15 research outputs found
Grand canonical diffusion-influenced reactions: a stochastic theory with applications to multiscale reaction-diffusion simulations
Smoluchowski-type models for diffusion-influenced reactions (A+B -> C) can be
formulated within two frameworks: the probabilistic-based approach for a pair
A, B of reacting particles and the concentration-based approach for systems in
contact with a bath that generates a concentration gradient of B particles that
interact with A. Although these two approaches are mathematically similar, it
is not straightforward to establish a precise mathematical relationship between
them. Determining this relationship is essential to derive particle-based
numerical methods that are quantitatively consistent with bulk concentration
dynamics. In this work, we determine the relationship between the two
approaches by introducing the grand canonical Smoluchowski master equation
(GC-SME), which consists of a continuous-time Markov chain that models an
arbitrary number of B particles, each one of them following Smoluchowski's
probabilistic dynamics. We show that the GC-SME recovers the
concentration-based approach by taking either the hydrodynamic or the large
copy number limit. In addition, we show that the GC-SME provides a clear
statistical mechanical interpretation of the concentration-based approach and
yields an emergent chemical potential for nonequilibrium spatially
inhomogeneous reaction processes. We further exploit the GC-SME robust
framework to accurately derive multiscale/hybrid numerical methods that couple
particle-based reaction-diffusion simulations with bulk concentration
descriptions, as described in detail through two computational implementations
MSM/RD: Coupling Markov state models of molecular kinetics with reaction-diffusion simulations
Molecular dynamics (MD) simulations can model the interactions between
macromolecules with high spatiotemporal resolution but at a high computational
cost. By combining high-throughput MD with Markov state models (MSMs), it is
now possible to obtain long-timescale behavior of small to intermediate
biomolecules and complexes. To model the interactions of many molecules at
large lengthscales, particle-based reaction-diffusion (RD) simulations are more
suitable but lack molecular detail. Thus, coupling MSMs and RD simulations
(MSM/RD) would be highly desirable, as they could efficiently produce
simulations at large time- and lengthscales, while still conserving the
characteristic features of the interactions observed at atomic detail. While
such a coupling seems straightforward, fundamental questions are still open:
Which definition of MSM states is suitable? Which protocol to merge and split
RD particles in an association/dissociation reaction will conserve the correct
bimolecular kinetics and thermodynamics? In this paper, we make the first step
towards MSM/RD by laying out a general theory of coupling and proposing a first
implementation for association/dissociation of a protein with a small ligand (A
+ B C). Applications on a toy model and CO diffusion into the heme cavity
of myoglobin are reported
Coupling particle-based reaction-diffusion simulations with reservoirs mediated by reaction-diffusion PDEs
Open biochemical systems of interacting molecules are ubiquitous in
life-related processes. However, established computational methodologies, like
molecular dynamics, are still mostly constrained to closed systems and
timescales too small to be relevant for life processes. Alternatively,
particle-based reaction-diffusion models are currently the most accurate and
computationally feasible approach at these scales. Their efficiency lies in
modeling entire molecules as particles that can diffuse and interact with each
other. In this work, we develop modeling and numerical schemes for
particle-based reaction-diffusion in an open setting, where the reservoirs are
mediated by reaction-diffusion PDEs. We derive two important theoretical
results. The first one is the mean-field for open systems of diffusing
particles; the second one is the mean-field for a particle-based
reaction-diffusion system with second-order reactions. We employ these two
results to develop a numerical scheme that consistently couples particle-based
reaction-diffusion processes with reaction-diffusion PDEs. This allows modeling
open biochemical systems in contact with reservoirs that are time-dependent and
spatially inhomogeneous, as in many relevant real-world applications
Chemical diffusion master equation: formulations of reaction--diffusion processes on the molecular level
The chemical diffusion master equation (CDME) describes the probabilistic
dynamics of reaction--diffusion systems at the molecular level [del Razo et
al., Lett. Math. Phys. 112:49, 2022]; it can be considered the master equation
for reaction--diffusion processes. The CDME consists of an infinite ordered
family of Fokker--Planck equations, where each level of the ordered family
corresponds to a certain number of particles and each particle represents a
molecule. The equations at each level describe the spatial diffusion of the
corresponding set of particles, and they are coupled to each other via reaction
operators --linear operators representing chemical reactions. These operators
change the number of particles in the system, and thus transport probability
between different levels in the family. In this work, we present three
approaches to formulate the CDME and show the relations between them. We
further deduce the non-trivial combinatorial factors contained in the reaction
operators, and we elucidate the relation to the original formulation of the
CDME, which is based on creation and annihilation operators acting on
many-particle probability density functions. Finally we discuss applications to
multiscale simulations of biochemical systems among other future prospects
toward modeling kinetics of biomolecular complexes
In order to advance the mission of in silico cell biology, modeling the interactions of large and complex biological systems becomes increasingly relevant. The combination of molecular dynamics (MD) and Markov state models (MSMs) have enabled the construction of simplified models of molecular kinetics on long timescales. Despite its success, this approach is inherently limited by the size of the molecular system. With increasing size of macromolecular complexes, the number of independent or weakly coupled subsystems increases, and the number of global system states increase exponentially, making the sampling of all distinct global states unfeasible. In this work, we present a technique called Independent Markov Decomposition (IMD) that leverages weak coupling between subsystems in order to compute a global kinetic model without requiring to sample all combinatorial states of subsystems. We give a theoretical basis for IMD and propose an approach for finding and validating such a decomposition. Using empirical few-state MSMs of ion channel models that are well established in electrophysiology, we demonstrate that IMD can reproduce experimental conductance measurements with a major reduction in sampling compared with a standard MSM approach. We further show how to find the optimal partition of all-atom protein simulations into weakly coupled subunits