GENE REGULATION IN THE LAC OPERON by Kathryn Grace Patterson A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics MONTANA STATE UNIVERSITY Bozeman, Montana July, 2009 c Copyright by Kathryn Grace Patterson 2009 All Rights Reserved ii APPROVAL of a dissertation submitted by Kathryn Grace Patterson This dissertation has been read by each member of the dissertation committee and has been found to be satisfactory regarding content, English usage, format, citations, bibliographic style, and consistency, and is ready for submission to the Division of Graduate Education. Dr. Tom´aˇsGedeon Approved for the Department of Mathematical Sciences Dr. Kenneth L. Bowers Approved for the Division of Graduate Education Dr. Carl A. Fox iii STATEMENT OF PERMISSION TO USE In presenting this dissertation in partial fulfillment of the requirements for a doc- toral degree at Montana State University, I agree that the Library shall make it available to borrowers under rules of the Library. I further agree that copying of this dissertation is allowable only for scholarly purposes, consistent with \fair use" as pre- scribed in the U.S. Copyright Law. Requests for extensive copying or reproduction of this dissertation should be referred to ProQuest Information and Learning, 300 North Zeeb Road, Ann Arbor, Michigan 48106, to whom I have granted \the exclusive right to reproduce and distribute my dissertation in and from microform along with the non-exclusive right to reproduce and distribute my abstract in any format in whole or in part." Kathryn Grace Patterson July, 2009 iv ACKNOWLEDGEMENTS I would like to thank my advisor Tom´aˇsGedeon for his guidance, support and all of the opportunities he has provided. I am also indebted my committee: Jack Dockery, Mark Pernarowski, Marcy Barge and Ken Bowers for their suggestions, encouragement and time reading this work. In addition, this dissertation would not have been possible without the Mathematics faculty at Montana State University. I am grateful for all of your time, patience and encouragement. I would also like to thank Konstantin Mischaikow for his guidance, support and inviting me to visit Rutgers. My semester there was a wonderful experience. NSF-IGERT provided funding and structure to begin my interdisciplinary stud- ies. NSF/NIH grant W0467 and NSF grant DMS-0818785 provided funding for this project. My family and friends provided encouragement and support throughout this pro- cess. I am particularly grateful to my husband Michael for doing everything else, so I could write. v TABLE OF CONTENTS 1. INTRODUCTION ........................................................................................1 Transcription................................................................................................1 Binding Control ........................................................................................2 Opening....................................................................................................4 Escape......................................................................................................4 Basic Biology of the lac Operon .....................................................................5 Significance of the lac Operon ....................................................................5 System Description....................................................................................8 lac Regulation......................................................................................... 10 Activator CAP .................................................................................... 10 Repressor ............................................................................................ 11 The lac Repressor and DNA Looping .................................................... 12 Feedback Regulation............................................................................ 15 Inducer Exclusion ................................................................................ 16 lac Operon on Artificial Inducers .......................................................... 17 Results ................................................................................................... 19 2. MODEL OVERVIEW .................................................................................22 Primary Components .................................................................................. 22 CAP Activation ...................................................................................... 23 The lac Repressor and DNA Looping........................................................ 23 The Inducer ............................................................................................ 25 Secondary Components ............................................................................... 26 Additional RNAP Binding Sites ............................................................... 26 Additional Repressor Binding Site O3*..................................................... 28 CAP Assisted DNA Looping .................................................................... 29 Steric Interference between CAP Bound to C1 and a Repressor Bound to O330 Binding of the Inducer-Impaired lac Repressor to the Operators................. 31 Binding of the Repressor to Deleted Operators.......................................... 31 3. TRANSCRIPTION OF THE LAC OPERON ...............................................32 Experimental Data...................................................................................... 32 Bistability Data ...................................................................................... 32 Repression Data...................................................................................... 34 Looping Data.......................................................................................... 38 Differential Equation Model of the lac Operon .............................................. 42 mRNA Concentration.............................................................................. 42 Protein Concentration ............................................................................. 43 vi TABLE OF CONTENTS { CONTINUED Internal Inducer Concentration................................................................. 43 Model Predicted Repression Value............................................................ 44 mRNA Transcription................................................................................... 45 States of F .......................................................................................... 46 Protein Concentrations ........................................................................ 48 Existing Models of the lac Operon ............................................................... 51 Bistability Models ................................................................................... 51 Repression Ratio Models.......................................................................... 55 Loop Formation Models........................................................................... 56 4. PARAMETERIZING THE MODEL AND FITTING THE DATA .................58 Parameter Selection and Estimation............................................................. 58 RNAP Concentration ([RNAP]) ............................................................... 58 Binding of RNAP to P1 (KP 1)................................................................. 58 RNAP-DNA Interaction in Presence of CAP (Kcoop).................................. 59 Binding of RNAP to P2 (KP 2)................................................................. 65 Transcription Initiation Rate Constants (kf and kfC1) ............................... 65 Activator CAP........................................................................................ 66 lac Repressor........................................................................................... 67 Loop Formation ...................................................................................... 73 Binding Energies of the Repressor ............................................................ 77 5. RESULTS ..................................................................................................83 The Complete Model................................................................................... 91 Secondary Component Experiments ............................................................. 96 NE1: The P2 Promoter ........................................................................... 96 NE2: The O3* Operator.......................................................................... 98 NE3: CAP assisted DNA loop formation ................................................ 101 NE4: C1-O3 Steric Interaction ............................................................... 104 NE5: Impaired Binding Domains............................................................ 107 Additional Experiments............................................................................. 109 NE6: Repressor bound to O2 Blocks Transcription.................................. 109 NE7: Low-Specific Binding to Deleted Sites ............................................ 111 NE8: Minimal Model............................................................................. 114 Conclusions .............................................................................................. 116 vii TABLE OF CONTENTS { CONTINUED REFERENCES CITED.................................................................................