Introduction to Biochemical Network Modelling by kmb15358

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									                            Biological modelling
                              Model calibration
                            Application projects
                             Bayesian inference
                        Summary and conclusions




    Introduction to Biochemical Network Modelling

                         Darren Wilkinson1,2
                 1 School of Mathematics & Statistics
  2 Centre for Integrated Systems Biology of Ageing and Nutrition

                       Newcastle University, UK


        SAMSI Undergraduate Workshop, 2nd–3rd March, 2007



Darren Wilkinson — SAMSI Undergraduate Workshop    Biochemical Network Modelling
                              Biological modelling
                                Model calibration
                              Application projects
                               Bayesian inference
                          Summary and conclusions


Overview



         Biological network modelling
         Model calibration
         Application projects — modelling and inference
         (Bayesian inference)
         Round-up




  Darren Wilkinson — SAMSI Undergraduate Workshop    Biochemical Network Modelling
                              Biological modelling
                                Model calibration    Introduction
                              Application projects   Modelling
                               Bayesian inference    Stochastic kinetics
                          Summary and conclusions


Computational Systems Biology (CSB)

         Much of CSB is concerned with building models of complex
         biological pathways, then validating and analysing those
         models using a variety of methods, including time-course
         simulation
         Most CSB researchers work with continuous deterministic
         models (coupled ODE and DAE systems)
         There is increasing evidence that much intra-cellular
         behaviour (including gene expression) is intrinsically
         stochastic, and that systems cannot be properly understood
         unless stochastic effects are incorporated into the models
         Stochastic models are harder to build, estimate, validate,
         analyse and simulate than deterministic models...

  Darren Wilkinson — SAMSI Undergraduate Workshop    Biochemical Network Modelling
                              Biological modelling
                                Model calibration    Introduction
                              Application projects   Modelling
                               Bayesian inference    Stochastic kinetics
                          Summary and conclusions


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 reactions
         Run some stochastic or deterministic computer simulator of
         the system dynamics
         Study the dynamics in a variety of ways to gain insight into
         the system
         Refine the model structure and/or parameters after comparing
         simulated dynamics with experimental observations



  Darren Wilkinson — SAMSI Undergraduate Workshop    Biochemical Network Modelling
                              Biological modelling
                                Model calibration    Introduction
                              Application projects   Modelling
                               Bayesian inference    Stochastic kinetics
                          Summary and conclusions


Example — genetic auto-regulation




                                                P
                                                                           r
               P2


                       RNAP




                            p      q                      g                    DNA




  Darren Wilkinson — SAMSI Undergraduate Workshop    Biochemical Network Modelling
                              Biological modelling
                                Model calibration    Introduction
                              Application projects   Modelling
                               Bayesian inference    Stochastic kinetics
                          Summary and conclusions


Biochemical reactions


  Simplified view:
  Reactions
   g + P2 ←→ g · P2                Repression
      g −→ g + r                   Transcription
      r −→ r + P                   Translation
      2P ←→ P2                     Dimerisation
        r −→ ∅                     mRNA degradation
        P −→ ∅                     Protein degradation

  But these aren’t as nice to look at as the picture...



  Darren Wilkinson — SAMSI Undergraduate Workshop    Biochemical Network Modelling
                              Biological modelling
                                Model calibration    Introduction
                              Application projects   Modelling
                               Bayesian inference    Stochastic kinetics
                          Summary and conclusions


Petri net representation




  Simple bipartite digraph representation of the reaction network —
  useful both for visualisation and computational analysis

  Darren Wilkinson — SAMSI Undergraduate Workshop    Biochemical Network Modelling
                              Biological modelling
                                Model calibration    Introduction
                              Application projects   Modelling
                               Bayesian inference    Stochastic kinetics
                          Summary and conclusions


Matrix representation of the Petri net


                                       Reactants (Pre)                          Products (Post)
               Species            g · P2 g r P P2                          g · P2 g r P P2
            Repression                     1           1                     1
    Reverse repression              1                                               1           1
          Transcription                    1                                        1 1
            Translation                        1                                        1 1
          Dimerisation                             2                                            1
           Dissociation                                1                                    2
   mRNA degradation                            1
   Protein degradation                             1

  But still need rate laws and reaction rates...

  Darren Wilkinson — SAMSI Undergraduate Workshop    Biochemical Network Modelling
                              Biological modelling
                                Model calibration    Introduction
                              Application projects   Modelling
                               Bayesian inference    Stochastic kinetics
                          Summary and conclusions


Mass-action stochastic kinetics

  Stochastic molecular approach:
         Statistical mechanics arguments lead to a Markov jump
         process in continuous time whose instantaneous reaction rates
         are directly proportional to the number of molecules of each
         reacting species
         Such dynamics can be simulated (exactly) on a computer
         using standard discrete-event simulation techniques
         Standard implementation of this strategy is known as the
         “Gillespie algorithm” (just discrete event simulation), but
         there are several exact and approximate variants of this basic
         approach


  Darren Wilkinson — SAMSI Undergraduate Workshop    Biochemical Network Modelling
                              Biological modelling
                                Model calibration    Introduction
                              Application projects   Modelling
                               Bayesian inference    Stochastic kinetics
                          Summary and conclusions


Lotka-Volterra system


  Reactions

                            X −→ 2X                                        (prey reproduction)
                    X + Y −→ 2Y                               (prey-predator interaction)
                            Y −→ ∅                                           (predator death)

         X – Prey, Y – Predator
         We can re-write this using matrix notation for the
         corresponding Petri net



  Darren Wilkinson — SAMSI Undergraduate Workshop    Biochemical Network Modelling
                              Biological modelling
                                Model calibration    Introduction
                              Application projects   Modelling
                               Bayesian inference    Stochastic kinetics
                          Summary and conclusions


Forming the matrix representation


  The L-V system in tabular form
                         Rate Law            LHS         RHS               Net-effect
                          h(·, c)           X Y         X Y                X      Y
                  R1        c1 x            1 0         2 0                 1      0
                  R2       c2 xy            1 1         0 2                -1      1
                  R3        c3 y            0 1         0 0                 0     -1

  Call the 3 × 2 net-effect (or reaction) matrix A. The matrix S = A
  is the stoichiometry matrix of the system. Typically both are
  sparse. The SVD of S (or A) is of interest for structural analysis of
  the system dynamics...


  Darren Wilkinson — SAMSI Undergraduate Workshop    Biochemical Network Modelling
                              Biological modelling
                                Model calibration    Introduction
                              Application projects   Modelling
                               Bayesian inference    Stochastic kinetics
                          Summary and conclusions


Petri net invariants

         A P-invariant is a non-zero solution to Ay = 0 (ie. y is in the
         null-space of A)
                P-invariants correspond to conservation laws in the network,
                and lead to rank-degeneracy of A
         A T -invariant is a non-zero, non-negative (integer-valued)
         solution to Sx = 0 (ie. x is in the null-space of S)
                T invariants correspond to sequences of reaction events that
                return the system to its original state
         The SVD of S (or A) characterises the null-space of S and A
         The Lotka-Volterra model is of full rank (so no P-invariants),
         and has one T -invariant, x = (1, 1, 1)


  Darren Wilkinson — SAMSI Undergraduate Workshop    Biochemical Network Modelling
                              Biological modelling
                                Model calibration     Introduction
                              Application projects    Modelling
                               Bayesian inference     Stochastic kinetics
                          Summary and conclusions


The Gillespie algorithm
     1   Initialise the system at t = 0 with rate constants c1 , c2 , . . . , cv and
         initial numbers of molecules for each species, x1 , x2 , . . . , xu .
     2   For each i = 1, 2, . . . , v , calculate hi (x, ci ) based on the current
         state, x.
                                         v
     3   Calculate h0 (x, c) ≡           i=1   hi (x, ci ), the combined reaction hazard.
     4   Simulate time to next event, t , as an Exp(h0 (x, c)) random
         quantity, and put t := t + t .
     5   Simulate the reaction index, j, as a discrete random quantity with
         probabilities hi (x, ci ) / h0 (x, c), i = 1, 2, . . . , v .
     6   Update x according to reaction j. That is, put x := x + S (j) , where
         S (j) denotes the jth column of the stoichiometry matrix S.
     7   Output x and t.
     8   If t < Tmax , return to step 2.
  Darren Wilkinson — SAMSI Undergraduate Workshop     Biochemical Network Modelling
                              Biological modelling
                                Model calibration    Introduction
                              Application projects   Modelling
                               Bayesian inference    Stochastic kinetics
                          Summary and conclusions


The continuous deterministic approximation

         If the discreteness and stochasticity are ignored, then by
         considering the reaction fluxes it is straightforward to deduce
         the mass-action ordinary differential equation (ODE) system:

  ODE Model
                                        dXt
                                            = Sh(Xt , c)
                                         dt

         Analytic solutions are rarely available, but good numerical
         solvers can generate time course behaviour
         Slight complications due to rank-degeneracy of S
         Also spatial versions — reaction-diffusion kinetics — PDE
         models — computationally intensive (slow)
  Darren Wilkinson — SAMSI Undergraduate Workshop    Biochemical Network Modelling
                                       Biological modelling
                                         Model calibration                            Introduction
                                       Application projects                           Modelling
                                        Bayesian inference                            Stochastic kinetics
                                   Summary and conclusions


The Lotka-Volterra model




                                                                                25
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              0   20        40          60        80               100                 0              2          4           6          8

                                 Time                                                                           [Y1]




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        400




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  Y

        200




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              0   5    10          15        20             25                             50   100       150    200   250       300   350

                                 Time                                                                           Y1

  Darren Wilkinson — SAMSI Undergraduate Workshop                                     Biochemical Network Modelling
                              Biological modelling
                                Model calibration    Introduction
                              Application projects   Modelling
                               Bayesian inference    Stochastic kinetics
                          Summary and conclusions


Key differences

         Deterministic solution is exactly periodic with perfectly
         repeating oscillations, carrying on indefinitely
         Stochastic solution oscillates, but in a random, unpredictable
         way (wandering from orbit to orbit in phase space)
         Stochastic solution will end in disaster! Either prey or
         predator numbers will hit zero...
         Either way, predators will end up extinct, so expected number
         of predators will tend to zero — qualitatively different to the
         deterministic solution
         So, in general the deterministic solution does not provide
         reliable information about either the stochastic process or its
         average behaviour

  Darren Wilkinson — SAMSI Undergraduate Workshop    Biochemical Network Modelling
                                    Biological modelling
                                      Model calibration      Introduction
                                    Application projects     Modelling
                                     Bayesian inference      Stochastic kinetics
                                Summary and conclusions


Simulated realisation of the auto-regulatory network
                 2.0
                 1.5
           Rna
                 1.0
                 0.5
                 0.0
                 50
                 30
           P
                 600 0 10
                 400
           P2
                 200
                 0




                            0         1000          2000          3000             4000      5000

                                                           Time



  Darren Wilkinson — SAMSI Undergraduate Workshop            Biochemical Network Modelling
                              Biological modelling
                                Model calibration    Introduction
                              Application projects   Likelihood-based fully Bayesian inference
                               Bayesian inference    “Likelihood-free” Bayesian inference
                          Summary and conclusions


Model calibration


         In its most basic form, model calibration is concerned with
         “tuning” the parameters of a computer model in order to
         make the output obtained by running it consistent with
         experimental observations
         In practice, this is only one aspect of the problem, as there
         will typically be a range of parameter values consistent with
         observations, and so the calibration exercise is part of a
         broader analysis, also concerning model validity and parameter
         identifiability and confounding



  Darren Wilkinson — SAMSI Undergraduate Workshop    Biochemical Network Modelling
                              Biological modelling
                                Model calibration     Introduction
                              Application projects    Likelihood-based fully Bayesian inference
                               Bayesian inference     “Likelihood-free” Bayesian inference
                          Summary and conclusions


Simple example: linear birth-death process


  Birth-death reactions

                                                 λX
                                            X −→ 2X
                                                 µX
                                            X −→ ∅

         Deterministic solution: Xt = X0 exp{(λ − µ)t}
         This is a function of (λ − µ) only!
         Stochastic solution is more interesting, and depends on both
         λ and µ...


  Darren Wilkinson — SAMSI Undergraduate Workshop     Biochemical Network Modelling
                              Biological modelling
                                Model calibration        Introduction
                              Application projects       Likelihood-based fully Bayesian inference
                               Bayesian inference        “Likelihood-free” Bayesian inference
                          Summary and conclusions


Birth-death realisations
               60
               50
               40
               30
          X

               20
               10
               0




                     0            1             2              3             4              5

                                                     t



  Darren Wilkinson — SAMSI Undergraduate Workshop        Biochemical Network Modelling
                              Biological modelling
                                Model calibration    Introduction
                              Application projects   Likelihood-based fully Bayesian inference
                               Bayesian inference    “Likelihood-free” Bayesian inference
                          Summary and conclusions


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, one
         can only ever identify (λ − µ) — never λ and µ separately
         Information about both λ and µ in the data...
         Need both λ and µ for reliable stochastic simulation
         Can’t fit parameters using a deterministic model, then run a
         stochastic simulation...



  Darren Wilkinson — SAMSI Undergraduate Workshop    Biochemical Network Modelling
                                Biological modelling
                                  Model calibration        Introduction
                                Application projects       Likelihood-based fully Bayesian inference
                                 Bayesian inference        “Likelihood-free” Bayesian inference
                            Summary and conclusions


Birth-death realisations

                         lambda=0, mu=1                                  lambda=3, mu=4
              60




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              40




                                                            40
          X




                                                       X
              20




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              0




                                                            0
                   0    1      2       3   4    5                0      1     2       3    4     5

                                   t                                              t



                         lambda=7, mu=8                                lambda=10, mu=11
              60




                                                            60
              40




                                                            40
          X




                                                       X
              20




                                                            20
              0




                                                            0




                   0    1      2       3   4    5                0      1     2       3    4     5

                                   t                                              t




  Darren Wilkinson — SAMSI Undergraduate Workshop          Biochemical Network Modelling
                              Biological modelling
                                Model calibration    Introduction
                              Application projects   Likelihood-based fully Bayesian inference
                               Bayesian inference    “Likelihood-free” Bayesian inference
                          Summary and conclusions


Fully Bayesian inference
         In principle it is possible to carry out rigorous statistical
         inference for the parameters of the stochastic process model
         Fairly detailed experimental data are required — eg.
         quantitative single-cell time-course data derived from live-cell
         imaging
         The standard procedure uses GFP labelling of key reporter
         proteins together with time-lapse confocal microscopy, but
         other approaches are also possible
         The statistical theory underlying the inference algorithms is
         fairly technical — the techniques are developed and illustrated
         in a sequence of papers. The main findings are summarised in:
         Golightly, A. & Wilkinson, D. J. (2006) Bayesian sequential
         inference for stochastic kinetic biochemical network models,
         Journal of Computational Biology, 13(3):838–851.
  Darren Wilkinson — SAMSI Undergraduate Workshop    Biochemical Network Modelling
                              Biological modelling
                                Model calibration    Introduction
                              Application projects   Likelihood-based fully Bayesian inference
                               Bayesian inference    “Likelihood-free” Bayesian inference
                          Summary and conclusions


“Likelihood-free” MCMC for Bayesian inference

         It is possible to develop a generic framework for Bayesian
         inference for model parameters applicable to both
         deterministic and stochastic models using the ideas of
         “likelihood-free MCMC”, which sacrifices some computational
         efficiency for considerable reduction in implementation
         complexity
         It exploits forward simulation from 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)
         For slow simulators (complex models), HPC facilities can be
         exploited in order to build a fast emulator of the slow
         simulator
  Darren Wilkinson — SAMSI Undergraduate Workshop    Biochemical Network Modelling
                              Biological modelling
                                Model calibration    Ageing
                              Application projects   Complex modelling
                               Bayesian inference    Bayesian calibration
                          Summary and conclusions


Mechanisms of ageing

         Ageing is caused by the gradual accumulation of unrepaired
         molecular damage, leading to an increasing fraction of
         damaged cells and eventually to functional impairment of
         tissues and organs
         One major cause of molecular damage is highly reactive
         oxygen species (ROS)
         Molecular damage may trigger cellular response programmes,
         so that the ageing process may also be seen to be governed by
         genetically determined pathways
         Many of the (random) damage and (imperfect) repair
         mechanisms important for understanding cellular ageing are
         intrinsically stochastic

  Darren Wilkinson — SAMSI Undergraduate Workshop    Biochemical Network Modelling
                              Biological modelling
                                Model calibration    Ageing
                              Application projects   Complex modelling
                               Bayesian inference    Bayesian calibration
                          Summary and conclusions


Network theory of ageing




  Darren Wilkinson — SAMSI Undergraduate Workshop    Biochemical Network Modelling
                              Biological modelling
                                Model calibration    Ageing
                              Application projects   Complex modelling
                               Bayesian inference    Bayesian calibration
                          Summary and conclusions


Modelling large biological systems

  BBSRC/MRC/DTI Grant (+ Unilever)
  BASIS — Biology of Ageing e-Science Integration and
  Simulation (4/02–3/06) — Kirkwood, Wilkinson, Boys, Gillespie,
  Proctor, Shanley

         Modelling large complex systems with many interacting
         components
         SBML model database (SBML encoded for discrete stochastic
         simulation)
         Discrete stochastic simulation service (and results database)
         Distributed computing infrastructure for routine use (web
         portal and web-service interface for GRID computing)

  Darren Wilkinson — SAMSI Undergraduate Workshop    Biochemical Network Modelling
                              Biological modelling
                                Model calibration    Ageing
                              Application projects   Complex modelling
                               Bayesian inference    Bayesian calibration
                          Summary and conclusions


SBML — The Systems Biology Markup Language

         SBML is an XML-based language for encoding and
         exchanging quantitative biochemical network models
         Encodes species, initial amounts, reactions, rate laws, etc.
         Original specification (Level 1) aimed mainly at continuous
         deterministic models
         Current specification (Level 2) perfectly capable of encoding
         discrete 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, and
         correctly encoding discrete stochastic models using
         off-the-shelf model-building tools

  Darren Wilkinson — SAMSI Undergraduate Workshop    Biochemical Network Modelling
                              Biological modelling
                                Model calibration    Ageing
                              Application projects   Complex modelling
                               Bayesian inference    Bayesian calibration
                          Summary and conclusions


Computer model technology


         BASIS features — service-oriented architecture (SOA)
                Controls access to models, data and computational resources
                Represents and encodes complex models using XML
                technology (SBML in this case)
                Simulation engine that can handle a broad class of models
                without recompilation
                Databases for models and simulation output
                Web interface for human-interaction
                SOAP web-services API for programmatical access
         Do we need a standard API for biological simulation services?



  Darren Wilkinson — SAMSI Undergraduate Workshop    Biochemical Network Modelling
                                                             Biological modelling
                                                               Model calibration    Ageing
                                                             Application projects   Complex modelling
                                                              Bayesian inference    Bayesian calibration
                                                         Summary and conclusions


BASIS Software –                                                      www.basis.ncl.ac.uk

                                                         UK e-Science GRID Pilot Project
           Web client        SOAP client (WS−Security)   SOAP client (SSL)




           Apache web server



                                     Tomcat
        Python
        Spyce/PSP
        and CGI
        scripts                      Axis




                                  Java
                                  WS−Security WSs




           Python − SOAP Web Services interface (SSL−based)


           Python
           Main BASIS API




      Postgres          Condor        libSBML       R              GraphViz
      Database          Job          SBML           Data           Network
                        sched        library        analysis       visualise




           C                                                   GSL
           Simulation code                                     Scientific
                                                               library



                         Debian GNU/Linux (sarge)


  Software architecture used to implement the BASIS system



  Darren Wilkinson — SAMSI Undergraduate Workshop                                   Biochemical Network Modelling
                              Biological modelling
                                Model calibration    Ageing
                              Application projects   Complex modelling
                               Bayesian inference    Bayesian calibration
                          Summary and conclusions


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) Modelling
         the 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 model
         repository, each with a unique ID — each can be copied,
         modified and simulated
         eg. urn:basis.ncl:model:518




  Darren Wilkinson — SAMSI Undergraduate Workshop    Biochemical Network Modelling
                              Biological modelling
                                Model calibration    Ageing
                              Application projects   Complex modelling
                               Bayesian inference    Bayesian calibration
                          Summary and conclusions


Outline CaliBayes architecture




  Darren Wilkinson — SAMSI Undergraduate Workshop    Biochemical Network Modelling
                              Biological modelling
                                Model calibration    Ageing
                              Application projects   Complex modelling
                               Bayesian inference    Bayesian calibration
                          Summary and conclusions


Distinctive features

         Although the outline architecture appears similar to that for
         many iterative parameter fitting algorithms, there are some
         fundamental differences
         This is not a hill-climbing algorithm, and is not searching for
         the “best fit”
         The calibration engine is using the information it receives
         from the statistical comparison service in order to randomly
         explore the posterior distribution for the parameters (the set
         of parameters consistent with the data and prior knowledge,
         weighted according to their probability)
         This posterior distribution can be used for a range of analyses,
         including calibration

  Darren Wilkinson — SAMSI Undergraduate Workshop    Biochemical Network Modelling
                                                Biological modelling
                                                  Model calibration                   Ageing
                                                Application projects                  Complex modelling
                                                 Bayesian inference                   Bayesian calibration
                                            Summary and conclusions


An example posterior distribution




                                                                                                      25
                                                          100
          0.25




                                                                                                      20
                                                          80




                                                                                                      15
          0.20




                                                          60
                                                Density




                                                                                            Density
  v’’dr




                                                                                                      10
                                                          40
          0.15




                                                                                                      5
                                                          20
          0.10




                                                          0




                                                                                                      0



                 0.00 0.01 0.02 0.03 0.04                       0.000 0.010 0.020 0.030                    0.05   0.15     0.25

                            v’d                                            v’d                                      v’’d



  Darren Wilkinson — SAMSI Undergraduate Workshop                                     Biochemical Network Modelling
                              Biological modelling
                                Model calibration    Ageing
                              Application projects   Complex modelling
                               Bayesian inference    Bayesian calibration
                          Summary and conclusions


Extensions

         Bayesian inference naturally integrates data from multiple
         sources, and may be assimilated simultaneously or sequentially
         depending on the context
         The architecture requires slight modification for complex
         models, as then the simulator is replaced by an emulator, built
         off-line using HPC facilities
         The framework can also be adapted to tackle experimental
         design questions such as: Given a limited budget, and our
         current 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 compare
         evidence for competing models for the same process
  Darren Wilkinson — SAMSI Undergraduate Workshop    Biochemical Network Modelling
                              Biological modelling
                                Model calibration    MCMC
                              Application projects   Future directions
                               Bayesian inference    Emulators
                          Summary and conclusions


MCMC-based fully Bayesian inference for fast computer
models

         Before worrying about the issues associated with slow
         simulators, it is worth thinking about the issues involved in
         calibrating fast deterministic and stochastic simulators, based
         only on the ability to forward-simulate from the model
         In this case it is often possible to construct MCMC algorithms
         for fully Bayesian inference using the ideas of likelihood-free
         MCMC (Marjoram et al 2003)
         Here an MCMC scheme is developed exploiting forward
         simulation from the model, and this causes problematic
         likelihood terms to drop out of the M-H acceptance
         probabilities

  Darren Wilkinson — SAMSI Undergraduate Workshop    Biochemical Network Modelling
                              Biological modelling
                                Model calibration    MCMC
                              Application projects   Future directions
                               Bayesian inference    Emulators
                          Summary and conclusions


Future directions

         In the presence of measurement error, the sequential
         likelihood-free scheme is effective, and is much simpler than a
         more efficient MCMC approach
         The likelihood-free approach is easier to tailor to non-standard
         models and data
         The essential problem is that of calibration of complex
         stochastic computer models
         Worth connecting with the literature on deterministic
         computer models
         For slow stochastic models, there is considerable interest in
         developing fast emulators and embedding these into MCMC
         algorithms

  Darren Wilkinson — SAMSI Undergraduate Workshop    Biochemical Network Modelling
                              Biological modelling
                                Model calibration    MCMC
                              Application projects   Future directions
                               Bayesian inference    Emulators
                          Summary and conclusions


Building emulators for slow simulators

         Use Gaussian process regression to build an emulator of a slow
         deterministic simulator
         Obtain runs on a carefully constructed set of design points
         (eg. a Latin hypercube) — easy to exploit parallel computing
         hardware here
         For a stochastic simulator, many approaches are possible
                (Mixtures of) Dirichlet processes (and related constructs) are
                potentially quite flexible
                Can also model output parametrically (say, Gaussian), with
                parameters modelled by (independent) Gaussian processes
                Will typically want more than one run per design point, in
                order to be able to estimate distribution


  Darren Wilkinson — SAMSI Undergraduate Workshop    Biochemical Network Modelling
                              Biological modelling
                                Model calibration    Biological computer models
                              Application projects   Problems
                               Bayesian inference    Reference
                          Summary and conclusions


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/states —
                even modelling the same biological process!
                Many parameters
                Structural uncertainty
                Genuine interest in the (posterior distribution of the)
                parameters — not just in prediction
         Data
                High-dimensional
                Diverse: high-resolution time-course data, coarse population
                averaged data, endpoint data, distributional data, individual
                specific parameters/data, covariates
                Multiple distinct sources of data for a given model
  Darren Wilkinson — SAMSI Undergraduate Workshop    Biochemical Network Modelling
                              Biological modelling
                                Model calibration    Biological computer models
                              Application projects   Problems
                               Bayesian inference    Reference
                          Summary and conclusions


Interesting methodological problems


         Calibration of fast and slow stochastic simulators, using
         individual, averaged and distributional data
         Dealing with heterogeneity — cell–cell, tissue–tissue, or
         organism–organism
         Emulation of slow stochastic simulators — good models and
         fitting procedures
         Experimental design for stochastic computer models — trade
         offs between repetition and space-filling, etc.
         Utilising fast stochastic or deterministic approximate
         simulators for a slow stochastic simulator


  Darren Wilkinson — SAMSI Undergraduate Workshop    Biochemical Network Modelling
                              Biological modelling
                                Model calibration    Biological computer models
                              Application projects   Problems
                               Bayesian inference    Reference
                          Summary and conclusions


Further information

                                Stochastic Modelling for Systems Biology
                                An accessible introduction to stochastic modelling
                                of 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 network
                                dynamics. ISBN: 1-58488-540-8


  Contact details...
  email: d.j.wilkinson@ncl.ac.uk
  www: http://www.staff.ncl.ac.uk/d.j.wilkinson/


  Darren Wilkinson — SAMSI Undergraduate Workshop    Biochemical Network Modelling

								
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