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Chae SJ, Shin S, Lee K, Lee S, Kim JK. From homogeneity to heterogeneity: Refining stochastic simulations of gene regulation. Comput Struct Biotechnol J 2025; 27:411-422. [PMID: 39906159 PMCID: PMC11791169 DOI: 10.1016/j.csbj.2025.01.004] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/12/2024] [Revised: 01/01/2025] [Accepted: 01/04/2025] [Indexed: 02/06/2025] Open
Abstract
Cellular processes are intricately controlled through gene regulation, which is significantly influenced by intrinsic noise due to the small number of molecules involved. The Gillespie algorithm, a widely used stochastic simulation method, is pervasively employed to model these systems. However, this algorithm typically assumes that DNA is homogeneously distributed throughout the nucleus, which is not realistic. In this study, we evaluated whether stochastic simulations based on the assumption of spatial homogeneity can accurately capture the dynamics of gene regulation. Our findings indicate that when transcription factors diffuse slowly, these simulations fail to accurately capture gene expression, highlighting the necessity to account for spatial heterogeneity. However, incorporating spatial heterogeneity considerably increases computational time. To address this, we explored various stochastic quasi-steady-state approximations (QSSAs) that simplify the model and reduce simulation time. While both the stochastic total quasi-steady state approximation (stQSSA) and the stochastic low-state quasi-steady-state approximation (slQSSA) reduced simulation time, only the slQSSA provided an accurate model reduction. Our study underscores the importance of utilizing appropriate methods for efficient and accurate stochastic simulations of gene regulatory dynamics, especially when incorporating spatial heterogeneity.
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Affiliation(s)
- Seok Joo Chae
- Department of Mathematical Sciences, KAIST, Daejeon, 34141, Republic of Korea
- Biomedical Mathematics group, Pioneer Research Center for Mathematical and Computational Sciences, Institute for Basic Science, Daejeon, 34126, Republic of Korea
- Department of Bioengineering, Rice University, Houston, 77005, TX, United States of America
| | - Seolah Shin
- Biomedical Mathematics group, Pioneer Research Center for Mathematical and Computational Sciences, Institute for Basic Science, Daejeon, 34126, Republic of Korea
- Department of Applied Mathematics, Korea University, Seoul, 02841, Republic of Korea
| | - Kangmin Lee
- Department of Mathematical Sciences, KAIST, Daejeon, 34141, Republic of Korea
- Biomedical Mathematics group, Pioneer Research Center for Mathematical and Computational Sciences, Institute for Basic Science, Daejeon, 34126, Republic of Korea
| | - Seunggyu Lee
- Biomedical Mathematics group, Pioneer Research Center for Mathematical and Computational Sciences, Institute for Basic Science, Daejeon, 34126, Republic of Korea
- Department of Applied Mathematics, Korea University, Seoul, 02841, Republic of Korea
| | - Jae Kyoung Kim
- Department of Mathematical Sciences, KAIST, Daejeon, 34141, Republic of Korea
- Biomedical Mathematics group, Pioneer Research Center for Mathematical and Computational Sciences, Institute for Basic Science, Daejeon, 34126, Republic of Korea
- Department of Medicine, College of Medicine, Korea University, Seoul, 02841, Republic of Korea
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Erban R, Winkelmann S. Multi-Grid Reaction-Diffusion Master Equation: Applications to Morphogen Gradient Modelling. Bull Math Biol 2024; 87:6. [PMID: 39601934 PMCID: PMC11602816 DOI: 10.1007/s11538-024-01377-y] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/03/2024] [Accepted: 10/21/2024] [Indexed: 11/29/2024]
Abstract
The multi-grid reaction-diffusion master equation (mgRDME) provides a generalization of stochastic compartment-based reaction-diffusion modelling described by the standard reaction-diffusion master equation (RDME). By enabling different resolutions on lattices for biochemical species with different diffusion constants, the mgRDME approach improves both accuracy and efficiency of compartment-based reaction-diffusion simulations. The mgRDME framework is examined through its application to morphogen gradient formation in stochastic reaction-diffusion scenarios, using both an analytically tractable first-order reaction network and a model with a second-order reaction. The results obtained by the mgRDME modelling are compared with the standard RDME model and with the (more detailed) particle-based Brownian dynamics simulations. The dependence of error and numerical cost on the compartment sizes is defined and investigated through a multi-objective optimization problem.
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Affiliation(s)
- Radek Erban
- Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK.
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Matveev VV. Close agreement between deterministic versus stochastic modeling of first-passage time to vesicle fusion. Biophys J 2022; 121:4569-4584. [PMID: 36815708 PMCID: PMC9748373 DOI: 10.1016/j.bpj.2022.10.033] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/30/2022] [Revised: 09/13/2022] [Accepted: 10/24/2022] [Indexed: 11/22/2022] Open
Abstract
Ca2+-dependent cell processes, such as neurotransmitter or endocrine vesicle fusion, are inherently stochastic due to large fluctuations in Ca2+ channel gating, Ca2+ diffusion, and Ca2+ binding to buffers and target sensors. However, previous studies revealed closer-than-expected agreement between deterministic and stochastic simulations of Ca2+ diffusion, buffering, and sensing if Ca2+ channel gating is not Ca2+ dependent. To understand this result more fully, we present a comparative study complementing previous work, focusing on Ca2+ dynamics downstream of Ca2+ channel gating. Specifically, we compare deterministic (mean-field/mass-action) and stochastic simulations of vesicle exocytosis latency, quantified by the probability density of the first-passage time (FPT) to the Ca2+-bound state of a vesicle fusion sensor, following a brief Ca2+ current pulse. We show that under physiological constraints, the discrepancy between FPT densities obtained using the two approaches remains small even if as few as ∼50 Ca2+ ions enter per single channel-vesicle release unit. Using a reduced two-compartment model for ease of analysis, we illustrate how this close agreement arises from the smallness of correlations between fluctuations of the reactant molecule numbers, despite the large magnitude of fluctuation amplitudes. This holds if all relevant reactions are heteroreaction between molecules of different species, as is the case for bimolecular Ca2+ binding to buffers and downstream sensor targets. In this case, diffusion and buffering effectively decorrelate the state of the Ca2+ sensor from local Ca2+ fluctuations. Thus, fluctuations in the Ca2+ sensor's state underlying the FPT distribution are only weakly affected by the fluctuations in the local Ca2+ concentration around its average, deterministically computable value.
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Affiliation(s)
- Victor V Matveev
- Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, New Jersey.
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Smith CA, Yates CA. Incorporating domain growth into hybrid methods for reaction-diffusion systems. J R Soc Interface 2021; 18:20201047. [PMID: 33849339 DOI: 10.1098/rsif.2020.1047] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
Reaction-diffusion mechanisms are a robust paradigm that can be used to represent many biological and physical phenomena over multiple spatial scales. Applications include intracellular dynamics, the migration of cells and the patterns formed by vegetation in semi-arid landscapes. Moreover, domain growth is an important process for embryonic growth and wound healing. There are many numerical modelling frameworks capable of simulating such systems on growing domains; however, each of these may be well suited to different spatial scales and particle numbers. Recently, spatially extended hybrid methods on static domains have been produced to bridge the gap between these different modelling paradigms in order to represent multi-scale phenomena. However, such methods have not been developed with domain growth in mind. In this paper, we develop three hybrid methods on growing domains, extending three of the prominent static-domain hybrid methods. We also provide detailed algorithms to allow others to employ them. We demonstrate that the methods are able to accurately model three representative reaction-diffusion systems accurately and without bias.
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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
| | - Christian A Yates
- Centre for Mathematical Biology, Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK
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KhudaBukhsh WR, Kang HW, Kenah E, Rempała GA. Incorporating age and delay into models for biophysical systems. Phys Biol 2021; 18:015002. [PMID: 33075757 PMCID: PMC9211760 DOI: 10.1088/1478-3975/abc2ab] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/01/2023]
Abstract
In many biological systems, chemical reactions or changes in a physical state are assumed to occur instantaneously. For describing the dynamics of those systems, Markov models that require exponentially distributed inter-event times have been used widely. However, some biophysical processes such as gene transcription and translation are known to have a significant gap between the initiation and the completion of the processes, which renders the usual assumption of exponential distribution untenable. In this paper, we consider relaxing this assumption by incorporating age-dependent random time delays (distributed according to a given probability distribution) into the system dynamics. We do so by constructing a measure-valued Markov process on a more abstract state space, which allows us to keep track of the 'ages' of molecules participating in a chemical reaction. We study the large-volume limit of such age-structured systems. We show that, when appropriately scaled, the stochastic system can be approximated by a system of partial differential equations (PDEs) in the large-volume limit, as opposed to ordinary differential equations (ODEs) in the classical theory. We show how the limiting PDE system can be used for the purpose of further model reductions and for devising efficient simulation algorithms. In order to describe the ideas, we use a simple transcription process as a running example. We, however, note that the methods developed in this paper apply to a wide class of biophysical systems.
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Affiliation(s)
- Wasiur R KhudaBukhsh
- Mathematical Biosciences Institute and the College of Public Health, The Ohio State University, 1735 Neil Avenue, Columbus OH 43210, United States of America
| | - Hye-Won Kang
- Department of Mathematics and Statistics, University of Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore MD 21250, United States of America
| | - Eben Kenah
- Division of Biostatistics, College of Public Health, The Ohio State University, 1841 Neil Avenue, Columbus OH 43210, United States of America
| | - Grzegorz A Rempała
- Mathematical Biosciences Institute and the College of Public Health, The Ohio State University, 1735 Neil Avenue, Columbus OH 43210, United States of America
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Cao Y, Linda P, Seitaridou E. Stochastic Simulation of Biochemical Systems: In Memory of Dan T. Gillespie's contributions. Bull Math Biol 2019; 81:2819-2821. [PMID: 31264134 DOI: 10.1007/s11538-019-00633-w] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/24/2022]
Affiliation(s)
- Yang Cao
- Department of Computer Science, Virginia Tech, Blacksburg, VA, 24061, USA.
| | - Petzold Linda
- Department of Computer Science and Mechanical Engineering, University of California at Santa Barbara, Santa Barbara, CA, 93106, USA
| | - Effrosyni Seitaridou
- Department of Physics, Oxford College of Emory University, 100 Hamill St., Oxford, GA, 30054, USA
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