Stochastic modelling for systems biology:
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| Main Author: | |
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| Format: | Book |
| Language: | English |
| Published: |
Boca Raton [u.a.]
Chapman & Hall
2006
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| Series: | Mathematical and computational biology series
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | 254 S. Ill., graph. Darst. |
| ISBN: | 1584885408 9781584885405 |
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MARC
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| 100 | 1 | |a Wilkinson, Darren James |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Stochastic modelling for systems biology |c Darren James Wilkinson |
| 264 | 1 | |a Boca Raton [u.a.] |b Chapman & Hall |c 2006 | |
| 300 | |a 254 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Mathematical and computational biology series | |
| 650 | 7 | |a Sistemas biológicos - Modelos matemáticos |2 lemb | |
| 650 | 4 | |a Stochastisches Modell - Systembiologie | |
| 650 | 4 | |a Mathematisches Modell | |
| 650 | 4 | |a Biological systems |x Mathematical models | |
| 650 | 4 | |a Kinetics | |
| 650 | 4 | |a Models, Biological | |
| 650 | 4 | |a Models, Statistical | |
| 650 | 4 | |a Stochastic Processes | |
| 650 | 4 | |a Systems Biology |x methods | |
| 650 | 4 | |a Systems biology | |
| 650 | 0 | 7 | |a Systembiologie |0 (DE-588)4809615-5 |2 gnd |9 rswk-swf |
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Record in the Search Index
| _version_ | 1804135485210099712 |
|---|---|
| adam_text | 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 Petrinets 21
2.4 Systems Biology Markup Language (SBML) 31
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 91
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 168
7.4 An auto regulatory genetic network 172
7.5 The/acoperon 176
7.6 Exercises 178
7.7 Further reading 179
8 Beyond the Gillespie algorithm 181
8.1 Introduction 181
8.2 Exact simulation methods 181
8.3 Approximate simulation strategies 186
8.4 Hybrid simulation strategies 190
8.5 Exercises 196
8.6 Further reading 196
9 Bayesian inference and MCMC 197
9.1 Likelihood and Bayesian inference 197
9.2 The Gibbs sampler 202
9.3 The Metropolis Hastings algorithm 212
9.4 Hybrid MCMC schemes 216
9.5 Exercises 216
9.6 Further reading 217
10 Inference for stochastic kinetic models 219
10.1 Introduction 219
10.2 Inference given complete data 220
10.3 Discrete time observations of the system state 223
10.4 Diffusion approximations for inference 230
10.5 Network inference 234
10.6 Exercises 234
10.7 Further reading 235
11 Conclusions 237
A SBML Models 239
A. 1 Auto regulatory network 239
A.2 Lotka Volterra reaction system 242
A.3 Dimerisation kinetics model 243
References 247
Index 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
List of figures
1.1 Five deterministic solutions of the linear birth death process for
values of A // given in the legend (x0 = 50) 4
1.2 Five realisations of a stochastic linear birth death process together
with the continuous deterministic solution (xo = 50, A = 3, yu = 4) 5
1.3 Five realisations of a stochastic linear birth death process together
with the continuous deterministic solution for four different (A, /x)
combinations, each with A — /x = —1 and xq = 50 6
1.4 Transcription of a single prokaryotic gene 9
1.5 A simple illustrative model of the transcription process in eukaryotic
cells 11
1.6 A simple prokaryotic transcription repression mechanism 12
1.7 A very simple model of a prokaryotic auto regulatory gene network.
Here dimers of a protein P coded for by a gene g repress their own
transcription by binding to a regulatory region q upstream of g and
downstream of the promoter p. 14
1.8 Key mechanisms involving the lac operon. Here an inhibitor protein
7 can repress transcription of the lac operon by binding to the
operator o. However, in the presence of lactose, the inhibitor
preferentially binds to it, and in the bound state can no longer bind
to the operator, thereby allowing transcription to proceed. 15
2.1 A simple graph of the auto regulatory reaction network 20
2.2 A simple digraph 21
2.3 A Petri net for the auto regulatory reaction network 22
2.4 A Petri net labelled with tokens 23
2.5 A Petri net with new numbers of tokens after reactions have taken
place 23
3.1 CDF for the sum of a pair of fair dice 5 8
3.2 PMF and CDF for a B(8,0.7) distribution 65
3.3 PMF and CDF for a Po(5) distribution 69
3.4 PDF and CDF for a f/(0,1) distribution 76
3.5 PDF and CDF for an Exp{l) distribution 78
3.6 PDF and CDF for a ^(0,1) distribution 83
3.7 Graph of T(a;) for small positive values of x 87
3.8 PDF and CDF for a T(3,1) distribution 88
4.1 Density of Y = exp(X), where X ~ 7V(2,1). 108
5.1 An R function to simulate a sample path of length n from a Markov
chain with transition matrix P and initial distribution pi0 115
5.2 A sample R session to simulate and analyse the sample path of a
finite Markov chain. The last two commands show how to use R to
directly compute the stationary distribution of a finite Markov chain. 116
5.3 SBML shorthand for the simple gene activation process with
a = 0.5 and [1 = 1 124
5.4 A simulated realisation of the simple gene activation process with
a = 0.5and/3=l 127
5.5 An R function to simulate a sample path with n events from a
continuous time Markov chain with transition rate matrix Q and
initial distribution p i 0 128
5.6 SBML shorthand for the immigration death process with A = 1 and
/i = 0.1 128
5.7 A single realisation of the immigration death process with param¬
eters A = 1 and /z = 0.1, initialised at X(0) = 0. Note that the
stationary distribution of this process is Poisson with mean 10. 130
5.8 R function for discrete event simulation of the immigration death
process 131
5.9 R function for simulation of a diffusion process using the Euler
method 134
5.10 A single realisation of the diffusion approximation to the immigration
death process with parameters A = 1 and /i = 0.1, initialised at
X(0)=0 135
5.11 R code for simulating the diffusion approximation to the immigration
death process 136
6.1 Lotka Volterra dynamics for [Yi](0) = 4, [Y2](0) = 10, ki =
1, k2 = 0.1, k3 = 0.1. Note that the equilibrium solution for this
combination of rate parameters is [Yi] = 1, [Y2] — 10. 141
6.2 Lotka Volterra dynamics in phase space for rate parameters fci =
1; k2 = 0.1, kz = 0.1 The dynamics for the initial condition
[Yi](0) = 4, [Y2}(0) = 10 are shown as the bold orbit. Note that
the system moves around this orbit in an anti clockwise direction.
Orbits for other initial conditions are shown as dotted curves. Note
that the equilibrium solution for this combination of rate parameters
is [Yi] = 1, [Y2] 10. 142
6.3 Dimerisation kinetics for [P](0) = 1, [P2}(0) = 0, kx = 1, k2 =
0.5. This combination of parameters gives Keq — 2, c = 1, and
hence equilibrium concentrations of [P] = 0.39, [P2] = 0.30. 143
6.4 An R function to numerically integrate a system of coupled ODEs
using a simple first order Euler method 145
6.5 An R function to implement the Gillespie algorithm for a stochastic
Petri net representation of a coupled chemical reaction system 150
6.6 Some R code to set up the LV system as a SPN and then simulate it
using the Gillespie algorithm. The state of the system is initialised
to 50 prey and 100 predators, and the stochastic rate constants are
c= (1,0.005,0.6) . 152
6.7 A single realisation of a stochastic LV process. The state of the
system is initialised to 50 prey and 100 predators, and the stochastic
rate constants are c = (1,0.005,0.6) . 152
6.8 A single realisation of a stochastic LV process in phase space. The
state of the system is initialised to 50 prey and 100 predators, and
the stochastic rate constants are c = (1,0.005,0.6) . 153
6.9 SBML shorthand for the stochastic Lotka Volterra system 154
6.10 An R function to discretise the output of gillespie onto a regular
grid of time points. The result is returned as an R multivariate time
series object. 154
6.11 An R function to implement the Gillespie algorithm for a SPN,
recording the state on a regular grid of time points. The result is
returned as an R multivariate time series object. 155
7.1 SBML shorthand for the dimerisation kinetics model (continuous
deterministic version) 164
7.2 Left: Simulated continuous deterministic dynamics of the dimerisa¬
tion kinetics model. Right: A simulated realisation of the discrete
stochastic dynamics of the dimerisation kinetics model. 164
7.3 SBML shorthand for the dimerisation kinetics model (discrete
stochastic version) 165
7.4 R code to build an SPN object representing the dimerisation kinetics
model 166
7.5 Left: A simulated realisation of the discrete stochastic dynamics of
the dimerisation kinetics model plotted on a concentration scale.
Right: The trajectories for levels of P from 20 runs overlaid. 167
7.6 Left: The mean trajectory of P together with some approximate
(point wise) confidence bounds based on 1,000 runs of the
simulator. Right: Density histogram of the simulated realisations of
P at time t = 10 based on 10,000 runs, giving an estimate of the
PMFforP(lO). 167
7.7 SBML shorthand for the Michaelis Menten kinetics model (contin¬
uous deterministic version) 169
7.8 Left: Simulated continuous deterministic dynamics of the Michaelis
Menten kinetics model. Right: Simulated continuous deterministic
dynamics of the Michaelis Menten kinetics model based on the
two dimensional representation. 170
7.9 SBML shorthand for the Michaelis Menten kinetics model (discrete
stochastic version) 171
7.10 Left: A simulated realisation of the discrete stochastic dynamics of
the Michaelis Menten kinetics model. Right: A simulated realisation
of the discrete stochastic dynamics of the reduced dimension
Michaelis Menten kinetics model. 172
7.11 SBML shorthand for the reduced dimension Michaelis Menten
kinetics model (discrete stochastic version) 172
7.12 Left: A simulated realisation of the discrete stochastic dynamics
of the prokaryotic genetic auto regulatory network model, for a
period of 5,000 seconds. Right: A close up on the first period of 250
seconds of the left plot. 174
7.13 Left: Close up showing the time evolution of the number of
molecules of P over a 10 second period. Right: Empirical PMP
for the number of molecules of P at time t = 10 seconds, based on
10,000 runs. 174
7.14 Left: Empirical PMF for the number of molecules of P at time
t = 10 seconds when k2 is changed from 0.01 to 0.02, based
on 10,000 runs. Right: Empirical PMF for the prior predictive
uncertainty regarding the observed value of P at time t = 10 based
on the prior distribution k2 ~ f/(0.005,0.03). 176
7.15 SBML shorthand for the /oc operon model (discrete stochastic
version) 177
7.16 A simulated realisation of the discrete stochastic dynamics of the
/ac operon model for a period of 50,000 seconds. An intervention is
applied at time t = 20,000, when 10,000 molecules of lactose are
added to the cell. 178
8.1 An R function to implement the first reaction method for a stochastic
Petri net representation of a coupled chemical reaction system It
is to be used in the same way as the gillespie function from
Figure 6.5. 183
8.2 An R function to implement the Poisson timestep method for a
stochastic Petri net representation of a coupled chemical reaction
system. It is to be used in the same way as the gillespied func¬
tion from Figure 6.11. 187
8.3 An R function to integrate the CLE using an Euler method for a
stochastic Petri net representation of a coupled chemical reaction
system. It is to be used in the same way as the pts function from
Figure 8.2. 190
9.1 Plot showing the prior and posterior for the Poisson rate example.
Note how the prior is modified to give a posterior more consistent
with the data (which has a sample mean of 3). 200
9.2 An R function to implement a Gibbs sampler for the simple normal
random sample model. Example code for using this function is given
in Figure 9.3. 207
9.3 Example R code illustrating the use of the function normgibbs
from Figure 9.2. The plots generated by running this code are
shown in Figure 9.4. In this example the prior took the form
fj, ~ N(10,100), t ~ T(3,11), and the sufficient statistics for the
data were n = 15, x = 25, s2 = 20. The sampler was run for
11,000 iterations with the first 1,000 discarded as burn in, and the
remaining 10,000 iterations used for the main monitoring run. 208
9.4 Figure showing the Gibbs sampler output resulting from running the
example code in Figure 9.3. The top two plots give an indication of
the bivariate posterior distribution. The second row shows trace plots
of the marginal distributions of interest, indicating a rapidly mixing
MCMC algorithm. The final row shows empirical marginal posterior
distributions for the parameters of interest. 209
9.5 An R function to implement a Metropolis sampler for a standard
normal random quantity based on U(—a, a) innovations. So,
metrop(10000,l) will execute a run of length 10,000 with an a
of 1. This a is close to optimal. Running with a = 0.1 gives a chain
that is too cold, and a = 100 gives a chain that is too hot. 215
|
| adam_txt |
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 Petrinets 21
2.4 Systems Biology Markup Language (SBML) 31
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 91
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 168
7.4 An auto regulatory genetic network 172
7.5 The/acoperon 176
7.6 Exercises 178
7.7 Further reading 179
8 Beyond the Gillespie algorithm 181
8.1 Introduction 181
8.2 Exact simulation methods 181
8.3 Approximate simulation strategies 186
8.4 Hybrid simulation strategies 190
8.5 Exercises 196
8.6 Further reading 196
9 Bayesian inference and MCMC 197
9.1 Likelihood and Bayesian inference 197
9.2 The Gibbs sampler 202
9.3 The Metropolis Hastings algorithm 212
9.4 Hybrid MCMC schemes 216
9.5 Exercises 216
9.6 Further reading 217
10 Inference for stochastic kinetic models 219
10.1 Introduction 219
10.2 Inference given complete data 220
10.3 Discrete time observations of the system state 223
10.4 Diffusion approximations for inference 230
10.5 Network inference 234
10.6 Exercises 234
10.7 Further reading 235
11 Conclusions 237
A SBML Models 239
A. 1 Auto regulatory network 239
A.2 Lotka Volterra reaction system 242
A.3 Dimerisation kinetics model 243
References 247
Index 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
List of figures
1.1 Five deterministic solutions of the linear birth death process for
values of A // given in the legend (x0 = 50) 4
1.2 Five realisations of a stochastic linear birth death process together
with the continuous deterministic solution (xo = 50, A = 3, yu = 4) 5
1.3 Five realisations of a stochastic linear birth death process together
with the continuous deterministic solution for four different (A, /x)
combinations, each with A — /x = —1 and xq = 50 6
1.4 Transcription of a single prokaryotic gene 9
1.5 A simple illustrative model of the transcription process in eukaryotic
cells 11
1.6 A simple prokaryotic transcription repression mechanism 12
1.7 A very simple model of a prokaryotic auto regulatory gene network.
Here dimers of a protein P coded for by a gene g repress their own
transcription by binding to a regulatory region q upstream of g and
downstream of the promoter p. 14
1.8 Key mechanisms involving the lac operon. Here an inhibitor protein
7 can repress transcription of the lac operon by binding to the
operator o. However, in the presence of lactose, the inhibitor
preferentially binds to it, and in the bound state can no longer bind
to the operator, thereby allowing transcription to proceed. 15
2.1 A simple graph of the auto regulatory reaction network 20
2.2 A simple digraph 21
2.3 A Petri net for the auto regulatory reaction network 22
2.4 A Petri net labelled with tokens 23
2.5 A Petri net with new numbers of tokens after reactions have taken
place 23
3.1 CDF for the sum of a pair of fair dice 5 8
3.2 PMF and CDF for a B(8,0.7) distribution 65
3.3 PMF and CDF for a Po(5) distribution 69
3.4 PDF and CDF for a f/(0,1) distribution 76
3.5 PDF and CDF for an Exp{l) distribution 78
3.6 PDF and CDF for a ^(0,1) distribution 83
3.7 Graph of T(a;) for small positive values of x 87
3.8 PDF and CDF for a T(3,1) distribution 88
4.1 Density of Y = exp(X), where X ~ 7V(2,1). 108
5.1 An R function to simulate a sample path of length n from a Markov
chain with transition matrix P and initial distribution pi0 115
5.2 A sample R session to simulate and analyse the sample path of a
finite Markov chain. The last two commands show how to use R to
directly compute the stationary distribution of a finite Markov chain. 116
5.3 SBML shorthand for the simple gene activation process with
a = 0.5 and [1 = 1 124
5.4 A simulated realisation of the simple gene activation process with
a = 0.5and/3=l 127
5.5 An R function to simulate a sample path with n events from a
continuous time Markov chain with transition rate matrix Q and
initial distribution p i 0 128
5.6 SBML shorthand for the immigration death process with A = 1 and
/i = 0.1 128
5.7 A single realisation of the immigration death process with param¬
eters A = 1 and /z = 0.1, initialised at X(0) = 0. Note that the
stationary distribution of this process is Poisson with mean 10. 130
5.8 R function for discrete event simulation of the immigration death
process 131
5.9 R function for simulation of a diffusion process using the Euler
method 134
5.10 A single realisation of the diffusion approximation to the immigration
death process with parameters A = 1 and /i = 0.1, initialised at
X(0)=0 135
5.11 R code for simulating the diffusion approximation to the immigration
death process 136
6.1 Lotka Volterra dynamics for [Yi](0) = 4, [Y2](0) = 10, ki =
1, k2 = 0.1, k3 = 0.1. Note that the equilibrium solution for this
combination of rate parameters is [Yi] = 1, [Y2] — 10. 141
6.2 Lotka Volterra dynamics in phase space for rate parameters fci =
1; k2 = 0.1, kz = 0.1 The dynamics for the initial condition
[Yi](0) = 4, [Y2}(0) = 10 are shown as the bold orbit. Note that
the system moves around this orbit in an anti clockwise direction.
Orbits for other initial conditions are shown as dotted curves. Note
that the equilibrium solution for this combination of rate parameters
is [Yi] = 1, [Y2] 10. 142
6.3 Dimerisation kinetics for [P](0) = 1, [P2}(0) = 0, kx = 1, k2 =
0.5. This combination of parameters gives Keq — 2, c = 1, and
hence equilibrium concentrations of [P] = 0.39, [P2] = 0.30. 143
6.4 An R function to numerically integrate a system of coupled ODEs
using a simple first order Euler method 145
6.5 An R function to implement the Gillespie algorithm for a stochastic
Petri net representation of a coupled chemical reaction system 150
6.6 Some R code to set up the LV system as a SPN and then simulate it
using the Gillespie algorithm. The state of the system is initialised
to 50 prey and 100 predators, and the stochastic rate constants are
c= (1,0.005,0.6)'. 152
6.7 A single realisation of a stochastic LV process. The state of the
system is initialised to 50 prey and 100 predators, and the stochastic
rate constants are c = (1,0.005,0.6)'. 152
6.8 A single realisation of a stochastic LV process in phase space. The
state of the system is initialised to 50 prey and 100 predators, and
the stochastic rate constants are c = (1,0.005,0.6)'. 153
6.9 SBML shorthand for the stochastic Lotka Volterra system 154
6.10 An R function to discretise the output of gillespie onto a regular
grid of time points. The result is returned as an R multivariate time
series object. 154
6.11 An R function to implement the Gillespie algorithm for a SPN,
recording the state on a regular grid of time points. The result is
returned as an R multivariate time series object. 155
7.1 SBML shorthand for the dimerisation kinetics model (continuous
deterministic version) 164
7.2 Left: Simulated continuous deterministic dynamics of the dimerisa¬
tion kinetics model. Right: A simulated realisation of the discrete
stochastic dynamics of the dimerisation kinetics model. 164
7.3 SBML shorthand for the dimerisation kinetics model (discrete
stochastic version) 165
7.4 R code to build an SPN object representing the dimerisation kinetics
model 166
7.5 Left: A simulated realisation of the discrete stochastic dynamics of
the dimerisation kinetics model plotted on a concentration scale.
Right: The trajectories for levels of P from 20 runs overlaid. 167
7.6 Left: The mean trajectory of P together with some approximate
(point wise) "confidence bounds" based on 1,000 runs of the
simulator. Right: Density histogram of the simulated realisations of
P at time t = 10 based on 10,000 runs, giving an estimate of the
PMFforP(lO). 167
7.7 SBML shorthand for the Michaelis Menten kinetics model (contin¬
uous deterministic version) 169
7.8 Left: Simulated continuous deterministic dynamics of the Michaelis
Menten kinetics model. Right: Simulated continuous deterministic
dynamics of the Michaelis Menten kinetics model based on the
two dimensional representation. 170
7.9 SBML shorthand for the Michaelis Menten kinetics model (discrete
stochastic version) 171
7.10 Left: A simulated realisation of the discrete stochastic dynamics of
the Michaelis Menten kinetics model. Right: A simulated realisation
of the discrete stochastic dynamics of the reduced dimension
Michaelis Menten kinetics model. 172
7.11 SBML shorthand for the reduced dimension Michaelis Menten
kinetics model (discrete stochastic version) 172
7.12 Left: A simulated realisation of the discrete stochastic dynamics
of the prokaryotic genetic auto regulatory network model, for a
period of 5,000 seconds. Right: A close up on the first period of 250
seconds of the left plot. 174
7.13 Left: Close up showing the time evolution of the number of
molecules of P over a 10 second period. Right: Empirical PMP
for the number of molecules of P at time t = 10 seconds, based on
10,000 runs. 174
7.14 Left: Empirical PMF for the number of molecules of P at time
t = 10 seconds when k2 is changed from 0.01 to 0.02, based
on 10,000 runs. Right: Empirical PMF for the prior predictive
uncertainty regarding the observed value of P at time t = 10 based
on the prior distribution k2 ~ f/(0.005,0.03). 176
7.15 SBML shorthand for the /oc operon model (discrete stochastic
version) 177
7.16 A simulated realisation of the discrete stochastic dynamics of the
/ac operon model for a period of 50,000 seconds. An intervention is
applied at time t = 20,000, when 10,000 molecules of lactose are
added to the cell. 178
8.1 An R function to implement the first reaction method for a stochastic
Petri net representation of a coupled chemical reaction system It
is to be used in the same way as the gillespie function from
Figure 6.5. 183
8.2 An R function to implement the Poisson timestep method for a
stochastic Petri net representation of a coupled chemical reaction
system. It is to be used in the same way as the gillespied func¬
tion from Figure 6.11. 187
8.3 An R function to integrate the CLE using an Euler method for a
stochastic Petri net representation of a coupled chemical reaction
system. It is to be used in the same way as the pts function from
Figure 8.2. 190
9.1 Plot showing the prior and posterior for the Poisson rate example.
Note how the prior is modified to give a posterior more consistent
with the data (which has a sample mean of 3). 200
9.2 An R function to implement a Gibbs sampler for the simple normal
random sample model. Example code for using this function is given
in Figure 9.3. 207
9.3 Example R code illustrating the use of the function normgibbs
from Figure 9.2. The plots generated by running this code are
shown in Figure 9.4. In this example the prior took the form
fj, ~ N(10,100), t ~ T(3,11), and the sufficient statistics for the
data were n = 15, x = 25, s2 = 20. The sampler was run for
11,000 iterations with the first 1,000 discarded as burn in, and the
remaining 10,000 iterations used for the main monitoring run. 208
9.4 Figure showing the Gibbs sampler output resulting from running the
example code in Figure 9.3. The top two plots give an indication of
the bivariate posterior distribution. The second row shows trace plots
of the marginal distributions of interest, indicating a rapidly mixing
MCMC algorithm. The final row shows empirical marginal posterior
distributions for the parameters of interest. 209
9.5 An R function to implement a Metropolis sampler for a standard
normal random quantity based on U(—a, a) innovations. So,
metrop(10000,l) will execute a run of length 10,000 with an a
of 1. This a is close to optimal. Running with a = 0.1 gives a chain
that is too cold, and a = 100 gives a chain that is too hot. 215 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Wilkinson, Darren James |
| author_facet | Wilkinson, Darren James |
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| callnumber-subject | QH - Natural History and Biology |
| classification_rvk | QH 252 WC 7000 WD 9200 |
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| ctrlnum | (OCoLC)255221355 (DE-599)BVBBV021669801 |
| dewey-full | 572.8 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 572 - Biochemistry |
| dewey-raw | 572.8 |
| dewey-search | 572.8 |
| dewey-sort | 3572.8 |
| dewey-tens | 570 - Biology |
| discipline | Biologie Wirtschaftswissenschaften |
| discipline_str_mv | Biologie Wirtschaftswissenschaften |
| format | Book |
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| id | DE-604.BV021669801 |
| illustrated | Illustrated |
| index_date | 2024-07-02T15:08:27Z |
| indexdate | 2024-07-09T20:41:15Z |
| institution | BVB |
| isbn | 1584885408 9781584885405 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-014884189 |
| oclc_num | 255221355 |
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| owner_facet | DE-20 DE-91G DE-BY-TUM DE-19 DE-BY-UBM DE-526 DE-634 DE-83 DE-11 DE-188 DE-29T |
| physical | 254 S. Ill., graph. Darst. |
| publishDate | 2006 |
| publishDateSearch | 2006 |
| publishDateSort | 2006 |
| publisher | Chapman & Hall |
| record_format | marc |
| series2 | Mathematical and computational biology series |
| spelling | Wilkinson, Darren James Verfasser aut Stochastic modelling for systems biology Darren James Wilkinson Boca Raton [u.a.] Chapman & Hall 2006 254 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Mathematical and computational biology series Sistemas biológicos - Modelos matemáticos lemb Stochastisches Modell - Systembiologie Mathematisches Modell Biological systems Mathematical models Kinetics Models, Biological Models, Statistical Stochastic Processes Systems Biology methods Systems biology Systembiologie (DE-588)4809615-5 gnd rswk-swf Stochastisches Modell (DE-588)4057633-4 gnd rswk-swf Stochastisches Modell (DE-588)4057633-4 s Systembiologie (DE-588)4809615-5 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014884189&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Wilkinson, Darren James Stochastic modelling for systems biology Sistemas biológicos - Modelos matemáticos lemb Stochastisches Modell - Systembiologie Mathematisches Modell Biological systems Mathematical models Kinetics Models, Biological Models, Statistical Stochastic Processes Systems Biology methods Systems biology Systembiologie (DE-588)4809615-5 gnd Stochastisches Modell (DE-588)4057633-4 gnd |
| subject_GND | (DE-588)4809615-5 (DE-588)4057633-4 |
| title | Stochastic modelling for systems biology |
| title_auth | Stochastic modelling for systems biology |
| title_exact_search | Stochastic modelling for systems biology |
| title_exact_search_txtP | Stochastic modelling for systems biology |
| title_full | Stochastic modelling for systems biology Darren James Wilkinson |
| title_fullStr | Stochastic modelling for systems biology Darren James Wilkinson |
| title_full_unstemmed | Stochastic modelling for systems biology Darren James Wilkinson |
| title_short | Stochastic modelling for systems biology |
| title_sort | stochastic modelling for systems biology |
| topic | Sistemas biológicos - Modelos matemáticos lemb Stochastisches Modell - Systembiologie Mathematisches Modell Biological systems Mathematical models Kinetics Models, Biological Models, Statistical Stochastic Processes Systems Biology methods Systems biology Systembiologie (DE-588)4809615-5 gnd Stochastisches Modell (DE-588)4057633-4 gnd |
| topic_facet | Sistemas biológicos - Modelos matemáticos Stochastisches Modell - Systembiologie Mathematisches Modell Biological systems Mathematical models Kinetics Models, Biological Models, Statistical Stochastic Processes Systems Biology methods Systems biology Systembiologie Stochastisches Modell |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014884189&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT wilkinsondarrenjames stochasticmodellingforsystemsbiology |