Neural Stochastic Forecasting for Lateral Molecular Diffusion Simulations

Jacob Ede Levine; Mario Hidalgo-Soria; Chen Yun Wen; Yun Lyna Luo; Sai Chandra Kosaraju

cs.LGNov 4, 2026

Neural Stochastic Forecasting for Lateral Molecular Diffusion Simulations

Authors: Jacob Ede Levine, Mario Hidalgo-Soria, Chen Yun Wen, Yun Lyna Luo, Sai Chandra Kosaraju

Organizations: Department of Computer Science, California State Polytechnic University, Pomona, 91768, California, USA. · Department of Biotechnology and Pharmaceutical Sciences, Western University of Health Sciences, Pomona, 91766, California, USA.

Abstract

Predicting the long-time diffusion of membrane proteins is challenging because molecular dynamics (MD) simulations are computationally expensive, while experimental single-particle tracking is often limited by finite observation times. Here, we introduce a neural stochastic (NS) model for forecasting the collective diffusion of membrane proteins from coarse-grained MD trajectories. The model combines a transformer-based representation of protein configurations, recurrent networks operating at multiple temporal scales, and a stochastic latent process with learned memory. A cross-attention decoder then generates correlated protein displacements while accounting for the instantaneous system geometry and periodic boundary conditions. We evaluate the model in two membrane systems: seven interacting PIEZO channels and a mixed system containing seven PIEZO and ten TREK channels. For the seven PIEZO system, the NS model reproduces the anomalous diffusion exponent measured in MD simulations $α\sim 1$ , substantially improving on a previously developed variational autoregressive model that exhibits superdiffusive behavior. The model also generates trajectories extending to 40,000 frames and captures the distinct diffusion characteristics of PIEZO and TREK populations in the mixed system. Comparisons of time-averaged mean-square displacements, Van Hove displacement distributions, and non-Gaussian parameters show that the model preserves key statistical features of the simulated dynamics. These results establish neural stochastic modeling as a promising approach for generating long trajectories of multi-protein membrane diffusion and extending the statistical analysis of molecular dynamics simulations.

Figures & tables

	PIEZO	PIEZO-TREK
Proteins $N$	7	17
Composition	7 PIEZO	7 PIEZO, 10 TREK
Species	1	2
Relabeling symmetry	all permutations	within each species
Replicas used	2	2
Frames, replica 1	12,578	6,001

Table 1: Coarse-grained systems. Frames are counted after the removal of duplicated frames described in the text.

L_{\text{raw}}

is the raw in-plane box edge,

\sigma_{\Delta}

the displacement scale defined after Equation 1 , and

L=L_{\text{raw}}/\sigma_{\Delta}

the dimensionless normalized box edge of Equation 2 .

	PIEZO	PIEZO-TREK
Geometry vector $\mathbf{e}^{(t)}$	462	1207
Bank input $\tilde{\mathbf{e}}^{(t)}_{i}$	84	199
LSTM banks (one per species)	1	2
Bank readout $\boldsymbol{\psi}^{(t)}$	3136	7616
Transition head hidden width	1568	3808
Latent state $\boldsymbol{\zeta}^{(t)}$	42	102

Table 2: Widths of the internal vectors of the model and parameter counts for each system.

Term	Weight	PIEZO	PIEZO-TREK
Displacement	$w_{\mathrm{disp}}$	1.836	7.712
Turning angle	$w_{\mathrm{ang}}$	181.67	858.79
Pairwise distance	$w_{\mathrm{pair}}$	18.125	18.125
Cohesion	$w_{\mathrm{coh}}$	1871.69	583.36
VACF distribution	$w_{\mathrm{vq}}$	12.705	2.229
VACF curve	$w_{\mathrm{vc}}$	166.43	166.43

Table 3: Loss weights of Equation 51 . The weights are dimensionless multipliers on observables in normalized units and are identical for the five training seeds of a system.

Setting	Value
Optimizer	Sophia with squared-gradient curvature,
	Equations 52 to 54
Learning rate $\eta$	$2.2117\times 10^{-5}$ , constant
Learning rate, latent dynamics	$0.06994$ , constant
Weight decay	$0.01$ , plain linear and attention weights only
Batch	64 windows

Table 4: Training settings, shared by both systems.

	PIEZO	PIEZO-TREK
Initial conditions from	replicas 1 and 2	replica 1
Frames available	19,681	11,001
Largest lag $\tau_{\max,s}$	5000	5000 for both species
Cells	204	$204+204=408$
Stand-in rollouts	$2\times 40$	$2\times 40$
Draws	600	960

Table 5: Evaluation settings. Initial conditions are drawn from the listed frames, and the reference pools both replicas of each system. The largest scored lag

\tau_{\max,s}

of species

s

is defined in the text, and the stand-in rollouts and the draws are defined in Section 2.13 .

Figure 1: Comparison between the ground truth (MD simulations) and predictions of our neural stochastic (NS) model and VAR-SDE model Thai et al. (2026) , for the 7 PIEZO multi-protein system. a) Trajectories of the 7 PIEZO channels (center of mass) obtained from MD simulations during a total time of 10,651 time frames. b) Predicted trajectories using the NS model of the PIEZO channels during a total time with 40,000 time frames. c) Predicted trajectories obtained by the VAR-SDE model for tracks with 10,000 time frames. d-f) TAMSD and ETAMSD for ground truth trajectories and predictions respectively. For convergence of the statistics, the TAMSD is computed in 10% of the trajectories (cite). g-i) Van Hove function Eq. 64 for selected lag-times $\Delta t=\{100,500,1000,2000\}$ , for MD simulations and predictions. And j-l) Non-Gaussian parameter Eq. 66 for lag times covering the whole trajectories of the ground truth and each prediction respectively.

Figure 2: Comparison between the ground truth (MD simulations) and predictions of our neural stochastic (NS) model , for the 7 PIEZO and 10 TREK multi-protein system. a) Trajectories of the 7 PIEZO channels (red lines) and 10 TREK channels (blue lines) obtained from MD simulations during a total time of 9,000 time frames. b) Predicted trajectories using the NS model of the PIEZO channels and TREK during a total time with 40,000 time frames.

Figure 3: TAMSD: comparison between the ground truth (MD simulations) and predictions of our neural stochastic (NS) model , for the 7 PIEZO and 10 TREK multi-protein system. a-b) Ground Truth TAMSD (gray lines) and ETAMSD (black line) for the 7 PIEZO channels and the 10 TREK proteins respectively. c-d) TAMSD and ETAMSD for the NS-Model for the 5 rollouts of the 7PIEZO and 10 TREK system, with 35 TAMSD curves for the PIEZO channels see c) and 50 TAMSD curves for TREK channels see d) .

Figure 4: Van Hove function: comparison between the ground truth (MD simulations) and predictions of our neural stochastic (NS) model , for the 7 PIEZO and 10 TREK multi-protein system. a-b) Ground Truth Van Hove function for the 7 PIEZO channels and the 10 TREK proteins respectively. c-d) Van Hove function for the NS-Model for the 5 rollouts of the 7PIEZO and 10 TREK system, for the PIEZO channels see c) and for TREK channels see d) .

Figure 5: Non Gaussian parameter: comparison between the ground truth (MD simulations) and predictions of our neural stochastic (NS) model , for the 7 PIEZO and 10 TREK multi-protein system. a-b) Ground Truth TAMSD for the 7 PIEZO channels and the 10 TREK proteins respectively. c-d) Non Gaussian parameter for the NS-Model employing 5 rollouts for the 7PIEZO and 10 TREK system, for the PIEZO channels see c) and for TREK channels see d) .

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Abstract

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