Mathematical Models of the Cell and Cell Associated Objects (eBook)

Front cover 1
Title page 4
Copyright page 5
Table of contents 8
Preface 16
List of abbreviations 19
List of notations 20
Introduction: Evolutionary Systems and Development Modeling 22
Evolutionary Systems (ES) 22
Artificial E S 22
Natural ES 25
Joined ES 27
Generalized Structure of ES 28
Generalized Structure of Two ES Interactions 29
Generalized Structure of CAO 31
On Development Modeling 33
Base simplest MM of development 33
Comparison with classical MM 35
A history, in short 37
Part I: SURVEY OF MM OF CAO 38
1 General Methods of Inlet and Analysis of MM 40
Methods of Simplification of the Systems of Kinetic Equations 40
On the Qualitative Theory for the Systems of the Second Order 42
MM of cell cycles, I 45
MM of hyper-cycle 48
MM of cell excitable membranes 49
MM of Spatial Systems 51
MM of tumor angiogenesis 54
Mechanical models for mesenchymal morphogenesis 56
Possibility for Description of Biological Events on Molecular Level 57
MM of traffic through cell membranes 60
Passive transport 60
Active transport 61
On the problem of structure and protein folding 62
Simple kinetic MM in protein folding 63
Theories of reaction rates 63
Application to the protein folding without barriers 64
Physical model of DNA 65
2 MM of Enzyme Reactions 68
Method of Graphs 68
MM of Open Reactions with Substratum Inhibition 70
MM of Two-Substratum Reactions with Inhibition 71
On Multi-enzyme Systems 73
General notions 73
Simple MM of energetic metabolism 74
Effects of pH and T on the Reaction Rates 77
MM of Cell-to-Cell Interactions 79
3 MM of Kinetic Cellular Theory 82
Assumption1 82
Assumption2 82
Assumption 3 82
Assumption 4 84
Assumption5 84
Assumption 6 84
Assumption 7 85
Assumptions8 87
Assumption9 87
Assumption 10 88
Assumption 11 88
On Generalized Kinetic Models in Applied Sciences 89
4 Some Other MM 90
Delay Differential Equations 90
Mathematics in Population Biology 90
Some Problems and MM in Computational Biology and Bioinformatics 93
On the Theory Catastrophes 96
On the Respective Software 99
Part II: MM OF DEVELOPMENT 100
5 Base MM 102
Base Minimal MM 102
More General MM 106
Additional MM 109
On Comparison with Well-Known MM 110
Methods for Estimation of MM Input Data 110
On the Problem of MM Completeness 112
Methods of CAO de- and aggregation 113
Detailing the CAO structure 113
Case of self-development of the indices alpha- and Beta-types 113
Case of the additional "executive" subsystem (E) 115
Case of the additional "recreation" subsystem (R) 116
Further factorization of the indices alpha- and Beta-types 117
On introduction of dimensionless relations 117
On using extreme concepts 117
Statement of the Modeling Problems 119
6 Examples of CAO and Their MM 122
On Hyper-Cycle, Origin of Life and DNA Code 122
Dehydration Synthesis and Hydrolysis 124
ATP Synthesis 125
Biosynthesis 130
On Pattern Formation 132
MM of Regulatory Processes in Cells 134
Biological background 134
Proteins turn genes on or off 134
Differentiated cells has a particular pattern of genes 135
Complex protein control transcription 137
RNA adds a cap and poly-A tail and removes non-coding segments 137
Multiple mechanisms control gene expression 138
Gene expression and cell-to-cell signaling 140
Signal-transduction pathways 140
MM of turning genes on or off 141
MM of DNA transcription and mRNA translation 143
MM of signal-transduction pathways 143
MM of molecular pumps 144
7 MM of the Cell 146
On Cell Biology 146
Molecular constituents of cells 146
Structure and functions of cells 148
On cell metabolism 152
On control of gene expression 153
Summing up 154
Base MM of a Cell 155
MM of cell structure and functions 156
MM of a cell with regard to its given spatial structure 157
MM of developing cell 158
MM of cell bio-field 160
On the Protein Folding Problem 161
MM of Cell Cycle, II 163
Biological background 163
The respective MM 166
MM of Morphogenesis 167
Biological background 167
The respective MM 170
8 MM of the Immune Network 172
Introduction 172
Cells of the Immune System 173
On the immune system 173
Properties of the immune network cells 175
Continuous MM 178
On the Classical and the New Approach 184
Some Results of Investigation 187
Conclusion 188
9 MM of Some Other CAO 190
Function of Special Cells 190
MM of Stem Cells 191
MM of Thymus and Spleen 193
MM of the Immune and Blood System Cells 194
MM of Endothelial Cells 196
Conclusion 196
Part III: INTRODUCTION TO APPLICATIONS 198
10 AIDS 200
Introduction 200
Immune Status of an Organism 200
Viruses 202
Main properties of viruses 202
On retrovirases and HTV 202
Pattern of HIV infection 204
MM of Viruses 204
Base MM 204
MM of HIV 205
Qualitative investigation of MM of HTV 206
More complicated MM of HIV and its qualitative investigation 209
MM of HIV population 210
MM of HIV population inside of the host cell 210
MM of HTV free population 211
MM of AIDS 213
Qualitative investigation of MM of AIDS 217
On AIDS immune therapy 219
Conclusion 221
11 Cancers 222
On Study of a Cancer at the Molecular Level 222
MM of a Cancer Cell 224
Base MM of a cell at the start of proliferation 224
More complicated MM 225
MM of developing daughter cell 227
MM of a cancer cell bio-field 228
Qualitative Investigation of MM of a Cancer Cell 229
Investigation of MM (1)-(5) 229
Investigation of MM (6)-(8) 231
Investigation of MM (9), (10) 233
MM of a Cancer 234
Base MM of interaction of a cancer with host organism 237
Qualitative investigation of MM of acancer 238
Investigation of MM (37)-(43) 238
Investigation of MM (44), (45) 242
On Immune Therapy of Cancers 245
On Gene Therapy of Cancers 246
Conclusion 248
12 On Life Longevity Problems 250
Clock of Cellular Aging 250
Error accumulation theory 250
Programmed dearth theory 250
Winding back the clock 251
Modeling of Cell Longevity Problems 252
Modeling of error accumulation 252
Modeling of programmed death 253
Modeling of winding back the clock 254
On Modeling of Organism Longevity Problems 255
Certain Perspectives 256
13 On MM of a Disease 258
Introduction 258
Conception of the Norm and Pathology 259
Norm Restoration Problems 260
On Optimization Problems 261
On the Problems of Individualization of MM. 262
On MM of Doctor's Business 263
Appendix: MATHEMATICS OF DEVELOPMENT 266
14 Investigation of Equations 268
Examples of Analytic Solutions 268
Existence and Uniqueness of Solutions 270
Cases of Non-Uniqueness of Solutions 280
On Stability of Solutions 281
Cases of Oscillating, Wave, and Some Other Solutions 283
Asymptotic Properties of Solutions 287
Some Additions and Remarks 288
15 Investigation of Optimization Problems 290
Preliminary Concepts and Results 290
Examples of Analytical Solutions 295
Existence and Uniqueness of Solutions 297
On Stability of Solutions 301
On Investigation of Solution Structure 304
Case of Non-Uniqueness of Solutions 307
Asymptotic Properties of Solutions 309
Additions and Remarks 311
16 Numerical Methods and Software 314
On the Solution with the Pre-Assigned Accuracy 314
Second kind of Volterra equations 314
Inherent errors 314
Method errors 318
Round-off errors 323
Analysis of the total errors 324
First kind of Volterra equations 325
Two main approaches 325
Quasi-solution method 326
Estimate of continuity modules for reciprocal operators 327
On regularization for linear equations 328
Reduction to Volterra-Type Equations 330
Mathematical programming problems 330
Some other problems 333
On Optimal Numerical Methods 334
Approximation of functions 334
On the solution of other applied problems 335
On Algorithms and Software 338
Summary 340
Bibliography 342
Index 350
About the Authors 354