The genome contains several high-affinity non-functional binding sites for transcription factors (TFs) creating a hidden and unexplored layer of gene regulation. We investigate the role of such “decoy sites” in controlling noise (random fluctuations) in the level of a TF that is synthesized in stochastic bursts. Prior studies have assumed that decoy-bound TFs are protected from degradation, and in this case decoys function to buffer noise. Relaxing this assumption to consider arbitrary degradation rates for both bound/unbound TF states, we find rich noise behaviors. For low-affinity decoys, noise in the level of unbound TF always monotonically decreases to the Poisson limit with increasing decoy numbers. In contrast, for high-affinity decoys, noise levels first increase with increasing decoy numbers, before decreasing back to the Poisson limit. Interestingly, while protection of bound TFs from degradation slows the time-scale of fluctuations in the unbound TF levels, the decay of bound TFs leads to faster fluctuations and smaller noise propagation to downstream target proteins. In summary, our analysis reveals stochastic dynamics emerging from nonspecific binding of TFs and highlights the dual role of decoys as attenuators or amplifiers of gene expression noise depending on their binding affinity and stability of the bound TF.

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Enhancement of gene expression noise from transcription factor binding to genomic decoy sites.

The chemical master equation (CME) is a fundamental description of interacting molecules commonly used to model chemical kinetics and noisy gene regulatory networks. Exact time-dependent solutions of the CME -- which typically consists of infinitely many coupled differential equations -- are rare, and are valuable for numerical benchmarking and getting intuition for the behavior of more complicated systems. Jahnke and Huisinga's landmark calculation of the exact time-dependent solution of the CME for monomolecular reaction systems is one of the most general analytic results known; however, it is hard to generalize, because it relies crucially on properties of monomolecular reactions. In this paper, we rederive Jahnke and Huisinga's result on the time-dependent probability distribution and moments of monomolecular reaction systems using the Doi-Peliti path integral approach, which reduces solving the CME to evaluating many integrals. While the Doi-Peliti approach is less intuitive, it is also more mechanical, and hence easier to generalize. To illustrate how the Doi-Peliti approach can go beyond the method of Jahnke and Huisinga, we also find an explicit and exact time-dependent solution to a problem involving an autocatalytic reaction that Jahnke and Huisinga identified as not solvable using their method. We also find a formal exact time-dependent solution for any CME whose list of reactions involves only zero and first order reactions, which may be the most general result currently known.

Solving the chemical master equation for monomolecular reaction systems analytically: a Doi-Peliti path integral view

The chemical birth-death process, whose chemical master equation (CME) is exactly solvable, is a paradigmatic toy problem often used to get intuition for how stochasticity affects chemical kinetics. In a certain limit, it can be approximated by an Ornstein-Uhlenbeck-like process which is also exactly solvable. In this paper, we use this system to showcase eight qualitatively different ways to exactly solve continuous stochastic systems: (i) integrating the stochastic differential equation; (ii) computing the characteristic function; (iii) eigenfunction expansion; (iv) using ladder operators; (v) the Martin-Siggia-Rose-Janssen-De Dominicis path integral; (vi) the Onsager-Machlup path integral; (vii) semiclassically approximating the Onsager-Machlup path integral; and (viii) approximating the solution to the corresponding CME.

The chemical birth-death process with additive noise

Heterogeneity in gene expression across isogenic cell populations can give rise to phenotypic diversity, even when cells are in homogenous environments. This diversity arises from the discrete, stochastic nature of biochemical reactions, which naturally arise due to the very small numbers of genes, RNA, or protein molecules in single cells. Modern measurements of single biomolecules have created a vast wealth of information about the fluctuations of these molecules, but a quantitative understanding of these complex, stochastic systems requires precise computational tools. In this article, we present modern tools necessary to model variability in biological system and to compare model results to experimental data. We review the Chemical Master Equation and approaches to solve for probability distributions of discrete numbers of biomolecules. We discuss how to fit probability distributions to single-cell data using likelihood based approaches. Finally, we provide examples of fitting discrete stochastic models to single-molecule fluorescent in-situ hybridization data that quantifies RNA levels across populations of cells in bacteria and yeast.

Stochasticity or Noise in Biochemical Reactions

Rhythms in gene regulatory networks are ubiquitous, from the bacterial circadian clock to the segmentation clock of vertebrates. There are many decoy binding sites in a genome where regulatory proteins bind and control the expression of a gene. The role decoys on oscillatory regulatory networks is not well understood. Here, in the presence of decoy binding sites, we investigate the stability and the precision of the well-known Goodwin oscillator, a minimal model for regulatory oscillators. We derive the stability criterion in the presence of decoys and find that decoy abundance increases the parameter space where oscillating solutions exist. If the Goodwin system does not show any oscillation without decoy binding sites, a sustained oscillation is possible in their presence. Finally, we study precision the oscillation using stochastic simulations and find that the decoy binding makes the oscillation more precise.

Genomic decoy sites enhance the oscillatory regime of a biomolecular clock.

An introduction to the quantitative modeling of biological processes, presenting modeling approaches, methodology, practical algorithms, software tools, and examples of current research. The quantitative modeling of biological processes promises to expand biological research from a science of observation and discovery to one of rigorous prediction and quantitative analysis. The rapidly growing field of quantitative biology seeks to use biology's emerging technological and computational capabilities to model biological processes. This textbook offers an introduction to the theory, methods, and tools of quantitative biology. The book first introduces the foundations of biological modeling, focusing on some of the most widely used formalisms. It then presents essential methodology for model-guided analyses of biological data, covering such methods as network reconstruction, uncertainty quantification, and experimental design; practical algorithms and software packages for modeling biological systems; and specific examples of current quantitative biology research and related specialized methods. Most chapters offer problems, progressing from simple to complex, that test the reader's mastery of such key techniques as deterministic and stochastic simulations and data analysis. Many chapters include snippets of code that can be used to recreate analyses and generate figures related to the text. Examples are presented in the three popular computing languages: Matlab, R, and Python. A variety of online resources supplement the the text. The editors are long-time organizers of the Annual q-bio Summer School, which was founded in 2007. Through the school, the editors have helped to train more than 400 visiting students in Los Alamos, NM, Santa Fe, NM, San Diego, CA, Albuquerque, NM, and Fort Collins, CO. This book is inspired by the school's curricula, and most of the contributors have participated in the school as students, lecturers, or both. Contributors John H. Abel, Roberto Bertolusso, Daniela Besozzi, Michael L. Blinov, Clive G. Bowsher, Fiona A. Chandra, Paolo Cazzaniga, Bryan C. Daniels, Bernie J. Daigle, Jr., Maciej Dobrzynski, Jonathan P. Doye, Brian Drawert, Sean Fancer, Gareth W. Fearnley, Dirk Fey, Zachary Fox, Ramon Grima, Andreas Hellander, Stefan Hellander, David Hofmann, Damian Hernandez, William S. Hlavacek, Jianjun Huang, Tomasz Jetka, Dongya Jia, Mohit Kumar Jolly, Boris N. Kholodenko, Markek Kimmel, Michal Komorowski, Ganhui Lan, Heeseob Lee, Herbert Levine, Leslie M Loew, Jason G. Lomnitz, Ard A. Louis, Grant Lythe, Carmen Molina-Pars, Ion I. Moraru, Andrew Mugler, Brian Munsky, Joe Natale, Ilya Nemenman, Karol Nienaltowski, Marco S. Nobile, Maria Nowicka, Sarah Olson, Alan S. Perelson, Linda R. Petzold, Sreenivasan Ponnambalam, Arya Pourzanjani, Ruy M. Ribeiro, William Raymond, William Raymond, Herbert M. Sauro, Michael A. Savageau, Abhyudai Singh, James C. Schaff, Boris M. Slepchenko, Thomas R. Sokolowski, Petr ulc, Andrea Tangherloni, Pieter Rein ten Wolde, Philipp Thomas, Karen Tkach Tuzman, Lev S. Tsimring, Dan Vasilescu, Margaritis Voliotis, Lisa Weber

Roberto Bertolusso

Papers

Quantitative Biology: Theory, Computational Methods, and Models