65C40 Computational Markov chains
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Finding metastable sets as dominant structures of Markov processes has been shown to be especially useful in modeling interesting slow dynamics of various real world complex processes. Furthermore, coarse graining of such processes based on their dominant structures leads to better understanding and dimension reduction of observed systems. However, in many cases, e.g. for nonreversible Markov processes, dominant structures are often not formed by metastable sets but by important cycles or mixture of both. This paper aims at understanding and identifying these different types of dominant structures for reversible as well as nonreversible ergodic Markov processes. Our algorithmic approach generalizes spectral based methods for reversible process by using Schur decomposition techniques which can tackle also nonreversible cases. We illustrate the mathematical construction of our new approach by numerical experiments.
The enormous time lag between fast atomic motion and complex pro- tein folding events makes it almost impossible to compute molecular dy- namics on a high resolution. A common way to tackle this problem is to model the system dynamics as a Markov process. Yet for large molec- ular systems the resulting Markov chains can hardly be handled due to the curse of dimensionality. Coarse graining methods can be used to re- duce the dimension of a Markov chain, but it is still unclear how far the coarse grained Markov chain resembles the original system. In order to answer this question, two different coarse-graining methods were analysed and compared: a classical set-based reduction method and an alternative subspace-based approach, which is based on membership vectors instead of sets. On the basis of a small toy system, it could be shown, that in con- trast to the subset-based approach, the subspace-based reduction method preserves the Markov property as well as the essential dynamics of the original system.
Gene Regulatory Networks are powerful models for describing the mechanisms and dynamics inside a cell. These networks are generally large in dimension and seldom yield analytical formulations. It was shown that studying the conditional expectations between dimensions (vertices or species) of a network could lead to drastic dimension reduction. These conditional expectations were classically given by solving equations of motions derived from the Chemical Master Equation. In this paper we deviate from this convention and take an Algebraic approach instead. That is, we explore the consequences of conditional expectations being described by a polynomial function. There are two main results in this work. Firstly: if the conditional expectation can be described by a polynomial function, then coefficients of this polynomial function can be reconstructed using the classical moments. And secondly: there are dimensions in Gene Regulatory Networks which inherently have conditional expectations with algebraic forms. We demonstrate through examples, that the theory derived in this work can be used to develop new and effective numerical schemes for forward simulation and parameter inference. The algebraic line of investigation of conditional expectations has considerable scope to be applied to many different aspects of Gene Regulatory Networks; this paper serves as a preliminary commentary in this direction.
Recent years have seen an increased interest in non-equilibrium molecular dynamics (NEMD) simulations, especially for molecular systems with periodic forcing by external fields, e.g., in the context of studying effects of electromagnetic radiation on the human body tissue. Lately, an NEMD methods with local thermostating has been proposed that allows for studying non-equilibrium processes in a statistically reliable and thermodynamically consistent way. In this article, we demonstrate how to construct Markov State Models (MSMs) for such NEMD simulations. MSM building has been well-established for systems in equilibrium where MSMs with just a few (macro-)states allow for accurate reproduction of the essential kinetics of the molecular system under consideration. Non-equilibrium MSMs have been lacking so far. The article presents how to construct such MSMs and illustrates their validity and usefulness for the case of conformation dynamics of alanine dipeptide in an external electric field.
The rebinding effect is a phenomenon which occurs when observing a ligand-receptor binding process.
On the macro scale this process comprises the Markov property.
This Makovian view is spoiled when switching to the atomistic scale of a binding process.
We therefore suggest a model which accurately describes the rebinding effect on the atomistic scale by allowing ''intermediate'' bound states.
This allows us to define an indicator for the magnitude of rebinding and to formulate an optimization problem.
The results form our examples show good agreement with data form laboratory.