Identification of almost invariant aggregates in reversible nearly uncoupled Markov chains

Deuflhard, Peter; Huisinga, Wilhelm; Fischer, Alexander; Schütte, Christof

doi:10.1016/s0024-3795(00)00095-1

Cited by 304 publications

(388 citation statements)

References 10 publications

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“…While some mathematicians, e.g. [15] or [8], are aware of the theory of Ando and Simon [22], it does not seem to be well known in the probability theory community. On the other hand, the mathematical theory of metastability following [5] is virtually unknown in the applied community, and the approaches by Young [29] and Wicks and Greenwald [28] appear to be completely disjoint from the others.…”

Section: Introductionmentioning

confidence: 99%

Multi-scale metastable dynamics and the asymptotic stationary distribution of perturbed Markov chains

Betz

Roux

2016

Stochastic Processes and their Applications

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Abstract. We consider a simple but important class of metastable discrete time Markov chains, which we call perturbed Markov chains. Basically, we assume that the transition matrices depend on a parameter ε, and converge as ε → 0. We further assume that the chain is irreducible for ε > 0 but may have several essential communicating classes when ε = 0. This leads to metastable behavior, possibly on multiple time scales. For each of the relevant time scales, we derive two effective chains. The first one describes the (possibly irreversible) metastable dynamics, while the second one is reversible and describes metastable escape probabilities. Closed probabilistic expressions are given for the asymptotic transition probabilities of these chains, but we also show how to compute them in a fast and numerically stable way. As a consequence, we obtain efficient algorithms for computing the committor function and the limiting stationary distribution.

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Section: Introductionmentioning

confidence: 99%

Multi-scale metastable dynamics and the asymptotic stationary distribution of perturbed Markov chains

Betz

Roux

2016

Stochastic Processes and their Applications

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“…The implied timescales should converge to an unchanged rate when Markovian behaviour is achieved, which corresponds to 4 ns for NtrC ( Supplementary Fig. 1) 56 . The first and second eigenvectors of the transition probability matrix T ij provide, respectively, the equilibrium conformational state populations and the slowest dynamical process on the conformational landscape.…”

Section: Methodsmentioning

confidence: 97%

A network of molecular switches controls the activation of the two-component response regulator NtrC

Vanatta

Shukla²,

Lawrenz

et al. 2015

Nat Commun

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Recent successes in simulating protein structure and folding dynamics have demonstrated the power of molecular dynamics to predict the long timescale behaviour of proteins. Here, we extend and improve these methods to predict molecular switches that characterize conformational change pathways between the active and inactive state of nitrogen regulatory protein C (NtrC). By employing unbiased Markov state model-based molecular dynamics simulations, we construct a dynamic picture of the activation pathways of this key bacterial signalling protein that is consistent with experimental observations and predicts new mutants that could be used for validation of the mechanism. Moreover, these results suggest a novel mechanistic paradigm for conformational switching.

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“…In other words, data points belonging to the same cluster are represented with exactly the same value in the eigenvectors. This property can be interpreted as a transformation from the original input space to a k-dimensional space spanned by the eigenvectors where the clusters are more evident and easier to identify (Meila and Shi, 2001;Deuflhard et al, 2000). Now, we establish the link between the spectral properties of P and the spectral properties of D −1 M D Ω.…”

Section: Choosing V and N Ementioning

confidence: 94%

Hierarchical kernel spectral clustering

Alzate

Suykens

2012

Neural Networks

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A methodology to reveal the underlying cluster hierarchy in kernel spectral clustering problems is presented. The formulation fits in a constrained optimization framework where the primal problem is expressed in terms of highdimensional feature maps and the dual problem is expressed in terms of kernel evaluations. Kernel spectral clustering can be seen a weighted form of kernel PCA where projections onto the eigenvectors constitute the clustering model. The kernel function acts here as the pairwise similarity function in classical spectral clustering. The clustering model can be evaluated at any data point allowing out-of-sample extensions which are useful for model selection in a learning setting. The parameters of the clustering problem can be found by optimizing the Fisher criterion on estimated eigenvectors for out-of-sample data. These eigenvectors display a localized structure when the clusters are wellformed. The Fisher criterion can also be used to reveal the hierarchical structure of the data. Simulations with toy data and image segmentation problems show the benefits of the proposed approach.

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Identification of almost invariant aggregates in reversible nearly uncoupled Markov chains

Cited by 304 publications

References 10 publications

Multi-scale metastable dynamics and the asymptotic stationary distribution of perturbed Markov chains

Multi-scale metastable dynamics and the asymptotic stationary distribution of perturbed Markov chains

A network of molecular switches controls the activation of the two-component response regulator NtrC

Hierarchical kernel spectral clustering

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