DOCSLIB.ORG

The Stochastic Dynamics of Biochemical Systems

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Stochastic Dynamics of Biochemical Systems THE STOCHASTIC DYNAMICS OF BIOCHEMICAL SYSTEMS A thesis submitted to the University of Manchester for the degree of Doctor of Philosophy in the Faculty of Engineering and Physical Sciences 2013 By Joseph Daniel Challenger School of Physics and Astronomy Contents Abstract 8 Declaration 9 Copyright 10 Acknowledgements 11 Publications 12 1 Introduction 13 1.1 Stochastic effects in biochemical systems . 15 1.2 The role of noise in oscillatory behaviour . 17 1.3 Theoretical and numerical techniques . 18 1.4 Our approach . 19 1.5 Introducing COPASI . 20 1.6 Outlook . 22 2 Background 23 2.1 The master equation . 23 2.1.1 An example . 26 2.2 Deterministic dynamics . 27 2.3 Simulating the master equation . 28 2.3.1 The Gillespie algorithm . 29 2.3.2 Other simulation algorithms . 33 2.4 The van Kampen expansion . 34 2.5 The Fokker-Planck equation . 37 2.5.1 Solving the Lyapunov equation . 39 2.6 The Langevin equation . 41 2 2.7 Stochastic oscillations & power spectra . 43 2.8 Summary . 46 3 Fluctuation Analysis and COPASI 50 3.1 Using COPASI . 50 3.2 Formalism . 53 3.3 Details of the implementation . 59 3.4 Conservation relations . 60 3.5 The linear noise approximation in COPASI . 63 3.6 A MAPK signalling system . 66 3.7 Summary . 69 4 Multi-compartment LNA 73 4.1 A two-compartment system . 74 4.2 The general case . 77 4.3 Relation between A^ and A~ ...................... 81 4.4 A model of metabolism in cardiac muscle . 83 4.5 A yeast glycolysis model . 86 4.6 Discussion . 91 5 Synchronisation of stochastic oscillators 92 5.1 Collective behaviour in Dictyostelium ................ 99 5.2 Synchronisation of glycolytic oscillations . 106 5.3 Discussion . 110 6 Conclusions 112 Bibliography 118 A Ill-Conditioned Systems 128 B Preliminary Code 133 C Form of the complex coherence function 140 Word Count: 28,981 3 4 List of Tables 3.1 The covariance matrix for the Michaelis-Menten reaction system . 65 3.2 Reaction scheme for the Kholodenko MAPK signalling model . 67 3.3 Covariance matrix for the MAPK signalling model . 70 3.4 A comparison of the optimisation algorithms in COPASI . 71 4.1 The three-compartment reaction system . 80 4.2 Covariances of the fluctuations for the three-compartment system 81 4.3 The creatine kinase model at the steady-state . 85 4.4 Covariances for the creatine kinase model . 86 4.5 Reaction scheme for the yeast glycolysis model . 88 4.6 Covariances for the yeast glycolysis model . 90 5 List of Figures 1.1 The user interface for the software package COPASI. 21 2.1 Flow chart illustrating the Gillespie algorithm. 31 2.2 Output from the Gillespie algorithm . 32 2.3 A comparison of the stochastic (dark blue) and deterministic (light blue) dynamics . 32 2.4 Time series of stochastic oscillations . 47 2.5 Power spectra for species Y1, Y2 and Y3................ 48 3.1 List of chemical reactions, as displayed in COPASI . 52 3.2 The COPASI display for the LNA task. 64 3.3 Reaction Scheme for the Kholodenko model . 67 3.4 Oscillations observed in MAPKK . 68 3.5 The variance of MKKK fluctuations around the steady state . 70 3.6 Two-dimensional parameter scan of the covariance of the fluctua- tions of MKKK and MKK-P . 71 4.1 The proposed geometry of the two-compartment model . 75 4.2 Reaction Scheme of the creatine kinase model . 84 4.3 Reaction Scheme of the yeast glycolysis model . 87 4.4 Stochastic oscillations in the yeast glycolysis model . 89 5.1 Absolute value of the CCF for species Y and Z . 97 5.2 Parametric plot of the CCF for species Y and Z . 98 5.3 Plot of the phase spectrum for species Y and Z . 98 5.4 Theoretical power spectra for the internal (larger peak) and exter- nal cAMP ............................... 100 5.5 Magnitude for the (theoretical) CCF for the internal and external cAMP fluctuations. 101 6 5.6 Parametric plot of the (theoretical) CCF for the internal and ex- ternal cAMP fluctuations . 101 5.7 Diagram from Kim et. al. illustrating the three-cell model . 102 5.8 Time series for cAMP i for a three-cell model . 103 5.9 Theoretical power spectra for cAMP i in each cell of the two-cell model . 103 5.10 The magnitude of the CCF for the cAMP i in each cell of the two cell model . 104 5.11 Parametric plot of the CCF for the cAMP i in each cell of the two-cell model . 104 5.12 Phase lag for oscillations of cAMP i in each cell for the two-cell model . 105 5.13 Results for a two-cell model, where the cell volumes are not identical107 5.14 Results for a two-cell model, where the cell volumes are not identical108 5.15 Phase spectrum for the cAMP fluctuations in a two-cell model for non-equal cell volumes . 109 5.16 Results for oscillations in species A3 ................. 110 5.17 Results for oscillations in species A3 for increased coupling strength 111 7 Abstract The stochastic dynamics of biochemical systems Joseph Daniel Challenger A thesis submitted to the University of Manchester for the degree of Doctor of Philosophy, 2013 The topic of this thesis is the stochastic dynamics of biochemical reaction systems. The importance of the intrinsic fluctuations in these systems has become more widely appreciated in recent years, and should be accounted for when modelling such systems mathematically. These models are described as continuous time Markov processes and their dynamics defined by a master equation. Analytical progress is made possible by the use of the van Kampen system-size expansion, which splits the dynamics into a macroscopic component plus stochastic correc- tions, statistics for which can then be obtained. In the first part of this thesis, the terms obtained from the expansion are written down for an arbitrary model, enabling the expansion procedure to be automated and implemented in the soft- ware package COPASI. This means that the fluctuation analysis may be used in tandem with other tools in COPASI, in particular parameter scanning and optimisation. This scheme is then extended so that models involving multiple compartments (e.g. cells) may be studied. This increases the range of models that can be evaluated in this fashion. The second part of this thesis also concerns these multi-compartment models, and examines how oscillations can synchronise across a population of cells. This has been observed in many biochemical processes, such as yeast glycolysis. How- ever, the vast majority of modelling of such systems has used the deterministic framework, which ignores the effect of fluctuations. It is now widely known that the type of models studied here can exhibit sustained temporal oscillations when formulated stochastically, despite no such oscillations being found in the deter- ministic version of the model. Using the van Kampen expansion as a starting point, multi-cell models are studied, to see how stochastic oscillations in one cell may influence, and be influenced by, oscillations in neighbouring cells. Analytical expressions are found, indicating whether or not the oscillations will synchronise across multiple cells and, if synchronisation does occur, whether the oscillations synchronise in phase, or with a phase lag. 8 Declaration No portion of the work referred to in this thesis has been submitted in support of an application for another degree or qualification of this or any other university or other institute of learning. 9 Copyright i. The author of this thesis (including any appendices and/or schedules to this thesis) owns certain copyright or related rights in it (the \Copyright") and s/he has given The University of Manchester certain rights to use such Copyright, including for administrative purposes. ii. Copies of this thesis, either in full or in extracts and whether in hard or electronic copy, may be made only in accordance with the Copyright, De- signs and Patents Act 1988 (as amended) and regulations issued under it or, where appropriate, in accordance with licensing agreements which the University has from time to time. This page must form part of any such copies made. iii. The ownership of certain Copyright, patents, designs, trade marks and other intellectual property (the \Intellectual Property") and any reproductions of copyright works in the thesis, for example graphs and tables (\Reproduc- tions"), which may be described in this thesis, may not be owned by the author and may be owned by third parties. Such Intellectual Property and Reproductions cannot and must not be made available for use without the prior written permission of the owner(s) of the relevant Intellectual Property and/or Reproductions. iv. Further information on the conditions under which disclosure, publication and commercialisation of this thesis, the Copyright and any Intellectual Property and/or Reproductions described in it may take place is available in the University IP Policy (see http://documents.manchester.ac.uk/ DocuInfo.aspx?DocID=487), in any relevant Thesis restriction declarations deposited in the University Library, The University Library's regulations (see http://www.manchester.ac.uk/library/aboutus/regulations) and in The University's policy on presentation of Theses 10 Acknowledgements First of all, I would like to thank my supervisor Alan McKane, for giving me the opportunity to work with him for the last three years. Alan's knowledge of the field and his intuitive ability to identify and refine research ideas have been extremely useful, and working with him has been a very rewarding experience. I would also like to thank Tobias Galla for his support, and for encouraging me to present my work wherever possible.
Load more

Recommended publications
  • The Effect of Compression and Expansion on Stochastic Reaction Networks
    IMT School for Advanced Studies, Lucca Lucca, Italy The Effect of Compression and Expansion on Stochastic Reaction Networks PhD Program in Institutions, Markets and Technologies Curriculum in Computer Science and Systems Engineering (CSSE) XXXI Cycle By Tabea Waizmann 2021 The dissertation of Tabea Waizmann is approved. Program Coordinator: Prof. Rocco De Nicola, IMT Lucca Supervisor: Prof. Mirco Tribastone, IMT Lucca The dissertation of Tabea Waizmann has been reviewed by: Dr. Catia Trubiani, Gran Sasso Science Institute Prof. Michele Loreti, University of Camerino IMT School for Advanced Studies, Lucca 2021 To everyone who believed in me. Contents List of Figures ix List of Tables xi Acknowledgements xiii Vita and Publications xv Abstract xviii 1 Introduction 1 2 Background 6 2.1 Multisets . .6 2.2 Reaction Networks . .7 2.3 Ordinary Lumpability . 11 2.4 Layered Queuing Networks . 12 2.5 PEPA . 16 3 Coarse graining mass-action stochastic reaction networks by species equivalence 19 3.1 Species Equivalence . 20 3.1.1 Species equivalence as a generalization of Markov chain ordinary lumpability . 27 3.1.2 Characterization of SE for mass-action networks . 27 3.1.3 Computation of the maximal SE and reduced net- work . 28 vii 3.2 Applications . 31 3.2.1 Computational systems biology . 31 3.2.2 Epidemic processes in networks . 33 3.3 Discussion . 36 4 DiffLQN: Differential Equation Analysis of Layered Queuing Networks 37 4.1 DiffLQN .............................. 38 4.1.1 Architecture . 38 4.1.2 Capabilities . 38 4.1.3 Syntax . 39 4.2 Case Study: Client-Server dynamics . 41 4.3 Discussion .
    [Show full text]
  • Intrinsic Noise Analyzer: a Software Package for the Exploration of Stochastic Biochemical Kinetics Using the System Size Expansion
    Intrinsic Noise Analyzer: A Software Package for the Exploration of Stochastic Biochemical Kinetics Using the System Size Expansion Philipp Thomas1,2,3., Hannes Matuschek4., Ramon Grima1,2* 1 School of Biological Sciences, University of Edinburgh, Edinburgh, United Kingdom, 2 SynthSys Edinburgh, University of Edinburgh, Edinburgh, United Kingdom, 3 Department of Physics, Humboldt University of Berlin, Berlin, Germany, 4 Institute of Physics and Astronomy, University of Potsdam, Potsdam, Germany Abstract The accepted stochastic descriptions of biochemical dynamics under well-mixed conditions are given by the Chemical Master Equation and the Stochastic Simulation Algorithm, which are equivalent. The latter is a Monte-Carlo method, which, despite enjoying broad availability in a large number of existing software packages, is computationally expensive due to the huge amounts of ensemble averaging required for obtaining accurate statistical information. The former is a set of coupled differential-difference equations for the probability of the system being in any one of the possible mesoscopic states; these equations are typically computationally intractable because of the inherently large state space. Here we introduce the software package intrinsic Noise Analyzer (iNA), which allows for systematic analysis of stochastic biochemical kinetics by means of van Kampen’s system size expansion of the Chemical Master Equation. iNA is platform independent and supports the popular SBML format natively. The present implementation is the first to adopt a complementary approach that combines state-of-the-art analysis tools using the computer algebra system Ginac with traditional methods of stochastic simulation. iNA integrates two approximation methods based on the system size expansion, the Linear Noise Approximation and effective mesoscopic rate equations, which to-date have not been available to non-expert users, into an easy-to-use graphical user interface.
    [Show full text]
  • System Size Expansion Using Feynman Rules and Diagrams
    System size expansion using Feynman rules and diagrams Philipp Thomas∗y, Christian Fleck,z Ramon Grimay, Nikola Popovi´c∗ October 2, 2018 Abstract Few analytical methods exist for quantitative studies of large fluctuations in stochastic systems. In this article, we develop a simple diagrammatic approach to the Chemical Master Equation that allows us to calculate multi-time correlation functions which are accurate to a any desired order in van Kampen's system size expansion. Specifically, we present a set of Feynman rules from which this diagrammatic perturbation expansion can be constructed algo- rithmically. We then apply the methodology to derive in closed form the leading order corrections to the linear noise approximation of the intrinsic noise power spectrum for general biochemical reaction networks. Finally, we illustrate our results by describing noise-induced oscillations in the Brusselator reaction scheme which are not captured by the common linear noise approximation. 1 Introduction The quantification of intrinsic fluctuations in biochemical networks is becoming increasingly important for the study of living systems, as some of the key molecular players are expressed in low numbers of molecules per cell [1]. The most commonly employed method for investigating this type of cell-to-cell variability is the stochastic simulation algorithm (SSA) [2]. While the SSA allows for the exact sampling of trajectories of discrete intracellular biochemistry, it is computationally expensive, which often prevents one from obtaining statistics for large ranges of the biological parameter space. A widely used approach to overcome these limitations has been the linear noise approximation (LNA) [3, 4, 5]. The LNA is obtained to leading order in van Kampen's system size expansion [6].
    [Show full text]
  • Approximate Probability Distributions of the Master Equation
    Approximate probability distributions of the master equation Philipp Thomas∗ School of Mathematics and School of Biological Sciences, University of Edinburgh Ramon Grimay School of Biological Sciences, University of Edinburgh Master equations are common descriptions of mesoscopic systems. Analytical solutions to these equations can rarely be obtained. We here derive an analytical approximation of the time-dependent probability distribution of the master equation using orthogonal polynomials. The solution is given in two alternative formulations: a series with continuous and a series with discrete support both of which can be systematically truncated. While both approximations satisfy the system size expansion of the master equation, the continuous distribution approximations become increasingly negative and tend to oscillations with increasing truncation order. In contrast, the discrete approximations rapidly converge to the underlying non-Gaussian distributions. The theory is shown to lead to particularly simple analytical expressions for the probability distributions of molecule numbers in metabolic reactions and gene expression systems. I. INTRODUCTION first few moments [12{14]; alternative methods are based on moment closure [15]. It is however the case that the Master equations are commonly used to describe fluc- knowledge of a limited number of moments does not al- tuations of particulate systems. In most instances, how- low to uniquely determine the underlying distribution ever, the number of reachable states is so large that their functions. Reconstruction of the probability distribution combinatorial complexity prevents one from obtaining therefore requires additional approximations such as the analytical solutions to these equations. Explicit solutions maximum entropy principle [16] or the truncated moment are known only for certain classes of linear birth-death generating function [17] which generally yield different processes [1], under detailed balance conditions [2], or results.
    [Show full text]
  • How Reliable Is the Linear Noise Approximation Of
    Thomas et al. BMC Genomics 2013, 14(Suppl 4):S5 http://www.biomedcentral.com/1471-2164/14/S4/S5 RESEARCH Open Access How reliable is the linear noise approximation of gene regulatory networks? Philipp Thomas1,2, Hannes Matuschek3, Ramon Grima1* From IEEE International Conference on Bioinformatics and Biomedicine 2012 Philadelphia, PA, USA. 4-7 October 2012 Abstract Background: The linear noise approximation (LNA) is commonly used to predict how noise is regulated and exploited at the cellular level. These predictions are exact for reaction networks composed exclusively of first order reactions or for networks involving bimolecular reactions and large numbers of molecules. It is however well known that gene regulation involves bimolecular interactions with molecule numbers as small as a single copy of a particular gene. It is therefore questionable how reliable are the LNA predictions for these systems. Results: We implement in the software package intrinsic Noise Analyzer (iNA), a system size expansion based method which calculates the mean concentrations and the variances of the fluctuations to an order of accuracy higher than the LNA. We then use iNA to explore the parametric dependence of the Fano factors and of the coefficients of variation of the mRNA and protein fluctuations in models of genetic networks involving nonlinear protein degradation, post-transcriptional, post-translational and negative feedback regulation. We find that the LNA can significantly underestimate the amplitude and period of noise-induced oscillations in genetic oscillators. We also identify cases where the LNA predicts that noise levels can be optimized by tuning a bimolecular rate constant whereas our method shows that no such regulation is possible.
    [Show full text]
  • Computation of Biochemical Pathway Fluctuations Beyond the Linear Noise Approximation Using
    Computation of biochemical pathway fluctuations beyond the linear noise approximation using iNA Philipp Thomas∗†‡ ¶, Hannes Matuschek§¶ and Ramon Grima∗† ∗School of Biological Sciences, University of Edinburgh, Edinburgh, United Kingdom †SynthSys Edinburgh, University of Edinburgh, Edinburgh, United Kingdom ‡Department of Physics, Humboldt University of Berlin, Berlin, Germany §Institute of Physics and Astronomy, University of Potsdam, Potsdam, Germany ¶These authors contributed equally. Abstract—The linear noise approximation is commonly used the first software package enabling a fluctuation analysis of to obtain intrinsic noise statistics for biochemical networks. biochemical networks via the Linear Noise Approximation These estimates are accurate for networks with large numbers (LNA) and Effective Mesoscopic Rate Equation (EMRE) of molecules. However it is well known that many biochemical networks are characterized by at least one species with a approximations of the CME. The former gives the variance small number of molecules. We here describe version 0.3 of and covariance of concentration fluctuations in the limit of the software intrinsic Noise Analyzer (iNA) which allows for large molecule numbers while the latter gives the mean accurate computation of noise statistics over wide ranges of concentrations for intermediate to large molecule numbers molecule numbers. This is achieved by calculating the next and is more accurate than the conventional Rate Equations order corrections to the linear noise approximation’s estimates of variance and covariance of concentration fluctuations. The (REs). efficiency of the methods is significantly improved by auto- In this proceeding we develop and efficiently implement mated just-in-time compilation using the LLVM framework in iNA, the Inverse Omega Square (IOS) method comple- leading to a fluctuation analysis which typically outperforms menting the EMRE method by providing estimates for the that obtained by means of exact stochastic simulations.
    [Show full text]
  • Stochastic Analysis of Biochemical Systems
    ON MOMENTS AND TIMING { STOCHASTIC ANALYSIS OF BIOCHEMICAL SYSTEMS by Khem Raj Ghusinga A dissertation submitted to the Faculty of the University of Delaware in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Electrical and Computer Engineering Summer 2018 c 2018 Khem Raj Ghusinga All Rights Reserved ON MOMENTS AND TIMING { STOCHASTIC ANALYSIS OF BIOCHEMICAL SYSTEMS by Khem Raj Ghusinga Approved: Kenneth E. Barner, Ph.D. Chair of the Department of Electrical and Computer Engineering Approved: Babatunde Ogunnaike, Ph.D. Dean of the College of Engineering Approved: Ann L. Ardis, Ph.D. Senior Vice Provost for Graduate and Professional Education I certify that I have read this dissertation and that in my opinion it meets the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy. Signed: Abhyudai Singh, Ph.D. Professor in charge of dissertation I certify that I have read this dissertation and that in my opinion it meets the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy. Signed: Ryan Zurakowski, Ph.D. Member of dissertation committee I certify that I have read this dissertation and that in my opinion it meets the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy. Signed: Gonzalo R. Arce, Ph.D. Member of dissertation committee I certify that I have read this dissertation and that in my opinion it meets the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy.
    [Show full text]
  • Nonequilibrium Critical Phenomena: Exact Langevin Equations, Erosion of Tilted Landscapes
    Nonequilibrium critical phenomena : exact Langevin equations, erosion of tilted landscapes. Charlie Duclut To cite this version: Charlie Duclut. Nonequilibrium critical phenomena : exact Langevin equations, erosion of tilted landscapes.. Physics [physics]. Université Pierre et Marie Curie - Paris VI, 2017. English. NNT : 2017PA066241. tel-01690438 HAL Id: tel-01690438 https://tel.archives-ouvertes.fr/tel-01690438 Submitted on 23 Jan 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. UNIVERSITÉ PIERRE ET MARIE CURIE ÉCOLE DOCTORALE PHYSIQUE EN ÎLE-DE-FRANCE THÈSE pour obtenir le titre de Docteur de l’Université Pierre et Marie Curie Mention : PHYSIQUE THÉORIQUE Présentée et soutenue par Charlie DUCLUT Nonequilibrium critical phenomena: exact Langevin equations, erosion of tilted landscapes. Thèse dirigée par Bertrand DELAMOTTE préparée au Laboratoire de Physique Théorique de la Matière Condensée soutenue le 11 septembre 2017 devant le jury composé de : M. Andrea GAMBASSI Rapporteur M. Michael SCHERER Rapporteur M. Giulio BIROLI Examinateur Mme Leticia CUGLIANDOLO Présidente du jury M. Andrei FEDORENKO Examinateur M. Bertrand DELAMOTTE Directeur de thèse Remerciements First of all, I would like to thank the referees Andrea Gambassi and Michael Scherer for taking the time to carefully read the manuscript.
    [Show full text]
  • An Alternative Route to the System-Size Expansion
    Home Search Collections Journals About Contact us My IOPscience An alternative route to the system-size expansion This content has been downloaded from IOPscience. Please scroll down to see the full text. 2017 J. Phys. A: Math. Theor. 50 395003 (http://iopscience.iop.org/1751-8121/50/39/395003) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 129.215.109.91 This content was downloaded on 05/09/2017 at 12:10 Please note that terms and conditions apply. You may also be interested in: Symmetry and Collective Fluctuations in Evolutionary Games: Symmetry and collective fluctuations: large deviations and scaling in population processes E Smith and S Krishnamurthy Approximation and inference methods for stochastic biochemical kinetics—a tutorial review David Schnoerr, Guido Sanguinetti and Ramon Grima System size expansion using Feynman rules and diagrams Philipp Thomas, Christian Fleck, Ramon Grima et al. Master equations and the theory of stochastic path integrals Markus F Weber and Erwin Frey Stochastic switching in biology: from genotype to phenotype Paul C Bressloff Multi-compartment linear noise approximation Joseph D Challenger, Alan J McKane and Jürgen Pahle Modeling stochasticity in biochemical reaction networks P H Constantino, M Vlysidis, P Smadbeck et al. Fundamentals of higher order stochastic equations P D Drummond How limit cycles and quasi-cycles are related in systems with intrinsic noise Richard P Boland, Tobias Galla and Alan J McKane IOP Journal of Physics A: Mathematical and Theoretical J. Phys. A: Math. Theor. Journal of Physics A: Mathematical and Theoretical J.
    [Show full text]
  • How Reliable Is the Linear Noise Approximation of Gene Regulatory Networks?
    Mathematisch-Naturwissenschaftliche Fakultät Philipp Thomas | Hannes Matuschek | Ramon Grima How reliable is the linear noise approximation of gene regulatory networks? Suggested citation referring to the original publication: BMC Genomics 14 (2013) S5 DOI https://doi.org/10.1186/1471-2164-14-S4-S5 ISSN (online) 1471-2164 Postprint archived at the Institutional Repository of the Potsdam University in: Postprints der Universität Potsdam Mathematisch-Naturwissenschaftliche Reihe ; 876 ISSN 1866-8372 https://nbn-resolving.org/urn:nbn:de:kobv:517-opus4-435028 DOI https://doi.org/10.25932/publishup-43502 Thomas et al. BMC Genomics 2013, 14(Suppl 4):S5 http://www.biomedcentral.com/1471-2164/14/S4/S5 RESEARCH Open Access How reliable is the linear noise approximation of gene regulatory networks? Philipp Thomas1,2, Hannes Matuschek3, Ramon Grima1* From IEEE International Conference on Bioinformatics and Biomedicine 2012 Philadelphia, PA, USA. 4-7 October 2012 Abstract Background: The linear noise approximation (LNA) is commonly used to predict how noise is regulated and exploited at the cellular level. These predictions are exact for reaction networks composed exclusively of first order reactions or for networks involving bimolecular reactions and large numbers of molecules. It is however well known that gene regulation involves bimolecular interactions with molecule numbers as small as a single copy of a particular gene. It is therefore questionable how reliable are the LNA predictions for these systems. Results: We implement in the software package intrinsic Noise Analyzer (iNA), a system size expansion based method which calculates the mean concentrations and the variances of the fluctuations to an order of accuracy higher than the LNA.
    [Show full text]
  • Optimal Experimental Design Applied to Models of Microbial Gene Regulation
    Optimal Experimental Design Applied to Models of Microbial Gene Regulation by Nathan Braniff A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Doctor of Philosophy in Applied Mathematics Waterloo, Ontario, Canada, 2020 c Nathan Braniff 2020 Examining Committee Membership The following served on the Examining Committee for this thesis. The decision of the Examining Committee is by majority vote. External Examiner: Julio Banga Professor, IIM-CISC Supervisor: Brian Ingalls Professor, Applied Mathematics, University of Waterloo Internal Members: Matt Scott Assoc. Professor, Applied Mathematics, University of Waterloo Kirsten Morris Professor, Applied Mathematics, University of Waterloo Internal-External Member: Trevor Charles Professor, Biology, University of Waterloo ii Authour's Declaration This thesis consists of material all of which I authored or co-authored: see Statement of Contributions included in the thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. iii Statement of Contributions Chapters1,2,7,&8 : I am the sole author of this material, under the supervision of Brian Ingalls. This material has not been previously published. Chapter3 : This material has been previously published as: \New opportunities for optimal design of dynamic experiments in systems and synthetic biology" by Nathan Braniff, and Brian Ingalls in Current Opinion in Systems Biology 9 (2018): 42-48 I co-authored this manuscript with Brian Ingalls. My contribution was gathering the candidate references and in selecting the initial areas of focus for the article.
    [Show full text]
  • Arxiv:1810.05368V2 [Q-Bio.PE] 20 Feb 2019 a Rusaeemergent, Are Groups Mal .Introduction 1
    Deriving mesoscopic models of collective behaviour for finite populations Jitesh Jhawar1 , Richard G. Morris2, Vishwesha Guttal1 1. Centre for Ecological Sciences, Indian Institute of Science, Bengaluru, 560012 2. National Centre for Biological Sciences, TIFR, Bengaluru, 560012 Abstract Animal groups exhibit many emergent properties that are a consequence of local interactions. Link- ing individual-level behaviour, which is often stochastic and local, to coarse-grained descriptions of animal groups has been a question of fundamental interest from both biological and mathe- matical perspectives. In this book chapter, we present two complementary approaches to derive coarse-grained descriptions of collective behaviour at so-called mesoscopic scales, which account for the stochasticity arising from the finite sizes of animal groups. We construct stochastic dif- ferential equations (SDEs) for a coarse-grained variable that describes the order/consensus within a group. The first method of construction is based on van Kampen’s system-size expansion of transition rates. The second method employs Gillespie’s chemical Langevin equations. We apply these two methods to two microscopic models from the literature, in which organisms stochasti- cally interact and choose between two directions/choices of foraging. These ‘binary-choice’ models differ only in the types of interactions between individuals, with one assuming simple pairwise interactions, and the other incorporating ternary effects. In both cases, the derived mesoscopic SDEs have multiplicative/state-dependent noise, i.e., the strength of the noise depends on the current state of the system. However, the different models demonstrate the contrasting effects of noise: increasing the order/consensus in the pairwise interaction model, whilst reducing the order/consensus in the higher-order interaction model.
    [Show full text]