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Costa AC, Ahamed T, Jordan D, Stephens GJ. A Markovian dynamics for Caenorhabditis elegans behavior across scales. Proc Natl Acad Sci U S A 2024; 121:e2318805121. [PMID: 39083417 PMCID: PMC11317559 DOI: 10.1073/pnas.2318805121] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/02/2023] [Accepted: 07/01/2024] [Indexed: 08/02/2024] Open
Abstract
How do we capture the breadth of behavior in animal movement, from rapid body twitches to aging? Using high-resolution videos of the nematode worm Caenorhabditis elegans, we show that a single dynamics connects posture-scale fluctuations with trajectory diffusion and longer-lived behavioral states. We take short posture sequences as an instantaneous behavioral measure, fixing the sequence length for maximal prediction. Within the space of posture sequences, we construct a fine-scale, maximum entropy partition so that transitions among microstates define a high-fidelity Markov model, which we also use as a means of principled coarse-graining. We translate these dynamics into movement using resistive force theory, capturing the statistical properties of foraging trajectories. Predictive across scales, we leverage the longest-lived eigenvectors of the inferred Markov chain to perform a top-down subdivision of the worm's foraging behavior, revealing both "runs-and-pirouettes" as well as previously uncharacterized finer-scale behaviors. We use our model to investigate the relevance of these fine-scale behaviors for foraging success, recovering a trade-off between local and global search strategies.
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Affiliation(s)
- Antonio C. Costa
- Department of Physics and Astronomy, Vrije Universiteit Amsterdam, Amsterdam1081HV, The Netherlands
| | | | - David Jordan
- Department of Biochemistry, University of Cambridge, CambridgeCB2 1GA, United Kingdom
| | - Greg J. Stephens
- Department of Physics and Astronomy, Vrije Universiteit Amsterdam, Amsterdam1081HV, The Netherlands
- Biological Physics Theory Unit, Okinawa Institute of Science and Technology Graduate University, Okinawa904-0495, Japan
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Costa AC, Ahamed T, Jordan D, Stephens GJ. Maximally predictive states: From partial observations to long timescales. CHAOS (WOODBURY, N.Y.) 2023; 33:023136. [PMID: 36859220 DOI: 10.1063/5.0129398] [Citation(s) in RCA: 5] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/04/2022] [Accepted: 01/31/2023] [Indexed: 06/18/2023]
Abstract
Isolating slower dynamics from fast fluctuations has proven remarkably powerful, but how do we proceed from partial observations of dynamical systems for which we lack underlying equations? Here, we construct maximally predictive states by concatenating measurements in time, partitioning the resulting sequences using maximum entropy, and choosing the sequence length to maximize short-time predictive information. Transitions between these states yield a simple approximation of the transfer operator, which we use to reveal timescale separation and long-lived collective modes through the operator spectrum. Applicable to both deterministic and stochastic processes, we illustrate our approach through partial observations of the Lorenz system and the stochastic dynamics of a particle in a double-well potential. We use our transfer operator approach to provide a new estimator of the Kolmogorov-Sinai entropy, which we demonstrate in discrete and continuous-time systems, as well as the movement behavior of the nematode worm C. elegans.
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Affiliation(s)
- Antonio C Costa
- Department of Physics and Astronomy, Vrije Universiteit Amsterdam, 1081HV Amsterdam, The Netherlands
| | - Tosif Ahamed
- Lunenfeld-Tanenbaum Research Institute, Mount Sinai Hospital, Toronto, Ontario M5G 1X5, Canada
| | - David Jordan
- Wellcome/CRUK Gurdon Institute, University of Cambridge, Cambridge CB2 1QN, United Kingdom
| | - Greg J Stephens
- Department of Physics and Astronomy, Vrije Universiteit Amsterdam, 1081HV Amsterdam, The Netherlands
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Sikorski A, Weber M, Schütte C. The Augmented Jump Chain. ADVANCED THEORY AND SIMULATIONS 2021. [DOI: 10.1002/adts.202000274] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
Affiliation(s)
- Alexander Sikorski
- Zuse Institute Berlin Mathematics for Life and Materials Sciences Takustr. 7 D‐14195 Berlin Germany
| | - Marcus Weber
- Zuse Institute Berlin Mathematics for Life and Materials Sciences Takustr. 7 D‐14195 Berlin Germany
| | - Christof Schütte
- Zuse Institute Berlin Mathematics for Life and Materials Sciences Takustr. 7 D‐14195 Berlin Germany
- Freie Universität Berlin Department of Mathematics and Computer Science Biocomputing Group Arnimallee 6 D‐14195 Berlin Germany
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Abstract
Machine learning (ML) is transforming all areas of science. The complex and time-consuming calculations in molecular simulations are particularly suitable for an ML revolution and have already been profoundly affected by the application of existing ML methods. Here we review recent ML methods for molecular simulation, with particular focus on (deep) neural networks for the prediction of quantum-mechanical energies and forces, on coarse-grained molecular dynamics, on the extraction of free energy surfaces and kinetics, and on generative network approaches to sample molecular equilibrium structures and compute thermodynamics. To explain these methods and illustrate open methodological problems, we review some important principles of molecular physics and describe how they can be incorporated into ML structures. Finally, we identify and describe a list of open challenges for the interface between ML and molecular simulation.
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Affiliation(s)
- Frank Noé
- Department of Mathematics and Computer Science, Freie Universität Berlin, 14195 Berlin, Germany; .,Department of Physics, Freie Universität Berlin, 14195 Berlin, Germany.,Department of Chemistry and Center for Theoretical Biological Physics, Rice University, Houston, Texas 77005, USA;
| | - Alexandre Tkatchenko
- Physics and Materials Science Research Unit, University of Luxembourg, 1511 Luxembourg, Luxembourg;
| | - Klaus-Robert Müller
- Department of Computer Science, Technical University Berlin, 10587 Berlin, Germany; .,Max-Planck-Institut für Informatik, 66123 Saarbrücken, Germany.,Department of Brain and Cognitive Engineering, Korea University, Seoul 136-713, South Korea
| | - Cecilia Clementi
- Department of Mathematics and Computer Science, Freie Universität Berlin, 14195 Berlin, Germany; .,Department of Chemistry and Center for Theoretical Biological Physics, Rice University, Houston, Texas 77005, USA; .,Department of Physics, Rice University, Houston, Texas 77005, USA
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Noé F. Machine Learning for Molecular Dynamics on Long Timescales. MACHINE LEARNING MEETS QUANTUM PHYSICS 2020. [DOI: 10.1007/978-3-030-40245-7_16] [Citation(s) in RCA: 13] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/21/2022]
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Fackeldey K, Koltai P, Névir P, Rust H, Schild A, Weber M. From metastable to coherent sets- Time-discretization schemes. CHAOS (WOODBURY, N.Y.) 2019; 29:012101. [PMID: 30709154 DOI: 10.1063/1.5058128] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/17/2018] [Accepted: 01/07/2019] [Indexed: 06/09/2023]
Abstract
Given a time-dependent stochastic process with trajectories x(t) in a space Ω, there may be sets such that the corresponding trajectories only very rarely cross the boundaries of these sets. We can analyze such a process in terms of metastability or coherence. Metastable setsM are defined in space M⊂Ω, and coherent setsM(t)⊂Ω are defined in space and time. Hence, if we extend the space Ω by the time-variable t, coherent sets are metastable sets in Ω×[0,∞) of an appropriate space-time process. This relation can be exploited, because there already exist spectral algorithms for the identification of metastable sets. In this article, we show that these well-established spectral algorithms (like PCCA+, Perron Cluster Cluster Analysis) also identify coherent sets of non-autonomous dynamical systems. For the identification of coherent sets, one has to compute a discretization (a matrix T) of the transfer operator of the process using a space-time-discretization scheme. The article gives an overview about different time-discretization schemes and shows their applicability in two different fields of application.
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Affiliation(s)
- Konstantin Fackeldey
- Institut für Mathematik, TU Berlin, Straße des 17, Juni 136, 10623 Berlin, Germany
| | - Péter Koltai
- Institut für Mathematik, FU Berlin, Arnimallee 6, 14195 Berlin, Germany
| | - Peter Névir
- Institut für Meteorologie, Freie Universität Berlin, Carl-Heinrich-Becker-Weg 6-10, 12165 Berlin, Germany
| | - Henning Rust
- Institut für Meteorologie, Freie Universität Berlin, Carl-Heinrich-Becker-Weg 6-10, 12165 Berlin, Germany
| | - Axel Schild
- Laboratorium für Physikalische Chemie, ETH Zürich, Vladimir-Prelog-Weg 2, 8093 Zürich, Switzerland
| | - Marcus Weber
- Zuse Institute Berlin (ZIB), Takustrasse 7, 14195 Berlin, Germany
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Koltai P, Renger DRM. From Large Deviations to Semidistances of Transport and Mixing: Coherence Analysis for Finite Lagrangian Data. JOURNAL OF NONLINEAR SCIENCE 2018; 28:1915-1957. [PMID: 30220792 PMCID: PMC6132839 DOI: 10.1007/s00332-018-9471-0] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 09/08/2017] [Accepted: 05/19/2018] [Indexed: 06/08/2023]
Abstract
One way to analyze complicated non-autonomous flows is through trying to understand their transport behavior. In a quantitative, set-oriented approach to transport and mixing, finite time coherent sets play an important role. These are time-parametrized families of sets with unlikely transport to and from their surroundings under small or vanishing random perturbations of the dynamics. Here we propose, as a measure of transport and mixing for purely advective (i.e., deterministic) flows, (semi)distances that arise under vanishing perturbations in the sense of large deviations. Analogously, for given finite Lagrangian trajectory data we derive a discrete-time-and-space semidistance that comes from the "best" approximation of the randomly perturbed process conditioned on this limited information of the deterministic flow. It can be computed as shortest path in a graph with time-dependent weights. Furthermore, we argue that coherent sets are regions of maximal farness in terms of transport and mixing, and hence they occur as extremal regions on a spanning structure of the state space under this semidistance-in fact, under any distance measure arising from the physical notion of transport. Based on this notion, we develop a tool to analyze the state space (or the finite trajectory data at hand) and identify coherent regions. We validate our approach on idealized prototypical examples and well-studied standard cases.
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Affiliation(s)
- Péter Koltai
- Institute of Mathematics, Freie Universität Berlin, Arnimallee 6, 14195 Berlin, Germany
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Reuter B, Weber M, Fackeldey K, Röblitz S, Garcia ME. Generalized Markov State Modeling Method for Nonequilibrium Biomolecular Dynamics: Exemplified on Amyloid β Conformational Dynamics Driven by an Oscillating Electric Field. J Chem Theory Comput 2018; 14:3579-3594. [DOI: 10.1021/acs.jctc.8b00079] [Citation(s) in RCA: 15] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/17/2023]
Affiliation(s)
- Bernhard Reuter
- University of Kassel, Institute of Physics, Theoretical Physics II, Heinrich-Plett-Str. 40, 34132 Kassel, Germany
- Zuse Institute Berlin (ZIB), Takustraße 7, 14195 Berlin, Germany
| | - Marcus Weber
- Zuse Institute Berlin (ZIB), Takustraße 7, 14195 Berlin, Germany
| | - Konstantin Fackeldey
- Zuse Institute Berlin (ZIB), Takustraße 7, 14195 Berlin, Germany
- Institute of Mathematics, Technical University Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany
| | - Susanna Röblitz
- Zuse Institute Berlin (ZIB), Takustraße 7, 14195 Berlin, Germany
| | - Martin E. Garcia
- University of Kassel, Institute of Physics, Theoretical Physics II, Heinrich-Plett-Str. 40, 34132 Kassel, Germany
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Optimal Data-Driven Estimation of Generalized Markov State Models for Non-Equilibrium Dynamics. COMPUTATION 2018. [DOI: 10.3390/computation6010022] [Citation(s) in RCA: 16] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/01/2023]
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Affiliation(s)
- Brooke E. Husic
- Department of Chemistry, Stanford University, Stanford, California 94305, United States
| | - Vijay S. Pande
- Department of Chemistry, Stanford University, Stanford, California 94305, United States
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Knoch F, Speck T. Nonequilibrium Markov state modeling of the globule-stretch transition. Phys Rev E 2017; 95:012503. [PMID: 28208388 DOI: 10.1103/physreve.95.012503] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/24/2016] [Indexed: 06/06/2023]
Abstract
We describe a systematic approach to construct coarse-grained Markov state models from molecular dynamics data of systems driven into a nonequilibrium steady state. We apply this method to study the globule-stretch transition of a single tethered model polymer in shear flow. The folding and unfolding rates of the coarse-grained model agree with the original detailed model. We demonstrate that the folding and unfolding proceeds through the same narrow region of configuration space but along different cycles.
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Affiliation(s)
- Fabian Knoch
- Institut für Physik, Johannes Gutenberg-Universität Mainz, Staudingerweg 7-9, 55128 Mainz, Germany
| | - Thomas Speck
- Institut für Physik, Johannes Gutenberg-Universität Mainz, Staudingerweg 7-9, 55128 Mainz, Germany
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