Molecular latent space simulators

Abstract

Small integration time steps limit molecular dynamics (MD) simulations to millisecond time scales. Markov state models (MSMs) and equation-free approaches learn low-dimensional kinetic models from MD simulation data by performing configurational or dynamical coarse-graining of the state space. The learned kinetic models enable the efficient generation of dynamical trajectories over vastly longer time scales than are accessible by MD, but the discretization of configurational space and/or absence of a means to reconstruct molecular configurations precludes the generation of continuous atomistic molecular trajectories. We propose latent space simulators (LSS) to learn kinetic models for continuous atomistic simulation trajectories by training three deep learning networks to (i) learn the slow collective variables of the molecular system, (ii) propagate the system dynamics within this slow latent space, and (iii) generatively reconstruct molecular configurations. We demonstrate the approach in an application to Trp-cage miniprotein to produce novel ultra-long synthetic folding trajectories that accurately reproduce atomistic molecular structure, thermodynamics, and kinetics at six orders of magnitude lower cost than MD. The dramatically lower cost of trajectory generation enables greatly improved sampling and greatly reduced statistical uncertainties in estimated thermodynamic averages and kinetic rates.

Fig. 1. Schematic diagram of the latent space simulator (LSS) comprising three back-to-back deep neural networks. A state-free reversible VAMPnet (SRV) learns an encoding E of molecular configurations into a latent space spanned by the leading eigenfunctions of the transfer operator (eqn (1)). A mixture density network (MDN) learns a propagator P to sample transition probabilities p_τ(ψ_t+τ|ψ_t) within the latent space. A conditional Wasserstein GAN (cWGAN) learns a generative decoding D of molecular configurations conditioned on the latent space coordinates. The trained LSS is used to generate ultra-long synthetic trajectories by projecting the initial configuration into the latent space using the SRV, sampling from the MDN to generate long latent space trajectories, and decoding to molecular configurations using the cWGAN.

Fig. 2. Validation of the LSS in a 1D four-well potential. The MDN propagator predicts (a) a stationary distribution, (b) kinetic transitions, and (c and d) transition densities in excellent accord with analytical and Brownian dynamics results.

Fig. 3. Free energy profiles for the MD and LSS trajectories projected into the slowest latent space coordinate ψ₁. Shaded backgrounds represent standard errors estimated by five-fold block averaging. The profiles agree within a 0.91k_BT root mean squared error. Ten representative structures from the MD and LSS ensembles are sampled from the folded (ψ₁ ≈ 0), unfolded (ψ₁ ≈ 0.9), and metastable (ψ₁ ≈ 0.45) regions.

Fig. 4. Free energy profiles of the MD and LSS trajectories projected into the leading three MD TICA coordinates.