Simple mathematical models of gene regulatory dynamics

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Lecture Notes on Mathematical Modelling in the Life Sciences Michael C Mackey Moisés Santillán Marta Tyran-Kamińska Eduardo S. Zeron Simple Mathematical Models of Gene Regulatory Dynamics Lecture Notes on Mathematical Modelling in the Life Sciences Editors-in-Chief: Michael C Mackey Angela Stevens Series editors Martin Burger Maurice Chacron Odo Diekmann Anita Layton Jinzhi Lei Mark Lewis Lakshminarayanan Mahadevan Philip Maini Masayasu Mimura Claudia Neuhauser Hans G Othmer Mark Peletier Alan S Perelson Charles S Peskin Luigi Preziosi Jonathan Rubin Moisés Santillán Christoph Schütte The rapid pace and development of the research in mathematics, biology and medicine has opened a niche for a new type of publication - short, up-to-date, readable lecture notes covering the breadth of mathematical modelling, analysis and computation in the life-sciences, at a high level, in both printed and electronic versions The volumes in this series are written in a style accessible to researchers, professionals and graduate students in the mathematical and biological sciences They can serve as an introduction to recent and emerging subject areas and/or as an advanced teaching aid at colleges, institutes and universities Besides monographs, we envision that this series will also provide an outlet for material less formally presented and more anticipatory of future needs, yet of immediate interest because of the novelty of its treatment of an application, or of the mathematics being developed in the context of exciting applications It is important to note that the LMML focuses on books by one or more authors, not on edited volumes The topics in LMML range from the molecular through the organismal to the population level, e.g genes and proteins, evolution, cell biology, developmental biology, neuroscience, organ, tissue and whole body science, immunology and disease, bioengineering and biofluids, population biology and systems biology Mathematical methods include dynamical systems, ergodic theory, partial differential equations, calculus of variations, numerical analysis and scientific computing, differential geometry, topology, optimal control, probability, stochastics, statistical mechanics, combinatorics, algebra, number theory, etc., which contribute to a deeper understanding of biomedical problems More information about this series at http://www.springer.com/series/10049 Michael C Mackey • Moisés Santillán • Marta Tyran-Kami´nska • Eduardo S Zeron Simple Mathematical Models of Gene Regulatory Dynamics 123 Michael C Mackey Department of Physiology McGill University Montreal, QC Canada Moisés Santillán Unidad Monterrey Cinvestav del IPN Apodaca, NL Mexico Marta Tyran-Kami´nska Institute of Mathematics University of Silesia Katowice Poland Eduardo S Zeron Departamento de Matemáticas Cinvestav del IPN Ciudad de México Mexico ISSN 2193-4789 ISSN 2193-4797 (electronic) Lecture Notes on Mathematical Modelling in the Life Sciences ISBN 978-3-319-45317-0 ISBN 978-3-319-45318-7 (eBook) DOI 10.1007/978-3-319-45318-7 Library of Congress Control Number: 2016956565 © Springer International Publishing Switzerland 2016 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland To students everywhere: past, present, and future Preface We survey work that has been carried out in the attempts of biomathematicians to understand the dynamic behavior of simple bacterial operons starting with the initial work of the 1960s We concentrate on the simplest of situations, discussing both repressible and inducible systems as well as the bistable switch and then turning to a discussion of the role of both extrinsic noise and the so-called intrinsic noise in the form of translational and/or transcriptional bursting We conclude with a consideration of the messier concrete examples of the lactose and tryptophan operons and the lysis-lysogeny switch of phage This survey has grown out of our work over the past 20 years and is an enlarged version of our review paper (Mackey et al 2015) Montreal, QC, Canada Apodaca, NL, Mexico Katowice, Poland Ciudad de México, Mexico June 2016 Michael C Mackey Moisés Santillán Marta Tyran-Kami´nska Eduardo S Zeron vii Acknowledgments We have benefited from the comments, suggestions, and criticisms of many colleagues over the years (you will know who you are) and from the institutional support of our home universities as well as the University of Oxford, the University of Bremen, Bergischen Universität Wuppertal, and the International Centre for Theoretical Physics MCM is especially grateful to a comment from Dr Jérôme Losson many years ago that directed attention to these fascinating problems This work was supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada, the Polish NCN grant no 2014/13/B/ST1/00224, and the Consejo Nacional de Ciencia y Tecnología (Conacyt) in México ix Contents Part I Deterministic Modeling Techniques Generic Deterministic Models of Prokaryotic Gene Regulation 1.1 Inducible Regulation 1.2 Repressible Regulation 3 General Dynamic Considerations 2.1 Operon Dynamics 2.1.1 No Control 2.1.2 Inducible Regulation 2.1.3 Repressible Regulation 2.1.4 Bistable Switches 2.2 The Appearance of Cell Growth Effects and Delays Due to Transcription and Translation 2.3 Fast and Slow Variables 7 9 13 13 Part II 23 26 Dealing with Noise Master Equation Modeling Approaches 3.1 The Chemical Master Equation 3.2 Relation to Deterministic Models 3.2.1 The Chemical Langevin Equation 3.3 Stability of the Chemical Master Equation 3.3.1 Algorithms to Find Steady State Density Functions 3.4 Application to a Simple Repressible Operon 31 32 34 36 37 40 43 Noise Effects in Gene Regulation: Intrinsic Versus Extrinsic 4.1 Dynamics with Bursting 4.1.1 Generalities 4.1.2 Distributions in the Presence of Bursting for Inducible and Repressible Systems 4.1.3 Bursting in a Switch 49 50 50 52 57 xi xii Contents 4.1.4 Recovering the Deterministic Case 4.1.5 A Discrete Space Bursting Model 4.2 Gaussian Distributed Noise in the Molecular Degradation Rate 4.3 Two Dominant Slow Genes with Bursting Part III 61 62 64 66 Specific Examples The Lactose Operon 5.1 The Lactose Operon Regulatory Pathway 5.2 Mathematical Modeling of the Lactose Operon 5.3 Quantitative Studies of the Lactose Operon Dynamics 73 73 77 83 The Tryptophan Operon 6.1 The Tryptophan Operon in E coli 6.2 Mathematical Modeling of the trp Operon 6.3 Quantitative Studies of the trp Operon 87 87 89 92 The Lysis-Lysogeny Switch 7.1 Phage Biology 7.2 The Lysis-Lysogeny Switch 7.3 Mathematical Modeling of the Phage Switch 7.4 Brief Review of Quantitative Studies on the Phage Switch 7.5 Closing Remarks 99 102 104 107 112 114 References 115 Index 123 ... 2016 M.C Mackey et al., Simple Mathematical Models of Gene Regulatory Dynamics, Lecture Notes on Mathematical Modelling in the Life Sciences, DOI 10.1007/978-3-319-45318-7_2 General Dynamic Considerations... copies of the gene regulatory network (e.g in a culture of many cells) where ‘large’ and ‘many’ mean something on the order of Avagadro’s number (' 1023 ) Chapter Generic Deterministic Models of. .. Mathematical Models of Gene Regulatory Dynamics, Lecture Notes on Mathematical Modelling in the Life Sciences, DOI 10.1007/978-3-319-45318-7_1 Generic Deterministic Models of Prokaryotic Gene Regulation
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