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  • Biomolecular Feedback Systems

    Domitilla Del Vecchio Richard M. Murray

    June 9, 2014

  • Contents

    Preface iii

    1 Introductory Concepts 1

    1.1 Systems biology: Modeling, analysis and role of feedback . . . . . . 1 1.2 The cell as a system . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Control and dynamical systems tools . . . . . . . . . . . . . . . . . 12 1.4 Input/Output modeling . . . . . . . . . . . . . . . . . . . . . . . . 18 1.5 From systems to synthetic biology . . . . . . . . . . . . . . . . . . 22 1.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

    2 Dynamic Modeling of Core Processes 29

    2.1 Modeling chemical reactions . . . . . . . . . . . . . . . . . . . . . 29 2.2 Transcription and translation . . . . . . . . . . . . . . . . . . . . . 44 2.3 Transcriptional regulation . . . . . . . . . . . . . . . . . . . . . . . 55 2.4 Post-transcriptional regulation . . . . . . . . . . . . . . . . . . . . 70 2.5 Cellular subsystems . . . . . . . . . . . . . . . . . . . . . . . . . . 80

    Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

    3 Analysis of Dynamic Behavior 89

    3.1 Analysis near equilibria . . . . . . . . . . . . . . . . . . . . . . . . 89 3.2 Robustness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 3.3 Oscillatory behavior . . . . . . . . . . . . . . . . . . . . . . . . . . 113 3.4 Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 3.5 Model reduction techniques . . . . . . . . . . . . . . . . . . . . . . 127

    Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

    4 Stochastic Modeling and Analysis 139

    4.1 Stochastic modeling of biochemical Systems . . . . . . . . . . . . . 139 4.2 Simulation of stochastic systems . . . . . . . . . . . . . . . . . . . 154

    Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

    5 Biological Circuit Components 161

    5.1 Introduction to Biological Circuit Design . . . . . . . . . . . . . . 161 5.2 Negative Autoregulation . . . . . . . . . . . . . . . . . . . . . . . 163

  • ii CONTENTS

    5.3 The Toggle Switch . . . . . . . . . . . . . . . . . . . . . . . . . . 169 5.4 The Repressilator . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 5.5 Activator-Repressor Clock . . . . . . . . . . . . . . . . . . . . . . 176 5.6 An Incoherent Feedforward Loop (IFFL) . . . . . . . . . . . . . . . 181 5.7 Bacterial Chemotaxis . . . . . . . . . . . . . . . . . . . . . . . . . 183

    Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

    6 Interconnecting Components 197

    6.1 Input/Output Modeling and the Modularity Assumption . . . . . . . 197 6.2 Introduction to Retroactivity . . . . . . . . . . . . . . . . . . . . . 198 6.3 Retroactivity in Gene Circuits . . . . . . . . . . . . . . . . . . . . 201 6.4 Retroactivity in Signaling Systems . . . . . . . . . . . . . . . . . . 206 6.5 Insulation Devices: Retroactivity Attenuation . . . . . . . . . . . . 211 6.6 A Case Study on the Use of Insulation Devices . . . . . . . . . . . 228

    Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

    7 Design Tradeoffs 235

    7.1 Competition for Shared Cellular Resources . . . . . . . . . . . . . 235 7.2 Stochastic Effects: Design Tradeoffs in Systems with Large Gains . 245

    Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

    Bibliography 251

    Index 258

  • Preface

    This text is intended for researchers interested in the application of feedback and control to biomolecular systems. The material has been designed so that it can be used in parallel with the textbook Feedback Systems [1] as part of a course on biomolecular feedback and control systems, or as a stand-alone reference for readers who have had a basic course in feedback and control theory. The full text for this book, along with additional supplemental material, is available on a companion Web site:

    http://www.cds.caltech.edu/˜murray/BFS

    The material in this book is intended to be useful to three overlapping audi- ences: graduate students in biology and bioengineering interested in understanding the role of feedback in natural and engineered biomolecular systems; advanced un- dergraduates and graduate students in engineering disciplines who are interested in the use of feedback in biological circuit design; and established researchers in the biological sciences who want to explore the potential application of principles and tools from control theory to biomolecular systems. We have written the text assum- ing some familiarity with basic concepts in feedback and control, but have tried to provide insights and specific results as needed, so that the material can be learned in parallel. We also assume some familiarity with cell biology, at the level of a first course for non-majors. The individual chapters in the text indicate the prerequisites in more detail, most of which are covered either in Åström and Murray [1] or in the supplemental information available from the companion Web site.

    Acknowledgments. Many colleagues and students provided feedback and advice on the book. We would particularly like to thank Mustafa Khammash, Eric Klavins, and Eduardo Sontag, who gave detailed comments on some of the early versions of the text. In addition, we would like to acknowledge Abdullah Amadeh, Andras Gyorgy, Narmada Herath, Yutaka Hori, Shridhar Jayanthi, Scott Livingston, Rob Phillips, Phillip Rivera, Vipul Singhal, Anandh Swaminathan, Eric Winfree, and Enoch Yeung for their support and comments along the way. Finally, we would like to thank Caltech, MIT and the University of Michigan for providing the many resources that were necessary to bring this book to fruition.

    Domitilla Del Vecchio Richard M. Murray Cambridge, Massachusetts Pasadena, California

    http://www.cds.caltech.edu/~murray/BFS

  • iv CONTENTS

  • Chapter 1

    Introductory Concepts

    This chapter provides a brief introduction to concepts from systems biology, tools from differential equations and control theory, and approaches to modeling, anal- ysis and design of biomolecular feedback systems. We begin with a discussion of the role of modeling, analysis and feedback in biological systems. This is followed by a short review of key concepts and tools from control and dynamical systems theory, intended to provide insight into the main methodology described in the text. Finally, we give a brief introduction to the field of synthetic biology, which is the primary topic of the latter portion of the text. Readers who are familiar with one or more of these areas can skip the corresponding sections without loss of continuity.

    1.1 Systems biology: Modeling, analysis and role of feedback

    At a variety of levels of organization—from molecular to cellular to organismal— biology is becoming more accessible to approaches that are commonly used in engineering: mathematical modeling, systems theory, computation and abstract ap- proaches to synthesis. Conversely, the accelerating pace of discovery in biological science is suggesting new design principles that may have important practical ap- plications in human-made systems. This synergy at the interface of biology and engineering offers many opportunities to meet challenges in both areas. The guid- ing principles of feedback and control are central to many of the key questions in biological science and engineering and can play an enabling role in understanding the complexity of biological systems.

    In this section we summarize our view on the role that modeling and analysis should (eventually) play in the study of biological systems, and discuss some of the ways in which an understanding of feedback principles in biology can help us better understand and design complex biomolecular circuits.

    There are a wide variety of biological phenomena that provide a rich source of examples for control, including gene regulation and signal transduction; hormonal, immunological, and cardiovascular feedback mechanisms; muscular control and locomotion; active sensing, vision, and proprioception; attention and conscious- ness; and population dynamics and epidemics. Each of these (and many more) pro- vide opportunities to figure out what works, how it works, and what can be done to affect it. Our focus here is at the molecular scale, but the principles and approach that we describe can also be applied at larger time and length scales.

  • 2 CHAPTER 1. INTRODUCTORY CONCEPTS

    Modeling and analysis

    Over the past several decades, there have been significant advances in modeling capabilities for biological systems that have provided new insights into the com- plex interactions of the molecular-scale processes that implement life. Reduced- order modeling has become commonplace as a mechanism for describing and doc- umenting experimental results, and high-dimensional stochastic models can now be simulated in reasonable periods of time to explore underlying stochastic effects. Coupled with our ability to collect large amounts of data from flow cytometry, micro-array analysis, single-cell microscopy, and other modern experimental tech- niques, our understanding of biomolecular processes is advancing at a rapid pace.

    Unfortunately, although models are becoming much more common in biolog- ical studies, they are still far from playing the central role in explaining complex biological phenomena. Although there are exceptions, the predominant use of mod- els is to “document” experimental results: a hypothesis is proposed and te

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