Optimal dimensionality reduction of Markov chains using graph transformation

Kannan, Deepti; Sharpe, Daniel; Swinburne, Thomas D.; Wales, David J.

doi:10.1063/5.0025174

Cited by 15 publications

(18 citation statements)

References 157 publications

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“…The system corresponds to a coarse-grained representation of an energy landscape, 57 with a discrete set of states connected via energy barriers. The internode transition rates have an Arrhenius form, 116 dependent on the temperature T , 35 and characterize the Markovian network dynamics in terms of branching probabilities 73 and mean waiting times for transitions from nodes. 30 A complete specification of this model system is given in Appendix A.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…77 The rewards associated with trajectories on the censored Markov chain are strictly an expectation with respect to the contributions of path segments that visit censored nodes. 77 Reducing the dimensionality of Markov chains by renormalization provides a strategy to facilitate the sampling of trajectories, 84 and Eq. 3 allows for the probability distributions of path rewards on the transformed network to be estimated within this framework.…”

Section: B Expected Rewards For Individual Paths On Censored Markov Chainsmentioning

confidence: 99%

“…Formally, the expected A ← B reward is a sum of contributions R[ξ] from all paths ξ of the first passage path ensemble 35,57,62…”

Section: Recursive Enumeration Algorithmmentioning

confidence: 99%

“…30 For a DTMC, 3 the fixed lag time associated with transitions is uniform for all nodes, τ j ≡ τ ∀ j. 35 The A ← B mean first passage time (MFPT), 36 T AB , is a sum of path times, taken over all possible first passage paths, weighted by the associated path probabilities. 35 The GT algorithm is numerically stable, and therefore valuable in many practical applications.…”

Section: Introductionmentioning

confidence: 99%

“…35 The A ← B mean first passage time (MFPT), 36 T AB , is a sum of path times, taken over all possible first passage paths, weighted by the associated path probabilities. 35 The GT algorithm is numerically stable, and therefore valuable in many practical applications. 31 Markov chains representing realistic dynamical processes are frequently observed to encompass a separation of characteristic timescales, and the corresponding transition probability or rate matrix is therefore ill-conditioned.…”

Section: Introductionmentioning

confidence: 99%

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Graph transformation and shortest paths algorithms for finite Markov chains

Sharpe

Wales

2021

Phys. Rev. E

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The graph transformation (GT) algorithm robustly computes the mean first passage time to an absorbing state in a finite Markov chain. Here we present a concise overview of the iterative and block formulations of the GT procedure and generalize the GT formalism to the case of any path property that is a sum of contributions from individual transitions. In particular, we examine the path action, which directly relates to the path probability, and analyze the first passage path ensemble for a model Markov chain that is metastable and therefore numerically challenging. We compare the mean first passage path action, obtained using GT, with the full path action probability distribution simulated efficiently using kinetic path sampling, and with values for the highest-probability paths determined by the recursive enumeration algorithm (REA). In Markov chains representing realistic dynamical processes, the probability distributions of first passage path properties are typically fat-tailed and therefore difficult to converge by sampling, which motivates the use of exact and numerically stable approaches to compute the expectation. We find that the kinetic relevance of the set of highest-probability paths depends strongly on the metastability of the Markov chain, and so the properties of the dominant first passage paths may be unrepresentative of the global dynamics. Use of a global measure for edge costs in the REA, based on net productive fluxes, allows the total reactive flux to be decomposed into a finite set of contributions from simple flux-paths. By considering transition flux-paths, a detailed quantitative analysis of the relative importance of competing dynamical processes is possible even in the metastable regime.

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Section: Numerical Resultsmentioning

confidence: 99%

Section: B Expected Rewards For Individual Paths On Censored Markov Chainsmentioning

confidence: 99%

“…Formally, the expected A ← B reward is a sum of contributions R[ξ] from all paths ξ of the first passage path ensemble 35,57,62…”

Section: Recursive Enumeration Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Graph transformation and shortest paths algorithms for finite Markov chains

Sharpe

Wales

2021

Phys. Rev. E

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Multiscale kinetic analysis of proteins

Swanson

2022

Current Opinion in Structural Biology

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Size-and-Shape Space Gaussian Mixture Models for Structural Clustering of Molecular Dynamics Trajectories

Klem

Hocky

McCullagh

2022

J. Chem. Theory Comput.

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Determining the optimal number and identity of structural clusters from an ensemble of molecular configurations continues to be a challenge. Recent structural clustering methods have focused on the use of internal coordinates due to the innate rotational and translational invariance of these features. The vast number of possible internal coordinates necessitates a feature space supervision step to make clustering tractable but yields a protocol that can be system type-specific. Particle positions offer an appealing alternative to internal coordinates but suffer from a lack of rotational and translational invariance, as well as a perceived insensitivity to regions of structural dissimilarity. Here, we present a method, denoted shape-GMM, that overcomes the shortcomings of particle positions using a weighted maximum likelihood alignment procedure. This alignment strategy is then built into an expectation maximization Gaussian mixture model (GMM) procedure to capture metastable states in the free-energy landscape. The resulting algorithm distinguishes between a variety of different structures, including those indistinguishable by root-mean-square displacement and pairwise distances, as demonstrated on several model systems. Shape-GMM results on an extensive simulation of the fast-folding HP35 Nle/Nle mutant protein support a four-state folding/unfolding mechanism, which is consistent with previous experimental results and provides kinetic details comparable to previous state-of-the art clustering approaches, as measured by the VAMP-2 score. Currently, training of shape-GMMs is recommended for systems (or subsystems) that can be represented by ≲200 particles and ≲100k configurations to estimate high-dimensional covariance matrices and balance computational expense. Once a shape-GMM is trained, it can be used to predict the cluster identities of millions of configurations.

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Optimal dimensionality reduction of Markov chains using graph transformation

Cited by 15 publications

References 157 publications

Graph transformation and shortest paths algorithms for finite Markov chains

Graph transformation and shortest paths algorithms for finite Markov chains

Multiscale kinetic analysis of proteins

Size-and-Shape Space Gaussian Mixture Models for Structural Clustering of Molecular Dynamics Trajectories

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