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Systems Analysis of Stochastic and Population Balance Models for Chemically Reacting Systems

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Systems Analysis of Stochastic and Population Balance Models for Chemically Reacting Systems Systems Analysis of Stochastic and Population Balance Models for Chemically Reacting Systems by Eric Lynn Haseltine A dissertation submitted in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY (Chemical Engineering) at the UNIVERSITY OF WISCONSIN–MADISON 2005 c Copyright by Eric Lynn Haseltine 2005 All Rights Reserved i To Lori and Grace, for their love and support ii Systems Analysis of Stochastic and Population Balance Models for Chemically Reacting Systems Eric Lynn Haseltine Under the supervision of Professor James B. Rawlings At the University of Wisconsin–Madison Chemical reaction models present one method of analyzing complex reaction pathways. Most models of chemical reaction networks employ a traditional, deterministic setting. The short- comings of this traditional framework, namely difficulty in accounting for population het- erogeneity and discrete numbers of reactants, motivate the need for more flexible modeling frameworks such as stochastic and cell population balance models. How to efficiently use models to perform systems-level tasks such as parameter estimation and feedback controller design is important in all frameworks. Consequently, this thesis focuses on three main areas: 1. improving the methods used to simulate and perform systems-level tasks using stochas- tic models, 2. formulating and applying cell population balance models to better account for experi- mental data, and 3. applying moving-horizon estimation to improve state estimates for nonlinear reaction systems. For stochastic models, we have derived and implemented techniques that improve simulation efficiency and perform systems-level tasks using these simulations. For discrete stochastic models, these systems-level tasks rely on approximate, biased sensitivities, whereas continuous models (i.e. stochastic differential equations) permit calculation of unbiased sen- sitivities. Numerous examples illustrate the efficiency of these methods, including an applica- tion to modeling of batch crystallization systems. We have also investigated using cell population balance models to incorporate both intracellular and extracellular levels of information in viral infections. Given experimental im- ages of the focal infection system for vesicular stomatitis virus, we have applied these models to better understand the dynamics of multiple rounds of virus infection and the interferon (antiviral) host response. The model provides estimates of key parameters and suggests that the experimental technique may cause salient features in the data. We have also proposed an iii efficient and accurate model decomposition that predicts population-level measurements of intracellular and extracellular species. Finally, we have assessed the capabilities of several state estimators, including moving- horizon estimation (MHE) and the extended Kalman filter (EKF). When multiple optima arise in the estimation problem, the judicious use of constraints and nonlinear optimization as em- ployed by MHE can lead to improved state estimates and closed-loop control performance than the EKF. This improvement comes at the price of the computational expense required to solve the MHE optimization. iv v Acknowledgments “Whatever you do, work at it with all your heart, as working for the Lord, not for men, since you know that you will receive an inheritance from the Lord as a reward.” -Colossians 3:23-24 I first thank God, creator of heaven and earth, by whose grace I have had the opportu- nity to complete the work comprising this thesis. I thank my wife Lori, for her love, patience, and support. I would not have had the courage to aim so high without your encouragement. Also, the years in Madison would not have been as special without your presence. I thank my daughter Grace, who has always been able to make me smile during this past year no matter how far graduation seemed away. I am grateful to my family: my parents, Doug and Lydia, and my brother, David. With- out your support and guidance through the years of my life, I would not be where I am today. I also wish to thank my in-laws, Carl and Linda Rutkowski, in particular for supporting my wife these past five years. I thank my extended church family at Mad City Church: the Billers, the Thompsons, the Smiths, the Sells, and the Konkols. In particular, I wish to acknowledge Shane and Karen Biller, who have loved, supported, and prayed for my family as if we were part of their own. There are many people in the chemical engineering department at the University of Wisconsin whom I must also acknowledge. First, I thank my advisor, Jim Rawlings, for giving me great latitude to exercise my creativity and to study interesting problems. I am always amazed by your ability to identify the important problems in a field. It has been a great honor to work with you and learn from you. I am also grateful to John Yin for first listening to my modeling ideas, then making ways for me to collaborate with his group. I am deeply indebted to Gabriele Pannocchia, who always made time to answer my questions, no matter how trivial. Since imitation is the highest form of flattery, I have tried to be as patient, kind, and understanding to my junior group members as you were to me. I could always count on either reasoning out research problems or taking a break for humor with Aswin Venkat (a.k.a. the British spy). Thank you, Matt Tenny, for your help in the office and the weight room, although perhaps I would have graduated sooner if you had not introduced me to Nethack. Brian Odelson and Daniel Patience always kept me from taking research too seriously, be it rounding everyone up for a game of darts, or getting MJ to drop by for an ice cream break. Thanks also to John Eaton for Octave and Linux support; who would have figured five years ago that I would install Linux on my laptop? It has been a pleasure getting to know Paul vi Larsen, Murali Rajamani, and especially Ethan Mastny, who listened to almost all of my ideas on stochastic simulation. I also thank former Rawlings group members Jenny Wang, Scott Middlebrooks, and Chris Rao for their help during my first years in the group. Finally, I have had the great pleasure of getting to know the Yin group over the past year. In particular, I thank Vy Lam for graciously putting up with of my experimental questions. I am also grateful to Patrick Suthers and Hwijin Kim for their friendship. ERIC LYNN HASELTINE University of Wisconsin–Madison February 2005 vii Contents Abstract ii Acknowledgments v List of Tables xiii List of Figures xv Chapter 1 Introduction 1 Chapter 2 Literature Review 5 2.1 Traditional Deterministic Reaction Models . 5 2.2 Systems Level Tasks for Deterministic Models . 8 2.2.1 Optimal Control . 8 2.2.2 State Estimation . 9 2.2.3 Parameter Estimation . 10 2.2.4 Sensitivities . 13 2.3 Stochastic Reaction Models . 15 2.3.1 Monte Carlo Simulation of the Stochastic Model . 16 2.3.2 Performing Systems Level Tasks with Stochastic Models . 25 2.4 Population Balance Models . 26 Chapter 3 Motivation 29 3.1 Current Limitations of Stochastic Models . 29 3.1.1 Integration Methods . 29 3.1.2 Systems Level Tasks . 31 3.2 Current Limitations of Traditional Deterministic Models . 33 3.3 Current Limitations of State Estimation Techniques . 34 Chapter 4 Approximations for Stochastic Reaction Models 35 4.1 Stochastic Partitioning . 35 4.1.1 Slow Reaction Subset . 38 4.1.2 Fast Reaction Subset . 39 4.1.3 The Combined System . 40 4.1.4 The Equilibrium Approximation . 40 viii 4.1.5 The Langevin and Deterministic Approximations . 41 4.2 Numerical Implementation of the Approximations . 44 4.2.1 Simulating the Equilibrium Approximation . 46 4.2.2 Simulating the Langevin and Deterministic Approximations: Exact Next Reaction Time . 47 4.2.3 Simulating the Langevin and Deterministic Approximations: Approxi- mate Next Reaction Time . 49 4.3 Practical Implementation . 50 4.4 Examples . 50 4.4.1 Enzyme Kinetics . 51 4.4.2 Simple Crystallization . 52 4.4.3 Intracellular Viral Infection . 59 4.5 Critical Analysis of the Stochastic Approximations . 62 Chapter 5 Sensitivities for Stochastic Models 69 5.1 The Chemical Master Equation . 69 5.2 Sensitivities for Stochastic Systems . 70 5.2.1 Approximate Methods for Generating Sensitivities . 71 5.2.2 Deterministic Approximation for the Sensitivity . 72 5.2.3 Finite Difference Sensitivities . 74 5.2.4 Examples . 75 5.3 Parameter Estimation With Approximate Sensitivities . 79 5.3.1 High-Order Rate Example Revisited . 80 5.4 Steady-State Analysis . 82 5.4.1 Lattice-Gas Example . 83 5.5 Conclusions . 83 Chapter 6 Sensitivity Analysis of Discrete Markov Chain Models 87 6.1 Smoothed Perturbation Analysis . 89 6.1.1 Coin Flip Example . 90 6.1.2 State-Dependent Simulation Example . 93 6.2 Smoothing by Integration . 97 6.3 Sensitivity Calculation for Stochastic Chemical Kinetics . 100 6.4 Conclusions and Future Directions . 100 Chapter 7 Sensitivity Analysis of Stochastic Differential Equation Models 103 7.1 The Master Equation . 104 7.2 Sensitivity Examples . 106 7.2.1 Simple Reversible Reaction . 106 7.2.2 Oregonator . 107 7.3 Applications of Parametric Sensitivities . 109 7.3.1 Parameter Estimation . 109 ix 7.3.2 Calculating Steady States . 113 7.3.3 Simple Dumbbell Model of a Polymer in Solution . 114 7.4 Conclusions . 116 Chapter 8 Stochastic Simulation of Particulate Systems 119 8.1 Introduction . 119 8.2 Stochastic Chemical Kinetics Overview . 121 8.2.1 Stochastic Formulation of Isothermal Chemical Kinetics . 121 8.2.2 Extension of the Problem Scope . 122 8.2.3 Interpretation of the Simulation Output . 123 8.3 Crystallization Model Assumptions . 124 8.4 Stochastic Simulation of Batch Crystallization . 126 8.4.1 Isothermal Nucleation and Growth .
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