biochemical network
TRANSCRIPT
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Summary and conclusions
Introduction to Biochemical Network Modelling
Darren Wilkinson1,21School of Mathematics & Statistics
2Centre for Integrated Systems Biology of Ageing and NutritionNewcastle University, UK
SAMSI Undergraduate Workshop, 2nd3rd March, 2007
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Biological modellingModel calibration
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Summary and conclusions
Overview
Biological network modelling
Model calibration
Application projects modelling and inference
(Bayesian inference)
Round-up
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Biological modellingModel calibration
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Summary and conclusions
IntroductionModellingStochastic kinetics
Computational Systems Biology (CSB)
Much of CSB is concerned with building models of complexbiological pathways, then validating and analysing thosemodels using a variety of methods, including time-course
simulationMost CSB researchers work with continuous deterministicmodels (coupled ODE and DAE systems)
There is increasing evidence that much intra-cellularbehaviour (including gene expression) is intrinsically
stochastic, and that systems cannot be properly understoodunless stochastic effects are incorporated into the models
Stochastic models are harder to build, estimate, validate,analyse and simulate than deterministic models...
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Biological modellingModel calibration
Application projectsBayesian inference
Summary and conclusions
IntroductionModellingStochastic kinetics
Modelling
Start with some kind of picture or diagram for a mechanism
Turn it into a set of (pseudo-)biochemical reactions
Specify the rate laws and rate parameters of the reactionsRun some stochastic or deterministic computer simulator ofthe system dynamics
Study the dynamics in a variety of ways to gain insight into
the systemRefine the model structure and/or parameters after comparingsimulated dynamics with experimental observations
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Bi l i l d lli
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Biological modellingModel calibration
Application projectsBayesian inference
Summary and conclusions
IntroductionModellingStochastic kinetics
Biochemical reactions
Simplified view:
Reactions
g+P2 g P2 Repressiong g+r Transcriptionr r+P Translation2P P2 Dimerisation
r mRNA degradationP Protein degradation
But these arent as nice to look at as the picture...
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Biological modelling
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IntroductionModellingStochastic kinetics
Petri net representation
Simple bipartite digraph representation of the reaction network useful both for visualisation and computational analysis
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Biological modellingModel calibration
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Summary and conclusions
IntroductionModellingStochastic kinetics
Matrix representation of the Petri net
Reactants (Pre) Products (Post)Species g P2 g r P P 2 g P2 g r P P 2
Repression 1 1 1Reverse repression 1 1 1
Transcription 1 1 1Translation 1 1 1
Dimerisation 2 1
Dissociation 1 2mRNA degradation 1Protein degradation 1
But still need rate laws and reaction rates...
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Biological modelling
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Biological modellingModel calibration
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Summary and conclusions
IntroductionModellingStochastic kinetics
Mass-action stochastic kinetics
Stochastic molecular approach:
Statistical mechanics arguments lead to aMarkov jumpprocessin continuous time whose instantaneous reaction rates
are directly proportional to the number of molecules of eachreacting species
Such dynamics can be simulated (exactly) on a computerusing standarddiscrete-event simulationtechniques
Standard implementation of this strategy is known as theGillespie algorithm (just discrete event simulation), butthere are several exact and approximate variants of this basicapproach
Darren Wilkinson SAMSI Undergraduate Workshop Biochemical Network Modelling
Biological modelling
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Biological modellingModel calibration
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Summary and conclusions
IntroductionModellingStochastic kinetics
Lotka-Volterra system
Reactions
X 2X (prey reproduction)X+Y 2Y (prey-predator interaction)
Y (predator death)
X Prey, Y PredatorWe can re-write this using matrix notation for thecorresponding Petri net
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g gModel calibration
Application projectsBayesian inference
Summary and conclusions
IntroductionModellingStochastic kinetics
Forming the matrix representation
The L-V system in tabular form
Rate Law LHS RHS Net-effecth(, c) X Y X Y X Y
R1 c1x 1 0 2 0 1 0R2 c2xy 1 1 0 2 -1 1R3 c3y 0 1 0 0 0 -1
Call the 3 2 net-effect (orreaction) matrix A. The matrix S=A
is thestoichiometry matrixof the system. Typically both aresparse. The SVD ofS (orA) is of interest for structural analysis ofthe system dynamics...
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g gModel calibration
Application projectsBayesian inference
Summary and conclusions
IntroductionModellingStochastic kinetics
Petri net invariants
A P-invariant is a non-zero solution to Ay= 0 (ie. y is in thenull-space ofA)
P-invariants correspond toconservation lawsin the network,
and lead to rank-degeneracy ofAA T-invariant is a non-zero, non-negative (integer-valued)solution to Sx= 0 (ie. x is in the null-space ofS)
T invariants correspond to sequences of reaction events thatreturn the system to its original state
The SVD ofS (orA) characterises the null-space ofS and A
The Lotka-Volterra model is of full rank (so no P-invariants),and has one T-invariant,x= (1, 1, 1)
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g gModel calibration
Application projectsBayesian inference
Summary and conclusions
IntroductionModellingStochastic kinetics
The Gillespie algorithm
1 Initialise the system at t= 0 with rate constants c1, c2, . . . , cv andinitial numbers of molecules for each species, x1, x2, . . . , xu.
2 For each i= 1, 2, . . . , v, calculate hi(x, ci) based on the currentstate, x.
3 Calculate h0(x, c)
vi=1hi(x, ci), the combined reaction hazard.
4 Simulate time to next event,t, as an Exp(h0(x, c)) randomquantity, and put t :=t+t.
5 Simulate the reaction index, j, as a discrete random quantity withprobabilitiesh
i(x, c
i) / h
0(x, c), i= 1, 2, . . . , v.
6 Update xaccording to reaction j. That is, put x :=x+S(j), whereS(j) denotes the jth column of the stoichiometry matrix S.
7 Output x and t.
8 Ift
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Model calibrationApplication projects
Bayesian inferenceSummary and conclusions
IntroductionModellingStochastic kinetics
The continuous deterministic approximation
If the discreteness and stochasticity are ignored, then byconsidering the reaction fluxes it is straightforward to deducethe mass-action ordinary differential equation (ODE) system:
ODE ModeldXt
dt =Sh(Xt, c)
Analytic solutions are rarely available, but good numericalsolvers can generate time course behaviour
Slight complications due to rank-degeneracy ofS
Also spatial versions reaction-diffusion kinetics PDEmodels computationally intensive (slow)
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Biological modellingM d l lib i I d i
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Model calibrationApplication projects
Bayesian inferenceSummary and conclusions
IntroductionModellingStochastic kinetics
The Lotka-Volterra model
Time
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Biological modellingM d l lib ti I t d ti
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Model calibrationApplication projects
Bayesian inferenceSummary and conclusions
IntroductionModellingStochastic kinetics
Key differences
Deterministic solution is exactly periodic with perfectlyrepeating oscillations, carrying on indefinitely
Stochastic solution oscillates, but in a random, unpredictable
way (wandering from orbit to orbit in phase space)Stochastic solutionwillend in disaster! Either prey orpredator numbers will hit zero...
Either way, predators will end up extinct, soexpectednumberof predators will tend to zero qualitatively differentto the
deterministic solution
So, in general the deterministic solution does not providereliable information about either the stochastic process or itsaverage behaviour
Darren Wilkinson SAMSI Undergraduate Workshop Biochemical Network Modelling
Biological modellingModel calibration Introduction
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Model calibrationApplication projects
Bayesian inferenceSummary and conclusions
IntroductionModellingStochastic kinetics
Simulated realisation of the auto-regulatory network
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Darren Wilkinson SAMSI Undergraduate Workshop Biochemical Network Modelling
Biological modellingModel calibration Introduction
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Model calibrationApplication projects
Bayesian inferenceSummary and conclusions
IntroductionLikelihood-based fully Bayesian inferenceLikelihood-free Bayesian inference
Model calibration
In its most basic form, model calibrationis concerned withtuning the parameters of a computer model in order to
make the output obtained by running it consistent withexperimental observations
In practice, this is only one aspect of the problem, as therewill typically be a range of parameter values consistent withobservations, and so the calibration exercise is part of a
broader analysis, also concerning modelvalidityand parameteridentifiabilityandconfounding
Darren Wilkinson SAMSI Undergraduate Workshop Biochemical Network Modelling
Biological modellingModel calibration Introduction
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Model calibrationApplication projects
Bayesian inferenceSummary and conclusions
IntroductionLikelihood-based fully Bayesian inferenceLikelihood-free Bayesian inference
Simple example: linear birth-death process
Birth-death reactions
X X 2X
X X
Deterministic solution: Xt=X0exp{( )t}
This is a function of ( ) only!Stochastic solution is more interesting, and depends on both and ...
Darren Wilkinson SAMSI Undergraduate Workshop Biochemical Network Modelling
Biological modellingModel calibration Introduction
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Model calibrationApplication projects
Bayesian inferenceSummary and conclusions
IntroductionLikelihood-based fully Bayesian inferenceLikelihood-free Bayesian inference
Birth-death realisations
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Darren Wilkinson SAMSI Undergraduate Workshop Biochemical Network Modelling
Biological modellingModel calibration Introduction
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Application projectsBayesian inference
Summary and conclusions
Likelihood-based fully Bayesian inferenceLikelihood-free Bayesian inference
Issues with the birth-death process
Stochastic variation: random distribution at each time point,correlations between time points, random time to extinction,
etc.Parameter identification: if a deterministic model is fitted, onecan onlyeveridentify ( ) never and separately
Information aboutboth and in the data...
Needboth and for reliable stochastic simulation
Cant fit parameters using a deterministic model, then run astochastic simulation...
Darren Wilkinson SAMSI Undergraduate Workshop Biochemical Network Modelling
Biological modellingModel calibration Introduction
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Application projectsBayesian inference
Summary and conclusions
Likelihood-based fully Bayesian inferenceLikelihood-free Bayesian inference
Birth-death realisations
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Darren Wilkinson SAMSI Undergraduate Workshop Biochemical Network Modelling
Biological modellingModel calibration Introduction
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Application projectsBayesian inference
Summary and conclusions
Likelihood-based fully Bayesian inferenceLikelihood-free Bayesian inference
Fully Bayesian inference
In principle it is possible to carry out rigorous statisticalinference for the parameters of the stochastic process model
Fairly detailed experimental data are required eg.quantitative single-cell time-course dataderived from live-cell
imagingThe standard procedure uses GFP labelling of key reporterproteins together with time-lapse confocal microscopy, butother approaches are also possible
The statistical theory underlying the inference algorithms is
fairly technical the techniques are developed and illustratedin a sequence of papers. The main findings are summarised in:Golightly, A. & Wilkinson, D. J. (2006)Bayesian sequentialinference for stochastic kinetic biochemical network models,Journal of Computational Biology, 13(3):838851.
Darren Wilkinson SAMSI Undergraduate Workshop Biochemical Network Modelling
Biological modellingModel calibration
A li i jIntroductionLik lih d b d f ll B i i f
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Application projectsBayesian inference
Summary and conclusions
Likelihood-based fully Bayesian inferenceLikelihood-free Bayesian inference
Likelihood-free MCMC for Bayesian inference
It is possible to develop a generic framework for Bayesianinference for model parameters applicable to bothdeterministic and stochastic models using the ideas oflikelihood-free MCMC, which sacrifices some computational
efficiency for considerable reduction in implementationcomplexity
It exploitsforward simulationfrom the computer model
Such an approach requires a very large number of simulation
runs, and is therefore most easily applied to fast simulators(simple models)
Forslow simulators(complex models), HPC facilities can beexploited in order to build a fastemulatorof the slowsimulator
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Biological modellingModel calibration
Application projectsAgeingComplex modelling
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Application projectsBayesian inference
Summary and conclusions
Complex modellingBayesian calibration
Network theory of ageing
Darren Wilkinson SAMSI Undergraduate Workshop Biochemical Network Modelling
Biological modellingModel calibration
Application projectsAgeingComplex modelling
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Application projectsBayesian inference
Summary and conclusions
Complex modellingBayesian calibration
Modelling large biological systems
BBSRC/MRC/DTI Grant (+ Unilever)
BASIS Biology of Ageing e-Science Integration and
Simulation(4/023/06) Kirkwood, Wilkinson, Boys, Gillespie,Proctor, Shanley
Modelling large complex systems with many interactingcomponents
SBML model database (SBML encoded for discrete stochastic
simulation)Discrete stochastic simulation service (and results database)
Distributed computing infrastructure for routine use (webportal and web-service interface for GRID computing)
Darren Wilkinson SAMSI Undergraduate Workshop Biochemical Network Modelling
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Application projectsAgeingComplex modelling
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Application projectsBayesian inference
Summary and conclusions
Complex modellingBayesian calibration
SBML The Systems Biology Markup Language
SBML is an XML-based language for encoding andexchanging quantitative biochemical network models
Encodes species, initial amounts, reactions, rate laws, etc.
Original specification (Level 1) aimed mainly at continuousdeterministic models
Current specification (Level 2) perfectly capable of encodingdiscrete stochastic models in an unambiguous way
Many tools for working with SBML models (model builders,
deterministic and stochastic simulators, etc.)
Issues with testing correctness of stochastic simulators, andcorrectly encoding discrete stochastic models usingoff-the-shelf model-building tools
Darren Wilkinson SAMSI Undergraduate Workshop Biochemical Network Modelling
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Application projectsAgeingComplex modelling
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Application projectsBayesian inference
Summary and conclusions
Complex modellingBayesian calibration
Computer model technology
BASIS features service-oriented architecture (SOA)
Controls access to models, data and computational resourcesRepresents and encodes complex models using XMLtechnology (SBML in this case)Simulation engine that can handle a broad class of modelswithout recompilationDatabases for models and simulation outputWeb interface for human-interaction
SOAP web-services API for programmatical access
Do we need a standard API for biological simulation services?
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pp p jBayesian inference
Summary and conclusions
p gBayesian calibration
BASIS Software www.basis.ncl.ac.uk
UK e-Science GRID Pilot Project
C
Simulation code
GSL
Scientific
library
Postgres
Database
Condor
Jobsched
libSBML
SBML
library
R
Data
analysis
Networkvisualise
Python
Main BASIS API
Python SOAP Web Services interface (SSLbased)
PythonSpyce/PSP
and CGI
scripts
Java
Axis
Tomcat
Apache web server
WSSecurity WSs
Web client
Debian GNU/Linux (sarge)
SOAP client (WSSecurity) SOAP client (SSL)
Software architecture used to implement the BASISsystem
GraphViz
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pp p jBayesian inference
Summary and conclusions
p gBayesian calibration
Example: Chaperones and their role in ageing
C. J. Proctor, C. Soti, R. J. Boys, C. S. Gillespie, D. P.Shanley, D. J. Wilkinson, T. B. L. Kirkwood (2005)Modellingthe actions of chaperones and their role in ageing,
Mechanisms of Ageing and Development, 126(1):119-131.Several versions of this model in the BASIS public modelrepository, each with a unique ID each can be copied,modified and simulated
eg. urn:basis.ncl:model:518
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Bayesian inferenceSummary and conclusions
Bayesian calibration
Outline CaliBayes architecture
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Biological modellingModel calibrationApplication projects
B i i f
AgeingComplex modellingB i lib ti
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Bayesian inferenceSummary and conclusions
Bayesian calibration
An example posterior distribution
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Darren Wilkinson SAMSI Undergraduate Workshop Biochemical Network Modelling
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Ba esian inference
AgeingComplex modellingBa esian calibration
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Bayesian inferenceSummary and conclusions
Bayesian calibration
Extensions
Bayesian inference naturally integrates data from multiplesources, and may be assimilated simultaneously or sequentiallydepending on the context
The architecture requires slight modification for complex
models, as then the simulator is replaced by an emulator, builtoff-line using HPC facilities
The framework can also be adapted to tackle experimentaldesign questions such as: Given a limited budget, and ourcurrent state of knowledge, what are the best new
experiments to carry out in order to learn most about the
model parameters of greatest interest?
It is also possible to extend the framework to compareevidence for competing models for the same process
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Bayesian inference
MCMCFuture directionsEmulators
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Bayesian inferenceSummary and conclusions
Emulators
MCMC-based fully Bayesian inference for fastcomputer
models
Before worrying about the issues associated withslowsimulators, it is worth thinking about the issues involved incalibratingfast deterministicandstochasticsimulators, basedonly on the ability to forward-simulatefrom the model
In this case it is often possible to construct MCMC algorithmsfor fully Bayesian inference using the ideas of likelihood-freeMCMC(Marjoram et al 2003)
Here an MCMC scheme is developed exploiting forwardsimulation from the model, and this causes problematiclikelihood terms to drop out of the M-H acceptanceprobabilities
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Bayesian inference
MCMCFuture directionsEmulators
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Bayesian inferenceSummary and conclusions
Emulators
Future directions
In the presence of measurement error, the sequentiallikelihood-free scheme is effective, and is muchsimpler than amore efficient MCMC approach
The likelihood-free approach is easier to tailor to non-standard
models and data
The essential problem is that ofcalibrationof complexstochastic computer models
Worth connecting with the literature on deterministic
computer modelsForslowstochastic models, there is considerable interest indeveloping fastemulatorsand embedding these into MCMCalgorithms
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Bayesian inference
MCMCFuture directionsEmulators
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Bayesian inferenceSummary and conclusions
Emulators
Building emulators for slowsimulators
UseGaussian process regressionto build an emulator of a slowdeterministic simulator
Obtain runs on a carefully constructed set of design points
(eg. a Latin hypercube) easy to exploit parallel computinghardware here
For a stochastic simulator, many approaches are possible
(Mixtures of) Dirichlet processes (and related constructs) arepotentially quite flexible
Can also model output parametrically (say, Gaussian), withparameters modelled by (independent) Gaussian processesWill typically want more than one run per design point, inorder to be able to estimate distribution
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Bayesian inference
Biological computer modelsProblemsReference
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ayes a e e ceSummary and conclusions
e e e ce
Why are Systems Biology models interesting examples of
computer models?Models
Diverse class of models: fast/slow,spatial/non-spatial,deterministic/stochastic,discrete/continuous time/stateseven modelling the same biological process!Many parametersStructural uncertaintyGenuine interest in the (posterior distribution of the)parameters not just in prediction
DataHigh-dimensionalDiverse: high-resolution time-course data, coarse populationaveraged data, endpoint data,distributional data, individualspecific parameters/data, covariatesMultiple distinct sources of data for a given model
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Bayesian inference
Biological computer modelsProblemsReference
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ySummary and conclusions
Interesting methodological problems
Calibration of fast and slowstochasticsimulators, usingindividual, averaged and distributional data
Dealing withheterogeneity cellcell, tissuetissue, or
organismorganismEmulationof slow stochastic simulators good models andfitting procedures
Experimental designfor stochastic computer models trade
offs between repetition and space-filling, etc.Utilising fast stochastic or deterministic approximatesimulators for a slow stochastic simulator
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Bayesian inference
Biological computer modelsProblemsReference
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Summary and conclusions
Further information
Stochastic Modelling for Systems BiologyAn accessible introduction to stochastic modellingof complex genetic and biochemical networks.Covers: biological modelling, biochemical reac-
tions, Petri nets, SBML, stochastic processes, sim-ulation algorithms (including Gillespie), case stud-ies, MCMC, and Bayesian inference for networkdynamics. ISBN: 1-58488-540-8
Contact details...
email: [email protected]: http://www.staff.ncl.ac.uk/d.j.wilkinson/
Darren Wilkinson SAMSI Undergraduate Workshop Biochemical Network Modelling
http://localhost/var/www/apps/conversion/tmp/scratch_10/[email protected]://www.staff.ncl.ac.uk/d.j.wilkinson/http://www.staff.ncl.ac.uk/d.j.wilkinson/http://localhost/var/www/apps/conversion/tmp/scratch_10/[email protected]