Wilkinson D.J. Stochastic Modelling for Systems Biology
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Chapman
&
HalljCRC
Mathematical and Computational Biology Series
Stochastic Modelling
for
Systems Biology
Darren
James
Wilkinson
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Contents
1
Introduction to biological modelling
1
1.1
What is modelling?
1
1.2
Aims
of
modelling
2
1.3
Why is stochastic modelling necessary?
2
1.4
Chemical reactions
6
1.5
Modelling genetic and biochemical netWorks
8
1.6
Modelling higher-level systems
16
1.7
Exercises
17
1.8
Further reading
17
2
Representation
of
biochemical networks
19
2.1
Coupled chemical reactions
19
2.2
Graphical representations
19
2.3
Petri nets
21
2.4
Systems Biology Markup Language (SBML)
3C
2.5
SBML-shorthand
36
2.6
Exercises
42
2.7
Further reading
43
3
Probability models
45
3.1
Probability
45
3.2
Discrete probability models
56
3.3
,The
discrete uniform distribution
64
3.4
The binomial distribution
64
3.5
The geometric distribution
65
3.6
The Poisson distribution
67
3.7 Continuous probability models
70
3.8 The uniform distribution
75
3.9
The exponential distribution
77
3.10 The normal/Gaussian distribution
82
3.11 The gamma distribution
86
3.12 Exercises
88
3.13 Further reading
89
4
Stochastic simulation
9T.
4.1
Introduction
91
4.2 Monte-Carlo integration
91
4.3
Uniform random number generation 92
4.4
Transformation methods
93
4.5
Lookup methods 97
4.6
Rejection samplers 98
4.7
The Poisson process
101
4.8
Using the statistical programming language, R
101
4.9
Analysis
of
simulation output 106
4.10
Exercises
107
4.11 Further reading
108
~~-
--
5
Markov
processes
109
5.1
Introduction
109
5.2
Finite discrete time Markov chains
109
5.3
Markov chains with continuous state space
115
5.4
Markov chains in continuous time
121
5.5
Diffusion processes
133
5.6
Exercises
135
5.7 Further reading
137
6
Chemical and biochemical kinetics 139
6.1
Classical continuous deterministic chemical kinetics
139
6.2 Molecular approach to kinetics
145
6.3
Mass-action stochastic kinetics 147
6.4 The Gillespie algorithm 149
6.5
Stochastic Petri nets (SPNs)
150
6.6
Rate constant conversion
153
6.7
The Master equation 157
6.8
Software for simulating stochastic kinetic networks 160
6.9
Exercises
161
6.10 Further reading
161
7 Case studies
163
7.1
Introduction
163
7.2
Dimerisation kinetics
163
7.3 Michaelis-Menten enzyme kinetics
7.4 An auto-regulatory genetic network
7.5 The
lac operon
7.6 Exercises
7. 7 Further reading
8 Beyond the Gillespie algorithm
8.1
Introduction
8.2 Exact simulation methods
8.3 Approximate simulation strategies
8.4 Hybrid simulation strategies
8.5 Exercises
8.6 Further reading
9 Bayesian inference and MCMC
9.1
Likelihood and Bayesian inference
9.2 The Gibbs sampler
9.3 The Metropolis-Hastings algorithm
9.4 Hybrid MCMC schemes
9.5 Exercises
9.6 Further reading
10 Inference for stochastic kinetic models
10.1 Introduction
10.2 Inference given complete data
10.3 Discrete-time observations
of
the system state
10.4 Diffusion approximations for inference
10.5 Network inference
10.6 Exercises
10.7 Further reading
11
Conclusions
A SBML Models
A.1
Auto-regulatory network
A.2 Lotka-Volterra reaction system
A.3 Dimerisation-kinetics model
References
Index
168
172
176
178
179
181
181
181
186
190
196
196
197
197
202
212
216
216
217
219
219
220
223
230
234
234
235
237
239
239
242
243
247
251
List
of
tables
2.1
The auto-regulatory system displayed in tabular (matrix) form (zero
stoichiometries omitted for clarity) 24
2.2 Table representing the overall effect
of
each transition (reaction) on
the marking (state)
of
the network
25