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9780387953403

Branching Processes in Biology

by ;
  • ISBN13:

    9780387953403

  • ISBN10:

    038795340X

  • Format: Hardcover
  • Copyright: 2002-08-01
  • Publisher: Springer Verlag
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Summary

This book provides a theoretical background of branching processes and discusses their biological applications. Branching processes are a well developed and powerful set of tools in the field of applied probability. The range of applications considered includes molecular biology, cellularbiology, human evolution and medicine. The branching processes discussed include Galton-Watson, Markov, Bellman-Harris, Multitype, and General Processes. As an aid to understanding specific examples, two introductory chapters, and two glossaries are included that provide background material in mathematics and in biology. The book will be of interest to scientists who work in quantitative modeling of biological systems, particularly probabilists, mathematical biologists, biostatisticians, cell biologists, molecular biologists, and bioinformaticians.The authors are a mathematician and cell biologist who have collaborated for more than a decade in the field of branching processes in biology.

Table of Contents

Preface vii
Guide to Applications, or How to Read This Book xvii
Motivating Examples and Other Preliminaries
1(18)
Some Motivating Examples
1(2)
Application: Polymerase Chain Reaction and Branching Processes
3(6)
Introduction to the mechanics of PCR
3(2)
Mathematical model
5(1)
Genealogical approach
5(2)
Statistical estimation of the mutation rate
7(1)
Mutagenic PCR and artificial evolution
8(1)
The Branching Property
9(2)
Probability Generating Functions and Analytical Methods
11(2)
Classifications of the Branching Processes
13(2)
Lifetime
13(1)
Type space
13(1)
Criticality
14(1)
Modeling with Branching Processes
15(4)
Biological Background
19(14)
Genomes: Changes in DNA and Chromosomes
19(5)
Genome
19(1)
DNA and genes
19(1)
Mutation
20(1)
Noncoding sequences of DNA
21(1)
Repeated sequences of DNA
21(1)
Gene amplification
21(1)
Chromosomes
22(1)
DNA replication
23(1)
Recombination
24(1)
Cells: Cell Cycle Kinetics and Cell Division
24(4)
Cells as the basic units of life
24(1)
Cell growth and division
25(2)
Cell cycle kinetics
27(1)
Cancer: Drug Resistance and Chemotherapy
28(1)
Cancer cells are immortal
28(1)
Tumor heterogeneity and instability
28(1)
Cell cycle and resistance to chemotherapy
29(1)
Mutations in cancer cells
29(1)
References
29(4)
Textbooks and monographs in biology
30(1)
Mathematical biology
30(1)
Arguments for mathematical modeling of biological phenomena
31(2)
The Galton-Watson Process
33(32)
Construction, Functional Equation, and Elementary Properties
34(3)
Backward equation
34(1)
Forward equation
35(1)
Moments
36(1)
The linear fractional case
36(1)
Application: Cell Cycle Model with Death and Quiescence
37(5)
The mathematical model
37(2)
Modeling biological data
39(3)
Extinction and Criticality
42(1)
Application: Complexity Threshold in the Evolution of Early Life
43(1)
Asymptotic properties
44(3)
Supercritical process
44(2)
Subcritical process
46(1)
Critical process
47(1)
Application: Gene Amplification
47(4)
Gene amplification and drug resistance
48(1)
Galton-Watson process model of gene amplification and deamplification
48(2)
Mathematical model of the loss of resistance
50(1)
Probabilities of gene amplification and deamplification from MTX data
51(1)
Application: Iterated Galton-Watson Process and Expansion of DNA Repeats
51(5)
Dynamics of DNA repeats in human pedigrees
52(1)
Definition of the process
52(2)
Example
54(1)
Properties
55(1)
Application: Galton-Watson Processes in a Random Environment and Macroevolution
56(3)
Reduced trees for subcritical GWBPRE
58(1)
Evolutionary interpretation
59(1)
Other Works and Applications
59(2)
Stochastic dependence
59(1)
Process state dependence
60(1)
Bisexual Galton-Watson process
60(1)
Age of the process
60(1)
Family trees and subtrees
61(1)
Problems
61(4)
The Age-Dependent Process: The Markov Case
65(22)
Differential Equation for the pgf and Its Elementary Properties
65(3)
Definition of the process
65(2)
Probability of extinction and moments
67(1)
Application: Clonal Resistance Theory of Cancer Cells
68(8)
Single-mutation case
69(4)
Two-mutation case
73(3)
Genealogies of Branching Processes
76(4)
``Near-critical'' processes
77(3)
Application: Estimation of the Age of the Mitochondrial Eve
80(3)
Population genetic model
80(2)
Numerical estimates
82(1)
Other Works and Applications
83(1)
Problems
84(3)
The Bellman-Harris Process
87(16)
Integral Equations for the pgf and Basic Properties
87(2)
Renewal Theory and Asymptotic of the Moments
89(2)
Basics of the renewal theory
89(1)
The moments
90(1)
Asymptotic Properties of the Process in the Supercritical Case
91(1)
Application: Analysis of the Stathmokinetic Experiment
91(6)
Age distributions
91(1)
The stathmokinetic experiment
92(1)
Model
93(3)
Estimation
96(1)
Other Works and Applications
97(4)
Cell populations
97(2)
Estimation of cell lifetimes
99(2)
Bifurcating autoregression
101(1)
Problems
101(2)
Multitype Processes
103(38)
Application: Two-Stage Mutations and Fluctuation Analysis
103(11)
Luria-Delbruck model
104(2)
The Markov branching process model
106(1)
The Galton-Watson process model
107(1)
The Galton-Watson process model with cell death
108(1)
Two-stage Galton-Watson process model
109(1)
The single-stage models versus data
110(2)
The two-stage model versus data
112(2)
The Positive Regular Case of the Multitype Galton-Watson Process
114(4)
Basics
115(2)
Positivity properties
117(1)
Asymptotic behavior in the supercritical case
117(1)
Probability of extinction
118(1)
Application: A Model of Two Cell Populations
118(1)
Application: Stochastic Model of the Cell Cycle with Chemotherapy
119(8)
Model of drug-perturbed stathmokinesis
120(3)
Model parameters
123(1)
Prediction of the effects of continuous exposure to the drug
124(1)
Results
124(2)
Discussion
126(1)
Application: Cell Surface Aggregation Phenomena
127(5)
Relationship between the Galton-Watson process and the aggregation process
129(1)
Progeny distributions
130(1)
Antigen size distribution on a cell surface
130(2)
Sampling Formulas for the Multitype Galton-Watson Process
132(2)
Formulas for mean and variance
133(1)
The Markov property
133(1)
Application: Deletions in Mitochondrial DNA
134(1)
Application: Polymerase Chain Reaction
135(2)
Other Works and Applications
137(4)
Hemopoiesis and clonal cell populations
137(1)
Gene amplification
138(1)
Modeling in varying environments
139(2)
Branching Processes with Infinitely Many Types
141(38)
Application: Stable Gene Amplification
141(6)
Assumptions
142(2)
Probability generating functions and expectations
144(2)
Model versus data
146(1)
Application: Mathematical Modeling of the Loss of Telomere Sequences
147(6)
Stochastic model
147(3)
Branching process
150(1)
Analysis in the Markov case
151(1)
Model versus data
152(1)
Further work on telomere modeling
153(1)
Branching Random Walk with an Absorbing Barrier
153(4)
Application: A Model of Unstable Gene Amplification
157(1)
Quasistationarity in a Branching Model of Division-Within-Division
158(4)
Definition of the process
158(2)
Quasistationarity
160(1)
Gene amplification
161(1)
Galton-Watson and Bellman-Harris Processes with Denumerably Many Types and Branching Random Walks
162(2)
Biological models with a denumerable infinity of types
163(1)
Application: Structured Cell Population Models
164(11)
A model of unequal division and growth regulation in cell colonies
165(4)
Cell cycle model with cell size control, unequal division of cells, and two cell types
169(6)
Application: Yule's Evolutionary Process
175(4)
References
179(18)
A Multivariate Probability Generating Functions 197(2)
B Probability Distributions for the Bellman-Harris Process 199(6)
Construction
199(3)
The families
199(1)
The number of objects at given time
200(1)
Probability measure
200(1)
The embedded Galton-Watson process and extinction probability
201(1)
Integral Equation
202(3)
Decomposition into subfamilies
202(1)
Generating functions
202(1)
Uniqueness of F(s, t) and finiteness of Z(t)
203(2)
C General Processes 205(10)
Introduction to the Jagers-Crump-Mode Process
205(5)
Definition of the general branching process
205(1)
Random characteristics and basic decomposition
206(1)
Expectations, Malthusian parameter, and exponential growth
207(1)
Abstract type spaces and composition of the process
208(2)
Application: Alexandersson's Cell Population Model Using a General Branching Process
210(5)
The model
211(1)
Existence of the stable birth size distribution
212(1)
Asymptotics of the cell model
213(2)
D Glossaries 215(12)
Biological Glossary for Mathematicians
215(4)
Mathematical Glossary for Biologists
219(8)
Index 227

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