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Multicellular Mathematical Models of Somitogenesis MULTICELLULAR MATHEMATICAL MODELS OF SOMITOGENESIS by Mark Benjamin Campanelli A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics MONTANA STATE UNIVERSITY Bozeman, Montana August, 2009 c Copyright by Mark Benjamin Campanelli 2009 All Rights Reserved ii APPROVAL of a dissertation submitted by Mark Benjamin Campanelli 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 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.” Mark Benjamin Campanelli August, 2009 iv DEDICATION I dedicate this dissertation to my family: To my wife Amber, for her enduring patience. To my daughter Ella, whose future I am trying to improve. v ACKNOWLEDGEMENTS I would like to thank my advisor, Dr. Tom`aˇsGedeon, for all of his help and guidance during this research project. I would also like to thank Jesse Berwald for his good cheer and generosity concerning all things computational. Lastly, I would like to thank Dr. Konstantin Mischaikow’s group at Rutgers University, for computational time on the conley2 computer cluster. vi TABLE OF CONTENTS 1. INTRODUCTION ........................................................................................1 Biological Pattern Formation .........................................................................1 Developmental Biology and Somitogenesis ......................................................2 Mathematical Insights into Somitogenesis .......................................................5 Purpose and Scope of the Present Work .........................................................9 2. SURVEY OF EXISTING MATHEMATICAL MODELS ...............................12 Early Models: Pattern Formation and Morphogenesis.................................... 12 Tissue-Based Reaction-Diffusion Models ....................................................... 14 Cell-Based Models....................................................................................... 16 Phase Oscillators..................................................................................... 16 Ordinary Differential Equation (ODE) Models .......................................... 17 Delay Differential Equation (DDE) Models ............................................... 19 Modeling Scopes and Multiple Scales ........................................................... 22 3. A MULTI-STABLE PHASE OSCILLATOR MODEL OF SOMITOGENESIS 25 Model Description....................................................................................... 25 Comparison to Existing Phase Oscillator Models........................................... 33 Lewis’s Phase Oscillator Model ................................................................ 33 Jaeger and Goodwin’s Cellular Oscillator Model........................................ 38 Discussion .................................................................................................. 39 4. A DELAY DIFFERENTIAL EQUATION MODEL OF POSTERIOR CLOCK- WAVE FORMATION .................................................................................41 The Biological Components of the Clock ...................................................... 43 The Clock............................................................................................... 44 The Control Protein ................................................................................ 46 The Coordinating Signal.......................................................................... 47 Modeling Posterior Clock-Wave Formation: Uncoupled Cells ......................... 49 PSM Growth........................................................................................... 49 Model Variables ...................................................................................... 49 The Control Protein ................................................................................ 50 The Intracellular Clock ............................................................................ 51 Clock-Gene Regulation by a Single Repressive Transcription Factor............ 55 Modeling Posterior Clock-Wave Formation: Coupled Cells............................. 59 Intercellular Signaling.............................................................................. 59 vii TABLE OF CONTENTS – CONTINUED Clock-Gene Regulation by Both Repressive and Activating Transcription Factors........................................................................................... 59 Interim Model Summary.............................................................................. 64 The Fast Dimerization Approximation.......................................................... 65 Algebraic Solution of the Fast Dimerization .............................................. 73 An Iterative Numerical Scheme for Computing the Fast Dimerization......... 75 Model Summary ......................................................................................... 78 5. MODEL VALIDATION: AN APPLICATION TO ZEBRAFISH SOMITOGE- NESIS........................................................................................................ 80 Computational Considerations for Validation ................................................ 81 Clock-Wave Formation in Zebrafish.............................................................. 83 Assignment of Model Components............................................................ 84 Model Validation Criteria ........................................................................ 85 Parameter Value and Range Selection....................................................... 87 Experimentally Determined Parameter Values ....................................... 88 Parameters Estimated from a Range of Values....................................... 91 Parameters for Model Scenarios I–IV .................................................... 93 Parameter Estimation and Model Selection................................................... 94 Stage One Validation............................................................................... 95 Parameter Sensitivities......................................................................... 98 Stage Two Validation ............................................................................ 102 Model Robustness.............................................................................. 103 Reproduction of Experiments..................................................................... 106 The Mechanism of Gradient Controlled Oscillation Rate.............................. 112 Comparison to Existing Zebrafish Models ............................................... 113 Applicability of the PCW Model ............................................................ 113 Further Analyses and Future Directions.................................................. 115 6. CONCLUSION ......................................................................................... 117 Future Directions ...................................................................................... 118 REFERENCES CITED.................................................................................. 120 APPENDICES .............................................................................................. 132 APPENDIX A: Impossibility of Nontrivial Periodic Solutions in Lewis’s Un- coupled DDE Model without Delays .......................................................... 133 APPENDIX B: Competitive Dimerization of Three Proteins ..................... 135 viii TABLE OF CONTENTS – CONTINUED APPENDIX C: Matlab Codes ................................................................. 140 ix LIST OF FIGURES Figure Page 1 Formed somites in a zebrafish embryo.....................................................3 2 Transverse schematic of the somitic mesoderm. .......................................3 3 Formed and forming somites in a zebrafish embryo. .................................4 4 Multiple gene expression during zebrafish somitogenesis...........................6 5 Maturity/susceptibility plots. ............................................................... 29 6 Phase portrait snapshots of the multi-stable phase oscillator. ................. 30 7 Computed solutions of the multi-stable phase oscillator model................ 32 8 Long-term computed solution behavior of the multi-stable phase oscil- lator model. .......................................................................................
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