A primer on mathematical models in biology:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Philadelphia
Society for Industrial and Applied Mathematics
2013
|
| Series: | Other titles in applied mathematics
129 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XXIII, 424 S. Ill., graph. Darst. |
| ISBN: | 9781611972498 |
Staff View
MARC
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Record in the Search Index
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|---|---|
| adam_text | Contents
List of Figures
ix
List of Tables
xv
Acknowledgments
xvii
Preface
xix
1
Introduction
1
1.1
When to model
.............................. 2
1.2
What is a model
............................. 2
1.3
Formulation of a mathematical model
.................. 3
1.4
Solving the model equations
....................... 5
1.5
Drawing qualitative conclusions
..................... 6
1.6
Choosing parameters
.......................... 7
1.7
Robustness
................................ 7
1.8
Analysis of results
............................ 9
1.9
Successes and failures of modeling
................... 9
1.10
Final remarks
.............................. 11
1.11
Other sources of information on mathematical modeling in
biology
.................................. 12
2
Introduction to biochemical kinetics
13
2.1
Transitions between states at the molecular level
............ 13
2.2
Transitions between states at the population level
............ 16
2.3
The law of mass action
......................... 23
2.4
Enzyme kinetics: Saturating and cooperative reactions
......... 24
2.5
Simple models for polymer growth dynamics
.............. 27
2.6
Discussion
................................ 36
Exercises
..................................... 37
3
Review of linear differential equations
43
3.1
First-order differential equations
.................... 44
3.2
Linear second-order equations
...................... 50
3.3
Linear second-order equations with constant coefficients
........ 52
3.4
Asystemof two linear equations
.................... 55
v¡
Contents
3.5
Summary of
solutions
to differential equations in this chapter
.....61
Exercises
..................................... 61
4
Introduction to nondimensionalization and scaling
67
4.1
Simple examples
............................. 67
4.2
Rescaling the dimerization model
.................... 72
4.3
Other examples
............................. 76
Exercises
..................................... 79
5
Qualitative behavior of simple differential equation models
83
5.1
Revisiting the simple linear ODEs
................... 83
5.2
Stability of steady states
......................... 86
5.3
Qualitative analysis of models with bifurcations
............ 88
Exercises
..................................... 99
6
Developing a model from the ground up: Case study of the spread of an
infection
103
6.1
Deriving a model for the spread of an infection
............. 103
6.2
Dimensional analysis applied to the model
............... 105
6.3
Analysis
................................. 107
6.4
Interpretation of the results
.......................
Ill
Exercises
.....................................
Ill
7
Phase plane analysis
115
7.1
Phase plane trajectories
......................... 115
7.2
Nuliclines
................................ 117
7.3
Steady states
............................... 118
7.4
Stability of steady states
......................... 120
7.5
Classification of steady state behavior
.................. 122
7.6
Qualitative behavior and phase plane analysis
............. 124
7.7
Limit cycles, attractors, and domains of attraction
........... 132
7.8
Bifurcations continued
.......................... 135
Exercises
..................................... 138
8
Quasi steady state and enzyme-mediated biochemical kinetics
145
8.1
Warm-up example: Transitions between three states
.......... 145
8.2
Enzyme-substrate complex and the quasi steady state approximation
. 152
8.3
Conditions for validity of the QSS
................... 158
8.4
Overview and discussion of the QSS
.................. 165
8.5
Related applications
........................... 167
Exercises
..................................... 168
9
Multiple
subunit
enzymes and proteins: Cooperativity
173
9.1
Preliminary model for rapid dimerization
................ 173
9.2
Dimer binding that induces conformational change:
Model formulation
............................ 176
9.3
Consequences of a QSS assumption
................... 178
9.4
Ligand binding to dimer
......................... 179
Contents
vii
9.5
Results for binding and their
interpretation:
Cooperati
vity
....... 183
9.6
Cooperativity in enzyme action
..................... 184
9.7
Monod-Wyman-Changeaux (MWC) cooperativity
........... 185
9.8
Discussion
................................ 190
Exercises
..................................... 190
10
Dynamic behavior of
neuronal
membranes
195
10.1
Introduction
...............................195
10.2
An informal preview of the Hodgkin-Huxley model
..........198
10.3
Working towards the Hodgkin-Huxley model
.............202
10.4
The full Hodgkin-Huxley model
....................214
10.5
Comparison between theory and experiment
..............216
10.6
Bifurcations in the Hodgkin-Huxley model
...............219
10.7
Discussion
................................222
Exercises
.....................................222
11
Excitable systems and the FitzHugh-Nagumo equations
227
11.1
A simple excitable system
........................227
11.2
Phase plane analysis of the model
....................229
11.3
Piecing together the qualitative behavior
................234
11.4
Simulations of the FitzHugh-Nagumo model
..............236
11.5
Connection to
neuronal
excitation
....................241
11.6
Other systems with excitable behavior
.................246
Exercises
.....................................247
12
Biochemical modules
251
12.1
Simple biochemical circuits with useful functions
...........251
12.2
Genetic switches
.............................257
12.3
Models for the cell division cycle
....................262
Exercises
.....................................277
13
Discrete networks of genes and cells
283
13.1
Some simple automata networks
....................284
13.2
Boolean algebra
.............................294
13.3
Lysis-lysogeny in bacteriophage
λ
...................299
13.4
Cell cycle, revisited
...........................304
13.5
Discussion
................................306
Exercises
.....................................308
14
For further study
311
14.1
Nondimensionalizing a functional relationship
.............311
14.2
Scaled dimensionless variables
.....................312
14.3
Mathematical development of the Michaelis-Menten QSS via scaled
variables
.................................316
14.4
Cooperativity in the Monod-Wyman-Changeaux theory for
binding
..................................318
14.5
Ultrasensitivity in covalent protein modification
............320
14.6
Fraction of open channels, Hodgkin-Huxley Model
..........324
viii Contents
14.7
Asynchronous
Boolean
networks
(kinetic
logic)
............327
Exercises
.....................................333
15
Extended exercises and projects
337
Exercises
.....................................337
A The Taylor approximation and Taylor series
355
Exercises
.....................................361
В
Complex numbers
363
Exercises
.....................................365
С
A review of basic theory of electricity
367
C.
1
Amps, coulombs, and volts
.......................367
C.2 Ohm s law
................................370
C.3 Capacitance
...............................371
C.4 Circuits
..................................373
C.5 The Nernst equation
...........................375
Exercises
.....................................377
D
Proofs of Boolean algebra rules
379
Exercises
.....................................381
E
Appendix: XPP files for models in this book
385
E.I Biochemical reactions
..........................385
E.2 Linear differential equations
.......................386
E.3 Simple differential equations and bifurcations
.............387
E.4 Disease dynamics models
........................389
E.5 Phase plane analysis
...........................389
E.6 Chemical reactions and the QSS
.....................391
E.7
Neuronal
excitation and excitable systems
...............392
E.8 Biochemical modules
..........................394
E.9 Cell division cycle models
........................396
E.10 Boolean network models
........................401
E.ll Odell-Oster model of Exercise
15.7...................403
Bibliography
405
Index
417
This textbook introduces differential equations, biological applications, and
simulations and emphasizes molecular events (biochemistry and enzyme kinetics),
excitable systems (neural signals), and small protein and genetic circuits. A Primer
on Mathematical Models in Biology will appeal to readers because it
•
represents the unique perspective developed by the popular and highly respected
applied mathematician
Lee Segel
in a course he taught at the Weizmann
Institute of Science;
•
combines clear and useful mathematical methods with applications that
illustrate the power of such tools; and
•
includes many exercises in reasoning, modeling, and simulations.
This book is intended for upper-level undergraduates in mathematics, graduate
students in biology, and lower-level graduate students in mathematics who would
like exposure to biological applications.
Lee
A. Segel
(1932-2005)
was a Professor at the Weizmann Institute of Science,
Rehovot, Israel, where he served as Chairman of Applied Mathematics, Dean
of Mathematical Sciences, and Chairman of the Scientific Council. He was an
Ulam
Scholar at the Los Alamos National Laboratory, a Fellow of the American
Association for the Advancement of Science, and a member of the Santa Fe Institute,
where he continued his work on complex adaptive systems. He served as editor or
editorial board member of six journals.
Leah Edelstein-Keshet is a Professor in the Department of Mathematics at the
University of British Columbia, Vancouver, Canada. Her book Mathematical Models
in Biology was republished in SIAM s Classics in Applied Mathematics series.
For more information about SIAM books, journals,
conferences, memberships, or activities, contact:
Society for Industrial and Applied Mathematics
3600
Market Street, 6th Floor
Philadelphia, PA
19104-2688
USA
+1-215-382-9800 ·
Fax
+1-215-386-7999
siam@siam.org
·
wvjw.siam.org
-—
OTU9
|
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| id | DE-604.BV040935629 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T00:35:43Z |
| institution | BVB |
| isbn | 9781611972498 |
| language | English |
| lccn | 2012044617 |
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| spelling | Segel, Lee A. 1932-2005 Verfasser (DE-588)141246294 aut A primer on mathematical models in biology Lee A. Segel, Leah Edelstein-Keshet Philadelphia Society for Industrial and Applied Mathematics 2013 XXIII, 424 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Other titles in applied mathematics 129 Includes bibliographical references and index Mathematisches Modell (DE-588)4114528-8 gnd rswk-swf Biologie (DE-588)4006851-1 gnd rswk-swf Biologie (DE-588)4006851-1 s Mathematisches Modell (DE-588)4114528-8 s DE-604 Edelstein-Keshet, Leah Verfasser (DE-588)1046879049 aut Other titles in applied mathematics 129 (DE-604)BV023088396 129 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025914499&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025914499&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Segel, Lee A. 1932-2005 Edelstein-Keshet, Leah A primer on mathematical models in biology Other titles in applied mathematics Mathematisches Modell (DE-588)4114528-8 gnd Biologie (DE-588)4006851-1 gnd |
| subject_GND | (DE-588)4114528-8 (DE-588)4006851-1 |
| title | A primer on mathematical models in biology |
| title_auth | A primer on mathematical models in biology |
| title_exact_search | A primer on mathematical models in biology |
| title_full | A primer on mathematical models in biology Lee A. Segel, Leah Edelstein-Keshet |
| title_fullStr | A primer on mathematical models in biology Lee A. Segel, Leah Edelstein-Keshet |
| title_full_unstemmed | A primer on mathematical models in biology Lee A. Segel, Leah Edelstein-Keshet |
| title_short | A primer on mathematical models in biology |
| title_sort | a primer on mathematical models in biology |
| topic | Mathematisches Modell (DE-588)4114528-8 gnd Biologie (DE-588)4006851-1 gnd |
| topic_facet | Mathematisches Modell Biologie |
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