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Zhang Y, Isaacson SA. Detailed balance for particle models of reversible reactions in bounded domains. J Chem Phys 2022; 156:204105. [DOI: 10.1063/5.0085296] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
In particle-based stochastic reaction–diffusion models, reaction rates and placement kernels are used to decide the probability per time a reaction can occur between reactant particles and to decide where product particles should be placed. When choosing kernels to use in reversible reactions, a key constraint is to ensure that detailed balance of spatial reaction fluxes holds at all points at equilibrium. In this work, we formulate a general partial-integral differential equation model that encompasses several of the commonly used contact reactivity (e.g., Smoluchowski-Collins-Kimball) and volume reactivity (e.g., Doi) particle models. From these equations, we derive a detailed balance condition for the reversible A + B ⇆ C reaction. In bounded domains with no-flux boundary conditions, when choosing unbinding kernels consistent with several commonly used binding kernels, we show that preserving detailed balance of spatial reaction fluxes at all points requires spatially varying unbinding rate functions near the domain boundary. Brownian dynamics simulation algorithms can realize such varying rates through ignoring domain boundaries during unbinding and rejecting unbinding events that result in product particles being placed outside the domain.
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Affiliation(s)
- Ying Zhang
- Department of Mathematics, Brandeis University, Waltham, Massachusetts 02453, USA
| | - Samuel A. Isaacson
- Department of Mathematics and Statistics, Boston University, Boston, Massachusetts 02215, USA
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Puchhammer F, Ben Abdellah A, L'Ecuyer P. Variance Reduction with Array-RQMC for Tau-Leaping Simulation of Stochastic Biological and Chemical Reaction Networks. Bull Math Biol 2021; 83:91. [PMID: 34236503 DOI: 10.1007/s11538-021-00920-5] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/01/2020] [Accepted: 06/09/2021] [Indexed: 10/20/2022]
Abstract
We explore the use of Array-RQMC, a randomized quasi-Monte Carlo method designed for the simulation of Markov chains, to reduce the variance when simulating stochastic biological or chemical reaction networks with [Formula: see text]-leaping. The task is to estimate the expectation of a function of molecule copy numbers at a given future time T by the sample average over n sample paths, and the goal is to reduce the variance of this sample-average estimator. We find that when the method is properly applied, variance reductions by factors in the thousands can be obtained. These factors are much larger than those observed previously by other authors who tried RQMC methods for the same examples. Array-RQMC simulates an array of realizations of the Markov chain and requires a sorting function to reorder these chains according to their states, after each step. The choice of sorting function is a key ingredient for the efficiency of the method, although in our experiments, Array-RQMC was never worse than ordinary Monte Carlo, regardless of the sorting method. The expected number of reactions of each type per step also has an impact on the efficiency gain.
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Affiliation(s)
- Florian Puchhammer
- Basque Center for Applied Mathematics, Alameda de Mazarredo 14, 48009, Bilbao, Basque Country, Spain.
- DIRO, Université de Montréal, C.P. 6128, Succ. Centre-Ville, Montréal, H3C 3J7, Canada.
| | - Amal Ben Abdellah
- DIRO, Université de Montréal, C.P. 6128, Succ. Centre-Ville, Montréal, H3C 3J7, Canada
| | - Pierre L'Ecuyer
- DIRO, Université de Montréal, C.P. 6128, Succ. Centre-Ville, Montréal, H3C 3J7, Canada
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Earnest TM, Cole JA, Luthey-Schulten Z. Simulating biological processes: stochastic physics from whole cells to colonies. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2018; 81:052601. [PMID: 29424367 DOI: 10.1088/1361-6633/aaae2c] [Citation(s) in RCA: 21] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
The last few decades have revealed the living cell to be a crowded spatially heterogeneous space teeming with biomolecules whose concentrations and activities are governed by intrinsically random forces. It is from this randomness, however, that a vast array of precisely timed and intricately coordinated biological functions emerge that give rise to the complex forms and behaviors we see in the biosphere around us. This seemingly paradoxical nature of life has drawn the interest of an increasing number of physicists, and recent years have seen stochastic modeling grow into a major subdiscipline within biological physics. Here we review some of the major advances that have shaped our understanding of stochasticity in biology. We begin with some historical context, outlining a string of important experimental results that motivated the development of stochastic modeling. We then embark upon a fairly rigorous treatment of the simulation methods that are currently available for the treatment of stochastic biological models, with an eye toward comparing and contrasting their realms of applicability, and the care that must be taken when parameterizing them. Following that, we describe how stochasticity impacts several key biological functions, including transcription, translation, ribosome biogenesis, chromosome replication, and metabolism, before considering how the functions may be coupled into a comprehensive model of a 'minimal cell'. Finally, we close with our expectation for the future of the field, focusing on how mesoscopic stochastic methods may be augmented with atomic-scale molecular modeling approaches in order to understand life across a range of length and time scales.
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Affiliation(s)
- Tyler M Earnest
- Department of Chemistry, University of Illinois, Urbana, IL, 61801, United States of America. National Center for Supercomputing Applications, University of Illinois, Urbana, IL, 61801, United States of America
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Hepburn I, Chen W, De Schutter E. Accurate reaction-diffusion operator splitting on tetrahedral meshes for parallel stochastic molecular simulations. J Chem Phys 2017; 145:054118. [PMID: 27497550 DOI: 10.1063/1.4960034] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Spatial stochastic molecular simulations in biology are limited by the intense computation required to track molecules in space either in a discrete time or discrete space framework, which has led to the development of parallel methods that can take advantage of the power of modern supercomputers in recent years. We systematically test suggested components of stochastic reaction-diffusion operator splitting in the literature and discuss their effects on accuracy. We introduce an operator splitting implementation for irregular meshes that enhances accuracy with minimal performance cost. We test a range of models in small-scale MPI simulations from simple diffusion models to realistic biological models and find that multi-dimensional geometry partitioning is an important consideration for optimum performance. We demonstrate performance gains of 1-3 orders of magnitude in the parallel implementation, with peak performance strongly dependent on model specification.
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Affiliation(s)
- I Hepburn
- Computational Neuroscience Unit, Okinawa Institute of Science and Technology Graduate University, Onna, Okinawa 904 0495, Japan
| | - W Chen
- Computational Neuroscience Unit, Okinawa Institute of Science and Technology Graduate University, Onna, Okinawa 904 0495, Japan
| | - E De Schutter
- Computational Neuroscience Unit, Okinawa Institute of Science and Technology Graduate University, Onna, Okinawa 904 0495, Japan
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Abstract
Stochastic simulation of cell signaling pathways and genetic regulatory networks has contributed to the understanding of cell function; however, investigation of larger, more complicated systems requires computationally efficient algorithms. τ-leaping methods, which improve efficiency when some molecules have high copy numbers, either use a fixed leap size, which does not adapt to changing state, or recalculate leap size at a heavy computational cost. We present a hybrid simulation method for reaction-diffusion systems which combines exact stochastic simulation and τ-leaping in a dynamic way. Putative times of events are stored in a priority queue, which reduces the cost of each step of the simulation. For every reaction and diffusion channel at each step of the simulation the more efficient of an exact stochastic event or a τ-leap is chosen. This new approach removes the inherent trade-off between speed and accuracy in stiff systems which was present in all τ-leaping methods by allowing each reaction channel to proceed at its own pace. Both directions of reversible reactions and diffusion are combined in a single event, allowing bigger leaps to be taken. This improves efficiency for systems near equilibrium where forward and backward events are approximately equally frequent. Comparison with existing algorithms and behaviour for five test cases of varying complexity shows that the new method is almost as accurate as exact stochastic simulation, scales well for large systems, and for various problems can be significantly faster than τ-leaping.
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Affiliation(s)
| | - Kim T Blackwell
- Krasnow Institute for Advanced Study, George Mason University, Fairfax, Virginia 22030, USA
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Strehl R, Ilie S. Hybrid stochastic simulation of reaction-diffusion systems with slow and fast dynamics. J Chem Phys 2015; 143:234108. [DOI: 10.1063/1.4937491] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Robert Strehl
- Department of Mathematics, Ryerson University, Toronto, Ontario M5B 2K3, Canada
| | - Silvana Ilie
- Department of Mathematics, Ryerson University, Toronto, Ontario M5B 2K3, Canada
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Bhattacharjee AK, Balakrishnan K, Garcia AL, Bell JB, Donev A. Fluctuating hydrodynamics of multi-species reactive mixtures. J Chem Phys 2015; 142:224107. [DOI: 10.1063/1.4922308] [Citation(s) in RCA: 28] [Impact Index Per Article: 3.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/29/2023] Open
Affiliation(s)
- Amit Kumar Bhattacharjee
- Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, New York 10012, USA
| | - Kaushik Balakrishnan
- Jet Propulsion Laboratory, 4800 Oak Grove Drive, Pasadena, California 91109, USA
| | - Alejandro L. Garcia
- Department of Physics and Astronomy, San Jose State University, 1 Washington Square, San Jose, California 95192, USA
| | - John B. Bell
- Computational Research Division, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, California 94720, USA
| | - Aleksandar Donev
- Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, New York 10012, USA
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