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Stochastic Analysis of Biochemical Systems

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Stochastic Analysis of Biochemical Systems ON MOMENTS AND TIMING { STOCHASTIC ANALYSIS OF BIOCHEMICAL SYSTEMS by Khem Raj Ghusinga A dissertation submitted to the Faculty of the University of Delaware in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Electrical and Computer Engineering Summer 2018 c 2018 Khem Raj Ghusinga All Rights Reserved ON MOMENTS AND TIMING { STOCHASTIC ANALYSIS OF BIOCHEMICAL SYSTEMS by Khem Raj Ghusinga Approved: Kenneth E. Barner, Ph.D. Chair of the Department of Electrical and Computer Engineering Approved: Babatunde Ogunnaike, Ph.D. Dean of the College of Engineering Approved: Ann L. Ardis, Ph.D. Senior Vice Provost for Graduate and Professional Education I certify that I have read this dissertation and that in my opinion it meets the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy. Signed: Abhyudai Singh, Ph.D. Professor in charge of dissertation I certify that I have read this dissertation and that in my opinion it meets the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy. Signed: Ryan Zurakowski, Ph.D. Member of dissertation committee I certify that I have read this dissertation and that in my opinion it meets the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy. Signed: Gonzalo R. Arce, Ph.D. Member of dissertation committee I certify that I have read this dissertation and that in my opinion it meets the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy. Signed: Pak{Wing Fok, Ph.D. Member of dissertation committee ACKNOWLEDGEMENTS I express my deepest gratitude to my advisor, Prof. Abhyudai Singh, as this thesis would not have been possible without his constant guidance and support. Thanks are also due to Prof. Ryan Zurakowski, Prof. Gonzalo Arce, and Prof. Pak-Wing Fok for being part of my PhD committee and for numerous discussions on this research and beyond. I am grateful to Prof. Andrew Lamperski, Prof. John J. Dennehy, Prof. Pak{ Wing Fok, and Prof. James Umen with whom I have collaborated on research projects. Working with them has exposed me to different approaches to research and has helped me formulate my own approach. I acknowledge all members of the Singh lab, both present and past, for their regular feedback and support. My colleagueship with C´esarand Mohammad developed into close friendships that I would cherish for lifetime. I thank all my friends who supported me in numerous ways throughout. My roommates provided a perfect environment to take my mind off research. Nistha mer- its a special mention for being my strongest critic as well as my ultimate supporter. Finally, I am grateful to my family for their unwavering support. I would strive to be at least a fraction as caring as they have been to me. iv TABLE OF CONTENTS LIST OF TABLES :::::::::::::::::::::::::::::::: viii LIST OF FIGURES ::::::::::::::::::::::::::::::: ix ABSTRACT ::::::::::::::::::::::::::::::::::: xvi Chapter 1 INTRODUCTION :::::::::::::::::::::::::::::: 1 1.1 Notation :::::::::::::::::::::::::::::::::: 3 1.2 Publication List :::::::::::::::::::::::::::::: 3 2 BOUNDING MOMENTS OF BIOCHEMICAL SYSTEMS :::: 5 2.1 Constraints on Moments ::::::::::::::::::::::::: 7 2.2 Bounds on Steady-State Moments :::::::::::::::::::: 10 2.2.1 Method :::::::::::::::::::::::::::::: 10 2.2.2 Stochastic Logistic Growth with Constant Immigration Rate : 11 2.2.3 Stochastic Gene Expression with Negative Auto{regulation :: 15 2.2.4 Activator-Repressor Gene Motif ::::::::::::::::: 19 2.3 Bounds on Transient Moments :::::::::::::::::::::: 22 2.3.1 Stochastic Logistic Model :::::::::::::::::::: 22 2.4 Conclusion ::::::::::::::::::::::::::::::::: 25 3 BOUNDING MOMENTS OF STOCHASTIC HYBRID SYSTEMS 26 3.1 Background on Stochastic Hybrid Systems ::::::::::::::: 27 3.1.1 Basic Setup :::::::::::::::::::::::::::: 27 v 3.1.2 Extended Generator ::::::::::::::::::::::: 28 3.2 Moment Analysis of Polynomial SHS :::::::::::::::::: 29 3.2.1 Moment Dynamics for Polynomial SHS with Single Discrete State :::::::::::::::::::::::::::::::: 29 3.2.2 Moment Dynamics for Polynomial SHS with Finite Number of Discrete States :::::::::::::::::::::::::: 30 3.3 Moment Analysis for Rational SHS ::::::::::::::::::: 33 3.3.1 Bounds on Moments via Semidefinite Programming :::::: 35 3.4 Numerical Examples of Moment Estimation :::::::::::::: 35 3.4.1 Network Control System ::::::::::::::::::::: 35 3.4.2 TCP On–Off ::::::::::::::::::::::::::: 37 3.4.3 Cell division :::::::::::::::::::::::::::: 40 3.5 Estimating Characteristic Functions via Semidefinite Programming : 42 3.5.1 Review of Relevant Properties of Characteristic Function ::: 43 3.5.2 Characteristic Function of a Stochastic Process :::::::: 45 3.5.3 Solving for Stationary Characteristic Function ::::::::: 46 3.5.4 Numerical Examples ::::::::::::::::::::::: 49 3.5.4.1 Logistic Growth with Continuous Dynamics ::::: 49 3.5.4.2 Protein Production and Decay :::::::::::: 52 3.6 Conclusion ::::::::::::::::::::::::::::::::: 53 4 MODELING EVENT TIMING IN SINGLE{CELLS :::::::: 55 4.1 Formulating Event Timing as a First{Passage Time Problem ::::: 56 4.2 Moments of First{Passage Time ::::::::::::::::::::: 60 4.2.1 When the Protein is Stable ::::::::::::::::::: 61 4.2.2 When Protein is Unstable :::::::::::::::::::: 62 4.2.2.1 Scale Invariance of FPT Distribution ::::::::: 62 vi 4.2.2.2 Effect of Model Parameters on FPT Statistics :::: 64 4.3 Optimal Feedback Strategy ::::::::::::::::::::::: 66 4.3.1 Optimal Feedback for a Stable Protein ::::::::::::: 66 4.3.2 Optimal Feedback for an Unstable Protein ::::::::::: 68 4.4 Biological Implications and Discussion ::::::::::::::::: 69 4.4.1 Connecting Theoretical Insights to λ Lysis Times ::::::: 71 4.4.2 Additional Mechanism for Noise Buffering ::::::::::: 72 5 A MECHANISTIC FRAMEWORK FOR BACTERIAL CELL DIVISION TIMING :::::::::::::::::::::::::::: 77 5.1 Model description ::::::::::::::::::::::::::::: 78 5.2 Cell Division Time as a First{Passage Time Problem ::::::::: 80 5.3 Distribution of the Volume Added Between Divisions ::::::::: 83 5.3.1 Mean of Added Volume ::::::::::::::::::::: 85 5.3.2 Higher Order Moments of Added Volume and Scale Invariance of Distributions :::::::::::::::::::::::::: 86 5.4 Biological Relevance and Discussion ::::::::::::::::::: 87 5.4.1 Potential Candidates for the Time{Keeper Protein :::::: 88 5.4.2 Deviations from the Adder Principle :::::::::::::: 89 BIBLIOGRAPHY :::::::::::::::::::::::::::::::: 90 Appendix A SEMIDEFINITENESS OF OUTER PRODUCTS AND MOMENT MATRICES :::::::::::::::::::::::::::::::::: 110 B SUPPLEMENTARY DETAILS FOR CHAPTER 4 ::::::::: 111 B.1 Some Properties of Matrix Λ ::::::::::::::::::::::: 111 B.2 Optimal Feedback when Protein Does Not Degrade :::::::::: 112 B.3 Optimal Feedback when Protein Does Not Degrade in Presence of Extrinsic Noise :::::::::::::::::::::::::::::: 115 C DISCLAIMER :::::::::::::::::::::::::::::::: 117 vii LIST OF TABLES 2.1 Description of reactions for the logistic growth model :::::::: 12 2.2 Description of reactions for the auto{regulating gene :::::::: 17 2.3 Description of reactions for the activator repressor motif :::::: 19 2.4 Description of reactions for the logistic growth model :::::::: 23 viii LIST OF FIGURES 2.1 Estimated bounds on steady-state moments for the logistic growth model. Left: Upper and lower bounds on the second moment of the population level are shown for different orders of truncation M. As M is increased, the bounds obtained get tighter. The exact mean value of 19:149 is obtained by averaging 100; 000 MC simulations performed using SSA [1]. Parameters in (2.15) taken as k = 1, r = 5, and Ω = 20. Right: The bounds on the coefficient of variation (standard deviation/mean) of the steady-state population level are shown as the relative immigration rate k=r is changed. Both lower and upper bounds decrease as the relative immigration rate increases. These bounds are obtained via a 5th order truncation. :::::::: 14 2.2 Estimated bounds on moments for a stochastic gene expression model. Left: The lower and upper bounds for the average gene activity (left axis)/protein level (right axis) for different orders of truncation M is shown. The bound for M = 1 correspond to (2.31). As M is increased, the bounds obtained get tighter. The exact mean values are obtained from the analytical solution of the system from [2]. Parameters in (2.26) taken as kon = 10, koff = 0:1, kp = 15, hBi = 5, and γp = 1. Right: The coefficient of variation (standard deviation/mean) of the steady-state protein level as a function of the gene activation rate kon is plotted. The steady-state protein level in the deterministic sense is kept constant at 50 molecules by varying kp with kon such that kp = 50γp (50koff + kon) =(kon hBi). The lower and upper bounds on the coefficient of variation are obtained for M = 5 and they exhibit U-shape profiles, thus showing that the noise is minimizing at a specific value of kon. Other parameters are taken as koff = 0:1, hBi = 5, and γp = 1. :::::::::::::::::::: 16 ix 2.3 Estimated bounds on steady-state moments for the activator repressor motif. Left: Upper and lower bounds on the mean of both the activator and repressor are shown for different orders of truncation M. As M is increased, the bounds obtained get tighter. The exact mean values of 1:23 for the activator and 4:24 for the repressor are obtained by averaging 30; 000 MC simulations
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