Simulation of Stochastic Chemical Systems: Applications in the Design and Construction of Synthetic Gene Networks A DISSERTATION SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE UNIVERSITY OF MINNESOTA BY Howard Michael Salis IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNDER THE GUIDANCE OF Yiannis Kaznessis Month/Year of Degree Clearance: February 2007. UMI Number: 3244458 Copyright 2006 by Salis, Howard Michael All rights reserved. INFORMATION TO USERS The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed-through, substandard margins, and improper alignment can adversely affect reproduction.
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(Howard Salis 2006-2007 Acknowledgements My doctoral research would not have been possible except for the continued help and support of many dear people. Writing a dissertation can be an especially isolating task and | am grateful to those who made it an enjoyable experience. | would like to thank my advisor, Yiannis Kaznessis, for his generous time and commitment. Throughout my doctoral work, he encouraged me to develop independent thinking and research skills and allowed me to explore new areas of mathematics with no guarantee of success.
He continually stimulated new thoughts and greatly assisted me with scientific writing. | also extend my thanks to my fellow graduate students and friends in the Kaznessis research group for helping me in many ways with my graduate studies. Jonathan Tomshine, Vassily Sotiropou- los, and John Barrett each greatly assisted me in advancing the rational design of synthetic gene networks. In addition, | am also grateful to Himanshu Khandelia, Spyros Vicatos, Allison Langham, Abdallah Sayyed, Chandrika Mulakala, and Dan Bolintineau for both scholarly and not-so-scholarly discussions on a wide variety of topics, for sharing tea times and cookies, for baking (thanks Allison!) and eating birthday cakes, and for making graduate school more than just a list of published papers.
| will miss our time together. | would like to thank Jennifer Maynard for allowing me to use her laboratory and equipment, and for teaching me a variety of genetic engineering techniques with her usual zeal and happiness. | also express my gratitude to Benjamin Roy, Ryan Myhre, Kavita Ramalingam, and Rakesh Motani for their patience with my many questions while working in their lab. | also thank David Morse, Hans Othmer, Marie Contou-Carrere, Chetan Gadgil, and Chang Hyeong Lee for intellectual stimulation, discussions, and thoughtful suggestions on manuscripts.
| would also like to extend my sincerest gratitude to Prabhas Moghe and Jane Tjia who mentored my research studies while at Rutgers University. They gave an inexperienced freshman a real job in their lab. | will always be grateful for their patience, their training, and their encouragement. | would also like to thank Troy Shinbrot and Stephen Conway at Rutgers for continuing that encouragement.
Of course, | have had many teachers over the years and | would not be where | am without them. | thank them all, but would like to especially thank Mr. Holmquist, who instilled a love of biology in me, and to Mrs. D'Esposito who taught me that “It’s not ‘I Know’, it’s ‘I Do’ ”.
| have been blessed with some great friends who have always been there to share in difficult and joyous occasions, to lend an ear, or to simply study the sometimes hilarious behavior of that well-known vociferous species of Leporidae Brachylagus.M, Rick, Andy, Jane, Kristen, and Alan, | thank you all for your friendship. | also extend my heart-felt thanks to Alexis who has brightened my life and given me great joy. This dissertation would not have been possible without your daily encouragement. Finally, | thank my parents for always loving me and being there to help solve problems big and small.
They are my greatest teachers in life. Funding for this research was provided by the United States National Institute of Health’s biotech- nology training grant (GM08347), the United States National Science Foundation (BES-0425882), and the Army High Performance Research Computing Center (AHPCR©) of the US Army Research Lab (Contract DAAD10-01-2-0014). Computational support was also provided by the University of Minnesota’s Digital Technology Center, the NSF Funded TeraGrid, and the National Center for Su- percomputing Applications (TG-MCA04N033). This dissertation is dedicated to my parents, Jan and Barry, who lovingly brought me into this world and taught me.
il Contents List of Tables vii List of Figures ix 1 Introduction mDune 1.1 An Overview of our Methodology. gu vn và và va 1.2 Key Results Inside This Dissertation.3 A Brief Introduction to the Mathematical Results .1 The Initial Motivation .2 The First Two Stochastic NumericalMethods. Onwards to Random DynamicalSystems.4 Numerical Methods for Stochastic Bifurcation Analysis. kg kg kg KV Kia 12 1.4 A Brief Introduction to the Biological Results.
Computer Aided Design of Synthetic Gene Networks.3 Protein Devices: A New Type of Synthetic Gene Network .4 Oscillatory Synthetic Gene Networks. Bottom-Up Mathematical Analysis of a Synthetic Promoter. g< 18 2 Stochastic Numerical Methods 2. cu nà kg à k kg Vi v v k k va 2.1 An Overview of the Chapter.2 A Brief Introduction to Probability and Stochastic Processes .2 Random Variables and Probability Distributons.
Commonly Used Random Variables .4 A Brief Overview of Stochastic Processes.3 The Numerical Simulation of Jump Markov and Poisson Processes.2 The Stochastic Simulation Algorthim. Poisson and Binomial Leaping.4 The Numerical Solution of Itô Stochastic Differential Equations .1 Definitions and Formal Solutions .2 Explicit Solutions of Some Stochastic Differential Equations .3 Strong and Weak Solutions.4 lô and Stratonovich Stochastic Integrals.5 The It6 Formula and It6-Taylor Expansions.6 Numerical Generation of Stochastic Integrals .7 It6-Taylor Explicit NumericalSchemes.8 Implicit Stochastic NumericalSchemes.9 Adaptive Time StepSchemes .5 HyJCMSS: The Hybrid Jump/Continuous Markov Stochastic Simulator. cuc c cv ng g v kg vi kg kg xà 71 2.4 Examples, Error Analysis, and Critical Comparisons .6 An Equation-Free Probabilistic Steady State Approximation. ee ee ee 96 2.
c c Q Q Q Q vu gu v ga và ga 99 2.7 Hy3S: Hybrid Stochastic Simulation for Supercomputes.73 Solution of a Hybrid Jump/Continuous Markov Process .4 The Fixed Euler-MaruyamaMethod.5 The Fixed Milstein Method. ce nu 1g và và kia 123 2. The Graphical User Interface. Q c c cu ng ng gà k kg vi v v kg k va 141 3 Design of Synthetic Gene Networks 142 3.1 An Overview of the Chapter.
An Overview of Regulated Bacterial Gene Expression. ng và kg kg tk kia 145 3. The Regulation of Transcriptional Ineracions.4 The Regulation of Translational Interactions.5 Messenger RNA and Protein Degradation and Dilution .3 The Modeling of Gene Networks 2.1 Kinetics and Equilibrium Data. The Chemical Partition Function and Equilibrium Holoenzyme Formation.5 mRNA and Protein Degradation and Dluton.6 Protein-Protein Interactions.4 The “AND” Protein Device: Logical Regulation of Gene Expression.
Molecular Design and Mathematical Methods.5 Conclusion and Outlook .6 Appendix Text: Notes on the Quantitative Model .5 An Oscillating Gene Network. The lac-tet-ara Gene Network .4 Results and Discussion. 209 Construction, Characterization, and Mathematical Analysis of a Synthetic Promoter 210 41 Introduction.2 Materials and Methods .1 Synthesizing and Cloning the ConsHUuct.2 Initial Confirmation of “AND”-like Promoter Activity .3 Sampling Inducer-Dependent Expression over Time .4 Characterization of Samples with FACS. Q Q Q HQ v n k KV sa 216 4.1 The “ON” Dynamics of the Synthetic Promoter Expression.2 The Steady-State Distribution of GFP Fluorescence over Varying Inducer Concentrations 2.
Cell Division Rates over Varying Inducer Concentration .4 A Steady-State Mathematical Model of the Synthetic Promoter.1 The Participating Molecular Interactions.2 The Steady-State Governing EqQUuAHO'NS.5 Combining Experimental and Model Resuls.1 Calculating an Unknown Parameter.2 Predicting the Behavior of Improved Synthetic Promoters .6 Discussion and Conclusions. ngà ga và 234 CONTENTS VI 5 Stochastic Bifurcation Analysis: New Numerical Methods 236 5.2 Conceptual Background on Random DynamicalSvstems. HQ nu kg va 237 5.3 Stochastic Stability Analysis.4 Stochastic Bifurcation Analysis .3 New Numerical Methods for Stochastic Bifurcation Analysis. Approximation the Action of the Forward and Reverse Time Cocycle 239 5.2 Reverse Stochastic Simulation.3 Iterative Forward-Reverse Samplng.4 Forward and Reverse Master Equations .5 Forward and Reverse Stochastic Simulation .6 Stationary and Non-Stationary Solutions.7 Iterative Forward-Reverse Sampling.
gà ki kia 246 Bibliography 247 List of Tables 21 Four important characteristics of five useful probability distribution functions .2 A list of the mass action rate laws for stochastic chemical kinetics. vu cv V2 V k KY k va 2.3 Cycle Test reactions and paramefffS.4 Ratios of Computational Run Times of Cycle Tests 2.5 A Simplified Model of the Pulse Generating Gene Network in Drosophila Circadian Rhythm 2. nu ng vn và vn kg k kg kg kg xa 2.6 The crystallization reaction system and kinetic parameters .7 The effect of time step on the run time and number of SDE integration steps of three hybrid stochastic methods.8 A Benchmark Model for Large-Scale Reaction Networks .9 Computational Run Times and SDE Calls of the Benchmark Models of Large-Scale SysteMS 20ee 2.10 The effect of Ø and ( on the probabilistic steady state approximation’s speed up when simulating the illustrative example reaction network. Parameter À is constant at 10.11 Accuracy and speed up of the probabilistic steady state approximation for the second example reaction network.
Parameter À is constant at30000.12 The reactions, kinetic constants, and initial conditions of the protein-protein interaction network eXampÌ€. cv HQ nu ng ng cv kg kg kg k va 2.13 The effect of increasing œ on the accuracy and efficiency of the stochastic simulation of the non-linear protein-protein interaction network.14 A diagnostic reaction network with multiple timescales is shown.15 An overview of the Hy3S numerical methods .16 A description of each command line argument and their defaultvalues.17 A Non-Linear Cycle Test 2. Q Q Q Q Quà vn vi kg kg va 2.18 A comparison of computational times of a large-scale system benchmark. The com- putational times of a large-scale system benchmark using the fixed Euler-Maruyama (EM) and Milstein implementations of the HyJCMSS algorithm and the Next Reaction variant of the stochastic simulation algorithm (SSA).
ND: Not Determined.19 A bistable biochemical network with multiple timescales and spontaneous escape .1 A list of consensus sequences for E.0 020004 vil LIST OF TABLES Vili 3.2 A list of the thermodynamic binding free energies between the Jac, tet, and ara tran- scription factors and their respective DNA operators at physiological temperature, pH, and salt concentration, 2.3 A list of the thermodynamic binding free energies between the inducer-bound /ac and tet repressors and their respective DNA operators.4 A selected list of ribosome binding sites (RBSs), The DNA sequences starting with AGGA and ending with a start codon are shown. The sequences are qualitatively ranked by their translation efficiency (average proteins per mRNA transcript with all other de- terminants equal) with one being the most efficient. The Gibbs free energies of the mRNA folding into a secondary structure (AG foiding) and the rRNA:mRNA hybridiza- tion (AGpyp-ia) are Shown for comparison.5 The enumeration and Gibbs free energies of the regulatory states of an example pro- moter with two operators and a single transcription factor.6 A list of potentially useful protein-protein interaction domains, their peptide ligands, and affinities, 2.7 The baseline and range of values foreach model parameter.8 The reaction network describing the protein-protein interactions between scaffold and scaffold-binding proteins. LH HQ ng vn kg k va 192 3.9 The reaction network describing production and degradation of scaffold and scaffold- binding proteins.