DOCSLIB.ORG

Intrinsic Noise in Systems with Switching Environments

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Intrinsic Noise in Systems with Switching Environments PHYSICAL REVIEW E 93, 052119 (2016) Intrinsic noise in systems with switching environments Peter G. Hufton,* YenTingLin,† Tobias Galla,‡ and Alan J. McKane§ Theoretical Physics, School of Physics and Astronomy, The University of Manchester, Manchester M13 9PL, United Kingdom (Received 2 December 2015; revised manuscript received 17 March 2016; published 10 May 2016) We study individual-based dynamics in finite populations, subject to randomly switching environmental conditions. These are inspired by models in which genes transition between on and off states, regulating underlying protein dynamics. Similarly, switches between environmental states are relevant in bacterial populations and in models of epidemic spread. Existing piecewise-deterministic Markov process approaches focus on the deterministic limit of the population dynamics while retaining the randomness of the switching. Here we go beyond this approximation and explicitly include effects of intrinsic stochasticity at the level of the linear-noise approximation. Specifically, we derive the stationary distributions of a number of model systems, in good agreement with simulations. This improves existing approaches which are limited to the regimes of fast and slow switching. DOI: 10.1103/PhysRevE.93.052119 I. INTRODUCTION Kramers-Moyal and van Kampen expansions. The latter is also known as the linear-noise approximation (LNA) [1,2]. There is now a broad consensus that noise plays a crucial These methods are now used routinely for the analysis role in most dynamical systems in biology, chemistry, and in of individual-based models with intrinsic noise in the weak- the social sciences. The theory with which to describe these noise limit. Most applications so far focus on systems in stochastic processes is well established and has its roots in which the reaction rates are set by constant model param- statistical physics. Modelling tools such as master equations, eters, and the only time dependence is in the evolution Fokker-Planck equations, and Langevin dynamics are standard of the population of individuals itself. Recently, exceptions and can be found in a number of textbooks [1–3]. Much of have gained attention [6,22–26]. In these models, reaction this work focuses on processes between discrete interacting rates vary continuously and deterministically in time, for individuals, which can be members of a population in example, to capture periodically varying infection rates to epidemiology [4–6], atoms or molecules in chemical reaction model seasonal variation in epidemic spreading. Crucially, systems [7,8], or proteins in the context of gene regulatory no additional stochasticity is introduced in these dynamics by networks [9,10]. Many of these models are Markovian and the environment, and the only discreteness in the dynamics their natural description is in terms of a master equation which is in the evolution of a finite population of individuals. Other describes the time evolution of the underlying probability authors have considered models with an environment which distribution of microstates. varies stochastically and continuously [27–29]. Solving the master equation analytically is often a difficult For many model systems it is more realistic to assume that task: Only very simple linear model dynamics allow further model parameters switch between different discrete states. treatment [1–3,11,12]. It is therefore common to employ This includes phases of antibiotic treatment in the context approximation techniques, most notably ones built around of bacteria [30,31], genetic switches [9,32–38], evolutionary the assumption that the size of the population is large but game theory [39], and predator-prey models in switching finite. The inverse system size (or its square root) then environmental conditions [40,41]. Such models describe two serves as a small parameter in which an expansion can be types of discreteness: that of the state of the environment and performed. The lowest order in this expansion corresponds that of the population of interacting individuals. In principle, to the limit of infinite systems and provides a deterministic the environmental switching can occur stochastically or follow description devoid of stochasticity. The second-order term in a deterministic pattern (e.g., prescribed periods of antibiotic the expansion introduces some stochastic effects of noise in the treatment, regularly interspersed with periods of no antibiosis). population but approximates the individual-level dynamics by In the present work we focus on stochastically switching a simpler Gaussian process on a continuum domain [13]. These environments. Assuming again a large but finite population, techniques have been very successful in capturing elements of the demographic noise in the population can be approximated noise-induced phenomena, for example, the weak selection using the above expansion techniques. The noise relating to of competing species [14–18], cyclic behavior, patterns, and switches in the environmental state, however, cannot be dealt waves [19–21]. The main techniques used to characterize these with in this way: There is no large parameter to expand in effects are system-size expansion methods, most notably the when the number of environmental states is small. In models with switching environments the lowest-order expansion in the strength of the intrinsic noise leads to a “piecewise-deterministic Markov process” (PDMP) [42,43]. *[email protected] In this approximation the dynamics of the population of †[email protected] individuals is described by deterministic rate equations be- ‡[email protected] tween stochastic switches of the environment. This approach §[email protected] neglects all intrinsic stochasticity from the reaction dynamics 2470-0045/2016/93(5)/052119(13) 052119-1 ©2016 American Physical Society HUFTON, LIN, GALLA, AND MCKANE PHYSICAL REVIEW E 93, 052119 (2016) within the population—the population scale is taken to be environment, often labeled as G0 and G1, correspond to infinite. The only type of randomness retained is that of regimes in which a certain promoter—a region on DNA which the switching of the environmental states. The application initiates transcription—is either inactive (G0) or activated of PDMPs has recently gained attention in the description (G1). A protein (A) is produced with rates b0 and b1 in the of genetic networks [32,44,45]. corresponding state. Independently of the state of the gene, The PDMP approximation has been surprisingly effective proteins degrade at a constant rate d. This simple model has in modeling systems with very large populations [32,45,46]; been studied, for example, in Refs. [9,37,38,45,47], and it can however, it does not produce accurate results outside this be written in the form limit. Recently, an alternative approach has incorporated some b0 b1 effects of demographic noise, but it is only valid if there is a G0 −−→ G0 + A, G1 −−→ G1 + A, very large separation between the time scales of environmental (2) G −λ→+ G G −λ→− G A −→d ∅ switching and that of the population dynamics [37,38]. Here, 0 1, 1 0, . we develop the theory further and construct a systematic The reaction rates of this model are linear in n. It is possible expansion in the noise strength about the PDMP. to develop exact solutions to linear models of this type using a The remainder of this paper is organized as follows: In generating-function approach [11,12,47–50]. However, such Sec. II we give a more formal introduction to the problem using solutions are often limited to simple model systems and a relatively simple linear model. We show how the system-size frequently they only provide limited insight into the actual expansion can be applied in the presence of switching envi- physical dynamics. We use this linear model to introduce our ronments and we analytically derive the resulting stationary approximation method. In later sections we will then apply distribution for the linear model. In Sec. III we construct a more this approach to cases with nonlinear reaction rates, where an general theory and describe how the method can be applied to a exact solution is no longer feasible. larger set of model dynamics. Section IV contains applications to a number of model systems with nonlinear reaction rates and/or other additional features. In Sec. V we summarize our B. Simulation of results and general behavior findings and give an outlook on future work. The Appendix To illustrate the general behavior of the model, we first contains further details of the relevant calculations. show the outcome of a set of characteristic simulations in Fig. 1. Figures 1(a), 1(c), and 1(e) depict individual simulation II. INTRODUCTORY EXAMPLE runs. In each of these panels a trajectory of the population density x(t) = n(t)/ is shown as a solid line, and the A. Model definition switching of the environmental states is indicated by the We first focus on a simple example with linear reaction shading of the background. Figure 1(a) shows an example rates. We consider a population of individuals of type A, and of relatively slow switching. In each environmental state, we write n for the number of individuals in the population at ∗ = the population tends to a fixed point, φσ bσ /d, specific to any given time. Individuals can be created and they can decay, the environment. It then fluctuates about this fixed point. The so the model describes a birth-death process. The death rate stationary distribution [Fig. 1(b)] is bimodal. As the switching per individual is assumed to be a constant d. The creation rate rate is increased, Figs. 1(c) and 1(d), the stochastic dynamics is taken to depend on the state of the external environment; spends more time in between the two fixed points and the individuals are born with rate bσ , where σ represents the bimodality of the stationary distribution is lost; we observe a state of the environment.
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]