36 research outputs found
On the Approximation Quality of Markov State Models
We consider a continuous-time Markov process on a large continuous or discrete state space. The process is assumed to have strong enough ergodicity properties and to exhibit a number of metastable sets. Markov state models (MSMs) are designed to represent the effective dynamics of such a process by a Markov chain that jumps between the metastable sets with the transition rates of the original process. MSMs have been used for a number of applications, including molecular dynamics, for more than a decade. Their approximation quality, however, has not yet been fully understood. In particular, it would be desirable to have a sharp error bound for the difference in propagation of probability densities between the MSM and the original process on long timescales. Here, we provide such a bound for a rather general class of Markov processes ranging from diffusions in energy landscapes to Markov jump processes on large discrete spaces. Furthermore, we discuss how this result provides formal support or shows the limitations of algorithmic strategies that have been found to be useful for the construction of MSMs. Our findings are illustrated by numerical experiments
Kinetic distance and kinetic maps from molecular dynamics simulation
Characterizing macromolecular kinetics from molecular dynamics (MD)
simulations requires a distance metric that can distinguish
slowly-interconverting states. Here we build upon diffusion map theory and
define a kinetic distance for irreducible Markov processes that quantifies how
slowly molecular conformations interconvert. The kinetic distance can be
computed given a model that approximates the eigenvalues and eigenvectors
(reaction coordinates) of the MD Markov operator. Here we employ the
time-lagged independent component analysis (TICA). The TICA components can be
scaled to provide a kinetic map in which the Euclidean distance corresponds to
the kinetic distance. As a result, the question of how many TICA dimensions
should be kept in a dimensionality reduction approach becomes obsolete, and one
parameter less needs to be specified in the kinetic model construction. We
demonstrate the approach using TICA and Markov state model (MSM) analyses for
illustrative models, protein conformation dynamics in bovine pancreatic trypsin
inhibitor and protein-inhibitor association in trypsin and benzamidine
A weak characterization of slow variables in stochastic dynamical systems
We present a novel characterization of slow variables for continuous Markov
processes that provably preserve the slow timescales. These slow variables are
known as reaction coordinates in molecular dynamical applications, where they
play a key role in system analysis and coarse graining. The defining
characteristics of these slow variables is that they parametrize a so-called
transition manifold, a low-dimensional manifold in a certain density function
space that emerges with progressive equilibration of the system's fast
variables. The existence of said manifold was previously predicted for certain
classes of metastable and slow-fast systems. However, in the original work, the
existence of the manifold hinges on the pointwise convergence of the system's
transition density functions towards it. We show in this work that a
convergence in average with respect to the system's stationary measure is
sufficient to yield reaction coordinates with the same key qualities. This
allows one to accurately predict the timescale preservation in systems where
the old theory is not applicable or would give overly pessimistic results.
Moreover, the new characterization is still constructive, in that it allows for
the algorithmic identification of a good slow variable. The improved
characterization, the error prediction and the variable construction are
demonstrated by a small metastable system
Two mathematical tools to analyze metastable stochastic processes
We present how entropy estimates and logarithmic Sobolev inequalities on the
one hand, and the notion of quasi-stationary distribution on the other hand,
are useful tools to analyze metastable overdamped Langevin dynamics, in
particular to quantify the degree of metastability. We discuss the interest of
these approaches to estimate the efficiency of some classical algorithms used
to speed up the sampling, and to evaluate the error introduced by some
coarse-graining procedures. This paper is a summary of a plenary talk given by
the author at the ENUMATH 2011 conference
Adaptive Resolution Simulation in Equilibrium and Beyond
In this paper, we investigate the equilibrium statistical properties of both
the force and potential interpolations of adaptive resolution simulation
(AdResS) under the theoretical framework of grand-canonical like AdResS
(GC-AdResS). The thermodynamic relations between the higher and lower
resolutions are derived by considering the absence of fundamental conservation
laws in mechanics for both branches of AdResS. In order to investigate the
applicability of AdResS method in studying the properties beyond the
equilibrium, we demonstrate the accuracy of AdResS in computing the dynamical
properties in two numerical examples: The velocity auto-correlation of pure
water and the conformational relaxation of alanine dipeptide dissolved in
water. Theoretical and technical open questions of the AdResS method are
discussed in the end of the paper
Data-driven model reduction and transfer operator approximation
In this review paper, we will present different data-driven dimension
reduction techniques for dynamical systems that are based on transfer operator
theory as well as methods to approximate transfer operators and their
eigenvalues, eigenfunctions, and eigenmodes. The goal is to point out
similarities and differences between methods developed independently by the
dynamical systems, fluid dynamics, and molecular dynamics communities such as
time-lagged independent component analysis (TICA), dynamic mode decomposition
(DMD), and their respective generalizations. As a result, extensions and best
practices developed for one particular method can be carried over to other
related methods
Analysis and optimization of weighted ensemble sampling
We give a mathematical framework for weighted ensemble (WE) sampling, a
binning and resampling technique for efficiently computing probabilities in
molecular dynamics. We prove that WE sampling is unbiased in a very general
setting that includes adaptive binning. We show that when WE is used for
stationary calculations in tandem with a coarse model, the coarse model can be
used to optimize the allocation of replicas in the bins.Comment: 22 pages, 3 figure
Soft versus hard metastable conformations in molecular simulations
Particle methods have become indispensible in conformation dynamics to compute transition rates in protein folding, binding processes and molecular design, to mention a few. Conformation dynamics requires at a decomposition of a molecule’s position space into metastable conformations. In this paper, we show how this decomposition can be obtained via the design of either “soft” or “hard” molecular conformations. We show, that the soft approach results in a larger metastabilitiy of the decomposition and is thus more advantegous. This is illustrated by a simulation of Alanine Dipeptide