Grand canonical diffusion-influenced reactions: A stochastic theory with applications to multiscale reaction-diffusion simulations

Data-driven dynamical coarse-graining for condensed matter systems

Simulations of condensed matter systems often focus on the dynamics of a few distinguished components but require integrating the full system. A prime example is a molecular dynamics simulation of a (macro)molecule in a solution, where the molecule(s) and the solvent dynamics need to be integrated, rendering the simulations computationally costly and often unfeasible for physically/biologically relevant time scales. Standard coarse graining approaches can reproduce equilibrium distributions and structural features but do not properly include the dynamics. In this work, we develop a general data-driven coarse-graining methodology inspired by the Mori-Zwanzig formalism, which shows that macroscopic systems with a large number of degrees of freedom can be described by a few relevant variables and additional noise and memory terms. Our coarse-graining method consists of numerical integrators for the distinguished components, where the noise and interaction terms with other system components are substituted by a random variable sampled from a data-driven model. The model is parameterized using data from multiple short-time full-system simulations, and then, it is used to run long-time simulations. Applying our methodology to three systems-a distinguished particle under a harmonic and a bistable potential and a dimer with two metastable configurations-the resulting coarse-grained models are capable of reproducing not only the equilibrium distributions but also the dynamic behavior due to temporal correlations and memory effects. Remarkably, our method even reproduces the transition dynamics between metastable states, which is challenging to capture correctly. Our approach is not constrained to specific dynamics and can be extended to systems beyond Langevin dynamics, and, in principle, even to non-equilibrium dynamics.

Integrative modeling of the cell

A whole-cell model represents certain aspects of the cell structure and/or function. Due to the high complexity of the cell, an integrative modeling approach is often taken to utilize all available information including experimental data, prior knowledge and prior models. In this review, we summarize an emerging workflow of whole-cell modeling into five steps: (i) gather information; (ii) represent the modeled system into modules; (iii) translate input information into scoring function; (iv) sample the whole-cell model; (v) validate and interpret the model. In particular, we propose the integrative modeling of the cell by combining available (whole-cell) models to maximize the accuracy, precision, and completeness. In addition, we list quantitative predictions of various aspects of cell biology from existing whole-cell models. Moreover, we discuss the remaining challenges and future directions, and highlight the opportunity to establish an integrative spatiotemporal multi-scale whole-cell model based on a community approach.

A neural network-assisted open boundary molecular dynamics simulation method

A neural network-assisted molecular dynamics method is developed to reduce the computational cost of open boundary simulations. Particle influxes and neural network-derived forces are applied at the boundaries of an open domain consisting of explicitly modeled Lennard-Jones atoms in order to represent the effects of the unmodeled surrounding fluid. Canonical ensemble simulations with periodic boundaries are used to train the neural network and to sample boundary fluxes. The method, as implemented in the LAMMPS, yields temperature, kinetic energy, potential energy, and pressure values within 2.5% of those calculated using periodic molecular dynamics and runs two orders of magnitude faster than a comparable grand canonical molecular dynamics system.

Multiscale molecular kinetics by coupling Markov state models and reaction-diffusion dynamics

A novel approach to simulate simple protein-ligand systems at large time and length scales is to couple Markov state models (MSMs) of molecular kinetics with particle-based reaction-diffusion (RD) simulations, MSM/RD. Currently, MSM/RD lacks a mathematical framework to derive coupling schemes, is limited to isotropic ligands in a single conformational state, and lacks multiparticle extensions. In this work, we address these needs by developing a general MSM/RD framework by coarse-graining molecular dynamics into hybrid switching diffusion processes. Given enough data to parameterize the model, it is capable of modeling protein-protein interactions over large time and length scales, and it can be extended to handle multiple molecules. We derive the MSM/RD framework, and we implement and verify it for two protein-protein benchmark systems and one multiparticle implementation to model the formation of pentameric ring molecules. To enable reproducibility, we have published our code in the MSM/RD software package.

Diffusion-influenced reaction rates in the presence of pair interactions

The kinetics of bimolecular reactions in solution depends, among other factors, on intermolecular forces such as steric repulsion or electrostatic interaction. Microscopically, a pair of molecules first has to meet by diffusion before the reaction can take place. In this work, we establish an extension of Doi's volume reaction model to molecules interacting via pair potentials, which is a key ingredient for interacting-particle-based reaction-diffusion (iPRD) simulations. As a central result, we relate model parameters and macroscopic reaction rate constants in this situation. We solve the corresponding reaction-diffusion equation in the steady state and derive semi-analytical expressions for the reaction rate constant and the local concentration profiles. Our results apply to the full spectrum from well-mixed to diffusion-limited kinetics. For limiting cases, we give explicit formulas, and we provide a computationally inexpensive numerical scheme for the general case, including the intermediate, diffusion-influenced regime. The obtained rate constants decompose uniquely into encounter and formation rates, and we discuss the effect of the potential on both subprocesses, exemplified for a soft harmonic repulsion and a Lennard-Jones potential. The analysis is complemented by extensive stochastic iPRD simulations, and we find excellent agreement with the theoretical predictions.

Transient probability currents provide upper and lower bounds on non-equilibrium steady-state currents in the Smoluchowski picture

Probability currents are fundamental in characterizing the kinetics of nonequilibrium processes. Notably, the steady-state current Jss for a source-sink system can provide the exact mean-first-passage time (MFPT) for the transition from the source to sink. Because transient nonequilibrium behavior is quantified in some modern path sampling approaches, such as the "weighted ensemble" strategy, there is strong motivation to determine bounds on Jss-and hence on the MFPT-as the system evolves in time. Here, we show that Jss is bounded from above and below by the maximum and minimum, respectively, of the current as a function of the spatial coordinate at any time t for one-dimensional systems undergoing overdamped Langevin (i.e., Smoluchowski) dynamics and for higher-dimensional Smoluchowski systems satisfying certain assumptions when projected onto a single dimension. These bounds become tighter with time, making them of potential practical utility in a scheme for estimating Jss and the long time scale kinetics of complex systems. Conceptually, the bounds result from the fact that extrema of the transient currents relax toward the steady-state current.