Branching processes in biology:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York ; Heidelberg ; Dordrecht ; London
Springer
[2015]
|
| Ausgabe: | Second edition |
| Schriftenreihe: | Interdisciplinary applied mathematics
volume 19 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | xx, 280 Seiten Illustrationen, Diagramme |
| ISBN: | 9781493915583 9781493938193 |
Internformat
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Datensatz im Suchindex
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| adam_text | Contents 1 Motivating Examples and Other Preliminaries ........................................ 1.1 Some Motivating Examples.................................................................. 1.2 Application: Polymerase Chain Reaction and Branching Processes . 1.2.1 Introduction About the Mechanics of PCR ............................ 1.2.2 Mathematical Model.................................................................. 1.2.3 Genealogical Approach............................................................ 1.2.4 Statistical Estimation of the Mutation Rate............................ 1.2.5 Mutagenic PCR and Artificial Evolution................................ 1.3 The Branching Property........................................................................ 1.4 Probability Generating Functions and Analytical Methods................ 1.5 Classifications of the Branching Processes.......................................... 1.5.1 Lifetime..................................................................................... 1.5.2 TypeSpace ............................................................................... 1.5.3 Criticality................................................................................... 1.6 Modeling with Branching Processes.................................................... 2 Biological Background..................................................................................... 2.1 2.2 Genomes: Changes in DNA and Chromosomes.................................. 2.1.1
Genome..................................................................................... 2.1.2 DNA and Genes....................................................................... 2.1.3 Mutation ................................................................................... 2.1.4 Noncoding Sequences of DNA............................................... 2.1.5 Repeated Sequences of DNA................................................... 2.1.6 Gene Amplification................................................................... 2.1.7 Chromosomes........................................................................... 2.1.8 DNA Replication ..................................................................... 2.1.9 Recombination......................................................................... Cells: Cell Cycle Kinetics and Cell Division...................................... 2.2.1 Cells as the Basic Units of Life............................................... 2.2.2 Cell Growth, Division, and Death........................................... 2.2.3 Stem Cells................................................................................. 2.2.4 Cell Cycle Kinetics................................................................... 1 1 3 3 4 5 7 7 9 11 13 13 13 13 15 19 19 19 19 20 21 21 22 22 23 24 25 25 26 27 28 XV
Contents xvi 3 2.3 Cancer........................................................................................................ 2.3.1 Cancer Cell Populations Are Immortal..................................... 2.3.2 Tumor Heterogeneity andInstability ....................................... 2.3.3 Cell Cycle and Resistance to Chemotherapy.......................... 2.3.4 Mutations in Cancer Cells......................................................... 2.3.5 Tumor Progression..................................................................... 2.4 Population Genetics and Evolution....................................................... 2.4.1 Wright-Fisher Model andCoalescent-Based Models.............. 2.4.2 Human Immunodeficiency Virus.............................................. 2.5 References................................................................................................ 2.5.1 Textbooks and Monographs in Biology................................... 2.5.2 Mathematical Biology................................................................ 2.5.3 Arguments for Mathematical Modeling Biological Phenomena, with Examples....................................... 36 28 28 29 29 30 30 31 31 31 32 32 34 The Galton-Watson Process........................................................................ 3.1 Construction, Functional Equation, and Elementary Properties........ 3.1.1 Backward Equation .................................................................. 3.1.2 Forward Equation...................................................................... 3.1.3
Moments.......................................................................... 3.1.4 The Linear-Fractional Case...................................................... 3.2 Application: Cell Cycle Model with Death and Quiescence ............ 3.2.1 The Mathematical Model.......................................................... 3.2.2 Modeling Biological Data........................................................ 3.3 Extinction and Criticality..................................................................... 3.4 Application: Complexity Threshold in the Evolution of Early Life .. 3.5 Asymptotic Properties........................................................................... 3.5.1 Supercritical Process ................................................................. 3.5.2 Subcritical Process..................................................................... 3.5.3 Critical Process........................................................................... 3.6 Application: Cancer Mutations ............................................................ 3.6.1 Modeling Driver and Passenger Mutations ............................ 3.6.2 Distribution of Mutational Events in Various Phases of Tumor Growth.............................................................. 52 3.7 Application: Gene Amplification.......................................................... 3.7.1 Gene Amplification and Drug Resistance................................. 3.7.2 Galton-Watson Process Model of Gene Amplification and Deamplification........................................................... 54 3.7.3
Mathematical Model of the Loss of Resistance....................... 3.7.4 Probabilities of Gene Amplification and Deamplification from MTX Data........................................................... 57 3.8 Application: Iterated Galton-Watson Process and Expansion of DNA Repeats ........................................................................................ 3.8.1 Dynamics of DNA Repeats in Human Pedigrees .................. 3.8.2 Definition of the Process.......................................................... 37 37 38 39 40 40 41 41 42 45 46 47 47 49 50 51 51 53 54 56 58 58 59
xvii Contents 3.8.3 Example................................................................................. 3.8.4 Properties............................................................................... 3.9 Application: Galton-Watson Processes in Random Environment and Macroevolution................................................................ 64 3.9.1 Reduced Trees for Subcriticai GWBPRE........................... 3.9.2 Evolutionary Interpretation................................................. 3.10 Other Works and Applications............................................................. 3.10.1 Stochastic Dependence ....................................................... 3.10.2 Process State Dependence................................................... 3.10.3 Bisexual Galton-Watson Process....................................... 3.10.4 Age of the Process............................................................... 3.10.5 Family Trees and Subtrees................................................... 3.10.6 Model of Next Generation Sequencing ............................. 3.11 Problems ............................................................................................... 61 61 4 The Age-Dependent Process: Markov Case................................................. 71 71 71 73 74 75 79 83 83 84 4.1 4.2 4.3 4.4 Differential Equation for the pgf and its Elementary Properties.... 4.1.1 Definition of the Process...................................................... 4.1.2 Probability of Extinction and Moments.............................. Application: Clonal
Resistance Theory of Cancer Cells................... 4.2.1 Single-Mutation Case............................................................ 4.2.2 Two-Mutations Case ............................................................ Other Works and Applications............................................................ 4.3.1 Fluctuation Analysis.............................................................. Problems .............................................................................................. 5 The Bellman-Harris Process .......................................................................... 5.1 5.2 5.3 5.4 5.5 5.6 Integral Equations for the pgf and Basic Properties.......................... 5.1.1 Heuristic Derivations............................................................ Renewal Theory and Asymptotics of the Moments.......................... 5.2.1 Basics of the Renewal Theory.............................................. 5.2.2 The Moments........................................................................ Asymptotic Properties of the Process in the Supercritical Case .... Application: Analysis of the Stathmokinetic Experiment................. 5.4.1 Age Distributions.................................................................. 5.4.2 The Stathmokinetic Experiment.......................................... 5.4.3 Model...................................................................................... 5.4.4 Estimation.............................................................................. Other Works and
Applications............................................................. 5.5.1 Cell Populations.................................................................... 5.5.2 Cell Proliferation.................................................................. 5.5.3 Estimation of Cell Lifetimes................................................. 5.5.4 Bifurcating Autoregression.................................................. 5.5.5 Branching Processes and Cancer Therapy........................... Problems ............................................................................................... 64 65 66 66 66 67 67 68 68 69 87 87 87 88 89 90 91 92 92 92 94 96 97 97 99 101 103 103 104
xviii Contents 6 Multitype Processes........................................................................................ 6.1 Application: Mutations and Fluctuation Analysis .............................. 6.1.1 Luria-Delbrück Model............................................................. 6.1.2 The Markov Branching Process Model................................. 6.1.3 The Galton-Watson Process Model........................................ 6.1.4 The Galton-Watson Process Model with Cell Death........... 6.1.5 Two-Stage Galton-Watson Process Model ............................ 6.1.6 The Single-Stage Models Versus Data................................... 6.1.7 The Two-Stage Model Versus Data........................................ 6.1.8 Modified Median Estimator of Mutation Rates..................... 6.1.9 Modified Median Estimator Versus Data............................... 6.1.10 Recent Developments in Theory and Application of Fluctuation Analysis................................................ 126 6.2 The Positive Regular Case of the Multitype Galton-Watson Process............................................................... 129 6.2.1 Basics.......................................................................................... 6.2.2 Positivity Properties.................................................................. 6.2.3 Asymptotic Behavior in the Supercritical Case..................... 6.2.4 Probability of Extinction.......................................................... 6.3 Application: A Model of Two-Cell Populations.................................. 6.4
Application: Stochastic Model of the Cell Cycle with Chemotherapy....................................................................... 133 6.4.1 Model of Drug-Perturbed Stathmokinesis............................ 6.4.2 Model Parameters..................................................................... 6.4.3 Prediction of the Effects of Continuous Exposure to the Drug................................................................ 6.4.4 Results...................................................................................... 6.4.5 Discussion................................................................................ 6.5 Application: Cell Surface Aggregation Phenomena........................... 6.5.1 Relationship Between the Galton-Watson Process and the Aggregation Process................................................ 143 6.5.2 Progeny Distributions.............................................................. 6.5.3 Antigen Size Distribution on a Cell Surface ........................ 6.6 Sampling Formulae for Multitype Galton-Watson Process.............. 6.6.1 Formulae for Mean and Variance.......................................... 6.6.2 The Markov Property.............................................................. 6.7 Application: Deletions in Mitochondria! DNA.................................... 6.8 Application: Polymerase Chain Reaction............................................ 6.9 Other Works and Applications.............................................................. 6.9.1 Hemopoiesis and Clonal Cell Populations............................. 6.9.2
Gene Amplification ................................................................ 6.9.3 Modeling in Varying Environments ....................................... 6.9.4 Model of Ovarian Cancer Progression and Metastasis......... 6.9.5 HIV Modeling.......................................................................... 107 107 108 110 Ill 112 113 115 118 119 123 129 131 131 132 132 134 138 139 139 142 142 144 144 146 147 147 148 150 151 151 152 153 153 154
xix Contents 7 Branching Processes with Infinitely Many Types..................................... Galton-Watson and Bellman-Harris Processes with Denumerably Many Types and Branching Random Walks...................................... 7.2 Generalized Linear-Fractional Distributions and Their Applications.................................................................... 157 7.2.1 Introduction............................................................................ 7.2.2 Definitions and Basic Properties.......................................... 7.2.3 Applications in Branching Processes.................................. 7.3 Biological Models with Denumerable Infinity of Types.................. 7.4 Application: A Model of Unstable Gene Amplification .................. 7.5 Application: Stable Gene Amplification............................................ 7.5.1 Assumptions.......................................................................... 7.5.2 Pgf’s and Expectations........................................................... 7.5.3 Model Versus Data................................................................. 7.6 Application: Quasistationarity in a Branching Model of Division-Within-Division.................................................. 171 7.6.1 Definition of the Process....................................................... 7.6.2 Quasistationarity..................................................................... 7.6.3 Gene Amplification ............................................................... 7.7 Application: Mathematical Modeling of the Loss of
Telomere Sequences ................................................................................ 175 7.7.1 Stochastic Model ................................................................... 7.7.2 Branching Process................................................................. 7.7.3 Analysis in the Markov Case................................................. 7.7.4 Model Versus Data................................................................. 7.7.5 Further Work on Telomere Modeling................................... 7.8 Application: Structured Cell Population Models.............................. 7.8.1 A Model of Unequal Division and Growth Regulation in Cell Colonies..................................................... 7.8.2 Cell Cycle Model with Cell-Size Control, Unequal Division of Cells, and Two Cell Types............... 190 7.9 Application: Yule’s Evolutionary Process.......................................... 7.10 Application: Branching Process with Infinite-Allele Mutations .... 7.10.1 Proliferation of Alu Repeats................................................ 155 7.1 8 155 157 158 160 161 162 165 166 168 170 172 174 175 175 178 180 181 183 185 186 196 198 198 Genealogies of Branching Processes and Their Applications................. 207 8.1 8.2 8.3 Genealogies of Branching Processes................................................... “Near-Critical” Processes.................................................................... Application: Estimation of the Age of the Mitochondrial Eve......... 8.3.1 Population Genetic
Model.................................................... 8.3.2 Numerical Estimates ............................................................ 8.3.3 Robustness of Mitochondrial Eve........................................ 207 209 212 212 215 216
Contents XX Appendix A Multivariate Probability Generating Functions....................... 221 Appendix В Probability Distributions for the Bellman-Harris Process ... 223 B.l Construction............................................................................................ B.l.l The Families............................................................................... В. 1.2 The Number of Objects at Given Time..................................... B.1.3 Probability Measure................................................................... B.l.4 The Embedded Galton-Watson Process and Extinction Probability.................................................................... 225 B.2 Integral Equation.................................................................................... B.2.1 Decomposition into Subfamilies............................................... B.2.2 Generating Functions................................................................. B.2.3 Uniqueness of F(s, t) and Finiteness of Z(t).......................... 223 223 224 224 Appendix C General Processes.............................................................................. 229 229 229 230 C.l Introduction to the Jagers-Crump-Mode Process................................ C.l.l Definition of the General Branching Process.......................... С. 1.2 Random Characteristics and Basic Decomposition................ С. 1.3 Expectations, Malthusian Parameter and Exponential Growth .................................................... С. 1.4 Abstract Type Spaces and Composition of the
Process.......... C.2 Application: Alexandersson’s Cell Population Model Using a General Branching Process ......................................................... 235 C.2.1 The Model................................................................................... C.2.2 Existence of the Stable Birth Size Distribution...................... C.2.3 Asymptotics of the Cell Model................................................ 226 226 226 227 231 233 235 237 237 239 Biological Glossary for Mathematicians ...................................................... 239 Mathematical Glossary for Biologists............................................................ 245 Glossary ..................................................................................................................... References................................................................................................................... 253 Author Index ............................................................................................................. 273 Subject Index............................................................................................................ 277
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| indexdate | 2024-07-10T01:14:02Z |
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| spelling | Kimmel, Marek 1953- Verfasser (DE-588)1146381581 aut Branching processes in biology Marek Kimmel, David E. Axelrod Second edition New York ; Heidelberg ; Dordrecht ; London Springer [2015] xx, 280 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Interdisciplinary applied mathematics volume 19 Mathematisches Modell Biology Mathematical models Branching processes Verzweigung Mathematik (DE-588)4078889-1 gnd rswk-swf Biologie (DE-588)4006851-1 gnd rswk-swf Biologie (DE-588)4006851-1 s Verzweigung Mathematik (DE-588)4078889-1 s DE-604 Axelrod, David E. Verfasser aut Erscheint auch als Online-Ausgabe 978-1-4939-1559-0 Interdisciplinary applied mathematics volume 19 (DE-604)BV004216726 19 Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027593279&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Kimmel, Marek 1953- Axelrod, David E. Branching processes in biology Interdisciplinary applied mathematics Mathematisches Modell Biology Mathematical models Branching processes Verzweigung Mathematik (DE-588)4078889-1 gnd Biologie (DE-588)4006851-1 gnd |
| subject_GND | (DE-588)4078889-1 (DE-588)4006851-1 |
| title | Branching processes in biology |
| title_auth | Branching processes in biology |
| title_exact_search | Branching processes in biology |
| title_full | Branching processes in biology Marek Kimmel, David E. Axelrod |
| title_fullStr | Branching processes in biology Marek Kimmel, David E. Axelrod |
| title_full_unstemmed | Branching processes in biology Marek Kimmel, David E. Axelrod |
| title_short | Branching processes in biology |
| title_sort | branching processes in biology |
| topic | Mathematisches Modell Biology Mathematical models Branching processes Verzweigung Mathematik (DE-588)4078889-1 gnd Biologie (DE-588)4006851-1 gnd |
| topic_facet | Mathematisches Modell Biology Mathematical models Branching processes Verzweigung Mathematik Biologie |
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