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Rao C, Lin W, Song Z. Analytical refractory period distribution for a class of time-variant biochemical systems with second-order reactions. J Chem Phys 2023; 159:124105. [PMID: 38127379 DOI: 10.1063/5.0156276] [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: 04/28/2023] [Accepted: 09/05/2023] [Indexed: 12/23/2023] Open
Abstract
Refractory period (RP), the waiting time between signals, can induce complex signaling dynamics, such as acceleration, adaptation, and oscillation, within many cellular biochemical networks. However, its underlying molecular mechanisms are still unclear. Rigorously estimating the RP distribution may be essential to identify its causal regulatory mechanisms. Traditional methods of estimating the RP distribution depend on solving the underlying Chemical Master Equations (CMEs), the dominant modeling formalism of biochemical systems. However, exact solutions of the CME are only known for simple reaction systems with zero- and first-order reactions or specific systems with second-order reactions. General solutions still need to be derived for systems with bimolecular reactions. It is even more challenging if large state-space and nonconstant reaction rates are involved. Here, we developed a direct method to gain the analytical RP distribution for a class of second-order reaction systems with nonconstant reaction rates and large state space. Instead of using the CME, we used an equivalent path-wise representation, which is the solution to a transformed martingale problem of the CME. This allowed us to bypass solving a CME. We then applied the method to derive the analytical RP distribution of a real complex biochemical network with second-order reactions, the Drosophila phototransduction cascade. Our approach provides an alternative to the CMEs in deriving the analytical RP distributions of a class of second-order reaction systems. Since the bimolecular reactions are common in biological systems, our approach could enhance understanding real-world biochemical processes.
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Affiliation(s)
- Changqian Rao
- School of Mathematical Sciences and Shanghai Center for Mathematical Sciences, Fudan University, 200433 Shanghai, China
- Research Institute of Intelligent Complex Systems, Fudan University, 200433 Shanghai, China
| | - Wei Lin
- School of Mathematical Sciences and Shanghai Center for Mathematical Sciences, Fudan University, 200433 Shanghai, China
- Research Institute of Intelligent Complex Systems, Fudan University, 200433 Shanghai, China
- Institute of Science and Technology for Brain-Inspired Intelligence, Fudan University, 200433 Shanghai, China
- State Key Laboratory of Medical Neurobiology and MOE Frontiers Center for Brain Science, Institute of Brain Science, Fudan University, 200032 Shanghai, China
| | - Zhuoyi Song
- Institute of Science and Technology for Brain-Inspired Intelligence, Fudan University, 200433 Shanghai, China
- State Key Laboratory of Medical Neurobiology and MOE Frontiers Center for Brain Science, Institute of Brain Science, Fudan University, 200032 Shanghai, China
- Key Laboratory of Computational Neuroscience and Brain-Inspired Intelligence (Fudan University), Ministry of Education, Shanghai, China
- Zhangjiang Fudan International Innovation Center, Shanghai, China
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2
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Waizmann T, Bortolussi L, Vandin A, Tribastone M. Improved estimations of stochastic chemical kinetics by finite-state expansion. Proc Math Phys Eng Sci 2021. [DOI: 10.1098/rspa.2020.0964] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
Stochastic reaction networks are a fundamental model to describe interactions between species where random fluctuations are relevant. The master equation provides the evolution of the probability distribution across the discrete state space consisting of vectors of population counts for each species. However, since its exact solution is often elusive, several analytical approximations have been proposed. The deterministic rate equation (DRE) gives a macroscopic approximation as a compact system of differential equations that estimate the average populations for each species, but it may be inaccurate in the case of nonlinear interaction dynamics. Here we propose finite-state expansion (FSE), an analytical method mediating between the microscopic and the macroscopic interpretations of a stochastic reaction network by coupling the master equation dynamics of a chosen subset of the discrete state space with the mean population dynamics of the DRE. An algorithm translates a network into an expanded one where each discrete state is represented as a further distinct species. This translation exactly preserves the stochastic dynamics, but the DRE of the expanded network can be interpreted as a correction to the original one. The effectiveness of FSE is demonstrated in models that challenge state-of-the-art techniques due to intrinsic noise, multi-scale populations and multi-stability.
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Affiliation(s)
| | - Luca Bortolussi
- Department of Mathematics and Geosciences, University of Trieste, Trieste 34127, Italy
| | - Andrea Vandin
- Sant’Anna School of Advanced Studies, Pisa 56127, Italy
- Department of Applied Mathematics and Computer Science, DTU Technical University of Denmark, Kgs. Lyngby 2800, Denmark
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3
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Loper J, Zhou G, Geman S. Capacities and efficient computation of first-passage probabilities. Phys Rev E 2020; 102:023304. [PMID: 32942394 DOI: 10.1103/physreve.102.023304] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/14/2019] [Accepted: 06/12/2020] [Indexed: 06/11/2023]
Abstract
A reversible diffusion process is initialized at position x_{0} and run until it first hits any of several targets. What is the probability that it terminates at a particular target? We propose a computationally efficient approach for estimating this probability, focused on those situations in which it takes a long time to hit any target. In these cases, direct simulation of the hitting probabilities becomes prohibitively expensive. On the other hand, if the timescales are sufficiently long, then the system will essentially "forget" its initial condition before it encounters a target. In these cases the hitting probabilities can be accurately approximated using only local simulations around each target, obviating the need for direct simulations. In empirical tests, we find that these local estimates can be computed in the same time it would take to compute a single direct simulation, but that they achieve an accuracy that would require thousands of direct simulation runs.
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Affiliation(s)
- Jackson Loper
- Data Science Institute, Columbia University, 10027 New York, New York, USA
| | - Guangyao Zhou
- Division of Applied Mathematics, Brown University, Providence, 02912 Rhode Island, USA
| | - Stuart Geman
- Division of Applied Mathematics, Brown University, Providence, 02912 Rhode Island, USA
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4
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Bast L, Calzolari F, Strasser MK, Hasenauer J, Theis FJ, Ninkovic J, Marr C. Increasing Neural Stem Cell Division Asymmetry and Quiescence Are Predicted to Contribute to the Age-Related Decline in Neurogenesis. Cell Rep 2019; 25:3231-3240.e8. [PMID: 30566852 DOI: 10.1016/j.celrep.2018.11.088] [Citation(s) in RCA: 26] [Impact Index Per Article: 5.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/04/2017] [Revised: 07/31/2018] [Accepted: 11/21/2018] [Indexed: 02/06/2023] Open
Abstract
Adult murine neural stem cells (NSCs) generate neurons in drastically declining numbers with age. How cellular dynamics sustain neurogenesis and how alterations with age may result in this decline are unresolved issues. We therefore clonally traced NSC lineages using confetti reporters in young and middle-aged adult mice. To understand the underlying mechanisms, we derived mathematical models that explain observed clonal cell type abundances. The best models consistently show self-renewal of transit-amplifying progenitors and rapid neuroblast cell cycle exit. In middle-aged mice, we identified an increased probability of asymmetric stem cell divisions at the expense of symmetric differentiation, accompanied by an extended persistence of quiescence between activation phases. Our model explains existing longitudinal population data and identifies particular cellular properties underlying adult NSC homeostasis and the aging of this stem cell compartment.
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Affiliation(s)
- Lisa Bast
- Institute of Computational Biology, Helmholtz Zentrum München-German Research Center for Environmental Health, Neuherberg, Germany; Department of Mathematics, Chair of Mathematical Modeling of Biological Systems, Technische Universität München, Garching, Germany
| | - Filippo Calzolari
- Institute for Physiological Chemistry, University Medical Center of the Johannes Gutenberg University Mainz, Mainz, Germany; Institute of Stem Cell Research, Helmholtz Zentrum München-German Research Center for Environmental Health, Neuherberg, Germany; Department of Physiological Genomics, Ludwig-Maximilians University Munich, Munich, Germany.
| | - Michael K Strasser
- Institute of Computational Biology, Helmholtz Zentrum München-German Research Center for Environmental Health, Neuherberg, Germany
| | - Jan Hasenauer
- Institute of Computational Biology, Helmholtz Zentrum München-German Research Center for Environmental Health, Neuherberg, Germany; Department of Mathematics, Chair of Mathematical Modeling of Biological Systems, Technische Universität München, Garching, Germany
| | - Fabian J Theis
- Institute of Computational Biology, Helmholtz Zentrum München-German Research Center for Environmental Health, Neuherberg, Germany; Department of Mathematics, Chair of Mathematical Modeling of Biological Systems, Technische Universität München, Garching, Germany
| | - Jovica Ninkovic
- Institute of Stem Cell Research, Helmholtz Zentrum München-German Research Center for Environmental Health, Neuherberg, Germany; Department of Physiological Genomics, Ludwig-Maximilians University Munich, Munich, Germany; Department for Cell Biology and Anatomy, Biomedical Center of LMU, Ludwig-Maximilians University Munich, Munich, Germany.
| | - Carsten Marr
- Institute of Computational Biology, Helmholtz Zentrum München-German Research Center for Environmental Health, Neuherberg, Germany.
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Muñoz-Cobo JL, Berna C. Chemical Kinetics Roots and Methods to Obtain the Probability Distribution Function Evolution of Reactants and Products in Chemical Networks Governed by a Master Equation. ENTROPY 2019; 21:e21020181. [PMID: 33266897 PMCID: PMC7514663 DOI: 10.3390/e21020181] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Subscribe] [Scholar Register] [Received: 01/24/2019] [Accepted: 02/11/2019] [Indexed: 11/16/2022]
Abstract
In this paper first, we review the physical root bases of chemical reaction networks as a Markov process in multidimensional vector space. Then we study the chemical reactions from a microscopic point of view, to obtain the expression for the propensities for the different reactions that can happen in the network. These chemical propensities, at a given time, depend on the system state at that time, and do not depend on the state at an earlier time indicating that we are dealing with Markov processes. Then the Chemical Master Equation (CME) is deduced for an arbitrary chemical network from a probability balance and it is expressed in terms of the reaction propensities. This CME governs the dynamics of the chemical system. Due to the difficulty to solve this equation two methods are studied, the first one is the probability generating function method or z-transform, which permits to obtain the evolution of the factorial moment of the system with time in an easiest way or after some manipulation the evolution of the polynomial moments. The second method studied is the expansion of the CME in terms of an order parameter (system volume). In this case we study first the expansion of the CME using the propensities obtained previously and splitting the molecular concentration into a deterministic part and a random part. An expression in terms of multinomial coefficients is obtained for the evolution of the probability of the random part. Then we study how to reconstruct the probability distribution from the moments using the maximum entropy principle. Finally, the previous methods are applied to simple chemical networks and the consistency of these methods is studied.
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Affiliation(s)
- José-Luis Muñoz-Cobo
- Department of Chemical and Nuclear Engineering, Universitat Politècnica de València, 46022 Valencia, Spain
- Instituto Universitario de Ingeniería Energética, Universitat Politècnica de València, 46022 Valencia, Spain
- Correspondence: ; Tel.: +34-96-387-7631
| | - Cesar Berna
- Instituto Universitario de Ingeniería Energética, Universitat Politècnica de València, 46022 Valencia, Spain
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Solving Stochastic Reaction Networks with Maximum Entropy Lagrange Multipliers. ENTROPY 2018; 20:e20090700. [PMID: 33265789 PMCID: PMC7513230 DOI: 10.3390/e20090700] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 08/03/2018] [Revised: 09/05/2018] [Accepted: 09/06/2018] [Indexed: 11/16/2022]
Abstract
The time evolution of stochastic reaction networks can be modeled with the chemical master equation of the probability distribution. Alternatively, the numerical problem can be reformulated in terms of probability moment equations. Herein we present a new alternative method for numerically solving the time evolution of stochastic reaction networks. Based on the assumption that the entropy of the reaction network is maximum, Lagrange multipliers are introduced. The proposed method derives equations that model the time derivatives of these Lagrange multipliers. We present detailed steps to transform moment equations to Lagrange multiplier equations. In order to demonstrate the method, we present examples of non-linear stochastic reaction networks of varying degrees of complexity, including multistable and oscillatory systems. We find that the new approach is as accurate and significantly more efficient than Gillespie’s original exact algorithm for systems with small number of interacting species. This work is a step towards solving stochastic reaction networks accurately and efficiently.
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Vlysidis M, Kaznessis YN. On Differences between Deterministic and Stochastic Models of Chemical Reactions: Schlögl Solved with ZI-Closure. ENTROPY 2018; 20:e20090678. [PMID: 33265767 PMCID: PMC7513203 DOI: 10.3390/e20090678] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 07/03/2018] [Revised: 08/29/2018] [Accepted: 09/04/2018] [Indexed: 11/16/2022]
Abstract
Deterministic and stochastic models of chemical reaction kinetics can give starkly different results when the deterministic model exhibits more than one stable solution. For example, in the stochastic Schlögl model, the bimodal stationary probability distribution collapses to a unimodal distribution when the system size increases, even for kinetic constant values that result in two distinct stable solutions in the deterministic Schlögl model. Using zero-information (ZI) closure scheme, an algorithm for solving chemical master equations, we compute stationary probability distributions for varying system sizes of the Schlögl model. With ZI-closure, system sizes can be studied that have been previously unattainable by stochastic simulation algorithms. We observe and quantify paradoxical discrepancies between stochastic and deterministic models and explain this behavior by postulating that the entropy of non-equilibrium steady states (NESS) is maximum.
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8
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Identifiability and Reconstruction of Biochemical Reaction Networks from Population Snapshot Data. Processes (Basel) 2018. [DOI: 10.3390/pr6090136] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/17/2022] Open
Abstract
Inference of biochemical network models from experimental data is a crucial problem in systems and synthetic biology that includes parameter calibration but also identification of unknown interactions. Stochastic modelling from single-cell data is known to improve identifiability of reaction network parameters for specific systems. However, general results are lacking, and the advantage over deterministic, population-average approaches has not been explored for network reconstruction. In this work, we study identifiability and propose new reconstruction methods for biochemical interaction networks. Focusing on population-snapshot data and networks with reaction rates affine in the state, for parameter estimation, we derive general methods to test structural identifiability and demonstrate them in connection with practical identifiability for a reporter gene in silico case study. In the same framework, we next develop a two-step approach to the reconstruction of unknown networks of interactions. We apply it to compare the achievable network reconstruction performance in a deterministic and a stochastic setting, showing the advantage of the latter, and demonstrate it on population-snapshot data from a simulated example.
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9
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Dowdy GR, Barton PI. Dynamic bounds on stochastic chemical kinetic systems using semidefinite programming. J Chem Phys 2018; 149:074103. [DOI: 10.1063/1.5029926] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/22/2023] Open
Affiliation(s)
- Garrett R. Dowdy
- Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
| | - Paul I. Barton
- Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
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10
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Vlysidis M, Kaznessis YN. A linearization method for probability moment equations. Comput Chem Eng 2018. [DOI: 10.1016/j.compchemeng.2018.01.015] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/18/2022]
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11
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Dowdy GR, Barton PI. Bounds on stochastic chemical kinetic systems at steady state. J Chem Phys 2018; 148:084106. [DOI: 10.1063/1.5009950] [Citation(s) in RCA: 14] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Garrett R. Dowdy
- Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
| | - Paul I. Barton
- Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
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12
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Price DJ, Breuzé A, Dybowski R, Mastroeni P, Restif O. An efficient moments-based inference method for within-host bacterial infection dynamics. PLoS Comput Biol 2017; 13:e1005841. [PMID: 29155811 PMCID: PMC5714343 DOI: 10.1371/journal.pcbi.1005841] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/18/2017] [Revised: 12/04/2017] [Accepted: 10/22/2017] [Indexed: 11/18/2022] Open
Abstract
Over the last ten years, isogenic tagging (IT) has revolutionised the study of bacterial infection dynamics in laboratory animal models. However, quantitative analysis of IT data has been hindered by the piecemeal development of relevant statistical models. The most promising approach relies on stochastic Markovian models of bacterial population dynamics within and among organs. Here we present an efficient numerical method to fit such stochastic dynamic models to in vivo experimental IT data. A common approach to statistical inference with stochastic dynamic models relies on producing large numbers of simulations, but this remains a slow and inefficient method for all but simple problems, especially when tracking bacteria in multiple locations simultaneously. Instead, we derive and solve the systems of ordinary differential equations for the two lower-order moments of the stochastic variables (mean, variance and covariance). For any given model structure, and assuming linear dynamic rates, we demonstrate how the model parameters can be efficiently and accurately estimated by divergence minimisation. We then apply our method to an experimental dataset and compare the estimates and goodness-of-fit to those obtained by maximum likelihood estimation. While both sets of parameter estimates had overlapping confidence regions, the new method produced lower values for the division and death rates of bacteria: these improved the goodness-of-fit at the second time point at the expense of that of the first time point. This flexible framework can easily be applied to a range of experimental systems. Its computational efficiency paves the way for model comparison and optimal experimental design.
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Affiliation(s)
- David J. Price
- Department of Veterinary Medicine, University of Cambridge, Cambridge, United Kingdom
| | - Alexandre Breuzé
- Department of Veterinary Medicine, University of Cambridge, Cambridge, United Kingdom
- ENSTA-ParisTech, Palaiseau, France
| | - Richard Dybowski
- Department of Veterinary Medicine, University of Cambridge, Cambridge, United Kingdom
| | - Piero Mastroeni
- Department of Veterinary Medicine, University of Cambridge, Cambridge, United Kingdom
| | - Olivier Restif
- Department of Veterinary Medicine, University of Cambridge, Cambridge, United Kingdom
- * E-mail:
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13
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Constantino PH, Kaznessis YN. Maximum Entropy Prediction of Non-Equilibrium Stationary Distributions for Stochastic Reaction Networks with Oscillatory Dynamics. Chem Eng Sci 2017; 171:139-148. [PMID: 30899124 DOI: 10.1016/j.ces.2017.05.029] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/19/2022]
Abstract
Many chemical reaction networks in biological systems present complex oscillatory dynamics. In systems such as regulatory gene networks, cell cycle, and enzymatic processes, the number of molecules involved is often far from the thermodynamic limit. Although stochastic models based on the probabilistic approach of the Chemical Master Equation (CME) have been proposed, studies in the literature have been limited by the challenges of solving the CME and the lack of computational power to perform large-scale stochastic simulations. In this paper, we show that the infinite set of stationary moment equations describing the stochastic Brusselator and Schnakenberg oscillatory reactions networks can be truncated and solved using maximization of the entropy of the distributions. The results from our numerical experiments compare with the distributions obtained from well-established kinetic Monte Carlo methods and suggest that the accuracy of the prediction increases exponentially with the closure order chosen for the system. We conclude that maximum entropy models can be used as an efficient closure scheme alternative for moment equations to predict the non-equilibrium stationary distributions of stochastic chemical reactions with oscillatory dynamics. This prediction is accomplished without any prior knowledge of the system dynamics and without imposing any biased assumptions on the mathematical relations among species involved.
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Affiliation(s)
- Pedro H Constantino
- Department of Chemical Engineering and Materials Science, University of Minnesota, 421 Washington Ave. SE, Minneapolis, MN 55455, USA.
| | - Yiannis N Kaznessis
- Department of Chemical Engineering and Materials Science, University of Minnesota, 421 Washington Ave. SE, Minneapolis, MN 55455, USA.
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14
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Smadbeck P, Kaznessis YN. On a theory of stability for nonlinear stochastic chemical reaction networks. J Chem Phys 2016; 142:184101. [PMID: 25978877 DOI: 10.1063/1.4919834] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We present elements of a stability theory for small, stochastic, nonlinear chemical reaction networks. Steady state probability distributions are computed with zero-information (ZI) closure, a closure algorithm that solves chemical master equations of small arbitrary nonlinear reactions. Stochastic models can be linearized around the steady state with ZI-closure, and the eigenvalues of the Jacobian matrix can be readily computed. Eigenvalues govern the relaxation of fluctuation autocorrelation functions at steady state. Autocorrelation functions reveal the time scales of phenomena underlying the dynamics of nonlinear reaction networks. In accord with the fluctuation-dissipation theorem, these functions are found to be congruent to response functions to small perturbations. Significant differences are observed in the stability of nonlinear reacting systems between deterministic and stochastic modeling formalisms.
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Affiliation(s)
- Patrick Smadbeck
- Department of Chemical Engineering and Materials Science, University of Minnesota, 421 Washington Ave. SE, Minneapolis, Minnesota 55455, USA
| | - Yiannis N Kaznessis
- Department of Chemical Engineering and Materials Science, University of Minnesota, 421 Washington Ave. SE, Minneapolis, Minnesota 55455, USA
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Abstract
With inexpensive DNA synthesis technologies, we can now construct biological systems by quickly piecing together DNA sequences. Synthetic biology is the promising discipline that focuses on the construction of these new biological systems. Synthetic biology is an engineering discipline, and as such, it can benefit from mathematical modeling. This chapter focuses on mathematical models of biological systems. These models take the form of chemical reaction networks. The importance of stochasticity is discussed and methods to simulate stochastic reaction networks are reviewed. A closure scheme solution is also presented for the master equation of chemical reaction networks. The master equation is a complete model of randomly evolving molecular populations. Because of its ambitious character, the master equation remained unsolved for all but the simplest of molecular interaction networks for over 70 years. With the first complete solution of chemical master equations, a wide range of experimental observations of biomolecular interactions may be mathematically conceptualized. We anticipate that models based on the closure scheme described herein may assist in rationally designing synthetic biological systems.
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Thomas P, Grima R. Approximate probability distributions of the master equation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:012120. [PMID: 26274137 DOI: 10.1103/physreve.92.012120] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/13/2014] [Indexed: 06/04/2023]
Abstract
Master equations are common descriptions of mesoscopic systems. Analytical solutions to these equations can rarely be obtained. We here derive an analytical approximation of the time-dependent probability distribution of the master equation using orthogonal polynomials. The solution is given in two alternative formulations: a series with continuous and a series with discrete support, both of which can be systematically truncated. While both approximations satisfy the system size expansion of the master equation, the continuous distribution approximations become increasingly negative and tend to oscillations with increasing truncation order. In contrast, the discrete approximations rapidly converge to the underlying non-Gaussian distributions. The theory is shown to lead to particularly simple analytical expressions for the probability distributions of molecule numbers in metabolic reactions and gene expression systems.
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Affiliation(s)
- Philipp Thomas
- School of Mathematics and School of Biological Sciences, University of Edinburgh, Edinburgh EH8 9YL, United Kingdom
| | - Ramon Grima
- School of Biological Sciences, University of Edinburgh, Edinburgh EH8 9YL, United Kingdom
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17
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Smadbeck P, Kaznessis YN. Solution of Chemical Master Equations for Nonlinear Stochastic Reaction Networks. Curr Opin Chem Eng 2014; 5:90-95. [PMID: 25215268 DOI: 10.1016/j.coche.2014.07.001] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/24/2022]
Abstract
Stochasticity in the dynamics of small reacting systems requires discrete-probabilistic models of reaction kinetics instead of traditional continuous-deterministic ones. The master probability equation is a complete model of randomly evolving molecular populations. Because of its ambitious character, the master equation remained unsolved for all but the simplest of molecular interaction networks. With the first solution of chemical master equations, a wide range of experimental observations of small-system interactions may be mathematically conceptualized.
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Affiliation(s)
- Patrick Smadbeck
- Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN 55455, USA
| | - Yiannis N Kaznessis
- Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN 55455, USA
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18
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Thomas P, Matuschek H, Grima R. How reliable is the linear noise approximation of gene regulatory networks? BMC Genomics 2013; 14 Suppl 4:S5. [PMID: 24266939 PMCID: PMC3849541 DOI: 10.1186/1471-2164-14-s4-s5] [Citation(s) in RCA: 29] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/06/2023] Open
Abstract
Background The linear noise approximation (LNA) is commonly used to predict how noise is regulated and exploited at the cellular level. These predictions are exact for reaction networks composed exclusively of first order reactions or for networks involving bimolecular reactions and large numbers of molecules. It is however well known that gene regulation involves bimolecular interactions with molecule numbers as small as a single copy of a particular gene. It is therefore questionable how reliable are the LNA predictions for these systems. Results We implement in the software package intrinsic Noise Analyzer (iNA), a system size expansion based method which calculates the mean concentrations and the variances of the fluctuations to an order of accuracy higher than the LNA. We then use iNA to explore the parametric dependence of the Fano factors and of the coefficients of variation of the mRNA and protein fluctuations in models of genetic networks involving nonlinear protein degradation, post-transcriptional, post-translational and negative feedback regulation. We find that the LNA can significantly underestimate the amplitude and period of noise-induced oscillations in genetic oscillators. We also identify cases where the LNA predicts that noise levels can be optimized by tuning a bimolecular rate constant whereas our method shows that no such regulation is possible. All our results are confirmed by stochastic simulations. Conclusion The software iNA allows the investigation of parameter regimes where the LNA fares well and where it does not. We have shown that the parametric dependence of the coefficients of variation and Fano factors for common gene regulatory networks is better described by including terms of higher order than LNA in the system size expansion. This analysis is considerably faster than stochastic simulations due to the extensive ensemble averaging needed to obtain statistically meaningful results. Hence iNA is well suited for performing computationally efficient and quantitative studies of intrinsic noise in gene regulatory networks.
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Abstract
Probability reigns in biology, with random molecular events dictating the fate of individual organisms, and propelling populations of species through evolution. In principle, the master probability equation provides the most complete model of probabilistic behavior in biomolecular networks. In practice, master equations describing complex reaction networks have remained unsolved for over 70 years. This practical challenge is a reason why master equations, for all their potential, have not inspired biological discovery. Herein, we present a closure scheme that solves the master probability equation of networks of chemical or biochemical reactions. We cast the master equation in terms of ordinary differential equations that describe the time evolution of probability distribution moments. We postulate that a finite number of moments capture all of the necessary information, and compute the probability distribution and higher-order moments by maximizing the information entropy of the system. An accurate order closure is selected, and the dynamic evolution of molecular populations is simulated. Comparison with kinetic Monte Carlo simulations, which merely sample the probability distribution, demonstrates this closure scheme is accurate for several small reaction networks. The importance of this result notwithstanding, a most striking finding is that the steady state of stochastic reaction networks can now be readily computed in a single-step calculation, without the need to simulate the evolution of the probability distribution in time.
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Ale A, Kirk P, Stumpf MPH. A general moment expansion method for stochastic kinetic models. J Chem Phys 2013; 138:174101. [DOI: 10.1063/1.4802475] [Citation(s) in RCA: 49] [Impact Index Per Article: 4.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Abstract
As stochastic simulations become increasingly common in biological research, tools for analysis of such systems are in demand. The deterministic analogue to stochastic models, a set of probability moment equations equivalent to the Chemical Master Equation (CME), offers the possibility of a priori analysis of systems without the need for computationally costly Monte Carlo simulations. Despite the drawbacks of the method, in particular non-linearity in even the simplest of cases, the use of moment equations combined with moment-closure techniques has been used effectively in many fields. The techniques currently available to generate moment equations rely upon analytical expressions that are not efficient upon scaling. Additionally, the resulting moment-dependent matrix is lower diagonal and demands massive memory allocation in extreme cases. Here it is demonstrated that by utilizing factorial moments and the probability generating function (the Z-transform of the probability distribution) a recursive algorithm is produced. The resulting method is scalable and particularly efficient when high-order moments are required. The matrix produced is banded and often demands substantially less memory resources.
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Affiliation(s)
- P. Smadbeck
- Department of Chemical Engineering and Materials Science, University of Minnesota, 421 Washington Ave SE, Minneapolis, MN 55455, USA
| | - Y. N. Kaznessis
- Department of Chemical Engineering and Materials Science, University of Minnesota, 421 Washington Ave SE, Minneapolis, MN 55455, USA
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