Computing long time scale biomolecular dynamics using quasi-stationary distribution kinetic Monte Carlo (QSD-KMC)

Abstract

It is a challenge to obtain an accurate model of the state-to-state dynamics of a complex biological system from molecular dynamics (MD) simulations. In recent years, Markov State Models have gained immense popularity for computing state-to-state dynamics from a pool of short MD simulations. However, the assumption that the underlying dynamics on the reduced space is Markovian induces a systematic bias in the model, especially in biomolecular systems with complicated energy landscapes. To address this problem, we have devised a new approach we call quasi-stationary distribution kinetic Monte Carlo (QSD-KMC) that gives accurate long time state-to-state evolution while retaining the entire time resolution even when the dynamics is highly non-Markovian.The proposed method is a kinetic Monte Carlo approach that takes advantage of two concepts: (i) the quasistationary distribution, the distribution that results when a trajectory remains in one state for a long time (the dephasing time), such that the next escape is Markovian, and (ii) dynamical corrections theory, which properly accounts for the correlated events that occur as a trajectory passes from state to state before it settles again.In practice, this is achieved by specifying, for each escape, the intermediate states and the final state that has resulted from the escape. Implementation of QSD-KMC imposes stricter requirements on the lengths of the trajectories than in a Markov State Model approach, as the trajectories must be long enough to dephase.However, the QSD-KMC model produces state-to-state trajectories that are statistically indistinguishable from an MD trajectory mapped onto the discrete set of states, for an arbitrary choice of state decomposition. Furthermore, the aforementioned concepts can be used to construct a Monte Carlo approach to optimize the state boundaries regardless of the initial choice of states. We demonstrate the QSD-KMC method on two one-dimensional model systems, one of which is a driven nonequilibrium system, and on two well-characterized biomolecular systems. * gnana@lanl.gov † afv@lanl.gov arXiv:1905.09975v2 [physics.chem-ph]

Estimating linear mass transport coefficients in solid solutions via correlation splitting and a law of total diffusion

“…derived from the reversibility property (3) and the law of total covariance [42], where χ ≡ χ 1 and d ≡ r 0→1 . Relationship (10) bridges between the LTV at the lower extremity and the LTD obtained for → ∞. The LTD entails that the diffusion matrix can be estimated from mean local quantities, by plugging the expected displacements and their conditional variances given the visited states into its extracorrelated and intracorrelated parts.…”

“…Besides, the LTD and bridging law are meaningful for reversible Markov chains and for any stochastic variable that is antisymmetric under chain reversal. Laws ( 7) and ( 9) also yield a Löwner partial ordering: (10) over a sample of l trajectories of L = max steps each. For statistical errors to be small, l must be large enough, standard deviations decaying as 1/ √ l [43].…”

“…Estimates D std X and D cnd X of the diffusion coefficients D X for both species and the vacancy (X = A, B, and A + B) are then obtained via D std l,L and D cnd l,L by averaging the corresponding elements over the two space directions. The optimal estimate is then obtained as (10) where Σ and Γ −1 are estimates of the corresponding intracorrelated and extracorrelated parts. Last, we define and evaluate the intracorrelation-toextracorrelation ratio φ = Σ L /Γ L and the reduction factor of the statistical variance η = var(D std X )/var(D opt X ).…”

“…They span a discrete space embedded into the continuous space which the dynamical system evolves in. Transition rates computed at the atomic scale possibly include anharmonic [8], quantum [9], and dynamical [10] corrections, whose respective contributions depend on temperature T . Dynamical corrections are to be included to account for memory effects when state-to-state transitions depend on past transitions [10].…”

“…Transition rates computed at the atomic scale possibly include anharmonic [8], quantum [9], and dynamical [10] corrections, whose respective contributions depend on temperature T . Dynamical corrections are to be included to account for memory effects when state-to-state transitions depend on past transitions [10]. They may be significant in simulations of dynamical systems evolving on complicated potential energy surfaces over long timescales when the encountered barrier heights E are similar or lower than the thermal energy k B T (k B stands for Boltzmann's constant).…”