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Mathematical Modelling in Systems Biology: An IntroductionBrian IngallsApplied MathematicsUniversity of [email protected] 18, 2012

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PrefaceSystems techniques are integral to current research in molecular cell biology. These systems approaches stand in contrast to the historically reductionist paradigm of molecular biology. The shifttoward a systems perspective was gradual; it passed a turning point at the end of the 20th century, when newly developed experimental techniques provided system-level observations of cellularnetworks. These observations revealed the full complexity of these networks, and made it clearthat traditional (largely qualitative) molecular biology techniques are ill-equipped for the investigation of these systems, which often exhibit non-intuitive behaviour. This point was illustrated in athought experiment proposed by Yuri Lazebnik (Lazebnik, 2002). He described a (failed) attemptto reverse-engineer a transistor radio using qualitative methods analogous to those used in traditional molecular biology. Lazebnik’s exercise demonstrates that without a quantitative frameworkto describe large networks of interacting components, the functioning of cellular networks cannotbe resolved. A quantitative approach to molecular biology allows traditional interaction diagramsto be extended to mechanistic mathematical models. These models serve as working hypotheses:they help us to understand and predict the behaviour of complex systems.The application of mathematical modelling to molecular cell biology is not a new endeavour;there is a long history of mathematical descriptions of biochemical and genetic networks. Successfulapplications include Alan Turing’s description of patterning in development (discussed by Murray,2003), the models of neuronal signalling developed by Alan Hodgkin and Andrew Huxley (reviewedby Rinzel, 1990), and Denis Noble’s mechanistic modelling of the heart (Noble, 2004). Despite thesesuccesses, this sort of mathematical work has not been considered central to (most of) molecularcell biology. That attitude is changing; system-level investigations are now frequently accompaniedby mathematical models, and such models may soon become requisites for describing the behaviourof cellular networks.What this book aims to achieveMathematical modelling is becoming an increasingly valuable tool for molecular cell biology. Consequently, it is important for life scientists to have a background in the relevant mathematical techniques, so that they can participate in the construction, analysis, and critique of published models.On the other hand, those with mathematical training—mathematicians, engineers and physicists—now have increased opportunity to participate in molecular cell biology research. This book aimsto provide both of these groups—readers with backgrounds in cell biology or mathematics—withan introduction to the key concepts that are needed for the construction and investigation of mathematical models in molecular systems biology.I hope that, after studying this book, the reader will be prepared to engage with publishedmodels of cellular networks. By ‘engage’, I mean not only to understand these models, but also toi

analyse them critically (both their construction and their interpretation). Readers should also bein a position to construct and analyse their own models, given appropriate experimental data.Who this book was written forThis book evolved from a course I teach to upper-level (junior/senior) undergraduate students.In my experience, the material is accessible to students in any science or engineering program,provided they have some background in calculus and are comfortable with mathematics. I alsoteach this material as a half-semester graduate course to students in math and engineering. Thetext could easily be adapted to a graduate course for life science students. Additionally, I hopethat interested researchers at all levels will find the book useful for self-study.The mathematical prerequisite for this text is a working knowledge of the derivative; this isusually reached after a first course in calculus, which should also bring a level of comfort withmathematical concepts and manipulations. A brief review of some fundamental mathematicalnotions is included as Appendix B. The models in this text are based on differential equations, buttraditional solution techniques are not covered. Models are developed directly from chemical andgenetic principles, and most of the model analysis is carried out via computational software. Toencourage interaction with the mathematical techniques, exercises are included throughout the text.The reader is urged to take the time to complete these exercises as they appear; they will confirmthat the concepts and techniques have been properly understood. (All of the in-text exercisescan be completed with pen-and-paper calculations; none are especially time-consuming. Completesolutions to these exercises are posted at the book’s website. ) More involved problems—mostlyinvolving computational software—are included in the end-of-chapter problem sets.An introduction to computational software is included as Appendix C. Two packages aredescribed: XPPAUT, a freely available program that that was written specifically for dynamicmodelling; and MATLAB, which is a more comprehensive computational tool. Readers with nobackground in computation will find XPPAUT more accessible.I have found that most students can grasp the necessary cell and molecular biology withouta prior university-level course. The required background is briefly reviewed in Appendix A; morespecialized topics are introduced throughout the text. The starting point for this material is a basicknowledge of (high-school) chemistry, which is needed for a discussion of molecular phenomena,such as chemical bonds.How this book is organizedThe first four chapters cover the basics of mathematical modelling in molecular systems biology.These should be read sequentially. The last four chapters address specific biological domains. Thematerial in these chapters is not cumulative; they can be studied in any order. After Chapter 2,each chapter ends with an optional section, marked with an asterisk (*). These optional sectionsaddress specialized modelling topics, some of which demand additional mathematical background(reviewed in Appendix B).Chapter 1 introduces molecular systems biology and describes some basic notions of mathematical modelling, concluding with four short case-studies. Chapter 2 introduces dynamic mathematicalmodels of chemical reaction networks. These are differential equation models based on mass-action www.math.uwaterloo.ca/ bingalls/MathModellingSysBioii

rate laws. Some basic methods for analysis and simulation are described. Chapter 3 covers biochemical kinetics, providing rate laws for biochemical processes (i.e. enzyme-catalysed reactionsand cooperative binding). An optional section treats common approximation methods. Chapter 4introduces techniques for analysis of differential equation models, including phase plane analysis,stability, bifurcations, and sensitivity analysis. The presentation in this chapter emphasizes the useof these techniques in model investigation; very little theory is covered. A final optional sectionbriefly introduces the calibration of models to experimental data.Chapter 5 covers modelling of metabolic networks. Sensitivity analysis plays a central role inthe investigation of these models. The optional section introduces stoichiometric modelling, whichis often applied to large-scale metabolic networks.Chapter 6 addresses modelling of signal transduction pathways. The examples taken up in thischapter survey a range of information-processing tasks performed by these pathways. An optionalsection introduces the use of frequency-response analysis for studying cellular input-output systems.Chapter 7 introduces modelling of gene regulatory networks. The chapter starts with a treatment of gene expression, then presents examples illustrating a range of gene-circuit functions. Thefinal optional section introduces stochastic modelling in molecular systems biology.Chapter 8 covers modelling of electrophysiology and neuronal action potentials. An optionalsection contains a brief introduction to spatial modelling using partial differential equations.The book closes with three Appendices. The first reviews basic concepts from molecular cellbiology. The second reviews mathematical concepts. The last contains tutorials for two computational software packages—XPPAUT and MATLAB—that can be used for model simulation andanalysis.The website www.math.uwaterloo.ca/ bingalls/MathModellingSysBio contains solutions to thein-text exercises, along with XPPAUT and MATLAB code for the models presented in the textand the end-of-chapter problem sets.AcknowledgmentsThe preparation of this book was a considerable effort, and I am grateful to students, colleagues,and friends who have helped me along the way. I would like to thank the students of AMATH/BIOL382 and AMATH 882 for being the test-subjects for the material, and for teaching me as much asI taught them. Colleagues in math, biology, and engineering have been invaluable in helping mesort out the details of all aspects of the text. In particular, I would like to thank Peter Swain,Bernie Duncker, Sue Ann Campbell, Ted Perkins, Trevor Charles, David Siegel, David McMillen,Jordan Ang, Madalena Chaves, Rahul, Abdullah Hamadeh, and Umar Aftab. Special thanks toBev Marshman, Mads Kaern, Herbert Sauro, and Matt Scott for reading early drafts and makingexcellent suggestions on improving the material and presentation. Thanks also to Bob Prior andSusan Buckley of the MIT Press for their support in bringing the book to completion.Finally, I thank my wife Angie and our children Logan, Alexa, Sophia, and Ruby for their loveand support. I dedicate this book to them.Brian IngallsWaterlooApril, 2012iii

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ContentsPrefacei1 Introduction1.1 Systems Biology and Synthetic Biology . . . . . . . . . . . . . . . . . . . . .1.2 What is a Dynamic Mathematical Model? . . . . . . . . . . . . . . . . . . .1.3 Why are Dynamic Mathematical Models Needed? . . . . . . . . . . . . . . .1.4 How are Dynamic Mathematical Models Used? . . . . . . . . . . . . . . . .1.5 Basic Features of Dynamic Mathematical Models . . . . . . . . . . . . . . .1.6 Dynamic Mathematical Models in Molecular Cell Biology . . . . . . . . . .1.6.1 Drug target prediction in Trypanosoma brucei metabolism . . . . . .1.6.2 Identifying the source of oscillatory behaviour in NF-κB signalling .1.6.3 Model-based design of an engineered genetic toggle switch . . . . . .1.6.4 Establishing the mechanism for neuronal action potential generation1.7 Suggestions for Further Reading . . . . . . . . . . . . . . . . . . . . . . . .11245688101213152 Modelling Chemical Reaction Networks2.1 Chemical Reaction Networks . . . . . . . . . . . . . . . . . . . . . .2.1.1 Closed and open networks . . . . . . . . . . . . . . . . . . . .2.1.2 Dynamic behaviour of reaction networks . . . . . . . . . . . .2.1.3 Simple network examples . . . . . . . . . . . . . . . . . . . .2.1.4 Numerical simulation of differential equations . . . . . . . . .2.2 Separation of Time-Scales and Model Reduction . . . . . . . . . . .2.2.1 Separation of time-scales: the rapid equilibrium assumption .2.2.2 Separation of time-scales: the quasi-steady state assumption2.3 Suggestions for Further Reading . . . . . . . . . . . . . . . . . . . .2.4 Problem Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17181819213033353841423 Biochemical Kinetics3.1 Enzyme Kinetics . . . . . . . . .3.1.1 Michaelis-Menten kinetics3.1.2 Two-substrate reactions .3.2 Regulation of Enzyme Activity .3.2.1 Competitive inhibition . .3.2.2 Allosteric regulation . . .3.3 Cooperativity . . . . . . . . . . .4747485355555658.v.

3.43.53.63.7Compartmental Modelling and Transport . . . . .3.4.1 Diffusion . . . . . . . . . . . . . . . . . . .3.4.2 Facilitated transport . . . . . . . . . . . . .*Generalized Mass Action and S-System ModellingSuggestions for Further Reading . . . . . . . . . .Problem Set . . . . . . . . . . . . . . . . . . . . . .4 Analysis of Dynamic Mathematical Models4.1 Phase Plane Analysis . . . . . . . . . . . . . . .4.1.1 Direction fields . . . . . . . . . . . . . .4.1.2 Nullclines . . . . . . . . . . . . . . . . .4.2 Stability . . . . . . . . . . . . . . . . . . . . . .4.2.1 Stable and unstable steady states . . . .4.2.2 Linearized stability analysis . . . . . . .4.3 Limit Cycle Oscillations . . . . . . . . . . . . .4.4 Bifurcation Analysis . . . . . . . . . . . . . . .4.5 Sensitivity Analysis . . . . . . . . . . . . . . . .4.5.1 Local sensitivity analysis . . . . . . . .4.5.2 Determining local sensitivity coefficients4.6 *Parameter Fitting . . . . . . . . . . . . . . . .4.7 Suggestions for Further Reading . . . . . . . .4.8 Problem Set . . . . . . . . . . . . . . . . . . . .636464666969.77777980828386939699991021031051055 Metabolic Networks5.1 Modelling Metabolism . . . . . . . . . . . . . . . . . . . . . . . . . .5.1.1 Example: a pathway model . . . . . . . . . . . . . . . . . . .5.