65C40 Computational Markov chains
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- Conformation Dynamics (1)
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- Rebinding (1)
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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 dynamic behavior of molecules can often be described by Markov processes. From computational molecular simulations one can derive transition rates or transition probabilities between subsets of the discretized conformational space. On the basis of this dynamic information, the spatial subsets are combined into a small number of so-called metastable molecular conformations. This is done by clustering methods like the Robust Perron Cluster Analysis (PCCA+). Up to now it is an open question how this coarse graining in space can be transformed to a coarse graining of the Markov chain while preserving the essential dynamic information. In the following article we aim at a consistent coarse graining of transition probabilities or rates on the basis of metastable conformations such that important physical and mathematical relations are preserved. This approach is new because PCCA+ computes molecular conformations as linear combinations of the dominant eigenvectors of the transition matrix which does not hold for other clustering methods.
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.
In this article we aim at an efficient sampling of the stationary distribution of dynamical systems in the presence of metastabilities. In the past decade many sophisticated algorithms have been inven ted in this field. We do not want to simply add a further one. We address the problem that one has applied a sampling algorithm for a dynamical system many times. This leads to different samplings which more or less represent the stationary distribution partially very well, but which are still far away from ergodicity or from the global stationary distribution. We will show how these samplings can be joined together in order to get one global sampling of the stationary distribution.