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<title>Biological Circuit Design</title>
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<span class="menu-text">Noise: Bug and feature</span></a>
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<span class="menu-text"><span class="chapter-number">16</span> <span class="chapter-title">The noisy, noisy nature of gene expression: How stochastic fluctuations create variation</span></span></a>
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<span class="menu-text"><span class="chapter-number">17</span> <span class="chapter-title">Bursty gene expression</span></span></a>
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<span class="menu-text"><span class="chapter-number">18</span> <span class="chapter-title">Cellular bet-hedging</span></span></a>
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<span class="menu-text"><span class="chapter-number">19</span> <span class="chapter-title">Excitability enables probabilistic, transient differentiation</span></span></a>
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<span class="menu-text">Patterning circuits</span></a>
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<span class="menu-text"><span class="chapter-number">21</span> <span class="chapter-title">Turing patterns</span></span></a>
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<span class="menu-text"><span class="chapter-number">22</span> <span class="chapter-title">Scaling reaction-diffusion patterns</span></span></a>
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<span class="menu-text">Technical appendices</span></a>
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<span class="menu-text">Technical Appendix 2a: Approximate solutions of autorepressive dynamics</span></a>
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<span class="menu-text">Technical Appendix 2b: Numerical solutions to ODEs with SciPy</span></a>
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<span class="menu-text">Technical Appendix 2c: Interactive plotting with Bokeh</span></a>
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<span class="menu-text">Technical Appendix 3a: Nondimensionalization</span></a>
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<span class="menu-text">Technical Appendix 4a: Numerical solution of FFLs</span></a>
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<span class="menu-text">Technical Appendix 6a: Michaelis-Menten enzyme kinetics</span></a>
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<span class="menu-text">Technical Appendix 8a: Signaling cascades</span></a>
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<span class="menu-text">Technical Appendix 8b: Cascades of multiply phosphorylated signaling molecules</span></a>
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<span class="menu-text">Technical Appendix 8c: Phosphorylation cascade with saturation</span></a>
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<span class="menu-text">Technical Appendix 10a: Fixed points and composite functions</span></a>
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<span class="menu-text">Technical Appendix 10b: Linear stability analysis</span></a>
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<span class="menu-text">Technical Appendix 10c: Numerical one-dimensional bounded root finding</span></a>
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<span class="menu-text">Technical Appendix 11a: Stability diagrams by numerical computation of eigenvalues</span></a>
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<span class="menu-text">Technical Appendix 11b: The Greshgorin circle theorem</span></a>
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<span class="menu-text">Technical Appendix 11c: Numerical solution of delay differential equations</span></a>
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<span class="menu-text">Technical Appendix 11d: Linear stability analysis of delay differential equations</span></a>
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<span class="menu-text">Technical Appendix 17a: Stochastic (Gillespie) simulation</span></a>
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<span class="menu-text">Technical Appendix 17b: Profiling code for speed and an application of the Gillespie algorithm</span></a>
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<span class="menu-text">Technical Appendix 20a: Lateral Inhibition: Spontaneous symmetry breaking to form a spatial pattern of cells</span></a>
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<span class="menu-text">Exercise 1.1: Strategies for controlling protein expression</span></a>
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<span class="menu-text">Exercise 1.2: Separation of time scales</span></a>
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<span class="menu-text">Exercise 1.3: Rate of production of gene product by an activator</span></a>
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<span class="menu-text">Exercise 1.4: Activators vs. repressors</span></a>
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<span class="menu-text">Exercise 1.5: Bound and unbound promoter regions</span></a>
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<span class="menu-text">Exercise 2.1: Cost of a steady state</span></a>
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<span class="menu-text">Exercise 2.2: Event handling for discontinuous derivatives</span></a>
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<span class="menu-text">Exercise 3.1: Modes of bistability</span></a>
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<span class="menu-text">Exercise 3.2: Controlling an autoactivation circuit</span></a>
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<span class="menu-text">Exercise 3.3: Autoactivation, bistability, and the importance of leakage</span></a>
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<span class="menu-text">Exercise 3.4: Modeling inducers in a toggle switch</span></a>
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<span class="menu-text">Exercise 3.5: Negative autoregulation including mRNA dynamics</span></a>
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<span class="menu-text">Exercise 3.6: Noncooperative bistability with a growth-determined positive feedback circuit</span></a>
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<span class="menu-text">Exercise 3.7: Design principles for toggles</span></a>
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<span class="menu-text">Exercise 4.1: Accelerated responses with FFLs</span></a>
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<span class="menu-text">Exercise 4.2: Statistics of random genetic circuits</span></a>
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<span class="menu-text">Exercise 4.3: XOR gates</span></a>
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<span class="menu-text">Exercise 5.1: Robustness in a C1-FFL</span></a>
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<span class="menu-text">Exercise 6.1: Co-substrate compensation</span></a>
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<span class="menu-text">Exercise 7.1: Interpretations of sensitivity</span></a>
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<span class="menu-text">Exercise 7.2: Signal propagation in a kinase cascade</span></a>
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<span class="menu-text">Exercise 7.3: Dependence of signaling response on total histidine kinase and response regulator concentrations</span></a>
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<span class="menu-text">Exercise 9.1: Kinetic proofreading in the immune system</span></a>
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<span class="menu-text">Exercise 10.1: Coupled repressilators</span></a>
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<span class="menu-text">Exercise 10.2: The KaiABC clock</span></a>
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<span class="menu-text">Exercise 10.3: Linear stability analysis of the repressilator with mRNA</span></a>
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<span class="menu-text">Exercise 11.1: Tuning delay oscillators with positive feedback</span></a>
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<span class="menu-text">Exercise 11.2: Controlling p53 levels</span></a>
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<span class="menu-text">Exercise 12.1: Temporal gene expression and inferring a circuit</span></a>
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<span class="menu-text">Exercise 14.1: Programming cellular response</span></a>
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<span class="menu-text">Exercise 15.1: A tri-stable cell-fate determinant circuit</span></a>
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<span class="menu-text">Exercise 17.1: Dynamics of noise</span></a>
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<span class="menu-text">Exercise 17.2: Noise in a switchable promoter</span></a>
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<span class="menu-text">Exercise 17.3: A possible mechanism for bursty gene expression</span></a>
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<span class="menu-text">Exercise 22.1: Turing patterns with expanders</span></a>
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<span class="menu-text"><span class="chapter-number">A</span> <span class="chapter-title">Configuring your computer to use Python for scientific computing</span></span></a>
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<span class="menu-text">Appendix B1: Hello, world.</span></a>
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<span class="menu-text">Appendix B2: Variables, operators, and types</span></a>
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<span class="menu-text">Appendix B3: Lists and tuples</span></a>
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<span class="menu-text">Appendix B4: Iteration</span></a>
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<span class="menu-text">Appendix B5: Introduction to functions</span></a>
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<span class="menu-text">Appendix B6: String methods</span></a>
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<span class="menu-text">Appendix B7: Dictionaries</span></a>
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<span class="menu-text">Appendix B8: Comprehensions</span></a>
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<span class="menu-text">Appendix B9: Packages and modules</span></a>
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<span class="menu-text">Appendix B10: Errors and exception handling</span></a>
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<span class="menu-text"><span class="chapter-number">C</span> <span class="chapter-title">Numerical solutions of ODES</span></span></a>
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<span class="menu-text"><span class="chapter-number">D</span> <span class="chapter-title">Regulatory functions and their derivatives</span></span></a>
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<h1 class="title">Biological Circuit Design</h1>
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<div class="quarto-title-meta-heading">Authors</div>
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<p class="author">Michael Elowitz </p>
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California Institute of Technology
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<p class="author">Justin Bois </p>
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California Institute of Technology
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<p class="date">April 12, 2026</p>
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<section id="welcome" class="level2 unnumbered">
<h2 class="unnumbered anchored" data-anchor-id="welcome">Welcome</h2>
<p>Welcome to <em>Biological Circuit Design</em>, a web-based book (in progress) focusing on quantitative studies of cellular and developmental systems in biology, including the architecture of specific genetic circuits controlling microbial behaviors and multicellular development in model organisms. Our goals are to build understanding of key principles of systems biology and to develop skills for modeling and analysis of genetic circuits.</p>
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<section id="pardon-our-dust" class="level2">
<h2 class="anchored" data-anchor-id="pardon-our-dust">Pardon our dust!</h2>
<p>This book is under construction. Furthermore, we recently switched to a new build system, so some (maybe many) links may be broken, figures may not all render properly, etc. So, please pardon our dust!</p>
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<h2 class="unnumbered anchored" data-anchor-id="copyright-and-license">Copyright and License</h2>
<p>Copyright 2017–2026, Michael Elowitz and Justin Bois.</p>
<p>With the exception of pasted graphics, where the source is noted, this work is licensed under a <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/deed.en">Creative Commons Attribution License CC BY-NC-SA 4.0</a>. All code contained herein is licensed under an <a href="https://opensource.org/license/MIT">MIT license</a>.</p>
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