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PreprintCited by 6Protein Structure and DynamicsAdvanced Physical and Chemical Molecular InteractionsModel Reduction and Neural Networks

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Addressing the Embeddability Problem in Transition Rate Estimation

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Goolsby, Curtis, Losey, James, Xu, Yuchen, Düker, Marie-Christine, Sherman, Mila Getmansky, Matteson, David S., Moradi, Mahmoud·2019·The Journal of Physical Chemistry A
DOI 10.1101/707919PMC PMC10350963

Preprint — published as 10.1021/acs.jpca.3c01367

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Abstract

Markov State Models (MSM) and related techniques have gained significant traction as a tool for analyzing and guiding molecular dynamics (MD) simulations due to their ability to extract structural, thermodynamic, and kinetic information on proteins using computationally feasible MD simulations. The MSM analysis often relies on spectral decomposition of empirically generated transition matrices. Here, we discuss an alternative approach for extracting the thermodynamic and kinetic information from the so-called rate/generator matrix rather than the transition matrix. Although the rate matrix itself is built from the empirical transition matrix, it provides an alternative approach for estimating both thermodynamic and kinetic quantities, particularly in diffusive processes. We particularly discuss a fundamental issue with this approach, known as the embeddability problem and offer ways to address this issue. We describe eight different methods to overcome the embeddability problem, including a novel approach developed for this work. The algorithms were tested on data from a one-dimensional toy model to show the workings of these methods and discuss the robustness of each method in terms of its dependence in lag time and trajectory length.

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Figure 1 Schematic representation of the proposed approach to analyze the MSM data. MD trajectories are first used within a clustering scheme to generate a Markov state model and build an empirical transition matrix using a given lag time; see MSM literature.21 Instead of spectral decomposition of the transition matrix, however, we propose to estimate a lag-time independent rate/generator matrix from the empirical transition matrix obtained from the MD trajectories.
Figure 1 Schematic representation of the proposed approach to analyze the MSM data. MD trajectories are first used within a clustering scheme to generate a Markov state model and build an empirical transition matrix using a given lag time; see MSM literature.21 Instead of spectral decomposition of the transition matrix, however, we propose to estimate a lag-time independent rate/generator matrix from the empirical transition matrix obtained from the MD trajectories.CC BY-NC-ND · PubMed Central
Figure 2 (A–D) Examples of the predicted free energy with a particular random seed as a function of x is shown for each algorithm using the extremes of trajectory length and τ. (A) L = 100% and τ = 1δt, (B) L = 100% and τ = 1δt, (C) L = 0.1% and τ = 1000δt, and (D) L = 0.1% and τ = 1000δt. Algorithms shown are DA (blue triangle), WA (orange ×), QOG (green square), CWO (red circle), EM (purple +), MLE (brown diamond), QP (pink hexagon), and PA (gray Y).
Figure 2 (A–D) Examples of the predicted free energy with a particular random seed as a function of x is shown for each algorithm using the extremes of trajectory length and τ. (A) L = 100% and τ = 1δt, (B) L = 100% and τ = 1δt, (C) L = 0.1% and τ = 1000δt, and (D) L = 0.1% and τ = 1000δt. Algorithms shown are DA (blue triangle), WA (orange ×), QOG (green square), CWO (red circle), EM (purple +), MLE (brown diamond), QP (pink hexagon), and PA (gray Y).CC BY-NC-ND · PubMed Central
Figure 3 Average predicted barrier height as a function of τ for each algorithm, where (A) L = 100% and (B) L = 0.1%. Also, average predicted barrier height as a function of trajectory length for constant τ is shown for each algorithm, where (C) τ = 1δt and (D) τ = 1000δt. The dashed line represents the analytic value for the well barrier of 2.5 (kcal/mol). The graph with error bars is given in Figure S2.
Figure 3 Average predicted barrier height as a function of τ for each algorithm, where (A) L = 100% and (B) L = 0.1%. Also, average predicted barrier height as a function of trajectory length for constant τ is shown for each algorithm, where (C) τ = 1δt and (D) τ = 1000δt. The dashed line represents the analytic value for the well barrier of 2.5 (kcal/mol). The graph with error bars is given in Figure S2.CC BY-NC-ND · PubMed Central
Figure 4 Average well symmetry as a function of τ for trajectory length is shown for each algorithm, where (A) L = 100% and (B) L = 0.1%. Well symmetry as a function of trajectory length for constant τ is shown for each algorithm, where (C) τ = 1δt and (D) τ = 1000δt. The graph with error bars is given in Figure S3.
Figure 4 Average well symmetry as a function of τ for trajectory length is shown for each algorithm, where (A) L = 100% and (B) L = 0.1%. Well symmetry as a function of trajectory length for constant τ is shown for each algorithm, where (C) τ = 1δt and (D) τ = 1000δt. The graph with error bars is given in Figure S3.CC BY-NC-ND · PubMed Central

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Metadata

Authors
Goolsby, C., Losey, J., Xu, Y., Düker, M.-C., Sherman, M. G., Matteson, D. S., & Moradi, M.
Affiliations
  • University of Arkansas at Fayetteville (US)
  • Cornell University (US)
  • University of Massachusetts Amherst (US)
Corresponding author
Moradi, Mahmoud
ORCID
Venue
The Journal of Physical Chemistry A
Publisher
openRxiv
Published
2019-08-02
Volume
127
Issue
27
Pages
5745-5759
ISSN
1089-5639 (ISSN-L)
DOI
10.1101/707919
PMC
PMC10350963
OpenAlex
W4382502848
Cited by
6
References
2
Citations by year
2026: 3 · 2025: 1 · 2021: 1 · 2019: 1
FWCI
0.35
Type
journal article
Language
en
Open access
Licence
cc-by-nc-nd
Locations
  • The Journal of Physical Chemistry A (publishedVersion, cc-by-nc-nd)
  • bioRxiv (Cold Spring Harbor Laboratory) (acceptedVersion)
  • PubMed Central (publishedVersion, cc-by-nc-nd)
  • PubMed (publishedVersion)
  • Journal of the Arkansas Academy of Science (submittedVersion, cc-by-nc-nd)
Funders
  • National Science Foundation (1940276, 1934985, 1945465, 1940188, 1940223, 1940124)
  • Arkansas Economic Development Commission
  • Arkansas High Performance Computing Center, University of Arkansas
  • Division of Chemistry (CHE 1945465, 1945465)
Topics
  • Life Sciences › Biochemistry, Genetics and Molecular Biology › Molecular Biology › Protein Structure and Dynamics
  • Physical Sciences › Chemistry › Physical and Theoretical Chemistry › Advanced Physical and Chemical Molecular Interactions
  • Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Model Reduction and Neural Networks
Keywords
transition rate estimation, Markov state models, rate matrix, generator matrix, transition matrix, spectral decomposition, protein kinetics, thermodynamic properties, diffusive dynamics, time delay, trajectory length

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CoChat. (2026). Citation check: Addressing the Embeddability Problem in Transition Rate Estimation. https://app.cochat.ai/papers/10.1101/707919
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Goolsby, C., Losey, J., Xu, Y., Düker, M.-C., Sherman, M. G., Matteson, D. S., & Moradi, M. (2019). Addressing the Embeddability Problem in Transition Rate Estimation. The Journal of Physical Chemistry A. https://doi.org/10.1101/707919