1
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Anderson DF, Howells AS. Stochastic Reaction Networks Within Interacting Compartments. Bull Math Biol 2023; 85:87. [PMID: 37624445 DOI: 10.1007/s11538-023-01185-w] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/25/2023] [Accepted: 06/30/2023] [Indexed: 08/26/2023]
Abstract
Stochastic reaction networks, which are usually modeled as continuous-time Markov chains on [Formula: see text], and simulated via a version of the "Gillespie algorithm," have proven to be a useful tool for the understanding of processes, chemical and otherwise, in homogeneous environments. There are multiple avenues for generalizing away from the assumption that the environment is homogeneous, with the proper modeling choice dependent upon the context of the problem being considered. One such generalization was recently introduced in Duso and Zechner (Proc Nat Acad Sci 117(37):22674-22683 , Duso and Zechner (2020)), where the proposed model includes a varying number of interacting compartments, or cells, each of which contains an evolving copy of the stochastic reaction system. The novelty of the model is that these compartments also interact via the merging of two compartments (including their contents), the splitting of one compartment into two, and the appearance and destruction of compartments. In this paper we begin a systematic exploration of the mathematical properties of this model. We (i) obtain basic/foundational results pertaining to explosivity, transience, recurrence, and positive recurrence of the model, (ii) explore a number of examples demonstrating some possible non-intuitive behaviors of the model, and (iii) identify the limiting distribution of the model in a special case that generalizes three formulas from an example in Duso and Zechner (Proc Nat Acad Sci 117(37):22674-22683 , Duso and Zechner (2020)).
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Affiliation(s)
- David F Anderson
- Department of Mathematics, University of Wisconsin-Madison, Madison, WI, USA
| | - Aidan S Howells
- Department of Mathematics, University of Wisconsin-Madison, Madison, WI, USA.
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2
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Emergence of Life-like properties from nonlinear spatial kinetics: Comment on "Unified representation of Life's basic properties by a 3-species Stochastic Cubic Autocatalytic Reaction-Diffusion system of equations" by A.P. Muñuzuri and J. Pérez-Mercader. Phys Life Rev 2023; 44:190-193. [PMID: 36773391 DOI: 10.1016/j.plrev.2023.01.020] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/17/2023] [Accepted: 01/30/2023] [Indexed: 02/05/2023]
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3
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Li C, Li J, Yang Y. A Feynman Path Integral-like Method for Deriving Reaction-Diffusion Equations. Polymers (Basel) 2022; 14:5156. [PMID: 36501551 PMCID: PMC9736185 DOI: 10.3390/polym14235156] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/28/2022] [Revised: 11/23/2022] [Accepted: 11/24/2022] [Indexed: 11/30/2022] Open
Abstract
This work is devoted to deriving a more accurate reaction-diffusion equation for an A/B binary system by summing over microscopic trajectories. By noting that an originally simple physical trajectory might be much more complicated when the reactions are incorporated, we introduce diffusion-reaction-diffusion (DRD) diagrams, similar to the Feynman diagram, to derive the equation. It is found that when there is no intermolecular interaction between A and B, the newly derived equation is reduced to the classical reaction-diffusion equation. However, when there is intermolecular interaction, the newly derived equation shows that there are coupling terms between the diffusion and the reaction, which will be manifested on the mesoscopic scale. The DRD diagram method can be also applied to derive a more accurate dynamical equation for the description of chemical reactions occurred in polymeric systems, such as polymerizations, since the diffusion and the reaction may couple more deeply than that of small molecules.
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Affiliation(s)
| | - Jianfeng Li
- The State Key Laboratory of Molecular Engineering of Polymers, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
| | - Yuliang Yang
- The State Key Laboratory of Molecular Engineering of Polymers, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
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4
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Subic T, Sbalzarini IF. A Gaussian jump process formulation of the reaction–diffusion master equation enables faster exact stochastic simulations. J Chem Phys 2022; 157:194110. [DOI: 10.1063/5.0123073] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022] Open
Abstract
We propose a Gaussian jump process model on a regular Cartesian lattice for the diffusion part of the Reaction–Diffusion Master Equation (RDME). We derive the resulting Gaussian RDME (GRDME) formulation from analogy with a kernel-based discretization scheme for continuous diffusion processes and quantify the limits of its validity relative to the classic RDME. We then present an exact stochastic simulation algorithm for the GRDME, showing that the accuracies of GRDME and RDME are comparable, but exact simulations of the GRDME require only a fraction of the computational cost of exact RDME simulations. We analyze the origin of this speedup and its scaling with problem dimension. The benchmarks suggest that the GRDME is a particularly beneficial model for diffusion-dominated systems in three dimensional spaces, often occurring in systems biology and cell biology.
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Affiliation(s)
- Tina Subic
- Technische Universität Dresden, Faculty of Computer Science, Dresden, Germany
- Max Planck Institute of Molecular Cell Biology and Genetics, Dresden, Germany
- Center for Systems Biology Dresden, Dresden, Germany
| | - Ivo F. Sbalzarini
- Technische Universität Dresden, Faculty of Computer Science, Dresden, Germany
- Max Planck Institute of Molecular Cell Biology and Genetics, Dresden, Germany
- Center for Systems Biology Dresden, Dresden, Germany
- Cluster of Excellence Physics of Life, TU Dresden, Dresden, Germany
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5
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Tran PD, Blanpied TA, Atzberger PJ. Protein drift-diffusion dynamics and phase separation in curved cell membranes and dendritic spines: Hybrid discrete-continuum methods. Phys Rev E 2022; 106:044402. [PMID: 36397472 DOI: 10.1103/physreve.106.044402] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/04/2021] [Accepted: 09/20/2022] [Indexed: 06/16/2023]
Abstract
We develop methods for investigating protein drift-diffusion dynamics in heterogeneous cell membranes and the roles played by geometry, diffusion, chemical kinetics, and phase separation. Our hybrid stochastic numerical methods combine discrete particle descriptions with continuum-level models for tracking the individual protein drift-diffusion dynamics when coupled to continuum fields. We show how our approaches can be used to investigate phenomena motivated by protein kinetics within dendritic spines. The spine geometry is hypothesized to play an important biological role regulating synaptic strength, protein kinetics, and self-assembly of clusters. We perform simulation studies for model spine geometries varying the neck size to investigate how phase-separation and protein organization is influenced by different shapes. We also show how our methods can be used to study the roles of geometry in reaction-diffusion systems including Turing instabilities. Our methods provide general approaches for investigating protein kinetics and drift-diffusion dynamics within curved membrane structures.
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Affiliation(s)
- Patrick D Tran
- Physics, College of Creative Studies, University of California, Santa Barbara, Santa Barbara, California 93106-3080, USA
| | - Thomas A Blanpied
- Department of Physiology, University of Maryland, Baltimore, Maryland 21201, USA
| | - Paul J Atzberger
- Department of Mathematics and Mechanical Engineering, University of California, Santa Barbara, Santa Barbara, California 93106-3080, USA
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6
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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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7
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Simulation of Biochemical Reactions with ANN-Dependent Kinetic Parameter Extraction Method. ELECTRONICS 2022. [DOI: 10.3390/electronics11020216] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/10/2022]
Abstract
The measurement of thermodynamic properties of chemical or biological reactions were often confined to experimental means, which produced overall measurements of properties being investigated, but were usually susceptible to pitfalls of being too general. Among the thermodynamic properties that are of interest, reaction rates hold the greatest significance, as they play a critical role in reaction processes where speed is of essence, especially when fast association may enhance binding affinity of reaction molecules. Association reactions with high affinities often involve the formation of a intermediate state, which can be demonstrated by a hyperbolic reaction curve, but whose low abundance in reaction mixture often preclude the possibility of experimental measurement. Therefore, we resorted to computational methods using predefined reaction models that model the intermediate state as the reaction progresses. Here, we present a novel method called AKPE (ANN-Dependent Kinetic Parameter Extraction), our goal is to investigate the association/dissociation rate constants and the concentration dynamics of lowly-populated states (intermediate states) in the reaction landscape. To reach our goal, we simulated the chemical or biological reactions as system of differential equations, employed artificial neural networks (ANN) to model experimentally measured data, and utilized Particle Swarm Optimization (PSO) algorithm to obtain the globally optimum parameters in both the simulation and data fitting. In the Results section, we have successfully modeled a protein association reaction using AKPE, obtained the kinetic rate constants of the reaction, and constructed a full concentration versus reaction time curve of the intermediate state during the reaction. Furthermore, judging from the various validation methods that the method proposed in this paper has strong robustness and accuracy.
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8
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Ramirez SA, Pablo M, Burk S, Lew DJ, Elston TC. A novel stochastic simulation approach enables exploration of mechanisms for regulating polarity site movement. PLoS Comput Biol 2021; 17:e1008525. [PMID: 34264926 PMCID: PMC8315557 DOI: 10.1371/journal.pcbi.1008525] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/03/2020] [Revised: 07/27/2021] [Accepted: 06/24/2021] [Indexed: 12/23/2022] Open
Abstract
Cells polarize their movement or growth toward external directional cues in many different contexts. For example, budding yeast cells grow toward potential mating partners in response to pheromone gradients. Directed growth is controlled by polarity factors that assemble into clusters at the cell membrane. The clusters assemble, disassemble, and move between different regions of the membrane before eventually forming a stable polarity site directed toward the pheromone source. Pathways that regulate clustering have been identified but the molecular mechanisms that regulate cluster mobility are not well understood. To gain insight into the contribution of chemical noise to cluster behavior we simulated clustering using the reaction-diffusion master equation (RDME) framework to account for molecular-level fluctuations. RDME simulations are a computationally efficient approximation, but their results can diverge from the underlying microscopic dynamics. We implemented novel concentration-dependent rate constants that improved the accuracy of RDME-based simulations, allowing us to efficiently investigate how cluster dynamics might be regulated. Molecular noise was effective in relocating clusters when the clusters contained low numbers of limiting polarity factors, and when Cdc42, the central polarity regulator, exhibited short dwell times at the polarity site. Cluster stabilization occurred when abundances or binding rates were altered to either lengthen dwell times or increase the number of polarity molecules in the cluster. We validated key results using full 3D particle-based simulations. Understanding the mechanisms cells use to regulate the dynamics of polarity clusters should provide insights into how cells dynamically track external directional cues. Cells localize polarity molecules in a small region of the plasma membrane forming a polarity cluster that directs functions such as migration, reproduction, and growth. Guided by external signals, these clusters move across the membrane allowing cells to reorient growth or motion. The polarity molecules continuously and randomly shuttle between the cluster and the cell cytosol and, as a result, the number and distribution of molecules at the cluster constantly changes. Here we present an improved stochastic simulation algorithm to investigate how such molecular-scale fluctuations induce cluster movement across the cell membrane. Unexpectedly, cluster mobility does not correlate with variations in total molecule abundance within the cluster, but rather with changes in the spatial distribution of molecules that form the cluster. Cluster motion is faster when polarity molecules are scarce and when they shuttle rapidly between the cluster and the cytosol. Our results suggest that cells control cluster mobility by regulating the abundance of polarity molecules and biochemical reactions that affect the time molecules spend at the cluster. We provide insights into how cells harness random molecular behavior to perform functions important for survival, such as detecting the direction of external signals.
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Affiliation(s)
- Samuel A. Ramirez
- Department of Pharmacology and Computational Medicine Program, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina, United States of America
- * E-mail: (SAR); (TCE)
| | - Michael Pablo
- Department of Chemistry, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina, United States of America
- Program in Molecular and Cellular Biophysics, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina, United States of America
| | - Sean Burk
- Department of Pharmacology and Computational Medicine Program, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina, United States of America
| | - Daniel J. Lew
- Department of Pharmacology and Cancer Biology, Duke University, Durham, North Carolina, United States of America
| | - Timothy C. Elston
- Department of Pharmacology and Computational Medicine Program, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina, United States of America
- * E-mail: (SAR); (TCE)
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9
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Johnson ME, Chen A, Faeder JR, Henning P, Moraru II, Meier-Schellersheim M, Murphy RF, Prüstel T, Theriot JA, Uhrmacher AM. Quantifying the roles of space and stochasticity in computer simulations for cell biology and cellular biochemistry. Mol Biol Cell 2021; 32:186-210. [PMID: 33237849 PMCID: PMC8120688 DOI: 10.1091/mbc.e20-08-0530] [Citation(s) in RCA: 10] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/12/2020] [Revised: 10/13/2020] [Accepted: 11/17/2020] [Indexed: 12/29/2022] Open
Abstract
Most of the fascinating phenomena studied in cell biology emerge from interactions among highly organized multimolecular structures embedded into complex and frequently dynamic cellular morphologies. For the exploration of such systems, computer simulation has proved to be an invaluable tool, and many researchers in this field have developed sophisticated computational models for application to specific cell biological questions. However, it is often difficult to reconcile conflicting computational results that use different approaches to describe the same phenomenon. To address this issue systematically, we have defined a series of computational test cases ranging from very simple to moderately complex, varying key features of dimensionality, reaction type, reaction speed, crowding, and cell size. We then quantified how explicit spatial and/or stochastic implementations alter outcomes, even when all methods use the same reaction network, rates, and concentrations. For simple cases, we generally find minor differences in solutions of the same problem. However, we observe increasing discordance as the effects of localization, dimensionality reduction, and irreversible enzymatic reactions are combined. We discuss the strengths and limitations of commonly used computational approaches for exploring cell biological questions and provide a framework for decision making by researchers developing new models. As computational power and speed continue to increase at a remarkable rate, the dream of a fully comprehensive computational model of a living cell may be drawing closer to reality, but our analysis demonstrates that it will be crucial to evaluate the accuracy of such models critically and systematically.
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Affiliation(s)
- M. E. Johnson
- Thomas C. Jenkins Department of Biophysics, Johns Hopkins University, Baltimore, MD, 21218
| | - A. Chen
- Thomas C. Jenkins Department of Biophysics, Johns Hopkins University, Baltimore, MD, 21218
| | - J. R. Faeder
- Department of Computational and Systems Biology, University of Pittsburgh School of Medicine, Pittsburgh, PA, 15260
| | - P. Henning
- Institute for Visual and Analytic Computing, University of Rostock, 18055 Rostock, Germany
| | - I. I. Moraru
- Department of Cell Biology, Center for Cell Analysis and Modeling, University of Connecticut Health Center, Farmington, CT 06030
| | - M. Meier-Schellersheim
- Laboratory of Immune System Biology, National Institute of Allergy and Infectious Diseases, National Institutes of Health, Bethesda, MD 20892
| | - R. F. Murphy
- Computational Biology Department, Department of Biological Sciences, Department of Biomedical Engineering, Machine Learning Department, Carnegie Mellon University, Pittsburgh, PA 15289
| | - T. Prüstel
- Laboratory of Immune System Biology, National Institute of Allergy and Infectious Diseases, National Institutes of Health, Bethesda, MD 20892
| | - J. A. Theriot
- Department of Biology and Howard Hughes Medical Institute, University of Washington, Seattle, WA 98195
| | - A. M. Uhrmacher
- Institute for Visual and Analytic Computing, University of Rostock, 18055 Rostock, Germany
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10
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Stepanova D, Byrne HM, Maini PK, Alarcón T. A multiscale model of complex endothelial cell dynamics in early angiogenesis. PLoS Comput Biol 2021; 17:e1008055. [PMID: 33411727 PMCID: PMC7817011 DOI: 10.1371/journal.pcbi.1008055] [Citation(s) in RCA: 24] [Impact Index Per Article: 8.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/10/2020] [Revised: 01/20/2021] [Accepted: 11/19/2020] [Indexed: 12/30/2022] Open
Abstract
We introduce a hybrid two-dimensional multiscale model of angiogenesis, the process by which endothelial cells (ECs) migrate from a pre-existing vascular bed in response to local environmental cues and cell-cell interactions, to create a new vascular network. Recent experimental studies have highlighted a central role of cell rearrangements in the formation of angiogenic networks. Our model accounts for this phenomenon via the heterogeneous response of ECs to their microenvironment. These cell rearrangements, in turn, dynamically remodel the local environment. The model reproduces characteristic features of angiogenic sprouting that include branching, chemotactic sensitivity, the brush border effect, and cell mixing. These properties, rather than being hardwired into the model, emerge naturally from the gene expression patterns of individual cells. After calibrating and validating our model against experimental data, we use it to predict how the structure of the vascular network changes as the baseline gene expression levels of the VEGF-Delta-Notch pathway, and the composition of the extracellular environment, vary. In order to investigate the impact of cell rearrangements on the vascular network structure, we introduce the mixing measure, a scalar metric that quantifies cell mixing as the vascular network grows. We calculate the mixing measure for the simulated vascular networks generated by ECs of different lineages (wild type cells and mutant cells with impaired expression of a specific receptor). Our results show that the time evolution of the mixing measure is directly correlated to the generic features of the vascular branching pattern, thus, supporting the hypothesis that cell rearrangements play an essential role in sprouting angiogenesis. Furthermore, we predict that lower cell rearrangement leads to an imbalance between branching and sprout elongation. Since the computation of this statistic requires only individual cell trajectories, it can be computed for networks generated in biological experiments, making it a potential biomarker for pathological angiogenesis. Angiogenesis, the process by which new blood vessels are formed by sprouting from the pre-existing vascular bed, plays a key role in both physiological and pathological processes, including tumour growth. The structure of a growing vascular network is determined by the coordinated behaviour of endothelial cells in response to various signalling cues. Recent experimental studies have highlighted the importance of cell rearrangements as a driver for sprout elongation. However, the functional role of this phenomenon remains unclear. We formulate a new multiscale model of angiogenesis which, by accounting explicitly for the complex dynamics of endothelial cells within growing angiogenic sprouts, is able to reproduce generic features of angiogenic structures (branching, chemotactic sensitivity, cell mixing, etc.) as emergent properties of its dynamics. We validate our model against experimental data and then use it to quantify the phenomenon of cell mixing in vascular networks generated by endothelial cells of different lineages. Our results show that there is a direct correlation between the time evolution of cell mixing in a growing vascular network and its branching structure, thus paving the way for understanding the functional role of cell rearrangements in angiogenesis.
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Affiliation(s)
- Daria Stepanova
- Centre de Recerca Matemàtica, Bellaterra (Barcelona), Spain
- Departament de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra (Barcelona), Spain
- * E-mail:
| | - Helen M. Byrne
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, UK
| | - Philip K. Maini
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, UK
| | - Tomás Alarcón
- Centre de Recerca Matemàtica, Bellaterra (Barcelona), Spain
- Departament de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra (Barcelona), Spain
- Institució Catalana de Recerca i Estudis Avançats (ICREA), Barcelona, Spain
- Barcelona Graduate School of Mathematics (BGSMath), Barcelona, Spain
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11
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Yates CA, George A, Jordana A, Smith CA, Duncan AB, Zygalakis KC. The blending region hybrid framework for the simulation of stochastic reaction-diffusion processes. J R Soc Interface 2020; 17:20200563. [PMID: 33081647 PMCID: PMC7653393 DOI: 10.1098/rsif.2020.0563] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022] Open
Abstract
The simulation of stochastic reaction–diffusion systems using fine-grained representations can become computationally prohibitive when particle numbers become large. If particle numbers are sufficiently high then it may be possible to ignore stochastic fluctuations and use a more efficient coarse-grained simulation approach. Nevertheless, for multiscale systems which exhibit significant spatial variation in concentration, a coarse-grained approach may not be appropriate throughout the simulation domain. Such scenarios suggest a hybrid paradigm in which a computationally cheap, coarse-grained model is coupled to a more expensive, but more detailed fine-grained model, enabling the accurate simulation of the fine-scale dynamics at a reasonable computational cost. In this paper, in order to couple two representations of reaction–diffusion at distinct spatial scales, we allow them to overlap in a ‘blending region’. Both modelling paradigms provide a valid representation of the particle density in this region. From one end of the blending region to the other, control of the implementation of diffusion is passed from one modelling paradigm to another through the use of complementary ‘blending functions’ which scale up or down the contribution of each model to the overall diffusion. We establish the reliability of our novel hybrid paradigm by demonstrating its simulation on four exemplar reaction–diffusion scenarios.
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Affiliation(s)
- Christian A Yates
- Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK
| | - Adam George
- Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK
| | - Armand Jordana
- Centre de Mathématiques et de Leurs Applications, CNRS, ENS Paris-Saclay, Université Paris-Saclay, 94235 Cachan cedex, France
| | - Cameron A Smith
- Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK
| | - Andrew B Duncan
- Department of Mathematics, Imperial College London, London SW7 2AZ, UK
| | - Konstantinos C Zygalakis
- School of Mathematics, University of Edinburgh, James Clerk Maxwell Building, The King's Buildings, Peter Guthrie Tait Road, Edinburgh EH9 3FD, UK
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12
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Abstract
Domain growth is a key process in many areas of biology, including embryonic development, the growth of tissue, and limb regeneration. As a result, mechanisms for incorporating it into traditional models for cell movement, interaction, and proliferation are of great importance. A previously well-used method to incorporate domain growth into on-lattice reaction-diffusion models causes a buildup of particles on the boundaries of the domain, which is particularly evident when diffusion is low in comparison to the rate of domain growth. Here we present an alternative method which addresses this unphysical buildup of particles at the boundaries and demonstrate that it is accurate for scenarios in which the previous method fails. Further, we discuss for which parameter regimes it is feasible to continue using the original method due to diffusion dominating the domain growth mechanism.
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Affiliation(s)
- Cameron A Smith
- Centre for Mathematical Biology, Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom
| | - Cécile Mailler
- Probability Laboratory, Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom
| | - Christian A Yates
- Centre for Mathematical Biology, Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom
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13
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Zhang Y, Clemens L, Goyette J, Allard J, Dushek O, Isaacson SA. The Influence of Molecular Reach and Diffusivity on the Efficacy of Membrane-Confined Reactions. Biophys J 2019; 117:1189-1201. [PMID: 31543263 PMCID: PMC6818170 DOI: 10.1016/j.bpj.2019.08.023] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/02/2019] [Revised: 07/30/2019] [Accepted: 08/22/2019] [Indexed: 11/15/2022] Open
Abstract
Signaling by surface receptors often relies on tethered reactions whereby an enzyme bound to the cytoplasmic tail of a receptor catalyzes reactions on substrates within reach. The overall length and stiffness of the receptor tail, the enzyme, and the substrate determine a biophysical parameter termed the molecular reach of the reaction. This parameter determines the probability that the receptor-tethered enzyme will contact the substrate in the volume proximal to the membrane when separated by different distances within the membrane plane. In this work, we develop particle-based stochastic reaction-diffusion models to study the interplay between molecular reach and diffusion. We find that increasing the molecular reach can increase reaction efficacy for slowly diffusing receptors, whereas for rapidly diffusing receptors, increasing molecular reach reduces reaction efficacy. In contrast, if reactions are forced to take place within the two-dimensional plasma membrane instead of the three-dimensional volume proximal to it or if molecules diffuse in three dimensions, increasing molecular reach increases reaction efficacy for all diffusivities. We show results in the context of immune checkpoint receptors (PD-1 dephosphorylating CD28), a standard opposing kinase-phosphatase reaction, and a minimal two-particle model. The work highlights the importance of the three-dimensional nature of many two-dimensional membrane-confined interactions, illustrating a role for molecular reach in controlling biochemical reactions.
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Affiliation(s)
- Ying Zhang
- Department of Mathematics and Statistics, Boston University, Boston, Massachusetts
| | - Lara Clemens
- Center for Complex Biological Systems, University of California-Irvine, Irvine, California
| | - Jesse Goyette
- School of Medical Sciences, University of New South Wales, Sydney, Australia
| | - Jun Allard
- Center for Complex Biological Systems, University of California-Irvine, Irvine, California
| | - Omer Dushek
- Sir William Dunn School of Pathology, University of Oxford, Oxford, United Kingdom.
| | - Samuel A Isaacson
- Department of Mathematics and Statistics, Boston University, Boston, Massachusetts.
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14
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Hofmann A, Mäkelä J, Sherratt DJ, Heermann D, Murray SM. Self-organised segregation of bacterial chromosomal origins. eLife 2019; 8:e46564. [PMID: 31397672 PMCID: PMC6701925 DOI: 10.7554/elife.46564] [Citation(s) in RCA: 20] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/05/2019] [Accepted: 08/09/2019] [Indexed: 01/12/2023] Open
Abstract
The chromosomal replication origin region (ori) of characterised bacteria is dynamically positioned throughout the cell cycle. In slowly growing Escherichia coli, ori is maintained at mid-cell from birth until its replication, after which newly replicated sister oris move to opposite quarter positions. Here, we provide an explanation for ori positioning based on the self-organisation of the Structural Maintenance of Chromosomes complex, MukBEF, which forms dynamically positioned clusters on the chromosome. We propose that a non-trivial feedback between the self-organising gradient of MukBEF complexes and the oris leads to accurate ori positioning. We find excellent agreement with quantitative experimental measurements and confirm key predictions. Specifically, we show that oris exhibit biased motion towards MukBEF clusters, rather than mid-cell. Our findings suggest that MukBEF and oris act together as a self-organising system in chromosome organisation-segregation and introduces protein self-organisation as an important consideration for future studies of chromosome dynamics.
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Affiliation(s)
- Andreas Hofmann
- Institute for Theoretical PhysicsHeidelberg UniversityHeidelbergGermany
| | - Jarno Mäkelä
- Department of BiochemistryUniversity of OxfordOxfordUnited Kingdom
| | - David J Sherratt
- Department of BiochemistryUniversity of OxfordOxfordUnited Kingdom
| | - Dieter Heermann
- Institute for Theoretical PhysicsHeidelberg UniversityHeidelbergGermany
| | - Seán M Murray
- Max Planck Institute for Terrestrial Microbiology, LOEWE Centre for Synthetic Microbiology (SYNMIKRO)MarburgGermany
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15
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Smith S, Grima R. Spatial Stochastic Intracellular Kinetics: A Review of Modelling Approaches. Bull Math Biol 2019; 81:2960-3009. [PMID: 29785521 PMCID: PMC6677717 DOI: 10.1007/s11538-018-0443-1] [Citation(s) in RCA: 27] [Impact Index Per Article: 5.4] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/14/2018] [Accepted: 05/03/2018] [Indexed: 01/22/2023]
Abstract
Models of chemical kinetics that incorporate both stochasticity and diffusion are an increasingly common tool for studying biology. The variety of competing models is vast, but two stand out by virtue of their popularity: the reaction-diffusion master equation and Brownian dynamics. In this review, we critically address a number of open questions surrounding these models: How can they be justified physically? How do they relate to each other? How do they fit into the wider landscape of chemical models, ranging from the rate equations to molecular dynamics? This review assumes no prior knowledge of modelling chemical kinetics and should be accessible to a wide range of readers.
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Affiliation(s)
- Stephen Smith
- School of Biological Sciences, University of Edinburgh, Mayfield Road, Edinburgh, EH9 3JR, Scotland, UK
| | - Ramon Grima
- School of Biological Sciences, University of Edinburgh, Mayfield Road, Edinburgh, EH9 3JR, Scotland, UK.
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16
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Carr EJ, Simpson MJ. New homogenization approaches for stochastic transport through heterogeneous media. J Chem Phys 2019; 150:044104. [PMID: 30709257 DOI: 10.1063/1.5067290] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/20/2022] Open
Abstract
The diffusion of molecules in complex intracellular environments can be strongly influenced by spatial heterogeneity and stochasticity. A key challenge when modelling such processes using stochastic random walk frameworks is that negative jump coefficients can arise when transport operators are discretized on heterogeneous domains. Often this is dealt with through homogenization approximations by replacing the heterogeneous medium with an effective homogeneous medium. In this work, we present a new class of homogenization approximations by considering a stochastic diffusive transport model on a one-dimensional domain containing an arbitrary number of layers with different jump rates. We derive closed form solutions for the kth moment of particle lifetime, carefully explaining how to deal with the internal interfaces between layers. These general tools allow us to derive simple formulae for the effective transport coefficients, leading to significant generalisations of previous homogenization approaches. Here, we find that different jump rates in the layers give rise to a net bias, leading to a non-zero advection, for the entire homogenized system. Example calculations show that our generalized approach can lead to very different outcomes than traditional approaches, thereby having the potential to significantly affect simulation studies that use homogenization approximations.
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Affiliation(s)
- Elliot J Carr
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia
| | - Matthew J Simpson
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia
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17
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Hoffmann M, Fröhner C, Noé F. ReaDDy 2: Fast and flexible software framework for interacting-particle reaction dynamics. PLoS Comput Biol 2019; 15:e1006830. [PMID: 30818351 PMCID: PMC6413953 DOI: 10.1371/journal.pcbi.1006830] [Citation(s) in RCA: 43] [Impact Index Per Article: 8.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/09/2018] [Revised: 03/12/2019] [Accepted: 01/16/2019] [Indexed: 12/30/2022] Open
Abstract
Interacting-particle reaction dynamics (iPRD) combines the simulation of dynamical trajectories of interacting particles as in molecular dynamics (MD) simulations with reaction kinetics, in which particles appear, disappear, or change their type and interactions based on a set of reaction rules. This combination facilitates the simulation of reaction kinetics in crowded environments, involving complex molecular geometries such as polymers, and employing complex reaction mechanisms such as breaking and fusion of polymers. iPRD simulations are ideal to simulate the detailed spatiotemporal reaction mechanism in complex and dense environments, such as in signalling processes at cellular membranes, or in nano- to microscale chemical reactors. Here we introduce the iPRD software ReaDDy 2, which provides a Python interface in which the simulation environment, particle interactions and reaction rules can be conveniently defined and the simulation can be run, stored and analyzed. A C++ interface is available to enable deeper and more flexible interactions with the framework. The main computational work of ReaDDy 2 is done in hardware-specific simulation kernels. While the version introduced here provides single- and multi-threading CPU kernels, the architecture is ready to implement GPU and multi-node kernels. We demonstrate the efficiency and validity of ReaDDy 2 using several benchmark examples. ReaDDy 2 is available at the https://readdy.github.io/ website.
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Affiliation(s)
- Moritz Hoffmann
- Department of Mathematics and Computer Science, Freie Universität Berlin, Berlin, Germany
| | - Christoph Fröhner
- Department of Mathematics and Computer Science, Freie Universität Berlin, Berlin, Germany
| | - Frank Noé
- Department of Mathematics and Computer Science, Freie Universität Berlin, Berlin, Germany
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18
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Del Razo MJ, Qian H, Noé F. Grand canonical diffusion-influenced reactions: A stochastic theory with applications to multiscale reaction-diffusion simulations. J Chem Phys 2018; 149:044102. [PMID: 30068197 DOI: 10.1063/1.5037060] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/21/2023] Open
Abstract
Smoluchowski-type models for diffusion-influenced reactions (A + B → C) can be formulated within two frameworks: the probabilistic-based approach for a pair A, B of reacting particles and the concentration-based approach for systems in contact with a bath that generates a concentration gradient of B particles that interact with A. Although these two approaches are mathematically similar, it is not straightforward to establish a precise mathematical relationship between them. Determining this relationship is essential to derive particle-based numerical methods that are quantitatively consistent with bulk concentration dynamics. In this work, we determine the relationship between the two approaches by introducing the grand canonical Smoluchowski master equation (GC-SME), which consists of a continuous-time Markov chain that models an arbitrary number of B particles, each one of them following Smoluchowski's probabilistic dynamics. We show that the GC-SME recovers the concentration-based approach by taking either the hydrodynamic or the large copy number limit. In addition, we show that the GC-SME provides a clear statistical mechanical interpretation of the concentration-based approach and yields an emergent chemical potential for nonequilibrium spatially inhomogeneous reaction processes. We further exploit the GC-SME robust framework to accurately derive multiscale/hybrid numerical methods that couple particle-based reaction-diffusion simulations with bulk concentration descriptions, as described in detail through two computational implementations.
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Affiliation(s)
- Mauricio J Del Razo
- Department of Mathematics and Computer Science, Freie Universität Berlin, Arnimallee 6, 14195 Berlin, Germany
| | - Hong Qian
- Department of Applied Mathematics, University of Washington, Seattle, Washington 98195-3925, USA
| | - Frank Noé
- Department of Mathematics and Computer Science, Freie Universität Berlin, Arnimallee 6, 14195 Berlin, Germany
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19
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Smith CA, Yates CA. The auxiliary region method: a hybrid method for coupling PDE- and Brownian-based dynamics for reaction-diffusion systems. ROYAL SOCIETY OPEN SCIENCE 2018; 5:180920. [PMID: 30225082 PMCID: PMC6124063 DOI: 10.1098/rsos.180920] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 06/18/2018] [Accepted: 06/28/2018] [Indexed: 06/08/2023]
Abstract
Reaction-diffusion systems are used to represent many biological and physical phenomena. They model the random motion of particles (diffusion) and interactions between them (reactions). Such systems can be modelled at multiple scales with varying degrees of accuracy and computational efficiency. When representing genuinely multiscale phenomena, fine-scale models can be prohibitively expensive, whereas coarser models, although cheaper, often lack sufficient detail to accurately represent the phenomenon at hand. Spatial hybrid methods couple two or more of these representations in order to improve efficiency without compromising accuracy. In this paper, we present a novel spatial hybrid method, which we call the auxiliary region method (ARM), which couples PDE- and Brownian-based representations of reaction-diffusion systems. Numerical PDE solutions on one side of an interface are coupled to Brownian-based dynamics on the other side using compartment-based 'auxiliary regions'. We demonstrate that the hybrid method is able to simulate reaction-diffusion dynamics for a number of different test problems with high accuracy. Furthermore, we undertake error analysis on the ARM which demonstrates that it is robust to changes in the free parameters in the model, where previous coupling algorithms are not. In particular, we envisage that the method will be applicable for a wide range of spatial multi-scales problems including filopodial dynamics, intracellular signalling, embryogenesis and travelling wave phenomena.
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Affiliation(s)
- Cameron A. Smith
- Centre for Mathematical Biology, Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK
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20
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Wilson DB, Byrne H, Bruna M. Reactions, diffusion, and volume exclusion in a conserved system of interacting particles. Phys Rev E 2018; 97:062137. [PMID: 30011580 DOI: 10.1103/physreve.97.062137] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/30/2018] [Indexed: 11/07/2022]
Abstract
Complex biological and physical transport processes are often described through systems of interacting particles. The effect of excluded volume on these transport processes has been well studied; however, the interplay between volume exclusion and reactions between heterogenous particles is less well studied. In this paper we develop a framework for modeling reaction-diffusion processes which directly incorporates volume exclusion. We consider simple reactions (unimolecular and bimolecular) that conserve the total number of particles. From an off-lattice microscopic individual-based model we use the Fokker-Planck equation and the method of matched asymptotic expansions to derive a low-dimensional macroscopic system of nonlinear partial differential equations describing the evolution of the particles. A biologically motivated, hybrid model of chemotaxis with volume exclusion is explored, where reactions occur at rates dependent upon the chemotactic environment. Further, we show that for reactions that require particle contact the appropriate reaction term in the macroscopic model is of lower order in the asymptotic expansion than the nonlinear diffusion term. However, we find that the next reaction term in the expansion is needed to ensure good agreement with simulations of the microscopic model. Our macroscopic model allows for more direct parametrization to experimental data than existing models.
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Affiliation(s)
- Daniel B Wilson
- Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, United Kingdom
| | - Helen Byrne
- Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, United Kingdom
| | - Maria Bruna
- Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, United Kingdom
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21
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Abstract
We develop a Split Reactive Brownian Dynamics (SRBD) algorithm for particle simulations of reaction-diffusion systems based on the Doi or volume reactivity model, in which pairs of particles react with a specified Poisson rate if they are closer than a chosen reactive distance. In our Doi model, we ensure that the microscopic reaction rules for various association and dissociation reactions are consistent with detailed balance (time reversibility) at thermodynamic equilibrium. The SRBD algorithm uses Strang splitting in time to separate reaction and diffusion and solves both the diffusion-only and reaction-only subproblems exactly, even at high packing densities. To efficiently process reactions without uncontrolled approximations, SRBD employs an event-driven algorithm that processes reactions in a time-ordered sequence over the duration of the time step. A grid of cells with size larger than all of the reactive distances is used to schedule and process the reactions, but unlike traditional grid-based methods such as reaction-diffusion master equation algorithms, the results of SRBD are statistically independent of the size of the grid used to accelerate the processing of reactions. We use the SRBD algorithm to compute the effective macroscopic reaction rate for both reaction-limited and diffusion-limited irreversible association in three dimensions and compare to existing theoretical predictions at low and moderate densities. We also study long-time tails in the time correlation functions for reversible association at thermodynamic equilibrium and compare to recent theoretical predictions. Finally, we compare different particle and continuum methods on a model exhibiting a Turing-like instability and pattern formation. Our studies reinforce the common finding that microscopic mechanisms and correlations matter for diffusion-limited systems, making continuum and even mesoscopic modeling of such systems difficult or impossible. We also find that for models in which particles diffuse off lattice, such as the Doi model, reactions lead to a spurious enhancement of the effective diffusion coefficients.
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Affiliation(s)
- Aleksandar Donev
- Courant Institute of Mathematical Sciences, New York University, New York, New York 10012, USA
| | - Chiao-Yu Yang
- Courant Institute of Mathematical Sciences, New York University, New York, New York 10012, USA
| | - Changho Kim
- Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
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22
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Kim C, Nonaka A, Bell JB, Garcia AL, Donev A. Stochastic simulation of reaction-diffusion systems: A fluctuating-hydrodynamics approach. J Chem Phys 2018; 146:124110. [PMID: 28388111 DOI: 10.1063/1.4978775] [Citation(s) in RCA: 14] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/12/2023] Open
Abstract
We develop numerical methods for stochastic reaction-diffusion systems based on approaches used for fluctuatinghydrodynamics (FHD). For hydrodynamicsystems, the FHD formulation is formally described by stochastic partial differential equations (SPDEs). In the reaction-diffusion systems we consider, our model becomes similar to the reaction-diffusion master equation (RDME) description when our SPDEs are spatially discretized and reactions are modeled as a source term having Poissonfluctuations. However, unlike the RDME, which becomes prohibitively expensive for an increasing number of molecules, our FHD-based description naturally extends from the regime where fluctuations are strong, i.e., each mesoscopic cell has few (reactive) molecules, to regimes with moderate or weak fluctuations, and ultimately to the deterministic limit. By treating diffusion implicitly, we avoid the severe restriction on time step size that limits all methods based on explicit treatments of diffusion and construct numerical methods that are more efficient than RDME methods, without compromising accuracy. Guided by an analysis of the accuracy of the distribution of steady-state fluctuations for the linearized reaction-diffusion model, we construct several two-stage (predictor-corrector) schemes, where diffusion is treated using a stochastic Crank-Nicolson method, and reactions are handled by the stochastic simulation algorithm of Gillespie or a weakly second-order tau leaping method. We find that an implicit midpoint tau leaping scheme attains second-order weak accuracy in the linearized setting and gives an accurate and stable structure factor for a time step size of an order of magnitude larger than the hopping time scale of diffusing molecules. We study the numerical accuracy of our methods for the Schlögl reaction-diffusion model both in and out of thermodynamic equilibrium. We demonstrate and quantify the importance of thermodynamicfluctuations to the formation of a two-dimensional Turing-like pattern and examine the effect of fluctuations on three-dimensional chemical front propagation. By comparing stochastic simulations to deterministic reaction-diffusion simulations, we show that fluctuations accelerate pattern formation in spatially homogeneous systems and lead to a qualitatively different disordered pattern behind a traveling wave.
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Affiliation(s)
- Changho Kim
- Computational Research Division, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, California 94720, USA
| | - Andy Nonaka
- Computational Research Division, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, California 94720, USA
| | - John B Bell
- Computational Research Division, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, California 94720, USA
| | - Alejandro L Garcia
- Department of Physics and Astronomy, San Jose State University, 1 Washington Square, San Jose, California 95192, USA
| | - Aleksandar Donev
- Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, New York 10012, USA
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23
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Hellander S, Petzold L. Reaction rates for reaction-diffusion kinetics on unstructured meshes. J Chem Phys 2018; 146:064101. [PMID: 28201913 DOI: 10.1063/1.4975167] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/03/2023] Open
Abstract
The reaction-diffusion master equation is a stochastic model often utilized in the study of biochemical reaction networks in living cells. It is applied when the spatial distribution of molecules is important to the dynamics of the system. A viable approach to resolve the complex geometry of cells accurately is to discretize space with an unstructured mesh. Diffusion is modeled as discrete jumps between nodes on the mesh, and the diffusion jump rates can be obtained through a discretization of the diffusion equation on the mesh. Reactions can occur when molecules occupy the same voxel. In this paper, we develop a method for computing accurate reaction rates between molecules occupying the same voxel in an unstructured mesh. For large voxels, these rates are known to be well approximated by the reaction rates derived by Collins and Kimball, but as the mesh is refined, no analytical expression for the rates exists. We reduce the problem of computing accurate reaction rates to a pure preprocessing step, depending only on the mesh and not on the model parameters, and we devise an efficient numerical scheme to estimate them to high accuracy. We show in several numerical examples that as we refine the mesh, the results obtained with the reaction-diffusion master equation approach those of a more fine-grained Smoluchowski particle-tracking model.
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Affiliation(s)
- Stefan Hellander
- Department of Computer Science, University of California, Santa Barbara, California 93106-5070, USA
| | - Linda Petzold
- Department of Computer Science, University of California, Santa Barbara, California 93106-5070, USA
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24
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Li F, Chen M, Erban R, Cao Y. Reaction time for trimolecular reactions in compartment-based reaction-diffusion models. J Chem Phys 2018; 148:204108. [PMID: 29865852 DOI: 10.1063/1.5024927] [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
Trimolecular reaction models are investigated in the compartment-based (lattice-based) framework for stochastic reaction-diffusion modeling. The formulae for the first collision time and the mean reaction time are derived for the case where three molecules are present in the solution under periodic boundary conditions. For the case of reflecting boundary conditions, similar formulae are obtained using a computer-assisted approach. The accuracy of these formulae is further verified through comparison with numerical results. The presented derivation is based on the first passage time analysis of Montroll [J. Math. Phys. 10, 753 (1969)]. Montroll's results for two-dimensional lattice-based random walks are adapted and applied to compartment-based models of trimolecular reactions, which are studied in one-dimensional or pseudo one-dimensional domains.
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Affiliation(s)
- Fei Li
- Department of Computer Science, Virginia Tech, Blacksburg, Virginia 24061, USA
| | - Minghan Chen
- Department of Computer Science, Virginia Tech, Blacksburg, Virginia 24061, USA
| | - Radek Erban
- Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom
| | - Yang Cao
- Department of Computer Science, Virginia Tech, Blacksburg, Virginia 24061, USA
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25
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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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26
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Nakamura H, Lee AA, Afshar AS, Watanabe S, Rho E, Razavi S, Suarez A, Lin YC, Tanigawa M, Huang B, DeRose R, Bobb D, Hong W, Gabelli SB, Goutsias J, Inoue T. Intracellular production of hydrogels and synthetic RNA granules by multivalent molecular interactions. NATURE MATERIALS 2018; 17:79-89. [PMID: 29115293 PMCID: PMC5916848 DOI: 10.1038/nmat5006] [Citation(s) in RCA: 78] [Impact Index Per Article: 13.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/31/2015] [Accepted: 09/08/2017] [Indexed: 05/06/2023]
Abstract
Some protein components of intracellular non-membrane-bound entities, such as RNA granules, are known to form hydrogels in vitro. The physico-chemical properties and functional role of these intracellular hydrogels are difficult to study, primarily due to technical challenges in probing these materials in situ. Here, we present iPOLYMER, a strategy for a rapid induction of protein-based hydrogels inside living cells that explores the chemically inducible dimerization paradigm. Biochemical and biophysical characterizations aided by computational modelling show that the polymer network formed in the cytosol resembles a physiological hydrogel-like entity that acts as a size-dependent molecular sieve. We functionalize these polymers with RNA-binding motifs that sequester polyadenine-containing nucleotides to synthetically mimic RNA granules. These results show that iPOLYMER can be used to synthetically reconstitute the nucleation of biologically functional entities, including RNA granules in intact cells.
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Affiliation(s)
- Hideki Nakamura
- Department of Cell Biology, School of Medicine, The Johns Hopkins University, Baltimore, MD, 21205
- Center for Cell Dynamics, Institute for Basic Biomedical Sciences, The Johns Hopkins University, Baltimore, MD, 21205
| | - Albert A. Lee
- Department of Cell Biology, School of Medicine, The Johns Hopkins University, Baltimore, MD, 21205
- Center for Cell Dynamics, Institute for Basic Biomedical Sciences, The Johns Hopkins University, Baltimore, MD, 21205
| | - Ali Sobhi Afshar
- Center for Imaging Science, Whitaker Biomedical Engineering Institute, The Johns Hopkins University, Baltimore, MD, 21218
- To whom correspondence regarding the computational analysis should be addressed: (A.S.A)
| | - Shigeki Watanabe
- Department of Cell Biology, School of Medicine, The Johns Hopkins University, Baltimore, MD, 21205
| | - Elmer Rho
- Center for Cell Dynamics, Institute for Basic Biomedical Sciences, The Johns Hopkins University, Baltimore, MD, 21205
| | - Shiva Razavi
- Department of Cell Biology, School of Medicine, The Johns Hopkins University, Baltimore, MD, 21205
- Department of Biomedical Engineering, Whitaker Biomedical Engineering Institute, The Johns Hopkins University, Baltimore, MD 21218
| | - Allison Suarez
- Department of Cell Biology, School of Medicine, The Johns Hopkins University, Baltimore, MD, 21205
- Center for Cell Dynamics, Institute for Basic Biomedical Sciences, The Johns Hopkins University, Baltimore, MD, 21205
| | - Yu-Chun Lin
- Department of Cell Biology, School of Medicine, The Johns Hopkins University, Baltimore, MD, 21205
- Center for Cell Dynamics, Institute for Basic Biomedical Sciences, The Johns Hopkins University, Baltimore, MD, 21205
| | - Makoto Tanigawa
- Department of Cell Biology, School of Medicine, The Johns Hopkins University, Baltimore, MD, 21205
- Department of Biomedical Engineering, Whitaker Biomedical Engineering Institute, The Johns Hopkins University, Baltimore, MD 21218
| | - Brian Huang
- Center for Cell Dynamics, Institute for Basic Biomedical Sciences, The Johns Hopkins University, Baltimore, MD, 21205
| | - Robert DeRose
- Department of Cell Biology, School of Medicine, The Johns Hopkins University, Baltimore, MD, 21205
- Center for Cell Dynamics, Institute for Basic Biomedical Sciences, The Johns Hopkins University, Baltimore, MD, 21205
| | - Diana Bobb
- Department of Cell Biology, School of Medicine, The Johns Hopkins University, Baltimore, MD, 21205
- Center for Cell Dynamics, Institute for Basic Biomedical Sciences, The Johns Hopkins University, Baltimore, MD, 21205
| | - William Hong
- Department of Biophysics and Biophysical Chemistry, School of Medicine, The Johns Hopkins University, Baltimore, MD, 21205
| | - Sandra B. Gabelli
- Department of Biophysics and Biophysical Chemistry, School of Medicine, The Johns Hopkins University, Baltimore, MD, 21205
- Department of Medicine, School of Medicine, Johns Hopkins University, Baltimore, MD, 21205
- Department of Oncology, School of Medicine, Johns Hopkins University, Baltimore, MD, 21205
| | - John Goutsias
- Center for Imaging Science, Whitaker Biomedical Engineering Institute, The Johns Hopkins University, Baltimore, MD, 21218
| | - Takanari Inoue
- Department of Cell Biology, School of Medicine, The Johns Hopkins University, Baltimore, MD, 21205
- Center for Cell Dynamics, Institute for Basic Biomedical Sciences, The Johns Hopkins University, Baltimore, MD, 21205
- Department of Biomedical Engineering, Whitaker Biomedical Engineering Institute, The Johns Hopkins University, Baltimore, MD 21218
- To whom general correspondence should be addressed: (T.I.)
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27
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Hellander S, Hellander A, Petzold L. Mesoscopic-microscopic spatial stochastic simulation with automatic system partitioning. J Chem Phys 2017; 147:234101. [PMID: 29272930 PMCID: PMC5732015 DOI: 10.1063/1.5002773] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/01/2017] [Accepted: 11/27/2017] [Indexed: 01/02/2023] Open
Abstract
The reaction-diffusion master equation (RDME) is a model that allows for efficient on-lattice simulation of spatially resolved stochastic chemical kinetics. Compared to off-lattice hard-sphere simulations with Brownian dynamics or Green's function reaction dynamics, the RDME can be orders of magnitude faster if the lattice spacing can be chosen coarse enough. However, strongly diffusion-controlled reactions mandate a very fine mesh resolution for acceptable accuracy. It is common that reactions in the same model differ in their degree of diffusion control and therefore require different degrees of mesh resolution. This renders mesoscopic simulation inefficient for systems with multiscale properties. Mesoscopic-microscopic hybrid methods address this problem by resolving the most challenging reactions with a microscale, off-lattice simulation. However, all methods to date require manual partitioning of a system, effectively limiting their usefulness as "black-box" simulation codes. In this paper, we propose a hybrid simulation algorithm with automatic system partitioning based on indirect a priori error estimates. We demonstrate the accuracy and efficiency of the method on models of diffusion-controlled networks in 3D.
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Affiliation(s)
- Stefan Hellander
- Department of Information Technology, Uppsala University, P.O.Box 337, SE-75105 Uppsala, Sweden
| | - Andreas Hellander
- Department of Information Technology, Uppsala University, P.O.Box 337, SE-75105 Uppsala, Sweden
| | - Linda Petzold
- Department of Computer Science, University of California, Santa Barbara, California 93106-5070, USA
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28
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de la Cruz R, Guerrero P, Calvo J, Alarcón T. Coarse-graining and hybrid methods for efficient simulation of stochastic multi-scale models of tumour growth. JOURNAL OF COMPUTATIONAL PHYSICS 2017; 350:974-991. [PMID: 29200499 PMCID: PMC5656096 DOI: 10.1016/j.jcp.2017.09.019] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/26/2017] [Revised: 09/08/2017] [Accepted: 09/09/2017] [Indexed: 05/10/2023]
Abstract
The development of hybrid methodologies is of current interest in both multi-scale modelling and stochastic reaction-diffusion systems regarding their applications to biology. We formulate a hybrid method for stochastic multi-scale models of cells populations that extends the remit of existing hybrid methods for reaction-diffusion systems. Such method is developed for a stochastic multi-scale model of tumour growth, i.e. population-dynamical models which account for the effects of intrinsic noise affecting both the number of cells and the intracellular dynamics. In order to formulate this method, we develop a coarse-grained approximation for both the full stochastic model and its mean-field limit. Such approximation involves averaging out the age-structure (which accounts for the multi-scale nature of the model) by assuming that the age distribution of the population settles onto equilibrium very fast. We then couple the coarse-grained mean-field model to the full stochastic multi-scale model. By doing so, within the mean-field region, we are neglecting noise in both cell numbers (population) and their birth rates (structure). This implies that, in addition to the issues that arise in stochastic-reaction diffusion systems, we need to account for the age-structure of the population when attempting to couple both descriptions. We exploit our coarse-graining model so that, within the mean-field region, the age-distribution is in equilibrium and we know its explicit form. This allows us to couple both domains consistently, as upon transference of cells from the mean-field to the stochastic region, we sample the equilibrium age distribution. Furthermore, our method allows us to investigate the effects of intracellular noise, i.e. fluctuations of the birth rate, on collective properties such as travelling wave velocity. We show that the combination of population and birth-rate noise gives rise to large fluctuations of the birth rate in the region at the leading edge of front, which cannot be accounted for by the coarse-grained model. Such fluctuations have non-trivial effects on the wave velocity. Beyond the development of a new hybrid method, we thus conclude that birth-rate fluctuations are central to a quantitatively accurate description of invasive phenomena such as tumour growth.
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Affiliation(s)
- Roberto de la Cruz
- Centre de Recerca Matemàtica, Edifici C, Campus de Bellaterra, 08193 Bellaterra (Barcelona), Spain
- Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain
| | - Pilar Guerrero
- Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK
| | - Juan Calvo
- Departmento de Matemática Aplicada, Universidad de Granada, Avda. Fuentenueva s/n, 18071 Granada, Spain
| | - Tomás Alarcón
- Centre de Recerca Matemàtica, Edifici C, Campus de Bellaterra, 08193 Bellaterra (Barcelona), Spain
- Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain
- ICREA, Pg. Lluís Companys 23, 08010 Barcelona, Spain
- Barcelona Graduate School of Mathematics (BGSMath), Barcelona, Spain
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29
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Weistuch C, Pressé S. Spatiotemporal Organization of Catalysts Driven by Enhanced Diffusion. J Phys Chem B 2017; 122:5286-5290. [DOI: 10.1021/acs.jpcb.7b06868] [Citation(s) in RCA: 20] [Impact Index Per Article: 2.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/18/2022]
Affiliation(s)
- C. Weistuch
- Department of Applied Mathematics and Statistics, Stony Brook University, Stony Brook, New York 11794, United States
- Laufer Center for Physical and Quantitative Biology, Stony Brook University, Stony Brook, New York 11794, United States
| | - S. Pressé
- Department of Physics, IUPUI Indianapolis, Indianapolis, Indiana 46202, United States
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30
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Chen M, Li F, Wang S, Cao Y. Stochastic modeling and simulation of reaction-diffusion system with Hill function dynamics. BMC SYSTEMS BIOLOGY 2017; 11:21. [PMID: 28361679 PMCID: PMC5374650 DOI: 10.1186/s12918-017-0401-9] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Indexed: 11/10/2022]
Abstract
Background Stochastic simulation of reaction-diffusion systems presents great challenges for spatiotemporal biological modeling and simulation. One widely used framework for stochastic simulation of reaction-diffusion systems is reaction diffusion master equation (RDME). Previous studies have discovered that for the RDME, when discretization size approaches zero, reaction time for bimolecular reactions in high dimensional domains tends to infinity. Results In this paper, we demonstrate that in the 1D domain, highly nonlinear reaction dynamics given by Hill function may also have dramatic change when discretization size is smaller than a critical value. Moreover, we discuss methods to avoid this problem: smoothing over space, fixed length smoothing over space and a hybrid method. Conclusion Our analysis reveals that the switch-like Hill dynamics reduces to a linear function of discretization size when the discretization size is small enough. The three proposed methods could correctly (under certain precision) simulate Hill function dynamics in the microscopic RDME system.
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Affiliation(s)
- Minghan Chen
- Department of Computer Science, Virginia Tech, Blacksburg, 24061, VA, USA
| | - Fei Li
- Department of Computer Science, Virginia Tech, Blacksburg, 24061, VA, USA
| | - Shuo Wang
- Department of Computer Science, Virginia Tech, Blacksburg, 24061, VA, USA
| | - Young Cao
- Department of Computer Science, Virginia Tech, Blacksburg, 24061, VA, USA.
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31
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Cianci C, Smith S, Grima R. Molecular finite-size effects in stochastic models of equilibrium chemical systems. J Chem Phys 2016; 144:084101. [PMID: 26931675 DOI: 10.1063/1.4941583] [Citation(s) in RCA: 24] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/19/2022] Open
Abstract
The reaction-diffusion master equation (RDME) is a standard modelling approach for understanding stochastic and spatial chemical kinetics. An inherent assumption is that molecules are point-like. Here, we introduce the excluded volume reaction-diffusion master equation (vRDME) which takes into account volume exclusion effects on stochastic kinetics due to a finite molecular radius. We obtain an exact closed form solution of the RDME and of the vRDME for a general chemical system in equilibrium conditions. The difference between the two solutions increases with the ratio of molecular diameter to the compartment length scale. We show that an increase in the fraction of excluded space can (i) lead to deviations from the classical inverse square root law for the noise-strength, (ii) flip the skewness of the probability distribution from right to left-skewed, (iii) shift the equilibrium of bimolecular reactions so that more product molecules are formed, and (iv) strongly modulate the Fano factors and coefficients of variation. These volume exclusion effects are found to be particularly pronounced for chemical species not involved in chemical conservation laws. Finally, we show that statistics obtained using the vRDME are in good agreement with those obtained from Brownian dynamics with excluded volume interactions.
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Affiliation(s)
- Claudia Cianci
- School of Biological Sciences, University of Edinburgh, Mayfield Road Edinburgh EH93JR Scotland,United Kingdom
| | - Stephen Smith
- School of Biological Sciences, University of Edinburgh, Mayfield Road Edinburgh EH93JR Scotland,United Kingdom
| | - Ramon Grima
- School of Biological Sciences, University of Edinburgh, Mayfield Road Edinburgh EH93JR Scotland,United Kingdom
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32
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Winkelmann S, Schütte C. The spatiotemporal master equation: Approximation of reaction-diffusion dynamics via Markov state modeling. J Chem Phys 2016; 145:214107. [DOI: 10.1063/1.4971163] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/25/2023] Open
Affiliation(s)
- Stefanie Winkelmann
- Zuse Institute Berlin (ZIB), Takustraße 7, 14195 Berlin, Germany
- Department of Mathematics, Freie Universität Berlin, Arnimallee 6, 14195 Berlin, Germany
| | - Christof Schütte
- Zuse Institute Berlin (ZIB), Takustraße 7, 14195 Berlin, Germany
- Department of Mathematics, Freie Universität Berlin, Arnimallee 6, 14195 Berlin, Germany
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33
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Li F, Subramanian K, Chen M, Tyson JJ, Cao Y. A stochastic spatiotemporal model of a response-regulator network in the Caulobacter crescentus cell cycle. Phys Biol 2016; 13:035007. [PMID: 27345750 DOI: 10.1088/1478-3975/13/3/035007] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/24/2022]
Abstract
The asymmetric cell division cycle in Caulobacter crescentus is controlled by an elaborate molecular mechanism governing the production, activation and spatial localization of a host of interacting proteins. In previous work, we proposed a deterministic mathematical model for the spatiotemporal dynamics of six major regulatory proteins. In this paper, we study a stochastic version of the model, which takes into account molecular fluctuations of these regulatory proteins in space and time during early stages of the cell cycle of wild-type Caulobacter cells. We test the stochastic model with regard to experimental observations of increased variability of cycle time in cells depleted of the divJ gene product. The deterministic model predicts that overexpression of the divK gene blocks cell cycle progression in the stalked stage; however, stochastic simulations suggest that a small fraction of the mutants cells do complete the cell cycle normally.
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Affiliation(s)
- Fei Li
- Departments of Computer Science, Virginia Tech, Blacksburg, VA 24061,USA
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34
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Smith S, Grima R. Breakdown of the reaction-diffusion master equation with nonelementary rates. Phys Rev E 2016; 93:052135. [PMID: 27300857 DOI: 10.1103/physreve.93.052135] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/21/2015] [Indexed: 11/07/2022]
Abstract
The chemical master equation (CME) is the exact mathematical formulation of chemical reactions occurring in a dilute and well-mixed volume. The reaction-diffusion master equation (RDME) is a stochastic description of reaction-diffusion processes on a spatial lattice, assuming well mixing only on the length scale of the lattice. It is clear that, for the sake of consistency, the solution of the RDME of a chemical system should converge to the solution of the CME of the same system in the limit of fast diffusion: Indeed, this has been tacitly assumed in most literature concerning the RDME. We show that, in the limit of fast diffusion, the RDME indeed converges to a master equation but not necessarily the CME. We introduce a class of propensity functions, such that if the RDME has propensities exclusively of this class, then the RDME converges to the CME of the same system, whereas if the RDME has propensities not in this class, then convergence is not guaranteed. These are revealed to be elementary and nonelementary propensities, respectively. We also show that independent of the type of propensity, the RDME converges to the CME in the simultaneous limit of fast diffusion and large volumes. We illustrate our results with some simple example systems and argue that the RDME cannot generally be an accurate description of systems with nonelementary rates.
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Affiliation(s)
- Stephen Smith
- School of Biological Sciences, University of Edinburgh, Mayfield Road, Edinburgh EH9 3JR, Scotland, United Kingdom
| | - Ramon Grima
- School of Biological Sciences, University of Edinburgh, Mayfield Road, Edinburgh EH9 3JR, Scotland, United Kingdom
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35
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Spill F, Maini PK, Byrne HM. Optimisation of simulations of stochastic processes by removal of opposing reactions. J Chem Phys 2016; 144:084105. [DOI: 10.1063/1.4942413] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Fabian Spill
- Department of Biomedical Engineering, Boston University, 44 Cummington Street, Boston, Massachusetts 02215, USA
- Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA
| | - Philip K. Maini
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom
| | - Helen M. Byrne
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom
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36
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Hellander S, Petzold L. Reaction rates for a generalized reaction-diffusion master equation. Phys Rev E 2016; 93:013307. [PMID: 26871190 DOI: 10.1103/physreve.93.013307] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/25/2015] [Indexed: 11/07/2022]
Abstract
It has been established that there is an inherent limit to the accuracy of the reaction-diffusion master equation. Specifically, there exists a fundamental lower bound on the mesh size, below which the accuracy deteriorates as the mesh is refined further. In this paper we extend the standard reaction-diffusion master equation to allow molecules occupying neighboring voxels to react, in contrast to the traditional approach, in which molecules react only when occupying the same voxel. We derive reaction rates, in two dimensions as well as three dimensions, to obtain an optimal match to the more fine-grained Smoluchowski model and show in two numerical examples that the extended algorithm is accurate for a wide range of mesh sizes, allowing us to simulate systems that are intractable with the standard reaction-diffusion master equation. In addition, we show that for mesh sizes above the fundamental lower limit of the standard algorithm, the generalized algorithm reduces to the standard algorithm. We derive a lower limit for the generalized algorithm which, in both two dimensions and three dimensions, is of the order of the reaction radius of a reacting pair of molecules.
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Affiliation(s)
- Stefan Hellander
- Department of Computer Science, University of California, Santa Barbara, Santa Barbara, California 93106-5070, USA
| | - Linda Petzold
- Department of Computer Science, University of California, Santa Barbara, Santa Barbara, California 93106-5070, USA
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37
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Spill F, Guerrero P, Alarcon T, Maini PK, Byrne H. Hybrid approaches for multiple-species stochastic reaction-diffusion models. JOURNAL OF COMPUTATIONAL PHYSICS 2015; 299:429-445. [PMID: 26478601 PMCID: PMC4554296 DOI: 10.1016/j.jcp.2015.07.002] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/12/2014] [Revised: 06/30/2015] [Accepted: 07/01/2015] [Indexed: 05/25/2023]
Abstract
Reaction-diffusion models are used to describe systems in fields as diverse as physics, chemistry, ecology and biology. The fundamental quantities in such models are individual entities such as atoms and molecules, bacteria, cells or animals, which move and/or react in a stochastic manner. If the number of entities is large, accounting for each individual is inefficient, and often partial differential equation (PDE) models are used in which the stochastic behaviour of individuals is replaced by a description of the averaged, or mean behaviour of the system. In some situations the number of individuals is large in certain regions and small in others. In such cases, a stochastic model may be inefficient in one region, and a PDE model inaccurate in another. To overcome this problem, we develop a scheme which couples a stochastic reaction-diffusion system in one part of the domain with its mean field analogue, i.e. a discretised PDE model, in the other part of the domain. The interface in between the two domains occupies exactly one lattice site and is chosen such that the mean field description is still accurate there. In this way errors due to the flux between the domains are small. Our scheme can account for multiple dynamic interfaces separating multiple stochastic and deterministic domains, and the coupling between the domains conserves the total number of particles. The method preserves stochastic features such as extinction not observable in the mean field description, and is significantly faster to simulate on a computer than the pure stochastic model.
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Affiliation(s)
- Fabian Spill
- Department of Biomedical Engineering, Boston University, 44 Cummington Street, Boston, MA 02215, USA
- Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
| | - Pilar Guerrero
- Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK
| | - Tomas Alarcon
- Centre de Recerca Matematica, Campus de Bellaterra, Edifici C, 08193 Bellaterra (Barcelona), Spain
- Departament de Matemàtiques, Universitat Atonòma de Barcelona, 08193 Bellaterra (Barcelona), Spain
| | - Philip K. Maini
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK
| | - Helen Byrne
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK
- Computational Biology Group, Department of Computer Science, University of Oxford, Oxford OX1 3QD, UK
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38
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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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39
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Hellander S, Hellander A, Petzold L. Reaction rates for mesoscopic reaction-diffusion kinetics. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:023312. [PMID: 25768640 PMCID: PMC4854576 DOI: 10.1103/physreve.91.023312] [Citation(s) in RCA: 26] [Impact Index Per Article: 2.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/01/2014] [Indexed: 05/25/2023]
Abstract
The mesoscopic reaction-diffusion master equation (RDME) is a popular modeling framework frequently applied to stochastic reaction-diffusion kinetics in systems biology. The RDME is derived from assumptions about the underlying physical properties of the system, and it may produce unphysical results for models where those assumptions fail. In that case, other more comprehensive models are better suited, such as hard-sphere Brownian dynamics (BD). Although the RDME is a model in its own right, and not inferred from any specific microscale model, it proves useful to attempt to approximate a microscale model by a specific choice of mesoscopic reaction rates. In this paper we derive mesoscopic scale-dependent reaction rates by matching certain statistics of the RDME solution to statistics of the solution of a widely used microscopic BD model: the Smoluchowski model with a Robin boundary condition at the reaction radius of two molecules. We also establish fundamental limits on the range of mesh resolutions for which this approach yields accurate results and show both theoretically and in numerical examples that as we approach the lower fundamental limit, the mesoscopic dynamics approach the microscopic dynamics. We show that for mesh sizes below the fundamental lower limit, results are less accurate. Thus, the lower limit determines the mesh size for which we obtain the most accurate results.
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Affiliation(s)
- Stefan Hellander
- Department of Computer Science, University of California, Santa Barbara, California 93106-5070, Santa Barbara, USA
| | - Andreas Hellander
- Department of Information Technology, Uppsala University, Box 337, SE-75105, Uppsala, Sweden
| | - Linda Petzold
- Department of Computer Science, University of California, Santa Barbara, California 93106-5070, Santa Barbara, USA
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40
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Taylor PR, Baker RE, Yates CA. Deriving appropriate boundary conditions, and accelerating position-jump simulations, of diffusion using non-local jumping. Phys Biol 2014; 12:016006. [PMID: 25514045 DOI: 10.1088/1478-3975/12/1/016006] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
Abstract
In this paper we explore lattice-based position-jump models of diffusion, and the implications of introducing non-local jumping; particles can jump to a range of nearby boxes rather than only to their nearest neighbours. We begin by deriving conditions for equivalence with traditional local jumping models in the continuum limit. We then generalize a previously postulated implementation of the Robin boundary condition for a non-local process of arbitrary maximum jump length, and present a novel implementation of flux boundary conditions, again generalized for a non-local process of arbitrary maximum jump length. In both these cases we validate our results using stochastic simulation. We then proceed to consider two variations on the basic diffusion model: a hybrid local/non-local scheme suitable for models involving sharp concentration gradients, and the implementation of biased jumping. In all cases we show that non-local jumping can deliver substantial time savings for stochastic simulations.
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Affiliation(s)
- P R Taylor
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK
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41
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MacNamara S. Multiscale modeling of dorsoventral patterning in Drosophila. Semin Cell Dev Biol 2014; 35:82-9. [PMID: 25047722 DOI: 10.1016/j.semcdb.2014.07.001] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/16/2014] [Revised: 07/02/2014] [Accepted: 07/02/2014] [Indexed: 10/25/2022]
Abstract
The role of mathematical models of signaling networks is showcased by examples from Drosophila development. Three models of consecutive stages in dorsoventral patterning are presented. We begin with a compartmental model of intracellular reactions that generates a gradient of nuclear-localized Dorsal, exhibiting constant shape and dynamic amplitude. A simple thermodynamic model of equilibrium binding explains how a spatially uniform transcription factor, Zelda, can act in combination with a graded factor, Dorsal, to cooperatively regulate gene expression borders. Finally, we formulate a dynamic and stochastic model that predicts spatiotemporal patterns of Sog expression based on known patterns of its transcription factor, Dorsal. The future of coupling multifarious models across multiple temporal and spatial scales is discussed.
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Affiliation(s)
- Shev MacNamara
- Massachusetts Institute of Technology, Cambridge, MA 02139, United States.
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42
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Agbanusi IC, Isaacson SA. A comparison of bimolecular reaction models for stochastic reaction-diffusion systems. Bull Math Biol 2013; 76:922-46. [PMID: 23579988 DOI: 10.1007/s11538-013-9833-6] [Citation(s) in RCA: 23] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/03/2013] [Accepted: 03/07/2013] [Indexed: 11/30/2022]
Abstract
Stochastic reaction-diffusion models have become an important tool in studying how both noise in the chemical reaction process and the spatial movement of molecules influences the behavior of biological systems. There are two primary spatially-continuous models that have been used in recent studies: the diffusion limited reaction model of Smoluchowski, and a second approach popularized by Doi. Both models treat molecules as points undergoing Brownian motion. The former represents chemical reactions between two reactants through the use of reactive boundary conditions, with two molecules reacting instantly upon reaching a fixed separation (called the reaction-radius). The Doi model uses reaction potentials, whereby two molecules react with a fixed probability per unit time, λ, when separated by less than the reaction radius. In this work, we study the rigorous relationship between the two models. For the special case of a protein diffusing to a fixed DNA binding site, we prove that the solution to the Doi model converges to the solution of the Smoluchowski model as λ→∞, with a rigorous [Formula: see text] error bound (for any fixed ϵ>0). We investigate by numerical simulation, for biologically relevant parameter values, the difference between the solutions and associated reaction time statistics of the two models. As the reaction-radius is decreased, for sufficiently large but fixed values of λ, these differences are found to increase like the inverse of the binding radius.
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Affiliation(s)
- I C Agbanusi
- Department of Mathematics and Statistics, Boston University, 111 Cummington St., Boston, MA, 02215, USA,
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