivity can be sensitive to the upper limit of integration. The position autocorrelation function decays slowly, in some cases (around z = ± 15 Å) it is still nonzero at a lag time of 500 ps. The running integrals of the position autocorrelation functions up to 500 ps for each window up are shown in Fig. S10 in the supporting information.

The MLP method is advantageous for several reasons. The first is that it is highly automated. It does not require the fitting procedure of Woolf-Roux due to its use of time-correlation functions in the time domain. Second, it does not depend on the position autocorrelation function, such that the slow decay ambiguity is avoided. A limitation of the original MLP optimization in the time domain is that sometimes apparently degenerate solutions of the memory function are obtained, which may or may not provide reasonable estimates of the diffusivity. By imposing boundaries set by the physically solid Hummer and Woolf-Roux approaches via the modified loss function (Eq. (10)), however, the MLP effectively incorporates the memory function’s integral, instead of strictly its time-dependent values, into the optimization.

To delve further into the origin of the deviations between the Woolf-Roux and Hummer methods, we consider the fundamental link between them within the framework of the GLE in a harmonic potential. In particular, Woolf-Roux depends on the velocity autocorrelation function, while Hummer depends on the position autocorrelation function. In Laplace space, these two can be interchanged with Eq. (14). In practice, this key relation is not strictly satisfied by finite sampling from a simulation. To examine the accuracy of Eq. (14), plots of the directly computed and formulaically interchanged position (${\tilde{C}}_{zz}(s)$) and velocity (${\tilde{C}}_{vv}(s)$) autocorrelation functions from several windows of the umbrella sampling MD simulations are shown in Fig. 5 A. and B., respectively. Importantly, ${\tilde{C}}_{vv}(s)$ does not identically vanish at s = 0 based on the sampling of the MD simulation. On the other hand, this condition is automatically satisfied if ${\tilde{C}}_{vv}(s)$ is converted formulaically from the position autocorrelation function via Eq. (14). The corresponding Laplace transforms of the memory functions based on ${\tilde{C}}_{zz}(s)$ and ${\tilde{C}}_{vv}(s)$ are shown in Fig. 5C. The directly computed ${\tilde{C}}_{zz}(s)$ memory function is effectively that of the Hummer formulation, while the directly computed ${\tilde{C}}_{vv}(s)$ memory function is effectively the Woolf-Roux formulation. As has been observed by others,⁴ the Woolf-Roux memory function diverges at small $s$, while the Hummer memory function is well-behaved. Observing the behavior of the formulaically converted ${\tilde{C}}_{vv}(s)$ and ${\tilde{C}}_{zz}(s)$ reveals that it is not inherently the Woolf-Roux formulation that leads to the characteristic divergence, but the initial calculation of ${\tilde{C}}_{vv}(s)$. When ${\tilde{C}}_{vv}(s)$ is converted formulaically to ${\tilde{C}}_{zz}(s)$, the Hummer memory function would also diverge at small $s$. Conversely, when ${\tilde{C}}_{zz}(s)$ is converted to ${\tilde{C}}_{vv}(z)$, the Woolf-Roux memory function would be well-behaved at small $s$, although the value of ${\tilde{C}}_{zz}(s)$ at small $s$ becomes very important.

FIG. 5.

Examining the accuracy of Eq. (14) based on three representative windows of the umbrella sampling MD simulations ($z=30,15$ and 0 Å). A. Laplace transform of the position autocorrelation function computed directly (solid green line) and converted from the velocity autocorrelation function via Eq. (14) (dashed black line). B. Laplace transform of the velocity autocorrelation function computed directly (solid blue line) and converted from the position autocorrelation function via Eq. (14) (dashed black line). C. Laplace transforms of the memory functions computed with the direct position autocorrelation functions, corresponding to the Hummer method (solid green line). Laplace transforms of the memory functions computed with the direct velocity autocorrelation functions, corresponding to the Woolf-Roux method (solid blue line). Laplace transforms of the MLP memory functions in Laplace space (Eq. (24)), shown for reference (dashed red line). For each plot, the shaded regions correspond to uncertainty intervals based on the standard deviation between the three replicates.

We provide this analysis as an addition to the previously discussed discrepancies between Hummer and Woolf-Roux from Rowley and coworkers.⁴ In their interpretation, it is the difference in timescales of the decay of the position-autocorrelation vs. velocity autocorrelation function that leads to the deviations in diffusivity. The position autocorrelation function, in certain membrane regions, may decay so slowly that it incorporates slow degrees of freedom that are not assumed by the underlying GLE. The Woolf-Roux method is expected to have higher accuracy due to the rapidly decaying velocity autocorrelation function. However, it has increased uncertainty at the critical small $s$ region due to divergence of the memory function when ${\tilde{C}}_{vv}(s)$ does not identically reach 0 when $s=0$. On the other hand, the MLP approach effectively regularizes the analysis by imposing an analytical form of the memory function (a sum of two decaying exponentials). It is interesting to include the novel MLP memory function in the discussion of the small $s$ behavior. The Laplace transform of the MLP memory function (Eq. (7)) is given analytically by,

$${\tilde{M}}^{MLP}(s)=\sum _{i=1}^2\frac{K_i}{s+1/{\tau }_i}$$ (24)

For reference, the MLP memory function in Laplace space (${\tilde{M}}^{MLP}(s)$) was plotted alongside the Hummer and Woolf-Roux memory functions in Fig. 5. Due to the modified loss function, the s = 0 value is constrained to fall between those of Woolf-Roux extrapolated to s = 0 and Hummer at s = 0. Interestingly, for both z = 0 and z = 15 Å, ${\tilde{M}}^{MLP}(s)$ begins to deviate from the other two memory functions at approximately s = 0.2 ps⁻¹ (5 ps), where Woolf-Roux begins to diverge and experience high uncertainty. At the same time, it ultimately reaches s = 0 at a smaller y-intercept than Hummer, which we expect is due to the reliance of the MLP on the quickly decaying velocity autocorrelation function. Therefore, the MLP method effectively serves as an automatic extrapolation of Woolf-Roux to s = 0, while working in the time domain.

Lastly, we use the non-Markovian GLE simulations to further illustrate the convergence of the Woolf-Roux and Hummer approaches for determining the diffusivity. As with the MD simulations, we carried out biased umbrella sampling with the numerical GLE simulations. The simulations were conducted using the same PMF, position-dependent memory function, and EM-3m integrator as the long equilibrium GLE simulations. The umbrella sampling was designed to reproduce the biased MD simulations in terms of force constants, total sampling time, and system conditions. By this construction, the Woolf-Roux and Hummer approaches can be systematically checked for accuracy, because the exact diffusivity of the GLE is known.

The Woolf-Roux and Hummer position-dependent diffusivity computed for these simulations are shown in Fig. 6. The agreement between Woolf-Roux and Hummer with the diffusivity input in the GLE. This is in contrast with the wide discrepancies observed from the analysis of the all-atom MD simulations (Fig. 4), especially at the critical z = 0 Å region. The Woolf-Roux is slightly larger than Hummer by a factor of 1.1 at this position, compared to a factor of 2.8 for the same point for the MD simulations. We have shown that both approaches can reliably extract the known position-dependent diffusivity of the underlying GLE. The extrapolation results for the Woolf-Roux diffusivity for each window are detailed in Fig. S11, Fig. S12, and Fig. S13, and the running integrals of the position autocorrelation functions in each window for the Hummer diffusivity are shown in Fig. S14 in the supporting information. For completeness, the velocity autocorrelation functions are shown in Fig. S15 and the force-velocity autocorrelation functions are shown in Fig. S16 in the supporting information.

FIG. 6.

Position-dependent diffusivity ($D(z)$) from umbrella sampling numerical GLE simulations with the EM-3m integrator. The two main approaches, Woolf-Roux (blue) and Hummer (green), are used, showing strong agreement with each other. Also shown is the $D(z)$ of the underlying GLE (dashed red). The plotted values are the mean of the three replicates for each window, which were first symmetrized about $z=0$. The localized error bars are plus and minus the standard deviation between the three replicates.

This numerical test based on umbrella sampling GLE simulations provides further support to the conjecture that the primary source of disagreement between the velocity (Woolf-Roux) and position (Hummer) routes is based on uncertainty in extrapolating (for Woolf-Roux) and the overlap of the position autocorrelation function with slow degrees of freedom (for Hummer). When the numerical GLE is used to model the dynamics, there is no longer a possibility of interference by slow degrees of freedom, and the agreement between the velocity and position routes improves dramatically. This analysis has also supports that the fitting and extrapolation criteria for Woolf-Roux is reasonable.

D. Evaluation of the permeability

The most direct and straightforward treatment of permeation consists of counting spontaneous membrane crossing events during unbiased trajectories.^17,18,26,43 Even for the GLE, this strategy remains critical, as there is no rigorous closed-form mathematical expression of the permeability coefficient for an arbitrary PMF and position-dependent memory function. For this reason, the permeability estimated from the long, equilibrium simulations serves as the reference for all of the other methods. The permeability based on counting membrane crossings is given by,^17,18,44

$$P_{\text{counting}}=\frac{N_c(t)}{2t\pi (-L/2)}$$ (25)

where $N_c(t)$ is the total number of crossings in either direction up to time $t$, and $\pi (-L/2)$ is the equilibrium probability at the edge of the membrane. In our previous work, we estimated the counting permeability of the water and DLPC membrane system to be 8.83±0.79×10⁻³ cm/s.¹⁸ Based on the 10,000 μs of sampling from the long, equilibrium GLE simulations, 295 crossings were observed in total. This rate of crossings corresponds a permeability of 9.96±0.61×10⁻³ cm/s. The trajectories of all of the observed permeation events in the GLE simulations are shown in Fig. 7. The beginning of each permeation was chosen to be when the reactive trajectory leaves region a or b (when $z$ becomes less than 30 Å or greater than –30 Å, respectively). Conversely, the end of the event is when the trajectory first touches the opposite side. Based on this analysis, the mean reactive transit time is then the average duration of each of the 295 permeation events, which was found to be 2.97 ns. Notably, this timescale is similar to that observed from the unbiased MD simulations of water, which was 2.93 ns.¹⁸ This analysis shows that the GLE with position-dependent memory provides a physically sound coarse-grained representation of water permeability across a lipid membrane.

FIG. 7.

All of the reactive trajectories from the equilibrium numerical GLE simulations. Crossings from side a to side b are colored blue, and crossings from b to a are colored red. Based on 1000 independent simulations of 10 μs each, there was a mean of 0.295 crossings with a variance of 0.322, which is in line with the expected Poisson distribution. The mean permeability is 0.00996 cm/s with a standard error of 0.00061 cm/s. The mean reactive transit time, or mean of duration of each crossing, is 2.97 ns.

The permeability coefficients based on the various approaches are summarized in Table II. The ISD permeability for the GLE system based on the analytical PMF and the exact position-dependent diffusivity of the underlying GLE is 9.38×10⁻³ cm/s. However, based on the PMF obtained from the p(z) that is actually sampled in the long, equilibrium simulations, the ISD permeability is 10.0×10⁻³ cm/s. We can also assess the relative accuracy of each position-dependent diffusivity approach to estimate the permeability. The ISD model MD permeability estimates were obtained by using each position-dependent diffusivity (Woolf-Roux, Hummer, MLP) as an input in Eq. (1), in conjunction with the single PMF from the equilibrium simulations. Because the same PMF is used in each case, we are able to evaluate each of the position-dependent diffusivity profiles for accuracy. While the Hummer diffusivity significantly underestimates the counting permeability, Woolf-Roux is only slightly larger. The MLP is in closest agreement. We expect that its increased robustness and accuracy is due to a combination of the automated optimization, avoidance of the position autocorrelation function, and introduction of the modified loss function.

TABLE II.

Permeability of DLPC to water, units of ×10⁻³ cm/s

MD simulations
	ISD, Woolf-Roux1	10.29 ± 0.20
	ISD, Hummer1	1.519 ± 0.074
	ISD, MLP1	8.72 ± 0.27
	counting2,3	8.83 ± 0.79
GLE simulations
	ISD, input parameters	9.38
	ISD, output PMF4	10.0
	counting3	9.96 ± 0.61

Notes:

^1.Uncertainty interval from standard deviation between three replicates of position-dependent diffusivity estimation.

^2.Taken from ref 18.

^3.Uncertainty interval based standard error in the number of crossings from equilibrium simulations.

^4.Permeability based on PMF from equilibrium p(z) sampling and the analytical position-dependent diffusivity.

An intriguing question concerns the importance of memory effects on the permeability of water through a lipid membrane. An all-atom model of a lipid membrane is a molecular system exhibiting a complex dynamics. Yet, the strong agreement between the ISD and counting permeability from the long, equilibrium GLE simulation suggests that memory effects do not have a significant impact on the rate of barrier crossing. To explore this issue further, we exploit the Grote-Hynes theory designed to model the non-Markovian barrier crossing in transition rate theory.²⁰ In this theory, the transmission coefficient, $\kappa$, which modulates the the rate from transition state theory, is given by, $\kappa =s/{\omega }_0$ where ${\omega }_0=\sqrt{-W^{\prime \prime }/m},W^{\prime \prime }$ is the (negative) second derivative of the inverted parabolic PMF, and $s$ is the real solution of the equation,²⁰

$$-W^{\prime \prime }-m{s}^2=s\tilde{M}(s)$$ (26)

Carrying out the Grote-Hynes analysis based on Eq. (26) with the MLP memory function yields a result corresponding to the non-Markovian dynamics of the GLE. On the other hand, neglecting the term $m{s}^2$ associated with inertial effects in Eq. (26) and replacing $\tilde{M}(s)$ by $\tilde{M}(0)$ yields a result that corresponds to the memory-less Markovian overdamped dynamics of the ISD approximation.¹⁸ By solving Eq. (26) numerically with the MLP memory function (Eq. (24)), it is found that the ratio of the ISD permeability to the non-Markovian permeability is 0.96, which confirms that water permeation is not significantly affected by memory effects. The full Grote-Hynes analysis is illustrated in Fig. S17 and the results are collected in Table S1 in the supporting information.

The physical reason is that the relaxation time over the inverted parabolic barrier is small compared to the short relaxation times of the memory function. This is explicitly shown starting from, $-W^{\prime \prime }=s(\tilde{M}(0)+s{\tilde{M}}^{\prime }(0)+\dots )$, and assuming that ${\tilde{M}}^{\prime }(0)$ is small. To zeroth order, the situation reduces to the ISD approximation, $s_{\text{ISD}}=-W^{\prime \prime }/\tilde{M}(0)$. The Grote-Hynes solution accounting for memory effect is written in terms of a perturbation $\delta s$ relative to $s_{\text{ISD}}$. To lowest order, the ratio of ISD to GH permeability is,

$$\frac{P_{\mathrm{I}\mathrm{S}\mathrm{D}}}{P_{\mathrm{G}\mathrm{H}}}\approx 1-\frac{W^{\prime \prime }{\tilde{M}}^{\prime }(0}{\tilde{M}(0{)}^2}=1-\frac{{\tau }_m}{{\tau }_b}$$ (27)

where ${\tau }_m=-{\tilde{M}}^{\prime }(0)/\tilde{M}(0)$ is the mean relaxation time of the memory function, and ${\tau }_b=-k_{\mathrm{B}}T/D{W}^{\prime \prime }$ is the relaxation time of the random motion over the energy barrier. In the present case, ${\tau }_m\ll {\tau }_b$. This finding is consistent with the good agreement between the ISD permeability and the counting permeability of the numerical GLE simulations. By extension, because the PMF and position-dependent memory function of the GLE system are quantitatively close to those of the MD system by construction, it is anticipated that memory effects have a negligible impact on water permeability through lipid membranes.

V. CONCLUSION

A theoretical framework based on the Generalized Langevin Equation (GLE) was used to examine the permeability of water molecules across lipid membranes. The multilayer perceptron (MLP) deep learning approach of Russo et al.¹⁹ was extended to extract the position-dependent memory function from biased umbrella sampling MD simulations of the permeant molecule across the membrane. The memory function was modeled as a sum of two positive, decaying exponential terms. It was shown that the MLP method can reliably extract the diffusivity from MD simulations. This is in contrast with the familiar Woolf and Roux¹⁴ and Hummer¹⁶ methods, which can display severe inaccuracies and numerical instabilities. One source of error associated with those methods appears to be the use of the Laplace transform of time-correlation functions to solve the GLE in closed form. In contrast, the present MLP approach solves for the memory function directly in the time domain. This procedure, together with the assumption of an analytical form of the memory function, provide a way to regularize the extraction of the space-dependent memory function and the diffusivity from umbrella sampling simulations.

The GLE parametrized from the MD data could be numerically integrated efficiently with Markovian embedding, which was possible due to the assumed form of the memory function.³⁰ A permeability of about 9.96×10⁻³ cm/s is estimated using unbiased simulations based on the GLE with a PMF and memory function extracted from all-atom MD simulations of water in a DLPC membrane. This result compares well with an estimate of 8.83 × 10⁻³ cm/s from from all-atom MD simulations of a DLPC membrane.¹⁸ Furthermore, the mean reactive transit time from the all-atom MD simulations and the GLE simulations are very similar, which are 2.93 and 2.97 ns, respectively. These results indicate that the GLE with position-dependent memory provides a physically sound and accurate representation of water permeation across a lipid membrane. In addition, the ISD permeability estimate from MD (with the MLP diffusivity) ranges between 8.72 and 10.0 ×10⁻³ cm/s based on the MD and GLE simulations. The near equivalence of the ISD and counting estimate for the permeability, together with an analysis based on the Grote-Hynes theory of barrier crossing demonstrate that memory effects have a negligible impact on this example of a permeation process.

The validation of the ISD model for this system, in addition to the development of an improved estimation of the position-dependent diffusivity from umbrella sampling MD simulations, is promising for the future of permeability calculations. In the field of drug design, the screening of small molecules by permeability can depend critically on the ISD model. Rigorous validation of its assumptions remains significant as small molecule screening becomes more widespread and automated.^45–47 In the future, we are interested in further assessing the accuracy of the numerical integration scheme described in this work for a GLE with position-dependent memory. It will be of interest to consider if more exact methods²⁸ will yield consistent results with the straightforward approach presented here. Future directions may also include tests of systems for which the ISD model is not valid, to determine if the GLE parameterized by the MLP can effectively solve for the exact permeability.

Supplementary Material

Additional information about the MLP structure and optimization, the numerical integration of the GLE, the time-correlation functions from each umbrella window of the MD and GLE simulations, the extraction of the Woolf-Roux and Hummer diffusivity from each umbrella window, and the Grote-Hynes analysis is contained in the supplementary material.

Appendix A: Derivation of the Woolf-Roux and Hummer diffusivity

Beginning with the time-correlation function representation of the GLE,

$$-m⟨\delta \ddot{z}(t)\delta \dot{z}(0)⟩-K⟨\delta z(t)\delta \dot{z}(0)⟩={\int }_0^{t}M\left(t-t^{\prime }\right)⟨\delta \dot{z}(t)\delta \dot{z}(0)⟩\mathrm{d}{t}^{\prime }$$ (A1)

We substitute, $C_{\nu v}(t)=⟨\delta \dot{z}(t)\delta \dot{z}(0)⟩$, $\frac{d}{dt}{C}_{\nu v}(t)=⟨\delta \ddot{z}(t)\delta \dot{z}(0)⟩$, and ${\int }_0^t{C}_{\nu \nu }(t)\mathrm{d}t=⟨\delta z(t)\delta \dot{z}(t)⟩$, and take the Laplace transform of both sides, yielding,

$$-ℒ\left\{m\frac{d}{dt}{C}_{vv}(t)+K{\int }_0^t{C}_{vv}(t)\mathrm{d}t\right\}=ℒ\left\{{\int }_0^{t}M\left(t-t^{\prime }\right)C_{vv}(t)\mathrm{d}{t}^{\prime }\right\}$$ (A2)

We invoke the following properties of the Laplace transform,

$$\begin{gathered} ℒ\left\{\frac{d}{dt}{C}_{vv}(t)\right\}=s{\tilde{C}}_{vv}(s)-⟨\delta {\dot{z}}^2⟩ \\ ℒ\left\{{\int }_0^t{C}_{vv}(t)\mathrm{d}t\right\}=\frac{1}{s}{\tilde{C}}_{vv}(s) \\ ℒ\left\{{\int }_0^{t}M\left(t-t^{\prime }\right)C_{vv}(t)\mathrm{d}{t}^{\prime }\right\}=\tilde{M}(s){\tilde{C}}_{vv}(s) \end{gathered}$$ (A3)

where $⟨\delta {\dot{z}}^2⟩=C_{vv}(0)$. We also substitute $m=k_{\mathrm{B}}T/⟨\delta {\dot{z}}^2⟩$ and $K=k_{\mathrm{B}}T/⟨\delta z^2⟩$, and solve for the memory function,

$$\begin{gathered} \tilde{M}(s){\tilde{C}}_{vv}(s)=-\frac{k_{\mathrm{B}}T}{⟨\delta {\dot{z}}^2⟩}\left[s{\tilde{C}}_{vv}(s)-⟨\delta {\dot{z}}^2⟩\right]-\frac{k_{\mathrm{B}}T}{s⟨\delta z^2⟩}{\tilde{C}}_{vv}(s) \\ \tilde{M}(s){\tilde{C}}_{vv}(s)⟨\delta {\dot{z}}^2⟩⟨\delta z^2⟩=-k_{\mathrm{B}}T\left(⟨\delta z^2⟩\left[s{\tilde{C}}_{vv}(s)-⟨\delta {\dot{z}}^2⟩\right]+\frac{⟨\delta {\dot{z}}^2⟩}{s}{\tilde{C}}_{vv}(s)\right) \\ \tilde{M}(s)=-k_{\mathrm{B}}T\frac{s⟨\delta z^2⟩{\tilde{C}}_{vv}(s)-⟨\delta z^2⟩⟨\delta {\dot{z}}^2⟩+\frac{⟨\delta {\dot{z}}^2⟩}{s}{\tilde{C}}_{vv}(s)}{⟨\delta {\dot{z}}^2⟩⟨\delta z^2⟩{\tilde{C}}_{vv}(s)} \\ \tilde{M}(s)=-k_{\mathrm{B}}T\frac{{\tilde{C}}_{vv}(s)\left[s⟨\delta z^2⟩+⟨\delta {\dot{z}}^2⟩/s\right]-⟨\delta z^2⟩⟨\delta {\dot{z}}^2⟩}{⟨\delta {\dot{z}}^2⟩⟨\delta z^2⟩{\tilde{C}}_{vv}(s)} \end{gathered}$$ (A4)

which is the memory function in Laplace space used for computing the Woolf-Roux diffusivity. In order to derive the memory function of the Hummer diffusivity, we first obtain ${\tilde{C}}_{vv}(s)$ in terms of ${\tilde{C}}_{zz}(s)$ by,

$$\begin{gathered} -ℒ\left\{C_{vv}(t)\right\}=ℒ\left\{\frac{d^2}{d{t}^2}{C}_{zz}(t)\right\} \\ =sℒ\left\{\frac{d}{dt}{C}_{zz}(t)\right\}-⟨\delta \dot{z}\delta z⟩ \\ =s\left(s{\tilde{C}}_{zz}(s)-⟨\delta z^2⟩\right) \\ {\tilde{C}}_{vv}(s)=-s^2{\tilde{C}}_{zz}(s)+s⟨\delta z^2⟩ \end{gathered}$$ (A5)

and then substitute this result into Eq. (A4). The memory function of the Hummer diffusivity in Laplace space is given by,

$$\begin{gathered} \tilde{M}(s)=-k_{\mathrm{B}}T\frac{\left(-s^2{\tilde{C}}_{zz}(s)+s⟨\delta z^2⟩\right)\left[s⟨\delta z^2⟩+⟨\delta {\dot{z}}^2⟩/s\right]-⟨\delta z^2⟩⟨\delta {\dot{z}}^2⟩}{⟨\delta {\dot{z}}^2⟩⟨\delta z^2⟩\left(-s^2{\tilde{C}}_{zz}(s)+s⟨\delta z^2⟩\right)} \\ =k_{\mathrm{B}}T\frac{{\tilde{C}}_{zz}(s)\left[s^2⟨\delta z^2⟩+⟨\delta {\dot{z}}^2⟩\right]-s{⟨\delta z^2⟩}^2}{⟨\delta {\dot{z}}^2⟩⟨\delta z^2⟩\left(⟨\delta z^2⟩-s{\tilde{C}}_{zz}(s)\right)} \end{gathered}$$ (A6)

DATA AVAILABILITY STATEMENT

An example implementation of the MLP with Pytorch, the time-correlation functions from the umbrella sampling MD simulations used for computing the position-dependent diffusivity and for deep learning the memory functions, and Python code for numerically integrating the GLE with positiondependent memory are available to download from Zenodo at DOI:10.5281/zenodo.21698905.