Carlo Cosentino, Declan Bates An Introduction to Feedback Control in Systems Biology CRC PRESS Boca Raton Ann Arbor London Tokyo To my parents, Rosa and Nicola, and to Cinzia. – C.C. To my parents, Br´ıd and Tony, and to Orlando and Lauren. – D.B. Contents 1 Introduction 7 1.1 Whatisfeedbackcontrol? . 7 1.2 Feedbackcontrolinbiologicalsystems . 10 1.2.1 The tryptophan operon feedback control system . 11 1.2.2 The polyamine feedback controlsystem . 12 1.2.3 Theheat shockfeedbackcontrolsystem . 14 1.3 Application of control theory to biological systems: a historical perspective ............................ 16 References ................................ 17 2 Linear Systems 23 2.1 Introduction ............................ 23 2.2 Statespacemodels ........................ 24 2.3 Linear time invariant systems and the frequency response .. 26 2.4 Fourieranalysis .......................... 32 2.5 Transfer functions and the Laplace transform . 36 2.6 Stability .............................. 39 2.7 Change of state variables and canonical representations ... 41 2.8 Characterising system dynamics in the time domain . 42 2.9 Characterising system dynamics in the frequency domain .. 46 2.10 Block diagram representations of interconnected systems ... 48 2.11 Case Study I: Characterising the frequency dependence of osmo–adaptation in Saccharomyces cerevisiae ......... 53 2.11.1 Introduction. 54 2.11.2 Frequencydomainanalysis . 54 2.11.3 Timedomainanalysis . 57 2.12 Case Study II: Characterising the dynamics of the Dictyostelium externalsignalreceptornetwork . 60 2.12.1 Introduction. 61 2.12.2 A generic structure for ligand/receptor interaction net- works ........................... 61 2.12.3 Structure of the ligand/receptor interaction network in aggregating Dictyostelium cells............ 63 2.12.4 Dynamic response of the Dictyostelium ligand/receptor interactionnetwork. 66 References ................................ 69 iii iv An Introduction to Feedback Control in Systems Biology 3 Nonlinear systems 73 3.1 Introduction ............................ 73 3.2 Equilibriumpoints ........................ 75 3.3 Linearisation around equilibrium points . 78 3.4 Stabilityandregionsofattractions . 84 3.4.1 Lyapunovstability . 84 3.4.2 Regionsofattraction. 87 3.5 Optimisation methods for nonlinear systems . 91 3.5.1 Localoptimisationmethods . 93 3.5.2 Global optimisation methods . 95 3.5.3 Linearmatrixinequalities . 97 3.6 Case Study III: Stability analysis of tumor dormancy equilib- rium ................................ 99 3.6.1 Introduction. .100 3.6.2 Modelofcancerdevelopment . 101 3.6.3 Stability of the equilibrium points . 102 3.6.4 Checking inclusion in the region of attraction . 103 3.6.5 Analysis of the tumor dormancy equilibrium . 106 3.7 Case Study IV: Global optimisation of a model of the trypto- phan control system against multiple experiment data . 111 3.7.1 Introduction. .112 3.7.2 Model of the tryptophan controlsystem . 112 3.7.3 Model analysis using global optimisation . 115 References ................................116 4 Negative feedback systems 121 4.1 Introduction ............................121 4.2 Stability of negative feedback systems . 125 4.3 Performanceofnegativefeedbacksystems . 128 4.4 Fundamental tradeoffs with negative feedback . 134 4.5 Case Study V: Analysis of stability and oscillations in the p53- Mdm2feedbacksystem . .138 4.6 Case Study VI: Perfect adaptation via integral feedback con- trolinbacterialchemotaxis . 143 4.6.1 A mathematical model of bacterial chemotaxis . 144 4.6.2 Analysis of the perfect adaptation mechanism . 147 4.6.3 Perfect adaptation requires demethylation only of ac- tivereceptors . .. .. .. .. .. .. .. .. .. .151 References ................................152 5 Positive feedback systems 155 5.1 Introduction ............................155 5.2 Bifurcations, bistability and limit cycles . 155 5.2.1 Bifurcations and bistability . 155 5.2.2 Limitcycles........................158 Table of Contents v 5.3 Monotonesystems ........................162 5.4 ChemicalReactionNetworkTheory . 165 5.4.1 Preliminaries on reaction network structure . 166 5.4.2 Networksofdeficiencyzero . 168 5.4.3 Networksofdeficiencyone. 170 5.5 Case Study VII: Positive feedback leads to multistability, bi- furcations, and hysteresis in a MAPK cascade . 172 5.6 Case Study VIII: Coupled positive and negative feedback loops intheyeastgalactosepathway . 179 References ................................186 6 Model validation using robustness analysis 189 6.1 Introduction ............................189 6.2 Robustness analysis tools for model validation . 191 6.2.1 Bifurcationdiagrams . 191 6.2.2 Sensitivityanalysis . 192 6.2.3 µ-analysis .........................196 6.2.4 Optimisation-based robustness analysis . 199 6.2.5 Sum-of-Squarespolynomials . 200 6.2.6 MonteCarlosimulation . 202 6.3 New robustness analysis tools for biological systems . 203 6.4 Case Study IX: Validating models of cAMP oscillations in ag- gregating Dictyostelium cells ...................206 6.5 Case Study X: Validating models of the p53-Mdm2 System . 208 References ................................210 7 ReverseEngineeringBiomolecularNetworks 215 7.1 Introduction ............................215 7.2 Inferring network interactions using linear models . 215 7.2.1 Discrete-time vs Continuous-time model . 217 7.3 Leastsquares ...........................220 7.3.1 Leastsquaresfor dynamicalsystems . 225 7.3.2 Methods based on least squares regression . 228 7.4 Exploitingpriorknowledge . 230 7.4.1 Network inference via LMI–based optimisation . 231 7.4.2 MAX-PARSE: An algorithm for pruning a fully con- nected network according to maximum parsimony . 234 7.4.3 CORE-Net: A network growth algorithm using pref- erentialattachment. 235 7.5 Dealingwithmeasurementnoise . 236 7.5.1 TotalLeastSquares . 236 7.5.2 ConstrainedTotalLeastSquares . 238 7.6 Exploitingtime-varyingmodels . 240 7.7 Case Study XI: Inferring regulatory interactions in the innate immune systemfromnoisy measurements . 244 vi An Introduction to Feedback Control in Systems Biology 7.8 Case Study XII: Reverse engineering a cell-cycle regulatory subnetwork of Saccharomyces cerevisiae from experimental micro- arraydata .............................248 7.8.1 PACTLS: An algorithm for reverse engineering par- tially known networks from noisy data . 250 7.8.2 Results ..........................252 References ................................255 8 Stochastic effects in biological control systems 261 8.1 Introduction ............................261 8.2 Stochastic modelling and simulation . 262 8.3 A framework for analysing the effect of stochastic noise on stability ..............................265 8.3.1 The effective stability approximation . 266 8.3.2 A computationally efficient approximation of the dom- inantstochasticperturbation . 267 8.3.3 Analysis using the Nyquist stability criterion . 269 8.4 Case Study XIII: Stochastic effects on the stability of cAMP oscillations in aggregating Dictyostelium cells .........272 8.5 Case Study XIV: Stochastic effects on the robustness of cAMP oscillations in aggregating Dictyostelium cells .........277 References ................................282 Preface The field of Systems Biology encompasses scientists with extremely diverse backgrounds, from biologists, biochemists, clinicians and physiologists to math- ematicians, physicists, computer scientists and engineers. Although many of these researchers have recently become interested in control-theoretic ideas such as feedback, stability, noise and disturbance attenuation, and robust- ness, it is still unfortunately the case that only researchers with an engineering background will usually have received any formal training in control theory. Indeed, our initial motivation to write this book arose from the difficulty we found in recommending an introductory text on feedback control to colleagues who were not from an engineering background, but who needed to understand control engineering methods to analyse complex biological systems. This difficulty stems from the fact that the traditional audience for control textbooks is made up of electrical, mechanical, process and aerospace engi- neers who require formal training in control system design methods for their respective applications. Systems Biologists, on the other hand, are more in- terested in the fundamental concepts and ideas which may be used to analyse the effects of feedback in evolved biological control systems. Researchers with a biological sciences background may often also lack the expertise in physical systems modelling (Newtonian mechanics, Kirchhoff’s electrical circuit laws, etc) that is typically assumed in the examples used in standard texts on feed- back control theory. The type of “control applications” in which a Systems Biologist is interested are systems such as metabolic and gene-regulatory net- works, not aircraft, robots or engines, and the type of mathematical models they are familiar with are typically derived from classical reaction kinetics, not classical mechanics. Another significant problem for Systems Biologists is that current under- graduate books on control theory (which introduce the basic concepts at great length) are uniformly restricted to linear systems, while nonlinear systems are usually only considered by specialist post-graduate texts which require advanced mathematical skills. Although it will always be appropriate to in- troduce basic ideas in control using linear systems, biological systems are in general highly nonlinear, and thus a clear understanding