1
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Loman TE, Locke JCW. The σB alternative sigma factor circuit modulates noise to generate different types of pulsing dynamics. PLoS Comput Biol 2023; 19:e1011265. [PMID: 37540712 PMCID: PMC10431680 DOI: 10.1371/journal.pcbi.1011265] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/10/2022] [Revised: 08/16/2023] [Accepted: 06/12/2023] [Indexed: 08/06/2023] Open
Abstract
Single-cell approaches are revealing a high degree of heterogeneity, or noise, in gene expression in isogenic bacteria. How gene circuits modulate this noise in gene expression to generate robust output dynamics is unclear. Here we use the Bacillus subtilis alternative sigma factor σB as a model system for understanding the role of noise in generating circuit output dynamics. σB controls the general stress response in B. subtilis and is activated by a range of energy and environmental stresses. Recent single-cell studies have revealed that the circuit can generate two distinct outputs, stochastic pulsing and a single pulse response, but the conditions under which each response is generated are under debate. We implement a stochastic mathematical model of the σB circuit to investigate this and find that the system's core circuit can generate both response types. This is despite one response (stochastic pulsing) being stochastic in nature, and the other (single response pulse) being deterministic. We demonstrate that the main determinant for whichever response is generated is the degree with which the input pathway activates the core circuit, although the noise properties of the input pathway also biases the system towards one or the other type of output. Thus, our work shows how stochastic modelling can reveal the mechanisms behind non-intuitive gene circuit output dynamics.
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Affiliation(s)
- Torkel E. Loman
- Sainsbury Laboratory, University of Cambridge, Cambridge, United Kingdom
| | - James C. W. Locke
- Sainsbury Laboratory, University of Cambridge, Cambridge, United Kingdom
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2
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Fintzi J, Wakefield J, Minin VN. A linear noise approximation for stochastic epidemic models fit to partially observed incidence counts. Biometrics 2022; 78:1530-1541. [PMID: 34374071 DOI: 10.1111/biom.13538] [Citation(s) in RCA: 4] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/11/2020] [Revised: 06/10/2021] [Accepted: 06/17/2021] [Indexed: 12/30/2022]
Abstract
Stochastic epidemic models (SEMs) fit to incidence data are critical to elucidating outbreak dynamics, shaping response strategies, and preparing for future epidemics. SEMs typically represent counts of individuals in discrete infection states using Markov jump processes (MJPs), but are computationally challenging as imperfect surveillance, lack of subject-level information, and temporal coarseness of the data obscure the true epidemic. Analytic integration over the latent epidemic process is impossible, and integration via Markov chain Monte Carlo (MCMC) is cumbersome due to the dimensionality and discreteness of the latent state space. Simulation-based computational approaches can address the intractability of the MJP likelihood, but are numerically fragile and prohibitively expensive for complex models. A linear noise approximation (LNA) that approximates the MJP transition density with a Gaussian density has been explored for analyzing prevalence data in large-population settings, but requires modification for analyzing incidence counts without assuming that the data are normally distributed. We demonstrate how to reparameterize SEMs to appropriately analyze incidence data, and fold the LNA into a data augmentation MCMC framework that outperforms deterministic methods, statistically, and simulation-based methods, computationally. Our framework is computationally robust when the model dynamics are complex and applies to a broad class of SEMs. We evaluate our method in simulations that reflect Ebola, influenza, and SARS-CoV-2 dynamics, and apply our method to national surveillance counts from the 2013-2015 West Africa Ebola outbreak.
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Affiliation(s)
- Jonathan Fintzi
- Biostatistics Research Branch, National Institute of Allergy and Infectious Diseases, Rockville, Maryland, USA
| | - Jon Wakefield
- Departments of Biostatistics and Statistics, University of Washington, Seattle, Washington, USA
| | - Vladimir N Minin
- Department of Statistics, University of California, Irvine, California, USA
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3
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Lunz D, Bonnans JF, Ruess J. Revisiting moment-closure methods with heterogeneous multiscale population models. Math Biosci 2022; 350:108866. [PMID: 35753520 DOI: 10.1016/j.mbs.2022.108866] [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: 12/14/2021] [Revised: 04/10/2022] [Accepted: 06/08/2022] [Indexed: 11/29/2022]
Abstract
Stochastic chemical kinetics at the single-cell level give rise to heterogeneous populations of cells even when all individuals are genetically identical. This heterogeneity can lead to nonuniform behaviour within populations, including different growth characteristics, cell-fate dynamics, and response to stimuli. Ultimately, these diverse behaviours lead to intricate population dynamics that are inherently multiscale: the population composition evolves based on population-level processes that interact with stochastically distributed single-cell states. Therefore, descriptions that account for this heterogeneity are essential to accurately model and control chemical processes. However, for real-world systems such models are computationally expensive to simulate, which can make optimisation problems, such as optimal control or parameter inference, prohibitively challenging. Here, we consider a class of multiscale population models that incorporate population-level mechanisms while remaining faithful to the underlying stochasticity at the single-cell level and the interplay between these two scales. To address the complexity, we study an order-reduction approximations based on the distribution moments. Since previous moment-closure work has focused on the single-cell kinetics, extending these techniques to populations models prompts us to revisit old observations as well as tackle new challenges. In this extended multiscale context, we encounter the previously established observation that the simplest closure techniques can lead to non-physical system trajectories. Despite their poor performance in some systems, we provide an example where these simple closures outperform more sophisticated closure methods in accurately, efficiently, and robustly solving the problem of optimal control of bioproduction in a microbial consortium model.
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Affiliation(s)
- Davin Lunz
- Inria Paris, 2 rue Simone Iff, 75012 Paris, France; Institut Pasteur, 28 rue du Docteur Roux, 75015 Paris, France.
| | - J Frédéric Bonnans
- Université Paris-Saclay, CNRS, CentraleSupélec, Inria, Laboratory of signals and systems, 91190, Gif-sur-Yvette, France
| | - Jakob Ruess
- Inria Paris, 2 rue Simone Iff, 75012 Paris, France; Institut Pasteur, 28 rue du Docteur Roux, 75015 Paris, France
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4
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Münch JL, Paul F, Schmauder R, Benndorf K. Bayesian inference of kinetic schemes for ion channels by Kalman filtering. eLife 2022; 11:e62714. [PMID: 35506659 PMCID: PMC9342998 DOI: 10.7554/elife.62714] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/02/2020] [Accepted: 04/22/2022] [Indexed: 11/16/2022] Open
Abstract
Inferring adequate kinetic schemes for ion channel gating from ensemble currents is a daunting task due to limited information in the data. We address this problem by using a parallelized Bayesian filter to specify hidden Markov models for current and fluorescence data. We demonstrate the flexibility of this algorithm by including different noise distributions. Our generalized Kalman filter outperforms both a classical Kalman filter and a rate equation approach when applied to patch-clamp data exhibiting realistic open-channel noise. The derived generalization also enables inclusion of orthogonal fluorescence data, making unidentifiable parameters identifiable and increasing the accuracy of the parameter estimates by an order of magnitude. By using Bayesian highest credibility volumes, we found that our approach, in contrast to the rate equation approach, yields a realistic uncertainty quantification. Furthermore, the Bayesian filter delivers negligibly biased estimates for a wider range of data quality. For some data sets, it identifies more parameters than the rate equation approach. These results also demonstrate the power of assessing the validity of algorithms by Bayesian credibility volumes in general. Finally, we show that our Bayesian filter is more robust against errors induced by either analog filtering before analog-to-digital conversion or by limited time resolution of fluorescence data than a rate equation approach.
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Affiliation(s)
- Jan L Münch
- Institut für Physiologie II, Universitätsklinikum Jena, Friedrich Schiller University JenaJenaGermany
| | - Fabian Paul
- Department of Biochemistry and Molecular Biology, University of ChicagoChicagoUnited States
| | - Ralf Schmauder
- Institut für Physiologie II, Universitätsklinikum Jena, Friedrich Schiller University JenaJenaGermany
| | - Klaus Benndorf
- Institut für Physiologie II, Universitätsklinikum Jena, Friedrich Schiller University JenaJenaGermany
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5
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HillTau: A fast, compact abstraction for model reduction in biochemical signaling networks. PLoS Comput Biol 2021; 17:e1009621. [PMID: 34843454 PMCID: PMC8659295 DOI: 10.1371/journal.pcbi.1009621] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/27/2021] [Revised: 12/09/2021] [Accepted: 11/08/2021] [Indexed: 12/03/2022] Open
Abstract
Signaling networks mediate many aspects of cellular function. The conventional, mechanistically motivated approach to modeling such networks is through mass-action chemistry, which maps directly to biological entities and facilitates experimental tests and predictions. However such models are complex, need many parameters, and are computationally costly. Here we introduce the HillTau form for signaling models. HillTau retains the direct mapping to biological observables, but it uses far fewer parameters, and is 100 to over 1000 times faster than ODE-based methods. In the HillTau formalism, the steady-state concentration of signaling molecules is approximated by the Hill equation, and the dynamics by a time-course tau. We demonstrate its use in implementing several biochemical motifs, including association, inhibition, feedforward and feedback inhibition, bistability, oscillations, and a synaptic switch obeying the BCM rule. The major use-cases for HillTau are system abstraction, model reduction, scaffolds for data-driven optimization, and fast approximations to complex cellular signaling. Chemical signals mediate many computations in cells, from housekeeping functions in all cells to memory and pattern selectivity in neurons. These signals form complex networks of interactions. Computer models are a powerful way to study how such networks behave, but it is hard to get all the chemical details for typical models, and it is slow to run them with standard numerical approaches to chemical kinetics. We introduce HillTau as a simplified way to model complex chemical networks. HillTau models condense multiple reaction steps into single steps defined by a small number of parameters for activation and settling time. As a result the models are simple, easy to find values for, and they run quickly. Remarkably, they fit the full chemical formulations rather well. We illustrate the utility of HillTau for modeling several signaling network functions, and for fitting complicated signaling networks.
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6
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Nguyen-Van-Yen B, Del Moral P, Cazelles B. Stochastic Epidemic Models inference and diagnosis with Poisson Random Measure Data Augmentation. Math Biosci 2021; 335:108583. [PMID: 33713696 DOI: 10.1016/j.mbs.2021.108583] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/21/2020] [Revised: 12/22/2020] [Accepted: 02/28/2021] [Indexed: 11/24/2022]
Abstract
We present a new Bayesian inference method for compartmental models that takes into account the intrinsic stochasticity of the process. We show how to formulate a SIR-type Markov jump process as the solution of a stochastic differential equation with respect to a Poisson Random Measure (PRM), and how to simulate the process trajectory deterministically from a parameter value and a PRM realization. This forms the basis of our Data Augmented MCMC, which consists of augmenting parameter space with the unobserved PRM value. The resulting simple Metropolis-Hastings sampler acts as an efficient simulation-based inference method, that can easily be transferred from model to model. Compared with a recent Data Augmentation method based on Gibbs sampling of individual infection histories, PRM-augmented MCMC scales much better with epidemic size and is far more flexible. It is also found to be competitive with Particle MCMC for moderate epidemics when using approximate simulations. PRM-augmented MCMC also yields a posteriori estimates of the PRM, that represent process stochasticity, and which can be used to validate the model. A pattern of deviation from the PRM prior distribution will indicate that the model underfits the data and help to understand the cause. We illustrate this by fitting a non-seasonal model to some simulated seasonal case count data. Applied to the Zika epidemic of 2013 in French Polynesia, our approach shows that a simple SEIR model cannot correctly reproduce both the initial sharp increase in the number of cases as well as the final proportion of seropositive. PRM augmentation thus provides a coherent story for Stochastic Epidemic Model inference, where explicitly inferring process stochasticity helps with model validation.
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Affiliation(s)
- Benjamin Nguyen-Van-Yen
- Institut Pasteur, Unité de Génétique Fonctionnelle des Maladies Infectieuses, UMR 2000 CNRS, Paris, France; Institut de Biologie de l'ENS (IBENS), Ecole Normale Supérieure, CNRS, INSERM, Université PSL, 75005 Paris, France.
| | | | - Bernard Cazelles
- Institut de Biologie de l'ENS (IBENS), Ecole Normale Supérieure, CNRS, INSERM, Université PSL, 75005 Paris, France; International Center for Mathematical and Computational Modeling of Complex Systems (UMMISCO), UMI 209, Sorbonne Université, France; iGLOBE, UMI CNRS 3157, University of Arizona, Tucson, AZ, United States of America
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7
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Hortsch SK, Kremling A. Stochastic Models for Studying the Role of Cellular Noise and Heterogeneity. SYSTEMS MEDICINE 2021. [DOI: 10.1016/b978-0-12-801238-3.11466-7] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022] Open
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8
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Vastola JJ, Holmes WR. Chemical Langevin equation: A path-integral view of Gillespie's derivation. Phys Rev E 2020; 101:032417. [PMID: 32289899 DOI: 10.1103/physreve.101.032417] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/22/2019] [Accepted: 02/25/2020] [Indexed: 12/16/2022]
Abstract
In 2000, Gillespie rehabilitated the chemical Langevin equation (CLE) by describing two conditions that must be satisfied for it to yield a valid approximation of the chemical master equation (CME). In this work, we construct an original path-integral description of the CME and show how applying Gillespie's two conditions to it directly leads to a path-integral equivalent to the CLE. We compare this approach to the path-integral equivalent of a large system size derivation and show that they are qualitatively different. In particular, both approaches involve converting many sums into many integrals, and the difference between the two methods is essentially the difference between using the Euler-Maclaurin formula and using Riemann sums. Our results shed light on how path integrals can be used to conceptualize coarse-graining biochemical systems and are readily generalizable.
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Affiliation(s)
- John J Vastola
- Department of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee, USA and Quantitative Systems Biology Center, Vanderbilt University, Nashville, Tennessee 37235, USA
| | - William R Holmes
- Department of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee, USA; Quantitative Systems Biology Center, Vanderbilt University, Nashville, Tennessee 37235, USA; and Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37235, USA
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9
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Nicholson LB, Blyuss KB, Fatehi F. Quantifying the Role of Stochasticity in the Development of Autoimmune Disease. Cells 2020; 9:E860. [PMID: 32252308 PMCID: PMC7226790 DOI: 10.3390/cells9040860] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/28/2020] [Revised: 03/11/2020] [Accepted: 03/26/2020] [Indexed: 12/11/2022] Open
Abstract
In this paper, we propose and analyse a mathematical model for the onset and development of autoimmune disease, with particular attention to stochastic effects in the dynamics. Stability analysis yields parameter regions associated with normal cell homeostasis, or sustained periodic oscillations. Variance of these oscillations and the effects of stochastic amplification are also explored. Theoretical results are complemented by experiments, in which experimental autoimmune uveoretinitis (EAU) was induced in B10.RIII and C57BL/6 mice. For both cases, we discuss peculiarities of disease development, the levels of variation in T cell populations in a population of genetically identical organisms, as well as a comparison with model outputs.
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Affiliation(s)
- Lindsay B. Nicholson
- School of Cellular and Molecular Medicine & School of Clinical Sciences, University of Bristol, University Walk, Bristol BS8 1TD, UK
| | | | - Farzad Fatehi
- Department of Mathematics, University of York, York YO10 5DD, UK;
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10
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Pucci F, Rooman M. Deciphering noise amplification and reduction in open chemical reaction networks. J R Soc Interface 2019; 15:20180805. [PMID: 30958227 DOI: 10.1098/rsif.2018.0805] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
The impact of fluctuations on the dynamical behaviour of complex biological systems is a longstanding issue, whose understanding would elucidate how evolutionary pressure tends to modulate intrinsic noise. Using the Itō stochastic differential equation formalism, we performed analytic and numerical analyses of model systems containing different molecular species in contact with the environment and interacting with each other through mass-action kinetics. For networks of zero deficiency, which admit a detailed- or complex-balanced steady state, all molecular species are uncorrelated and their Fano factors are Poissonian. Systems of higher deficiency have non-equilibrium steady states and non-zero reaction fluxes flowing between the complexes. When they model homo-oligomerization, the noise on each species is reduced when the flux flows from the oligomers of lowest to highest degree, and amplified otherwise. In the case of hetero-oligomerization systems, only the noise on the highest-degree species shows this behaviour.
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Affiliation(s)
- Fabrizio Pucci
- 2 Department of BioModeling, BioInformatics and BioProcesses, Université Libre de Bruxelles , 50 Roosevelt Ave, 1050 Brussels , Belgium
| | - Marianne Rooman
- 1 Department of Theoretical Physics, BioInformatics and BioProcesses, Université Libre de Bruxelles , 50 Roosevelt Ave, 1050 Brussels , Belgium.,2 Department of BioModeling, BioInformatics and BioProcesses, Université Libre de Bruxelles , 50 Roosevelt Ave, 1050 Brussels , Belgium
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11
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Leite SC, Williams RJ. A constrained Langevin approximation for chemical reaction networks. ANN APPL PROBAB 2019. [DOI: 10.1214/18-aap1421] [Citation(s) in RCA: 5] [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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Iida K, Obata N, Kimura Y. Quantifying heterogeneity of stochastic gene expression. J Theor Biol 2019; 465:56-62. [PMID: 30611711 DOI: 10.1016/j.jtbi.2019.01.003] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/18/2018] [Revised: 12/27/2018] [Accepted: 01/03/2019] [Indexed: 11/20/2022]
Abstract
The heterogeneity of stochastic gene expression, which refers to the temporal fluctuation in a gene product and its cell-to-cell variation, has attracted considerable interest from biologists, physicists, and mathematicians. The dynamics of protein production and degradation have been modeled as random processes with transition probabilities. However, there is a gap between theory and phenomena, particularly in terms of analytical formulation and parameter estimation. In this study, we propose a theoretical framework in which we present a basic model of a gene regulatory system, derive a steady-state solution, and provide a Bayesian approach for estimating the model parameters from single-cell experimental data. The proposed framework is demonstrated to be applicable for various scales of single-cell experiments at both the mRNA and protein levels and is useful for comparing kinetic parameters across species, genomes, and cell strains.
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Affiliation(s)
- Keita Iida
- Graduate School of Medicine, Tohoku University, Sendai 980-8575, Japan.
| | - Nobuaki Obata
- Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan.
| | - Yoshitaka Kimura
- Graduate School of Medicine, Tohoku University, Sendai 980-8575, Japan.
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13
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Vu TV, Hasegawa Y. An algebraic method to calculate parameter regions for constrained steady-state distribution in stochastic reaction networks. CHAOS (WOODBURY, N.Y.) 2019; 29:023123. [PMID: 30823706 DOI: 10.1063/1.5047579] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/09/2018] [Accepted: 01/25/2019] [Indexed: 06/09/2023]
Abstract
Steady state is an essential concept in reaction networks. Its stability reflects fundamental characteristics of several biological phenomena such as cellular signal transduction and gene expression. Because biochemical reactions occur at the cellular level, they are affected by unavoidable fluctuations. Although several methods have been proposed to detect and analyze the stability of steady states for deterministic models, these methods cannot be applied to stochastic reaction networks. In this paper, we propose an algorithm based on algebraic computations to calculate parameter regions for constrained steady-state distribution of stochastic reaction networks, in which the means and variances satisfy some given inequality constraints. To evaluate our proposed method, we perform computer simulations for three typical chemical reactions and demonstrate that the results obtained with our method are consistent with the simulation results.
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Affiliation(s)
- Tan Van Vu
- Department of Information and Communication Engineering, Graduate School of Information Science and Technology, The University of Tokyo, Tokyo 113-8656, Japan
| | - Yoshihiko Hasegawa
- Department of Information and Communication Engineering, Graduate School of Information Science and Technology, The University of Tokyo, Tokyo 113-8656, Japan
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14
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Peralta AF, Toral R. System-size expansion of the moments of a master equation. CHAOS (WOODBURY, N.Y.) 2018; 28:106303. [PMID: 30384646 DOI: 10.1063/1.5039817] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/11/2018] [Accepted: 07/02/2018] [Indexed: 06/08/2023]
Abstract
We study an expansion method of the general multidimensional master equation, based on a system-size expansion of the time evolution equations of the moments. The method turns out to be more accurate than the traditional van Kampen expansion for the first and second moments, with an error that scales with system-size similar to an alternative expansion, also applied to the equations of the moments, called Gaussian approximation, with the advantage that it has less systematic errors. Besides, we analyze a procedure to find the solution of the expansion method and we show different cases where it greatly simplifies. This includes the analytical solution of the average value and fluctuations in the number of infected nodes of the SIS epidemic model in complex networks, under the degree-based approximation.
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Affiliation(s)
- A F Peralta
- IFISC (Instituto de Física Interdisciplinar y Sistemas Complejos), Universitat de les Illes Balears-CSIC, 07122 Palma de Mallorca, Spain
| | - R Toral
- IFISC (Instituto de Física Interdisciplinar y Sistemas Complejos), Universitat de les Illes Balears-CSIC, 07122 Palma de Mallorca, Spain
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15
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Gao Z, Sun H, Qin S, Yang X, Tang C. A systematic study of the determinants of protein abundance memory in cell lineage. Sci Bull (Beijing) 2018; 63:1051-1058. [PMID: 36755457 DOI: 10.1016/j.scib.2018.07.010] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/19/2018] [Revised: 06/12/2018] [Accepted: 07/02/2018] [Indexed: 10/28/2022]
Abstract
Proteins are essential players of life activities. Intracellular protein levels directly affect cellular functions and cell fate. Upon cell division, the proteins in the mother cell are inherited by the daughters. However, what factors and by how much they affect this epigenetic inheritance of protein abundance remains unclear. Using both computational and experimental approaches, we systematically investigated this problem. We derived an analytical expression for the dependence of protein inheritance on various factors and showed that it agreed with numerical simulations of protein production and experimental results. Our work provides a framework for quantitative studies of protein inheritance and for the potential application of protein memory manipulation.
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Affiliation(s)
- Zongmao Gao
- Center for Quantitative Biology, Peking University, Beijing 100871, China
| | - Haoyuan Sun
- Peking-Tsinghua Center for Life Sciences, Peking University, Beijing 100871, China
| | - Shanshan Qin
- Center for Quantitative Biology, Peking University, Beijing 100871, China
| | - Xiaojing Yang
- Center for Quantitative Biology, Peking University, Beijing 100871, China.
| | - Chao Tang
- Center for Quantitative Biology, Peking University, Beijing 100871, China; Peking-Tsinghua Center for Life Sciences, Peking University, Beijing 100871, China; School of Physics, Peking University, Beijing 100871, China.
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16
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Cañizo JA, Carrillo JA, Pájaro M. Exponential equilibration of genetic circuits using entropy methods. J Math Biol 2018; 78:373-411. [PMID: 30120513 PMCID: PMC6437139 DOI: 10.1007/s00285-018-1277-z] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/24/2018] [Revised: 07/16/2018] [Indexed: 02/05/2023]
Abstract
We analyse a continuum model for genetic circuits based on a partial integro-differential equation initially proposed in Friedman et al. (Phys Rev Lett 97(16):168302, 2006) as an approximation of a chemical master equation. We use entropy methods to show exponentially fast convergence to equilibrium for this model with explicit bounds. The asymptotic equilibration for the multidimensional case of more than one gene is also obtained under suitable assumptions on the equilibrium stationary states. The asymptotic equilibration property for networks involving one and more than one gene is investigated via numerical simulations.
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Affiliation(s)
- José A Cañizo
- Departamento de Matemática Aplicada, Universidad de Granada, 18071, Granada, Spain
| | - José A Carrillo
- Department of Mathematics, Imperial College London, London, SW7 2AZ, UK.
| | - Manuel Pájaro
- BioProcess Engineering Group, IIM-CSIC, Spanish Council for Scientific Research, Eduardo Cabello 6, 36208, Vigo, Spain
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17
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Qin S, Tang C. Early-warning signals of critical transition: Effect of extrinsic noise. Phys Rev E 2018; 97:032406. [PMID: 29776126 DOI: 10.1103/physreve.97.032406] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/08/2018] [Indexed: 06/08/2023]
Abstract
Complex dynamical systems often have tipping points and exhibit catastrophic regime shift. Despite the notorious difficulty of predicting such transitions, accumulating studies have suggested the existence of generic early-warning signals (EWSs) preceding upcoming transitions. However, previous theories and models were based on the effect of the intrinsic noise (IN) when a system is approaching a critical point, and did not consider the pervasive environmental fluctuations or the extrinsic noise (EN). Here, we extend previous theory to investigate how the interplay of EN and IN affects EWSs. Stochastic simulations of model systems subject to both IN and EN have verified our theory and demonstrated that EN can dramatically alter and diminish the EWS. This effect is stronger with increasing amplitude and correlation time scale of the EN. In the presence of EN, the EWS can fail to predict or even give a false alarm of critical transitions.
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Affiliation(s)
- Shanshan Qin
- Center for Quantitative Biology, Peking University, Beijing 100871, China
| | - Chao Tang
- Center for Quantitative Biology, Peking University, Beijing 100871, China
- School of Physics and Peking-Tsinghua Center for Life Sciences, Peking University, Beijing, 10087, China
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18
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Fatehi F, Kyrychko SN, Ross A, Kyrychko YN, Blyuss KB. Stochastic Effects in Autoimmune Dynamics. Front Physiol 2018; 9:45. [PMID: 29456513 PMCID: PMC5801658 DOI: 10.3389/fphys.2018.00045] [Citation(s) in RCA: 14] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/19/2017] [Accepted: 01/15/2018] [Indexed: 01/05/2023] Open
Abstract
Among various possible causes of autoimmune disease, an important role is played by infections that can result in a breakdown of immune tolerance, primarily through the mechanism of “molecular mimicry”. In this paper we propose and analyse a stochastic model of immune response to a viral infection and subsequent autoimmunity, with account for the populations of T cells with different activation thresholds, regulatory T cells, and cytokines. We show analytically and numerically how stochasticity can result in sustained oscillations around deterministically stable steady states, and we also investigate stochastic dynamics in the regime of bi-stability. These results provide a possible explanation for experimentally observed variations in the progression of autoimmune disease. Computations of the variance of stochastic fluctuations provide practically important insights into how the size of these fluctuations depends on various biological parameters, and this also gives a headway for comparison with experimental data on variation in the observed numbers of T cells and organ cells affected by infection.
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Affiliation(s)
- Farzad Fatehi
- Department of Mathematics, University of Sussex, Brighton, United Kingdom
| | | | - Aleksandra Ross
- Department of Mathematics, University of Sussex, Brighton, United Kingdom
| | - Yuliya N Kyrychko
- Department of Mathematics, University of Sussex, Brighton, United Kingdom
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19
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Lin YT, Hufton PG, Lee EJ, Potoyan DA. A stochastic and dynamical view of pluripotency in mouse embryonic stem cells. PLoS Comput Biol 2018; 14:e1006000. [PMID: 29451874 PMCID: PMC5833290 DOI: 10.1371/journal.pcbi.1006000] [Citation(s) in RCA: 32] [Impact Index Per Article: 5.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/25/2017] [Revised: 03/01/2018] [Accepted: 01/19/2018] [Indexed: 12/26/2022] Open
Abstract
Pluripotent embryonic stem cells are of paramount importance for biomedical sciences because of their innate ability for self-renewal and differentiation into all major cell lines. The fateful decision to exit or remain in the pluripotent state is regulated by complex genetic regulatory networks. The rapid growth of single-cell sequencing data has greatly stimulated applications of statistical and machine learning methods for inferring topologies of pluripotency regulating genetic networks. The inferred network topologies, however, often only encode Boolean information while remaining silent about the roles of dynamics and molecular stochasticity inherent in gene expression. Herein we develop a framework for systematically extending Boolean-level network topologies into higher resolution models of networks which explicitly account for the promoter architectures and gene state switching dynamics. We show the framework to be useful for disentangling the various contributions that gene switching, external signaling, and network topology make to the global heterogeneity and dynamics of transcription factor populations. We find the pluripotent state of the network to be a steady state which is robust to global variations of gene switching rates which we argue are a good proxy for epigenetic states of individual promoters. The temporal dynamics of exiting the pluripotent state, on the other hand, is significantly influenced by the rates of genetic switching which makes cells more responsive to changes in extracellular signals.
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Affiliation(s)
- Yen Ting Lin
- Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico, United States of America
- School of Physics and Astronomy, The University of Manchester, Manchester, United Kingdom
| | - Peter G. Hufton
- School of Physics and Astronomy, The University of Manchester, Manchester, United Kingdom
| | - Esther J. Lee
- Department of Bioengineering, Rice University, Houston, Texas, United States of America
| | - Davit A. Potoyan
- Department of Chemistry, Iowa State University, Ames, Iowa, United States of America
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20
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Yang J, Axelrod DE, Komarova NL. Determining the control networks regulating stem cell lineages in colonic crypts. J Theor Biol 2017; 429:190-203. [PMID: 28669884 PMCID: PMC5689466 DOI: 10.1016/j.jtbi.2017.06.033] [Citation(s) in RCA: 16] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/01/2016] [Revised: 05/18/2017] [Accepted: 06/25/2017] [Indexed: 12/27/2022]
Abstract
The question of stem cell control is at the center of our understanding of tissue functioning, both in healthy and cancerous conditions. It is well accepted that cellular fate decisions (such as divisions, differentiation, apoptosis) are orchestrated by a network of regulatory signals emitted by different cell populations in the lineage and the surrounding tissue. The exact regulatory network that governs stem cell lineages in a given tissue is usually unknown. Here we propose an algorithm to identify a set of candidate control networks that are compatible with (a) measured means and variances of cell populations in different compartments, (b) qualitative information on cell population dynamics, such as the existence of local controls and oscillatory reaction of the system to population size perturbations, and (c) statistics of correlations between cell numbers in different compartments. Using the example of human colon crypts, where lineages are comprised of stem cells, transit amplifying cells, and differentiated cells, we start with a theoretically known set of 32 smallest control networks compatible with tissue stability. Utilizing near-equilibrium stochastic calculus of stem cells developed earlier, we apply a series of tests, where we compare the networks' expected behavior with the observations. This allows us to exclude most of the networks, until only three, very similar, candidate networks remain, which are most compatible with the measurements. This work demonstrates how theoretical analysis of control networks combined with only static biological data can shed light onto the inner workings of stem cell lineages, in the absence of direct experimental assessment of regulatory signaling mechanisms. The resulting candidate networks are dominated by negative control loops and possess the following properties: (1) stem cell division decisions are negatively controlled by the stem cell population, (2) stem cell differentiation decisions are negatively controlled by the transit amplifying cell population.
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Affiliation(s)
- Jienian Yang
- Department of Mathematics, University of California, Irvine, Irvine, CA 92697 USA
| | - David E Axelrod
- Department of Genetics and Cancer Institute of New Jersey, Rutgers University, Piscataway, NJ 08854-8082, USA
| | - Natalia L Komarova
- Department of Ecology and Evolutionary Biology, University of California, Irvine, Irvine, CA 92697, USA.
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21
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Cardelli L, Kwiatkowska M, Laurenti L. Stochastic analysis of Chemical Reaction Networks using Linear Noise Approximation. Biosystems 2016; 149:26-33. [PMID: 27816736 DOI: 10.1016/j.biosystems.2016.09.004] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/27/2015] [Revised: 07/08/2016] [Accepted: 09/01/2016] [Indexed: 10/20/2022]
Abstract
Stochastic evolution of Chemical Reactions Networks (CRNs) over time is usually analyzed through solving the Chemical Master Equation (CME) or performing extensive simulations. Analysing stochasticity is often needed, particularly when some molecules occur in low numbers. Unfortunately, both approaches become infeasible if the system is complex and/or it cannot be ensured that initial populations are small. We develop a probabilistic logic for CRNs that enables stochastic analysis of the evolution of populations of molecular species. We present an approximate model checking algorithm based on the Linear Noise Approximation (LNA) of the CME, whose computational complexity is independent of the population size of each species and polynomial in the number of different species. The algorithm requires the solution of first order polynomial differential equations. We prove that our approach is valid for any CRN close enough to the thermodynamical limit. However, we show on four case studies that it can still provide good approximation even for low molecule counts. Our approach enables rigorous analysis of CRNs that are not analyzable by solving the CME, but are far from the deterministic limit. Moreover, it can be used for a fast approximate stochastic characterization of a CRN.
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Affiliation(s)
- Luca Cardelli
- Department of Computer Science, University of Oxford, United Kingdom; Microsoft Research, Cambridge, United Kingdom.
| | - Marta Kwiatkowska
- Department of Computer Science, University of Oxford, United Kingdom.
| | - Luca Laurenti
- Department of Computer Science, University of Oxford, United Kingdom.
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22
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Mc Mahon SS, Lenive O, Filippi S, Stumpf MPH. Information processing by simple molecular motifs and susceptibility to noise. J R Soc Interface 2016; 12:0597. [PMID: 26333812 DOI: 10.1098/rsif.2015.0597] [Citation(s) in RCA: 22] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
Biological organisms rely on their ability to sense and respond appropriately to their environment. The molecular mechanisms that facilitate these essential processes are however subject to a range of random effects and stochastic processes, which jointly affect the reliability of information transmission between receptors and, for example, the physiological downstream response. Information is mathematically defined in terms of the entropy; and the extent of information flowing across an information channel or signalling system is typically measured by the 'mutual information', or the reduction in the uncertainty about the output once the input signal is known. Here, we quantify how extrinsic and intrinsic noise affects the transmission of simple signals along simple motifs of molecular interaction networks. Even for very simple systems, the effects of the different sources of variability alone and in combination can give rise to bewildering complexity. In particular, extrinsic variability is apt to generate 'apparent' information that can, in extreme cases, mask the actual information that for a single system would flow between the different molecular components making up cellular signalling pathways. We show how this artificial inflation in apparent information arises and how the effects of different types of noise alone and in combination can be understood.
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Affiliation(s)
- Siobhan S Mc Mahon
- Centre for Integrative Systems Biology and Bioinformatics, Department of Life Sciences, Biosciences, Imperial College London, South Kensington, London SW7 2AZ, UK
| | - Oleg Lenive
- Centre for Integrative Systems Biology and Bioinformatics, Department of Life Sciences, Biosciences, Imperial College London, South Kensington, London SW7 2AZ, UK
| | - Sarah Filippi
- Department of Statistics, University of Oxford, Oxford OX1 3TG, UK
| | - Michael P H Stumpf
- Centre for Integrative Systems Biology and Bioinformatics, Department of Life Sciences, Biosciences, Imperial College London, South Kensington, London SW7 2AZ, UK Institute of Chemical Biology, Imperial College London, South Kensington, London SW7 2AZ, UK
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23
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Abstract
Cells operate in noisy molecular environments via complex regulatory networks. It is possible to understand how molecular counts are related to noise in specific networks, but it is not generally clear how noise relates to network complexity, because different levels of complexity also imply different overall number of molecules. For a fixed function, does increased network complexity reduce noise, beyond the mere increase of overall molecular counts? If so, complexity could provide an advantage counteracting the costs involved in maintaining larger networks. For that purpose, we investigate how noise affects multistable systems, where a small amount of noise could lead to very different outcomes; thus we turn to biochemical switches. Our method for comparing networks of different structure and complexity is to place them in conditions where they produce exactly the same deterministic function. We are then in a good position to compare their noise characteristics relatively to their identical deterministic traces. We show that more complex networks are better at coping with both intrinsic and extrinsic noise. Intrinsic noise tends to decrease with complexity, and extrinsic noise tends to have less impact. Our findings suggest a new role for increased complexity in biological networks, at parity of function.
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24
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Hilfinger A, Norman TM, Vinnicombe G, Paulsson J. Constraints on Fluctuations in Sparsely Characterized Biological Systems. PHYSICAL REVIEW LETTERS 2016; 116:058101. [PMID: 26894735 PMCID: PMC4834202 DOI: 10.1103/physrevlett.116.058101] [Citation(s) in RCA: 22] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/27/2015] [Indexed: 06/05/2023]
Abstract
Biochemical processes are inherently stochastic, creating molecular fluctuations in otherwise identical cells. Such "noise" is widespread but has proven difficult to analyze because most systems are sparsely characterized at the single cell level and because nonlinear stochastic models are analytically intractable. Here, we exactly relate average abundances, lifetimes, step sizes, and covariances for any pair of components in complex stochastic reaction systems even when the dynamics of other components are left unspecified. Using basic mathematical inequalities, we then establish bounds for whole classes of systems. These bounds highlight fundamental trade-offs that show how efficient assembly processes must invariably exhibit large fluctuations in subunit levels and how eliminating fluctuations in one cellular component requires creating heterogeneity in another.
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Affiliation(s)
- Andreas Hilfinger
- Department of Systems Biology, Harvard University, 200 Longwood Avenue, Boston, Massachusetts 02115, USA
| | - Thomas M. Norman
- Department of Systems Biology, Harvard University, 200 Longwood Avenue, Boston, Massachusetts 02115, USA
| | - Glenn Vinnicombe
- Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, United Kingdom
| | - Johan Paulsson
- Department of Systems Biology, Harvard University, 200 Longwood Avenue, Boston, Massachusetts 02115, USA
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25
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Approximation of Probabilistic Reachability for Chemical Reaction Networks Using the Linear Noise Approximation. QUANTITATIVE EVALUATION OF SYSTEMS 2016. [DOI: 10.1007/978-3-319-43425-4_5] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/12/2022]
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26
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Grima R. Linear-noise approximation and the chemical master equation agree up to second-order moments for a class of chemical systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:042124. [PMID: 26565185 DOI: 10.1103/physreve.92.042124] [Citation(s) in RCA: 24] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/05/2015] [Indexed: 06/05/2023]
Abstract
It is well known that the linear-noise approximation (LNA) agrees with the chemical master equation, up to second-order moments, for chemical systems composed of zero and first-order reactions. Here we show that this is also a property of the LNA for a subset of chemical systems with second-order reactions. This agreement is independent of the number of interacting molecules.
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Affiliation(s)
- Ramon Grima
- School of Biological Sciences, University of Edinburgh, United Kingdom
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27
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Lakatos E, Ale A, Kirk PDW, Stumpf MPH. Multivariate moment closure techniques for stochastic kinetic models. J Chem Phys 2015; 143:094107. [DOI: 10.1063/1.4929837] [Citation(s) in RCA: 55] [Impact Index Per Article: 6.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Eszter Lakatos
- Department of Life Sciences, Centre for Integrative Systems Biology and Bioinformatics, Imperial College London, London SW7 2AZ, United Kingdom
| | - Angelique Ale
- Department of Life Sciences, Centre for Integrative Systems Biology and Bioinformatics, Imperial College London, London SW7 2AZ, United Kingdom
| | - Paul D. W. Kirk
- Department of Life Sciences, Centre for Integrative Systems Biology and Bioinformatics, Imperial College London, London SW7 2AZ, United Kingdom
| | - Michael P. H. Stumpf
- Department of Life Sciences, Centre for Integrative Systems Biology and Bioinformatics, Imperial College London, London SW7 2AZ, United Kingdom
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28
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Yang J, Sun Z, Komarova NL. Analysis of stochastic stem cell models with control. Math Biosci 2015; 266:93-107. [PMID: 26073965 DOI: 10.1016/j.mbs.2015.06.001] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/30/2014] [Revised: 05/28/2015] [Accepted: 06/03/2015] [Indexed: 12/11/2022]
Abstract
Understanding the dynamics of stem cell lineages is of central importance both for healthy and cancerous tissues. We study stochastic population dynamics of stem cells and differentiated cells, where cell decisions, such as proliferation vs. differentiation decisions, or division and death decisions, are under regulation from surrounding cells. The goal is to understand how different types of control mechanisms affect the means and variances of cell numbers. We use the assumption of weak dependencies of the regulatory functions (the controls) on the cell populations near the equilibrium to formulate moment equations. We then study three different methods of closure, showing that they all lead to the same results for the highest order terms in the expressions for the moments. We derive simple explicit expressions for the means and the variances of stem cell and differentiated cell numbers. It turns out that the variance is expressed as an algebraic function of partial derivatives of the controls with respect to the population sizes at the equilibrium. We demonstrate that these findings are consistent with the results previously obtained in the context of particular systems, and also present two novel examples with negative and positive control of division and differentiation decisions. This methodology is formulated without any specific assumptions on the functional form of the controls, and thus can be used for any biological system.
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Affiliation(s)
- Jienian Yang
- Department of Mathematics, University of California Irvine, Irvine, CA 92617, United States
| | - Zheng Sun
- Department of Mathematics, University of California Irvine, Irvine, CA 92617, United States
| | - Natalia L Komarova
- Department of Mathematics, University of California Irvine, Irvine, CA 92617, United States.
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29
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Schnoerr D, Sanguinetti G, Grima R. The complex chemical Langevin equation. J Chem Phys 2015; 141:024103. [PMID: 25027995 DOI: 10.1063/1.4885345] [Citation(s) in RCA: 46] [Impact Index Per Article: 5.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
The chemical Langevin equation (CLE) is a popular simulation method to probe the stochastic dynamics of chemical systems. The CLE's main disadvantage is its break down in finite time due to the problem of evaluating square roots of negative quantities whenever the molecule numbers become sufficiently small. We show that this issue is not a numerical integration problem, rather in many systems it is intrinsic to all representations of the CLE. Various methods of correcting the CLE have been proposed which avoid its break down. We show that these methods introduce undesirable artefacts in the CLE's predictions. In particular, for unimolecular systems, these correction methods lead to CLE predictions for the mean concentrations and variance of fluctuations which disagree with those of the chemical master equation. We show that, by extending the domain of the CLE to complex space, break down is eliminated, and the CLE's accuracy for unimolecular systems is restored. Although the molecule numbers are generally complex, we show that the "complex CLE" predicts real-valued quantities for the mean concentrations, the moments of intrinsic noise, power spectra, and first passage times, hence admitting a physical interpretation. It is also shown to provide a more accurate approximation of the chemical master equation of simple biochemical circuits involving bimolecular reactions than the various corrected forms of the real-valued CLE, the linear-noise approximation and a commonly used two moment-closure approximation.
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Affiliation(s)
- David Schnoerr
- School of Biological Sciences, University of Edinburgh, United Kingdom
| | | | - Ramon Grima
- School of Biological Sciences, University of Edinburgh, United Kingdom
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30
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Hey KL, Momiji H, Featherstone K, Davis JRE, White MRH, Rand DA, Finkenstädt B. A stochastic transcriptional switch model for single cell imaging data. Biostatistics 2015; 16:655-69. [PMID: 25819987 PMCID: PMC4570576 DOI: 10.1093/biostatistics/kxv010] [Citation(s) in RCA: 27] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/12/2014] [Accepted: 02/21/2015] [Indexed: 12/03/2022] Open
Abstract
Gene expression is made up of inherently stochastic processes within single cells and can be modeled through stochastic reaction networks (SRNs). In particular, SRNs capture the features of intrinsic variability arising from intracellular biochemical processes. We extend current models for gene expression to allow the transcriptional process within an SRN to follow a random step or switch function which may be estimated using reversible jump Markov chain Monte Carlo (MCMC). This stochastic switch model provides a generic framework to capture many different dynamic features observed in single cell gene expression. Inference for such SRNs is challenging due to the intractability of the transition densities. We derive a model-specific birth–death approximation and study its use for inference in comparison with the linear noise approximation where both approximations are considered within the unifying framework of state-space models. The methodology is applied to synthetic as well as experimental single cell imaging data measuring expression of the human prolactin gene in pituitary cells.
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Affiliation(s)
- Kirsty L Hey
- Department of Statistics, University of Warwick, Coventry CV4 7AL, UK
| | - Hiroshi Momiji
- Warwick Systems Biology, University of Warwick, Coventry CV4 7AL, UK
| | - Karen Featherstone
- Centre for Endocrinology and Diabetes, University of Manchester, Manchester M13 9PT, UK
| | - Julian R E Davis
- Centre for Endocrinology and Diabetes, University of Manchester, Manchester M13 9PT, UK
| | - Michael R H White
- Systems Biology Centre, University of Manchester, Manchester M13 9PL, UK
| | - David A Rand
- Warwick Systems Biology, University of Warwick, Coventry CV4 7AL, UK
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31
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Stochastic Analysis of Chemical Reaction Networks Using Linear Noise Approximation. COMPUTATIONAL METHODS IN SYSTEMS BIOLOGY 2015. [DOI: 10.1007/978-3-319-23401-4_7] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/12/2023]
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32
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Székely T, Burrage K. Stochastic simulation in systems biology. Comput Struct Biotechnol J 2014; 12:14-25. [PMID: 25505503 PMCID: PMC4262058 DOI: 10.1016/j.csbj.2014.10.003] [Citation(s) in RCA: 37] [Impact Index Per Article: 3.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/10/2014] [Accepted: 10/13/2014] [Indexed: 11/03/2022] Open
Abstract
Natural systems are, almost by definition, heterogeneous: this can be either a boon or an obstacle to be overcome, depending on the situation. Traditionally, when constructing mathematical models of these systems, heterogeneity has typically been ignored, despite its critical role. However, in recent years, stochastic computational methods have become commonplace in science. They are able to appropriately account for heterogeneity; indeed, they are based around the premise that systems inherently contain at least one source of heterogeneity (namely, intrinsic heterogeneity). In this mini-review, we give a brief introduction to theoretical modelling and simulation in systems biology and discuss the three different sources of heterogeneity in natural systems. Our main topic is an overview of stochastic simulation methods in systems biology. There are many different types of stochastic methods. We focus on one group that has become especially popular in systems biology, biochemistry, chemistry and physics. These discrete-state stochastic methods do not follow individuals over time; rather they track only total populations. They also assume that the volume of interest is spatially homogeneous. We give an overview of these methods, with a discussion of the advantages and disadvantages of each, and suggest when each is more appropriate to use. We also include references to software implementations of them, so that beginners can quickly start using stochastic methods for practical problems of interest.
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Affiliation(s)
- Tamás Székely
- Department of Computer Science, University of Oxford, Oxford, United Kingdom
| | - Kevin Burrage
- Department of Computer Science, University of Oxford, Oxford, United Kingdom ; Department of Mathematics, Queensland University of Technology, Brisbane, Queensland, Australia
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33
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Ghavami S, Wolkenhauer O, Lahouti F, Ullah M, Linnebacher M. Accounting for randomness in measurement and sampling in studying cancer cell population dynamics. IET Syst Biol 2014; 8:230-41. [PMID: 25257023 DOI: 10.1049/iet-syb.2013.0031] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/20/2022] Open
Abstract
Knowing the expected temporal evolution of the proportion of different cell types in sample tissues gives an indication about the progression of the disease and its possible response to drugs. Such systems have been modelled using Markov processes. We here consider an experimentally realistic scenario in which transition probabilities are estimated from noisy cell population size measurements. Using aggregated data of FACS measurements, we develop MMSE and ML estimators and formulate two problems to find the minimum number of required samples and measurements to guarantee the accuracy of predicted population sizes. Our numerical results show that the convergence mechanism of transition probabilities and steady states differ widely from the real values if one uses the standard deterministic approach for noisy measurements. This provides support for our argument that for the analysis of FACS data one should consider the observed state as a random variable. The second problem we address is about the consequences of estimating the probability of a cell being in a particular state from measurements of small population of cells. We show how the uncertainty arising from small sample sizes can be captured by a distribution for the state probability.
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Affiliation(s)
- Siavash Ghavami
- Department of Systems Biology and Bioinformatics, University of Rostock, Rostock, Germany.
| | - Olaf Wolkenhauer
- Stellenbosch Institute for Advanced Study (STIAS), Wallenberg Research Centre at Stellenbosch University, Stellenbosch, South Africa
| | - Farshad Lahouti
- Center for Wireless Multimedia Communications, Center of Excellence in Applied Electromagnetic Systems, School of Electrical & Computer Engineering, College of Engineering, University of Tehran, Tehran, Iran
| | - Mukhtar Ullah
- Department of Systems Biology and Bioinformatics, University of Rostock, Rostock, Germany
| | - Michael Linnebacher
- Department of General, Thoracic, Vascular and Transplantation Surgery, University of Rostock, Rostock, Germany
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34
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Mc Mahon SS, Sim A, Filippi S, Johnson R, Liepe J, Smith D, Stumpf MPH. Information theory and signal transduction systems: from molecular information processing to network inference. Semin Cell Dev Biol 2014; 35:98-108. [PMID: 24953199 DOI: 10.1016/j.semcdb.2014.06.011] [Citation(s) in RCA: 33] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/14/2014] [Revised: 06/04/2014] [Accepted: 06/10/2014] [Indexed: 01/05/2023]
Abstract
Sensing and responding to the environment are two essential functions that all biological organisms need to master for survival and successful reproduction. Developmental processes are marshalled by a diverse set of signalling and control systems, ranging from systems with simple chemical inputs and outputs to complex molecular and cellular networks with non-linear dynamics. Information theory provides a powerful and convenient framework in which such systems can be studied; but it also provides the means to reconstruct the structure and dynamics of molecular interaction networks underlying physiological and developmental processes. Here we supply a brief description of its basic concepts and introduce some useful tools for systems and developmental biologists. Along with a brief but thorough theoretical primer, we demonstrate the wide applicability and biological application-specific nuances by way of different illustrative vignettes. In particular, we focus on the characterisation of biological information processing efficiency, examining cell-fate decision making processes, gene regulatory network reconstruction, and efficient signal transduction experimental design.
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Affiliation(s)
- Siobhan S Mc Mahon
- Centre for Integrative Systems Biology and Bioinformatics, Department of Life Sciences, Imperial College London, London SW7 2AZ, UK
| | - Aaron Sim
- Centre for Integrative Systems Biology and Bioinformatics, Department of Life Sciences, Imperial College London, London SW7 2AZ, UK
| | - Sarah Filippi
- Centre for Integrative Systems Biology and Bioinformatics, Department of Life Sciences, Imperial College London, London SW7 2AZ, UK
| | - Robert Johnson
- Centre for Integrative Systems Biology and Bioinformatics, Department of Life Sciences, Imperial College London, London SW7 2AZ, UK
| | - Juliane Liepe
- Centre for Integrative Systems Biology and Bioinformatics, Department of Life Sciences, Imperial College London, London SW7 2AZ, UK
| | - Dominic Smith
- Centre for Integrative Systems Biology and Bioinformatics, Department of Life Sciences, Imperial College London, London SW7 2AZ, UK
| | - Michael P H Stumpf
- Centre for Integrative Systems Biology and Bioinformatics, Department of Life Sciences, Imperial College London, London SW7 2AZ, UK.
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35
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Biancalani T, Dyson L, McKane AJ. Noise-induced bistable states and their mean switching time in foraging colonies. PHYSICAL REVIEW LETTERS 2014; 112:038101. [PMID: 24484166 DOI: 10.1103/physrevlett.112.038101] [Citation(s) in RCA: 30] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/18/2013] [Indexed: 06/03/2023]
Abstract
We investigate a type of bistability occurring in population systems where noise not only causes transitions between stable states, but also constructs the states themselves. We focus on the experimentally well-studied system of ants choosing between two food sources to illustrate the essential points, but the ideas are more general. The mean time for switching between the two bistable states of the system is calculated. This suggests a procedure for estimating, in a real system, the critical population size above which bistability ceases to occur.
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Affiliation(s)
- Tommaso Biancalani
- Theoretical Physics Division, School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom
| | - Louise Dyson
- Theoretical Physics Division, School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom
| | - Alan J McKane
- Theoretical Physics Division, School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom
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36
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Finkenstädt B, Woodcock DJ, Komorowski M, Harper CV, Davis JRE, White MRH, Rand DA. Quantifying intrinsic and extrinsic noise in gene transcription using the linear noise approximation: An application to single cell data. Ann Appl Stat 2013. [DOI: 10.1214/13-aoas669] [Citation(s) in RCA: 26] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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37
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Jetka T, Charzyńska A, Gambin A, Stumpf MPH, Komorowski M. StochDecomp--Matlab package for noise decomposition in stochastic biochemical systems. ACTA ACUST UNITED AC 2013; 30:137-8. [PMID: 24191070 DOI: 10.1093/bioinformatics/btt631] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/25/2023]
Abstract
MOTIVATION Stochasticity is an indispensable aspect of biochemical processes at the cellular level. Studies on how the noise enters and propagates in biochemical systems provided us with non-trivial insights into the origins of stochasticity, in total, however, they constitute a patchwork of different theoretical analyses. RESULTS Here we present a flexible and widely applicable noise decomposition tool that allows us to calculate contributions of individual reactions to the total variability of a system's output. With the package it is, therefore, possible to quantify how the noise enters and propagates in biochemical systems. We also demonstrate and exemplify using the JAK-STAT signalling pathway that the noise contributions resulting from individual reactions can be inferred from data experimental data along with Bayesian parameter inference. The method is based on the linear noise approximation, which is assumed to provide a reasonable representation of analyzed systems. AVAILABILITY AND IMPLEMENTATION http://sourceforge.net/p/stochdecomp/
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Affiliation(s)
- Tomasz Jetka
- Institute of Fundamental Technological Research, Institute of Computer Science, Polish Academy of Sciences, Warsaw, Poland, Faculty of Mathematics Informatics and Mechanics, Institute of Informatics, University of Warsaw, Warsaw, Poland and Division of Molecular Biosciences, Imperial College London, London, UK
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38
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Komorowski M, Miękisz J, Stumpf MPH. Decomposing noise in biochemical signaling systems highlights the role of protein degradation. Biophys J 2013; 104:1783-93. [PMID: 23601325 DOI: 10.1016/j.bpj.2013.02.027] [Citation(s) in RCA: 36] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/23/2012] [Revised: 01/30/2013] [Accepted: 02/08/2013] [Indexed: 11/17/2022] Open
Abstract
Stochasticity is an essential aspect of biochemical processes at the cellular level. We now know that living cells take advantage of stochasticity in some cases and counteract stochastic effects in others. Here we propose a method that allows us to calculate contributions of individual reactions to the total variability of a system's output. We demonstrate that reactions differ significantly in their relative impact on the total noise and we illustrate the importance of protein degradation on the overall variability for a range of molecular processes and signaling systems. With our flexible and generally applicable noise decomposition method, we are able to shed new, to our knowledge, light on the sources and propagation of noise in biochemical reaction networks; in particular, we are able to show how regulated protein degradation can be employed to reduce the noise in biochemical systems.
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Affiliation(s)
- Michał Komorowski
- Division of Molecular Biosciences, Imperial College London, London, United Kingdom.
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39
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Widmer LA, Stelling J, Doyle FJ. Note: Parameter-independent bounding of the stochastic Michaelis-Menten steady-state intrinsic noise variance. J Chem Phys 2013; 139:166102. [DOI: 10.1063/1.4827496] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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40
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Thomas P, Straube AV, Timmer J, Fleck C, Grima R. Signatures of nonlinearity in single cell noise-induced oscillations. J Theor Biol 2013; 335:222-34. [DOI: 10.1016/j.jtbi.2013.06.021] [Citation(s) in RCA: 41] [Impact Index Per Article: 3.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/10/2013] [Revised: 05/20/2013] [Accepted: 06/18/2013] [Indexed: 01/10/2023]
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Abstract
Identifying the exact regulatory circuits that can stably maintain tissue homeostasis is critical for our basic understanding of multicellular organisms, and equally critical for identifying how tumors circumvent this regulation, thus providing targets for treatment. Despite great strides in the understanding of the molecular components of stem-cell regulation, the overall mechanisms orchestrating tissue homeostasis are still far from being understood. Typically, tissue contains the stem cells, transit amplifying cells, and terminally differentiated cells. Each of these cell types can potentially secrete regulatory factors and/or respond to factors secreted by other types. The feedback can be positive or negative in nature. This gives rise to a bewildering array of possible mechanisms that drive tissue regulation. In this paper, we propose a novel method of studying stem cell lineage regulation, and identify possible numbers, types, and directions of control loops that are compatible with stability, keep the variance low, and possess a certain degree of robustness. For example, there are exactly two minimal (two-loop) control networks that can regulate two-compartment (stem and differentiated cell) tissues, and 20 such networks in three-compartment tissues. If division and differentiation decisions are coupled, then there must be a negative control loop regulating divisions of stem cells (e.g. by means of contact inhibition). While this mechanism is associated with the highest robustness, there could be systems that maintain stability by means of positive divisions control, coupled with specific types of differentiation control. Some of the control mechanisms that we find have been proposed before, but most of them are new, and we describe evidence for their existence in data that have been previously published. By specifying the types of feedback interactions that can maintain homeostasis, our mathematical analysis can be used as a guide to experimentally zero in on the exact molecular mechanisms in specific tissues.
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Affiliation(s)
- Natalia L. Komarova
- Department of Mathematics, University of California Irvine, Irvine, California, United States of America
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42
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Jenkinson G, Goutsias J. Statistically testing the validity of analytical and computational approximations to the chemical master equation. J Chem Phys 2013; 138:204108. [PMID: 23742455 DOI: 10.1063/1.4807390] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
The master equation is used extensively to model chemical reaction systems with stochastic dynamics. However, and despite its phenomenological simplicity, it is not in general possible to compute the solution of this equation. Drawing exact samples from the master equation is possible, but can be computationally demanding, especially when estimating high-order statistical summaries or joint probability distributions. As a consequence, one often relies on analytical approximations to the solution of the master equation or on computational techniques that draw approximative samples from this equation. Unfortunately, it is not in general possible to check whether a particular approximation scheme is valid. The main objective of this paper is to develop an effective methodology to address this problem based on statistical hypothesis testing. By drawing a moderate number of samples from the master equation, the proposed techniques use the well-known Kolmogorov-Smirnov statistic to reject the validity of a given approximation method or accept it with a certain level of confidence. Our approach is general enough to deal with any master equation and can be used to test the validity of any analytical approximation method or any approximative sampling technique of interest. A number of examples, based on the Schlögl model of chemistry and the SIR model of epidemiology, clearly illustrate the effectiveness and potential of the proposed statistical framework.
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Affiliation(s)
- Garrett Jenkinson
- Whitaker Biomedical Engineering Institute, The Johns Hopkins University, Baltimore, Maryland 21218, USA
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43
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Gillespie DT, Hellander A, Petzold LR. Perspective: Stochastic algorithms for chemical kinetics. J Chem Phys 2013; 138:170901. [PMID: 23656106 PMCID: PMC3656953 DOI: 10.1063/1.4801941] [Citation(s) in RCA: 165] [Impact Index Per Article: 15.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/05/2013] [Accepted: 03/25/2013] [Indexed: 11/14/2022] Open
Abstract
We outline our perspective on stochastic chemical kinetics, paying particular attention to numerical simulation algorithms. We first focus on dilute, well-mixed systems, whose description using ordinary differential equations has served as the basis for traditional chemical kinetics for the past 150 years. For such systems, we review the physical and mathematical rationale for a discrete-stochastic approach, and for the approximations that need to be made in order to regain the traditional continuous-deterministic description. We next take note of some of the more promising strategies for dealing stochastically with stiff systems, rare events, and sensitivity analysis. Finally, we review some recent efforts to adapt and extend the discrete-stochastic approach to systems that are not well-mixed. In that currently developing area, we focus mainly on the strategy of subdividing the system into well-mixed subvolumes, and then simulating diffusional transfers of reactant molecules between adjacent subvolumes together with chemical reactions inside the subvolumes.
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Affiliation(s)
- Daniel T Gillespie
- Dan T Gillespie Consulting, 30504 Cordoba Pl., Castaic, California 91384, USA.
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44
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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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45
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Wallace E. Where is that Noise Coming from? Biophys J 2013; 104:1637-8. [DOI: 10.1016/j.bpj.2013.02.031] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/18/2013] [Accepted: 02/08/2013] [Indexed: 10/27/2022] Open
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46
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Stathopoulos V, Girolami MA. Markov chain Monte Carlo inference for Markov jump processes via the linear noise approximation. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2013; 371:20110541. [PMID: 23277599 DOI: 10.1098/rsta.2011.0541] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/01/2023]
Abstract
Bayesian analysis for Markov jump processes (MJPs) is a non-trivial and challenging problem. Although exact inference is theoretically possible, it is computationally demanding, thus its applicability is limited to a small class of problems. In this paper, we describe the application of Riemann manifold Markov chain Monte Carlo (MCMC) methods using an approximation to the likelihood of the MJP that is valid when the system modelled is near its thermodynamic limit. The proposed approach is both statistically and computationally efficient whereas the convergence rate and mixing of the chains allow for fast MCMC inference. The methodology is evaluated using numerical simulations on two problems from chemical kinetics and one from systems biology.
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Affiliation(s)
- Vassilios Stathopoulos
- Department of Statistical Science, Centre for Computational Statistics and Machine Learning, University College London, Gower Street, London WC1E 6BT, UK
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