Transition Path Theory for Markov Jump Processes

Metzner, Philipp; Schütte, Christof; Vanden‐Eijnden, Eric

doi:10.1137/070699500

Cited by 450 publications

(612 citation statements)

References 23 publications

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“…[9] (which is the same as the quantity f AB ij − f AB ji in Ref. [37]). The reactive current F R ij is conserved at every node of the network [37] except for the specially designated subsets of source states A and sink states B.…”

Section: Eigencurrentsmentioning

confidence: 99%

“…For example, the reactive current is one of the key concepts of the Transition Path Theory [15,16,37,9]. In the context of spectral analysis, E. Vanden-Eijnden proposed to consider eigencurrents [50,9].…”

Section: Eigencurrentsmentioning

confidence: 99%

“…It is instructive to compare the key concepts of the spectral analysis with those of the Transition Path Theory (TPT) [15,16,37,35]. We will discuss similarities and differences between the escape process quantified by the left and right eigenvectors, the eigenvalue, and the eigencurrent and the Transition Path Process [35,9] described by the committor, the transition rate, and the reactive current.…”

mentioning

confidence: 99%

“…Section 4 is devoted to the application to the LJ 38 network. The analyses of the LJ 38 network by means of the proposed spectral approach and the Transition Path Theory [15,16,37,35,9] are compared in Section 5. Metastability issues and a connection with experimental works are discussed in Section 6.…”

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confidence: 99%

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Metastability, spectrum, and eigencurrents of the Lennard-Jones-38 network

Cameron

2014

The Journal of Chemical Physics

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We develop computational tools for spectral analysis of stochastic networks representing energy landscapes of atomic and molecular clusters. Physical meaning and some properties of eigenvalues, left and right eigenvectors, and eigencurrents are discussed. We propose an approach to compute a collection of eigenpairs and corresponding eigencurrents describing the most important relaxation processes taking place in the system on its way to the equilibrium. It is suitable for large and complex stochastic networks where pairwise transition rates, given by the Arrhenius law, vary by orders of magnitude. The proposed methodology is applied to the network representing the Lennard-Jones-38 cluster created by Wales's group. Its energy landscape has a double funnel structure with a deep and narrow face-centered cubic funnel and a shallower and wider icosahedral funnel. Contrary to the expectations, there is no appreciable spectral gap separating the eigenvalue corresponding to the escape from the icosahedral funnel. We provide a detailed description of the escape process from the icosahedral funnel using the eigencurrent and demonstrate a superexponential growth of the corresponding eigenvalue. The proposed spectral approach is compared to the methodology of the Transition Path Theory. Finally, we discuss whether the Lennard-Jones-38 cluster is metastable from the points of view of a mathematician and a chemical physicist, and make a connection with experimental works. Modeling by means of stochastic networks has become widespread in chemical physics. Efficient methods for the conversion of energy landscapes into stochastic networks have been developed. Wales's group constructed a large number of networks representing energy landscapes of atomic and molecular clusters and proteins [55,58,53]. A number of researchers developed efficient techniques for constructing continuous-time Markov chains representing coarse grained dynamics of biomolecules and protein folding called Markov State Models [44,47,20,10,39,40,43,46]. Stochastic networks are attractive for analysis. On one hand, they are more mathematically tractable than the original continuous systems. On the other hand, they are designed to preserve important features of the underlying systems. Nevertheless, analysis of large and complex networks still presents a challenge. The numbers of states (vertices) and edges can be of the order of 10 n , n = 3, 4, 5, 6, ..., and the pairwise transition rates may vary by tens of orders of magnitude. The study of such networks motivates development of new techniques and leads to a new level of understanding of complex systems.In this work, we focus on developing a methodology for computing eigenvalues, eigenvectors, and eigencurrents for networks representing energy landscapes whose pairwise transition rates are given by the Arrhenius law. The spectral decomposition of the generator matrix of a

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Section: Eigencurrentsmentioning

confidence: 99%

Section: Eigencurrentsmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Metastability, spectrum, and eigencurrents of the Lennard-Jones-38 network

Cameron

2014

The Journal of Chemical Physics

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“…Interestingly, these four metastable states obtained from hidden MSM correspond to the energy minima basin (ms1-4) obtained using REMD data. Furthermore, we used transition path theory (TPT) [53][54][55] as implemented in PyEMMA 56 package to identify the folding paths from unfolded state. The TPT analysis suggests that transition from unfolded state (ms1) to folded state (ms4) is more preferable through intermediate ms3 than ms2 (helical state).…”

Section: Markov State Model Along Optimized Dimensions Kineticallymentioning

confidence: 99%

Assessment and Optimization of Collective Variables for Protein Conformational Landscape: GB1 β-hairpin as a Case Study

Ahalawat

Mondal

2018

Preprint

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Collective variables (CV), when chosen judiciously, can play an important role in recognizing rate-limiting processes and rare events in any biomolecular systems. However, high dimensionality and inherent complexities associated with such biochemical systems render the identification of an optimal CV a challenging task, which in turn precludes the elucidation of underlying conformational landscape in su cient details. In this context, a relevant model system is presented by 16-residue -hairpin of GB1 protein. Despite being the target of numerous theoretical and computational studies for understanding the protein folding, the set of CVs optimally characterizing the conformational landscape of -hairpin of GB1 protein has remained elusive, resulting in a lack of consensus on its folding mechanism. Here we address this by proposing a pair of optimal CVs which can resolve the underlying free energy landscape of GB1 hairpin quite e ciently. Expressed as a linear combination of a number of traditional CVs, the optimal CV for this system is derived by employing recently introduced Time-structured Independent Component Analysis (TICA) approach on a large number of independent unbiased simulations. By projecting the replica-exchange simulated trajectories along these pair of optimized CVs, the resulting free energy landscape of this system are able to resolve four distinct well-separated metastable states encompassing the extensive ensembles of folded,unfolded and molten globule states. Importantly, the optimized CVs were found to be capable of automatically recovering a novel partial helical state of this protein, without needing to explicitly invoke helicity as a constituent CV. Furthermore, a quantitative sensitivity analysis of each constituent in the optimized CV provided key insights on the relative contributions of the constituent CVs in the overall free energy landscapes. Finally, the kinetic pathways connecting these metastable states, constructed using a Markov State Model, provide an optimum description of underlying folding mechanism of the peptide. Taken together, this work o↵ers a quantitatively robust approach towards comprehensive mapping of the underlying folding landscape of a quintessential model system along its optimized collective variables.

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Markov State Models of Protein–Protein Encounters

Olsson

2022

Protein Interactions

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Transition Path Theory for Markov Jump Processes

Cited by 450 publications

References 23 publications

Metastability, spectrum, and eigencurrents of the Lennard-Jones-38 network

Metastability, spectrum, and eigencurrents of the Lennard-Jones-38 network

Assessment and Optimization of Collective Variables for Protein Conformational Landscape: GB1 β-hairpin as a Case Study

Markov State Models of Protein–Protein Encounters

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