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Buckwar E, Conte M, Meddah A. A stochastic hierarchical model for low grade glioma evolution. J Math Biol 2023; 86:89. [PMID: 37147527 PMCID: PMC10163130 DOI: 10.1007/s00285-023-01909-5] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/22/2022] [Revised: 03/17/2023] [Accepted: 03/22/2023] [Indexed: 05/07/2023]
Abstract
A stochastic hierarchical model for the evolution of low grade gliomas is proposed. Starting with the description of cell motion using a piecewise diffusion Markov process (PDifMP) at the cellular level, we derive an equation for the density of the transition probability of this Markov process based on the generalised Fokker-Planck equation. Then, a macroscopic model is derived via parabolic limit and Hilbert expansions in the moment equations. After setting up the model, we perform several numerical tests to study the role of the local characteristics and the extended generator of the PDifMP in the process of tumour progression. The main aim focuses on understanding how the variations of the jump rate function of this process at the microscopic scale and the diffusion coefficient at the macroscopic scale are related to the diffusive behaviour of the glioma cells and to the onset of malignancy, i.e., the transition from low-grade to high-grade gliomas.
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Affiliation(s)
- Evelyn Buckwar
- Institute of Stochastics, Johannes Kepler University, Altenberger Straße 69, 4040, Linz, Austria
- Centre for Mathematical Sciences, Lund University, 221 00, Lund, Sweden
| | - Martina Conte
- Department of Mathematical Sciences "G. L. Lagrange", Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Torino, Italy
| | - Amira Meddah
- Institute of Stochastics, Johannes Kepler University, Altenberger Straße 69, 4040, Linz, Austria.
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2
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Shao J, Wang L, Wu Q. Ergodicity and stability of hybrid systems with piecewise constant type state-dependent switching. Stoch Process Their Appl 2023. [DOI: 10.1016/j.spa.2023.03.014] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 03/29/2023]
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3
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Fluctuating-rate model with multiple gene states. J Math Biol 2020; 81:1099-1141. [PMID: 33000313 DOI: 10.1007/s00285-020-01538-2] [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] [Received: 02/20/2019] [Revised: 08/28/2020] [Indexed: 10/23/2022]
Abstract
Multiple phenotypic states of single cells often co-exist in the presence of positive feedbacks. Stochastic gene-state switchings and low copy numbers of proteins in single cells cause considerable fluctuations. The chemical master equation (CME) is a powerful tool that describes the dynamics of single cells, but it may be overly complicated. Among many simplified models, a fluctuating-rate (FR) model has been proposed recently to approximate the full CME model in the realistic intermediate region of gene-state switchings. However, only the scenario with two gene states has been carefully analysed. In this paper, we generalise the FR model to the case with multiple gene states, in which the mathematical derivation becomes more complicated. The leading order of fluctuations around each phenotypic state, as well as the transition rates between phenotypic states, in the intermediate gene-state switching region is characterized by the rate function of the stationary distribution of the FR model in the Freidlin-Wentzell-type large deviation principle (LDP). Under certain reasonable assumptions, we show that the derivative of the rate function is equal to the unique nontrivial solution of a dominant generalised eigenvalue problem, leading to a new numerical algorithm for obtaining the LDP rate function directly. Furthermore, we prove the Lyapunov property of the rate function for the corresponding deterministic mean-field dynamics. Finally, through a tristable example, we show that the local fluctuations (the asymptotic variance of the stationary distribution at each phenotypic state) in the intermediate and rapid regions of gene-state switchings are different. Finally, a tri-stable example is constructed to illustrate the validity of our theory.
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Debussche A, Nguepedja Nankep MJ. A Law of Large Numbers in the Supremum Norm for a Multiscale Stochastic Spatial Gene Network. Int J Biostat 2019; 15:ijb-2017-0091. [DOI: 10.1515/ijb-2017-0091] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/14/2017] [Accepted: 04/10/2019] [Indexed: 11/15/2022]
Abstract
Abstract
We study the asymptotic behavior of multiscale stochastic spatial gene networks. Multiscaling takes into account the difference of abundance between molecules, and captures the dynamic of rare species at a mesoscopic level. We introduce an assumption of spatial correlations for reactions involving rare species and a new law of large numbers is obtained. According to the scales, the whole system splits into two parts with different but coupled dynamics. The high scale component converges to the usual spatial model which is the solution of a partial differential equation, whereas the low scale component converges to the usual homogeneous model which is the solution of an ordinary differential equation. Comparisons are made in the supremum norm.
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Affiliation(s)
- Arnaud Debussche
- Institut de Recherche Mathematique de Rennes , Rennes , Bretagne , France
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Löcherbach E. Convergence to Equilibrium for Time-Inhomogeneous Jump Diffusions with State-Dependent Jump Intensity. J THEOR PROBAB 2019. [DOI: 10.1007/s10959-019-00947-4] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
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6
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Poincaré-Type Inequalities for Compact Degenerate Pure Jump Markov Processes. MATHEMATICS 2019. [DOI: 10.3390/math7060518] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
Abstract
We aim to prove Poincaré inequalities for a class of pure jump Markov processes inspired by the model introduced by Galves and Löcherbach to describe the behavior of interacting brain neurons. In particular, we consider neurons with degenerate jumps, i.e., which lose their memory when they spike, while the probability of a spike depends on the actual position and thus the past of the whole neural system. The process studied by Galves and Löcherbach is a point process counting the spike events of the system and is therefore non-Markovian. In this work, we consider a process describing the membrane potential of each neuron that contains the relevant information of the past. This allows us to work in a Markovian framework.
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Bally V, Goreac D, Rabiet V. Regularity and stability for the semigroup of jump diffusions with state-dependent intensity. ANN APPL PROBAB 2018. [DOI: 10.1214/18-aap1382] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Bronstein L, Koeppl H. Marginal process framework: A model reduction tool for Markov jump processes. Phys Rev E 2018; 97:062147. [PMID: 30011601 DOI: 10.1103/physreve.97.062147] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/10/2018] [Indexed: 06/08/2023]
Abstract
Markov jump process models have many applications across science. Often these models are defined on a state space of product form and only one of the components of the process is of direct interest. In this paper we extend the marginal process framework, which provides a marginal description of the component of interest, to the case of fully coupled processes. We use entropic matching to obtain a finite-dimensional approximation of the filtering equation, which governs the transition rates of the marginal process. The resulting equations can be seen as a combination of two projection operations applied to the full master equation so that we obtain a principled model reduction framework. We demonstrate the resulting reduced description on the totally asymmetric exclusion process. An important class of Markov jump processes are stochastic reaction networks, which have applications in chemical and biomolecular kinetics, ecological models, and models of social networks. We obtain a particularly simple instantiation of the marginal process framework for mass-action systems by using product Poisson distributions for the approximate solution of the filtering equation. We investigate the resulting approximate marginal process analytically and numerically.
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Affiliation(s)
- Leo Bronstein
- Department of Electrical Engineering and Information Technology, Technische Universität Darmstadt, 64283 Darmstadt, Germany
| | - Heinz Koeppl
- Department of Electrical Engineering and Information Technology, Technische Universität Darmstadt, 64283 Darmstadt, Germany
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9
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Liu Y, Yang D, Zhang C. Relaxed conditions for convergence analysis of online back-propagation algorithm with L2 regularizer for Sigma-Pi-Sigma neural network. Neurocomputing 2018. [DOI: 10.1016/j.neucom.2017.06.057] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/19/2022]
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10
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Herbach U, Bonnaffoux A, Espinasse T, Gandrillon O. Inferring gene regulatory networks from single-cell data: a mechanistic approach. BMC SYSTEMS BIOLOGY 2017; 11:105. [PMID: 29157246 PMCID: PMC5697158 DOI: 10.1186/s12918-017-0487-0] [Citation(s) in RCA: 25] [Impact Index Per Article: 3.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 05/12/2017] [Accepted: 11/09/2017] [Indexed: 01/13/2023]
Abstract
Background The recent development of single-cell transcriptomics has enabled gene expression to be measured in individual cells instead of being population-averaged. Despite this considerable precision improvement, inferring regulatory networks remains challenging because stochasticity now proves to play a fundamental role in gene expression. In particular, mRNA synthesis is now acknowledged to occur in a highly bursty manner. Results We propose to view the inference problem as a fitting procedure for a mechanistic gene network model that is inherently stochastic and takes not only protein, but also mRNA levels into account. We first explain how to build and simulate this network model based upon the coupling of genes that are described as piecewise-deterministic Markov processes. Our model is modular and can be used to implement various biochemical hypotheses including causal interactions between genes. However, a naive fitting procedure would be intractable. By performing a relevant approximation of the stationary distribution, we derive a tractable procedure that corresponds to a statistical hidden Markov model with interpretable parameters. This approximation turns out to be extremely close to the theoretical distribution in the case of a simple toggle-switch, and we show that it can indeed fit real single-cell data. As a first step toward inference, our approach was applied to a number of simple two-gene networks simulated in silico from the mechanistic model and satisfactorily recovered the original networks. Conclusions Our results demonstrate that functional interactions between genes can be inferred from the distribution of a mechanistic, dynamical stochastic model that is able to describe gene expression in individual cells. This approach seems promising in relation to the current explosion of single-cell expression data. Electronic supplementary material The online version of this article (doi:10.1186/s12918-017-0487-0) contains supplementary material, which is available to authorized users.
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Affiliation(s)
- Ulysse Herbach
- Univ Lyon, ENS de Lyon, Univ Claude Bernard, CNRS UMR 5239, INSERM U1210, Laboratory of Biology and Modelling of the Cell, 46 allée d'Italie Site Jacques Monod, Lyon, F-69007, France.,Inria Team Dracula, Inria Center Grenoble Rhône-Alpes, Lyon, France.,Univ Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, 43 blvd. du 11 novembre 1918, Villeurbanne Cedex, F-6962, France
| | - Arnaud Bonnaffoux
- Univ Lyon, ENS de Lyon, Univ Claude Bernard, CNRS UMR 5239, INSERM U1210, Laboratory of Biology and Modelling of the Cell, 46 allée d'Italie Site Jacques Monod, Lyon, F-69007, France.,Inria Team Dracula, Inria Center Grenoble Rhône-Alpes, Lyon, France.,The CoSMo company, 5 passage du Vercors, Lyon, 69007, France
| | - Thibault Espinasse
- Univ Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, 43 blvd. du 11 novembre 1918, Villeurbanne Cedex, F-6962, France
| | - Olivier Gandrillon
- Univ Lyon, ENS de Lyon, Univ Claude Bernard, CNRS UMR 5239, INSERM U1210, Laboratory of Biology and Modelling of the Cell, 46 allée d'Italie Site Jacques Monod, Lyon, F-69007, France. .,Inria Team Dracula, Inria Center Grenoble Rhône-Alpes, Lyon, France.
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Höpfner R, Löcherbach E, Thieullen M. Strongly degenerate time inhomogeneous SDEs: Densities and support properties. Application to Hodgkin–Huxley type systems. BERNOULLI 2017. [DOI: 10.3150/16-bej820] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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12
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Winkelmann S, Schütte C. Hybrid models for chemical reaction networks: Multiscale theory and application to gene regulatory systems. J Chem Phys 2017; 147:114115. [PMID: 28938803 DOI: 10.1063/1.4986560] [Citation(s) in RCA: 19] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Stefanie Winkelmann
- Zuse Institute Berlin (ZIB), Takustr. 7, 14195 Berlin, Germany
- Department of Mathematics, Freie Universität Berlin, Arnimallee 6, 14195 Berlin, Germany
| | - Christof Schütte
- Zuse Institute Berlin (ZIB), Takustr. 7, 14195 Berlin, Germany
- Department of Mathematics, Freie Universität Berlin, Arnimallee 6, 14195 Berlin, Germany
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14
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Hodara P, Krell N, Löcherbach E. Non-parametric estimation of the spiking rate in systems of interacting neurons. STATISTICAL INFERENCE FOR STOCHASTIC PROCESSES 2016. [DOI: 10.1007/s11203-016-9150-4] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 10/20/2022]
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15
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Gene expression noise is affected differentially by feedback in burst frequency and burst size. J Math Biol 2016; 74:1483-1509. [PMID: 27665109 DOI: 10.1007/s00285-016-1059-4] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/03/2016] [Revised: 08/08/2016] [Indexed: 12/31/2022]
Abstract
Inside individual cells, expression of genes is stochastic across organisms ranging from bacterial to human cells. A ubiquitous feature of stochastic expression is burst-like synthesis of gene products, which drives considerable intercellular variability in protein levels across an isogenic cell population. One common mechanism by which cells control such stochasticity is negative feedback regulation, where a protein inhibits its own synthesis. For a single gene that is expressed in bursts, negative feedback can affect the burst frequency or the burst size. In order to compare these feedback types, we study a piecewise deterministic model for gene expression of a self-regulating gene. Mathematically tractable steady-state protein distributions are derived and used to compare the noise suppression abilities of the two feedbacks. Results show that in the low noise regime, both feedbacks are similar in term of their noise buffering abilities. Intriguingly, feedback in burst size outperforms the feedback in burst frequency in the high noise regime. Finally, we discuss various regulatory strategies by which cells implement feedback to control burst sizes of expressed proteins at the level of single cells.
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Influence of gene copy number on self-regulated gene expression. J Theor Biol 2016; 408:222-236. [PMID: 27528448 DOI: 10.1016/j.jtbi.2016.08.018] [Citation(s) in RCA: 20] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/09/2016] [Revised: 07/15/2016] [Accepted: 08/11/2016] [Indexed: 02/06/2023]
Abstract
Using an analytically solvable stochastic model, we study the properties of a simple genetic circuit consisting of multiple copies of a self-regulating gene. We analyse how the variation in gene copy number and the mutations changing the auto-regulation strength affect the steady-state distribution of protein concentration. We predict that one-reporter assay, an experimental method where the extrinsic noise level is inferred from the comparison of expression variance of a single and duplicated reporter gene, may give an incorrect estimation of the extrinsic noise contribution when applied to self-regulating genes. We also show that an imperfect duplication of an auto-activated gene, changing the regulation strength of one of the copies, may lead to a hybrid, binary+graded response of these genes to external signal. The analysis of relative changes in mean gene expression before and after duplication suggests that evolutionary accumulation of gene duplications may, at a given mean burst size, non-trivially depend on the inherent noisiness of a given gene, quantified by the inverse of the maximal mean frequency of bursts. Moreover, we find that the dependence of gene expression noise on gene copy number and auto-regulation strength may qualitatively differ, e.g. in monotonicity, depending on whether the noise is measured by Fano factor or coefficient of variation. Thus, experimentally-based hypotheses linking gene expression noise and evolutionary optimisation in the context of gene copy number variation may be ambiguous as they are dependent on the particular function chosen to quantify noise.
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Genadot A, Thieullen M. Averaging for a Fully Coupled Piecewise-Deterministic Markov Process in Infinite Dimensions. ADV APPL PROBAB 2016. [DOI: 10.1239/aap/1346955263] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
In this paper we consider the generalized Hodgkin-Huxley model introduced in Austin (2008). This model describes the propagation of an action potential along the axon of a neuron at the scale of ion channels. Mathematically, this model is a fully coupled piecewise-deterministic Markov process (PDMP) in infinite dimensions. We introduce two time scales in this model in considering that some ion channels open and close at faster jump rates than others. We perform a slow-fast analysis of this model and prove that, asymptotically, this ‘two-time-scale’ model reduces to the so-called averaged model, which is still a PDMP in infinite dimensions, for which we provide effective evolution equations and jump rates.
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18
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Azaïs R, Muller-Gueudin A. Optimal choice among a class of nonparametric estimators of the jump rate for piecewise-deterministic Markov processes. Electron J Stat 2016. [DOI: 10.1214/16-ejs1207] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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19
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Innocentini GCP, Forger M, Radulescu O, Antoneli F. Protein Synthesis Driven by Dynamical Stochastic Transcription. Bull Math Biol 2015; 78:110-31. [PMID: 26670316 DOI: 10.1007/s11538-015-0131-3] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/12/2014] [Accepted: 11/18/2015] [Indexed: 10/22/2022]
Abstract
In this manuscript, we propose a mathematical framework to couple transcription and translation in which mRNA production is described by a set of master equations, while the dynamics of protein density is governed by a random differential equation. The coupling between the two processes is given by a stochastic perturbation whose statistics satisfies the master equations. In this approach, from the knowledge of the analytical time-dependent distribution of mRNA number, we are able to calculate the dynamics of the probability density of the protein population.
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Affiliation(s)
- Guilherme C P Innocentini
- Departamento de Matemática Aplicada, Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão, 1010, Cidade Universitária, São Paulo, SP, 05508-090, Brazil. .,DIMNP, UMR 5235, Université de Montpellier 2, Pl. E. Bataillon, Bat. 24, 34095, Montpellier Cedex 5, France.
| | - Michael Forger
- Departamento de Matemática Aplicada, Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão, 1010, Cidade Universitária, São Paulo, SP, 05508-090, Brazil.
| | - Ovidiu Radulescu
- DIMNP, UMR 5235, Université de Montpellier 2, Pl. E. Bataillon, Bat. 24, 34095, Montpellier Cedex 5, France.
| | - Fernando Antoneli
- Laboratório de Genômica Evolutiva e Biocomplexidade & DIS, Escola Paulista de Medicina, Universidade Federal de São Paulo, Rua Pedro de Toledo, 669, 4th floor, São Paulo, SP, 04039-032, Brazil.
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Rudnicki R, Tomski A. On a stochastic gene expression with pre-mRNA, mRNA and protein contribution. J Theor Biol 2015; 387:54-67. [PMID: 26434618 DOI: 10.1016/j.jtbi.2015.09.012] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/13/2015] [Revised: 08/31/2015] [Accepted: 09/10/2015] [Indexed: 12/17/2022]
Abstract
In this paper we develop a model of stochastic gene expression, which is an extension of the model investigated in the paper [T. Lipniacki, P. Paszek, A. Marciniak-Czochra, A.R. Brasier, M. Kimmel, Transcriptional stochasticity in gene expression, J. Theor. Biol. 238 (2006) 348-367]. In our model, stochastic effects still originate from random fluctuations in gene activity status, but we precede mRNA production by the formation of pre-mRNA, which enriches classical transcription phase. We obtain a stochastically regulated system of ordinary differential equations (ODEs) describing evolution of pre-mRNA, mRNA and protein levels. We perform mathematical analysis of a long-time behavior of this stochastic process, identified as a piece-wise deterministic Markov process (PDMP). We check exact results using numerical simulations for the distributions of all three types of particles. Moreover, we investigate the deterministic (adiabatic) limit state of the process, when depending on parameters it can exhibit two specific types of behavior: bistability and the existence of the limit cycle. The latter one is not present when only two kinds of gene expression products are considered.
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Affiliation(s)
- Ryszard Rudnicki
- Institute of Mathematics, Polish Academy of Sciences, Bankowa 14, 40-007 Katowice, Poland.
| | - Andrzej Tomski
- Institute of Mathematics, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland.
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22
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23
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Hepp B, Gupta A, Khammash M. Adaptive hybrid simulations for multiscale stochastic reaction networks. J Chem Phys 2015; 142:034118. [DOI: 10.1063/1.4905196] [Citation(s) in RCA: 35] [Impact Index Per Article: 3.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/15/2022] Open
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Adiabatic reduction of a model of stochastic gene expression with jump Markov process. J Math Biol 2013; 68:1051-70. [PMID: 23460478 DOI: 10.1007/s00285-013-0661-y] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/20/2012] [Revised: 02/20/2013] [Indexed: 10/27/2022]
Abstract
This paper considers adiabatic reduction in a model of stochastic gene expression with bursting transcription considered as a jump Markov process. In this model, the process of gene expression with auto-regulation is described by fast/slow dynamics. The production of mRNA is assumed to follow a compound Poisson process occurring at a rate depending on protein levels (the phenomena called bursting in molecular biology) and the production of protein is a linear function of mRNA numbers. When the dynamics of mRNA is assumed to be a fast process (due to faster mRNA degradation than that of protein) we prove that, with appropriate scalings in the burst rate, jump size or translational rate, the bursting phenomena can be transmitted to the slow variable. We show that, depending on the scaling, the reduced equation is either a stochastic differential equation with a jump Poisson process or a deterministic ordinary differential equation. These results are significant because adiabatic reduction techniques seem to have not been rigorously justified for a stochastic differential system containing a jump Markov process. We expect that the results can be generalized to adiabatic methods in more general stochastic hybrid systems.
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