  Computational modeling of assembly is challenging for many systems because
their timescales vastly exceed those accessible to simulations. This article
describes the MultiMSM, which is a general framework that uses Markov state
models (MSMs) to enable simulating self-assembly and self-organization on
timescales that are orders of magnitude longer than those accessible to brute
force dynamics simulations. In contrast to previous MSM approaches to
simulating assembly, the framework describes simultaneous assembly of many
clusters and the consequent depletion of free subunits or other small
oligomers. The algorithm accounts for changes in transition rates as
concentrations of monomers and intermediates evolve over the course of the
reaction. Using two model systems, we show that the MultiMSM accurately
predicts the concentrations of the full ensemble of intermediates on the long
timescales required for reactions to reach equilibrium. Importantly, after
constructing a MultiMSM for one system concentration, a wide range of other
concentrations can be simulated without any further sampling. This capability
allows for orders of magnitude additional speed up. In addition, the method
enables highly efficient calculation of quantities such as free energy
profiles, nucleation timescales, flux along the ensemble of assembly pathways,
and entropy production rates. Identifying contributions of individual
transitions to entropy production rates reveals sources of kinetic traps. The
method is broadly applicable to systems with equilibrium or nonequilibrium
dynamics, and is trivially parallelizable and thus highly scalable.
