Model-based External Forcing of Nonlinear Dynamics in Chemical and Biochemical Reaction via Optimal Control (PhD thesis 2008)

1 Introduction and Overview......Page 8
2.1.1 Limit sets, linear stability, and bifurcations......Page 13
2.1.2 Excitable, bistable and oscillatory systems......Page 16
2.2 DAE-Constrained Optimal Control Problems......Page 18
2.3.1 Indirect Method......Page 19
2.3.2 Direct Method......Page 21
2.3.3 Direct Multiple Shooting (DMS)......Page 22
2.4 Optimal Feedback Control......Page 23
2.4.1 Model Predictive Control......Page 25
2.5.1 Mixed-Integer Nonlinear Programming......Page 26
3.1.1 Control Parameterization......Page 27
3.1.2 State Parameterization......Page 28
3.1.3 Path Constraints in DMS......Page 29
3.2 The Nonlinear Programming Problem (NLP)......Page 30
3.3 Sequential Quadratic Programming (SQP)......Page 31
3.4.2 Backward Differentiation Formulae (BDF)......Page 32
3.5 Nonlinear Model Predictive Control......Page 33
3.5.1 Real-time Iteration Scheme......Page 34
3.5.2 Extended Kalman Filter......Page 35
3.6.1 Mixed Integer Optimization Algorithm......Page 36
4.1 Introduction......Page 37
4.1.1 Health issues......Page 38
4.2 Molecular biology of circadian rhythms......Page 40
4.2.2 Mammalian circadian clock......Page 41
4.2.3 Resetting the clock......Page 43
4.3 Modeling of circadian rhythms in Drosophila......Page 44
4.3.1 Drosophila model-equations and rate-constants......Page 46
4.4 Mathematical analysis of the circadian model......Page 47
4.4.2 Control of circadian oscillations by light......Page 48
4.4.3 Sensitivity analysis of the circadian model......Page 51
4.4.4 Bifurcation Analysis......Page 52
4.5 Control of circadian rhythms......Page 53
4.5.1 Formulation of optimal control problems......Page 54
4.6 Optimal phase resetting of circadian rhythms by critical light pulses......Page 55
4.7 Optimal phase tracking of circadian rhythms......Page 58
4.8 Restoration of altered circadian rhythms......Page 64
4.8.1 Restoration of altered perl mutant rhythm......Page 65
4.8.2 Restoration of altered pers mutant rhythm......Page 66
4.9.1 Control of circadian rhythms with NMPC......Page 70
4.9.2 Restoration of altered perl mutant rhythm by NMPC......Page 72
4.9.3 Restoration of altered pers mutant rhythm by NMPC......Page 75
5.1 Cell Cycle......Page 78
5.1.1 A Computational model for the cell cycle......Page 79
5.2 A Computational model for the mammalian circadian clock......Page 80
5.3 Coupling the circadian cycle and the cell cycle......Page 83
5.3.1 Linking the cell cycle and the circadian cycle via WEE1 kinase......Page 85
6.1 Construction of a detailed photo-BZ model......Page 88
6.2 Control of BZ reaction system......Page 91
6.2.1 Stabilization of unstable steady states......Page 93
6.2.2 Driving the BZ system at a desired frequency and amplitude......Page 95
6.3 Reduction of the detailed model......Page 97
6.3.1 Complex oscillations in the 8-variable model......Page 103
6.4.1 Finding a periodic orbit......Page 105
6.4.2 Response to optimal periodic stimuli in case of 8-variable model......Page 108
6.4.3 Comparions with 3-variable modified Oregonator model......Page 110
7 Summary, Conclusion and Outlook......Page 115
A Cell cycle model equations and rate constants......Page 117
B Circadian mammalian model-equations and rate-constants......Page 121
Bibliography......Page 123
List of Figures......Page 140
Symbols......Page 142