Bayesian inference for stochastic kinetic models. intracellular reaction networks

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1 for stochastic models of intracellular reaction networks Darren Wilkinson School of Mathematics & Statistics and Centre for Integrated Systems Biology of Ageing and Nutrition Newcastle University, UK IMA Workshop: Stochastic models for intracellular reaction networks IMA, Minneapolis, U.S.A. May 13, 2008

2 Overview Introduction Overview Biochemical network models Stochastic kinetic models Experimental data Markov process models of biochemical network dynamics for model parameters (CLE) An efficient MCMC algorithm for diffusions Likelihood-free and emulation based approaches to Bayesian calibration of complex models If time permits: Sparse VAR(1) models for HTP data Linking VAR(1) models to the CLE

3 Overview Biochemical network models Stochastic kinetic models Experimental data 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...

4 Stochastic Chemical Kinetics Overview Biochemical network models Stochastic kinetic models Experimental data Stochastic molecular approach: Statistical mechanical 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

5 Lotka-Volterra system Introduction Overview Biochemical network models Stochastic kinetic models Experimental data Trivial (familiar) example from population dynamics (in reality, the reactions will be elementary biochemical reactions taking place inside a cell) Reactions X 2X X + Y 2Y Y (prey reproduction) (prey-predator interaction) (predator death) X Prey, Y Predator We can re-write this using matrix notation

6 Forming the matrix representation Overview Biochemical network models Stochastic kinetic models Experimental data The L-V system in tabular form Rate Law LHS RHS Net-effect h(, c) X Y X Y X Y R 1 c 1 x R 2 c 2 xy R 3 c 3 y Call the 3 2 net-effect (or reaction) matrix N. The matrix S = N is the stoichiometry matrix of the system. Typically both are sparse. The SVD of S (or N) is of interest for structural analysis of the system dynamics...

7 The Lotka-Volterra model Overview Biochemical network models Stochastic kinetic models Experimental data [ Y ] [Y1] [Y2] [Y2] Time [Y1] Y Y1 Y2 Y Time Y1

8 Single cell fluorescence microscopy Overview Biochemical network models Stochastic kinetic models Experimental data p53-cfp and Mdm2-YFP p53/mdm2 oscillations subsequent to gamma irradiation

9 Single cell time course data Overview Biochemical network models Stochastic kinetic models Experimental data M3C4 M3C8 flourescence flourescence time (hours) time (hours) M3C10 M3C1 flourescence flourescence time (hours) time (hours) Geva-Zatorsky et al (2006), Mol. Sys. Bio. [Uri Alon s lab]

10 Stochastic kinetic model Overview Biochemical network models Stochastic kinetic models Experimental data Stochastic kinetic model developed at Newcastle (by C. J. Proctor) for the key biomolecular interactions between p53, Mdm2 and their response to DNA damage induced by irradiation More complex than the simple Lotka-Volterra system (17 species and 20 reactions), but essentially the same regulatory feedback mechanism (Mdm2 synthesis depends on the level of free p53, and Mdm2 encourages degradation of p53) Some information about most kinetic parameters, but considerable uncertainty for several ideal for a Bayesian analysis

11 Model structure and sample output Overview Biochemical network models Stochastic kinetic models Experimental data Molecules Molecules Time Time

12 Introduction for diffusions Tuning model parameters so that output from the model better matches experimental data is a standard optimisation problem, but is problematic and unsatisfactory for a number of reasons: Defining an appropriate objective function is not straightforward if the model is stochastic or the measurement error has a complex structure (not IID Gaussian) The statistical concept of likelihood provides the correct way of measuring the evidence in favour of a set of model parameters, but typically requires computationally intensive Monte Carlo procedures for evaluation in complex settings Simple optimisation of the likelihood (the maximum likelihood approach) is also unsatisfactory, as there are typically many parameter combinations with very similar likelihoods (and the likelihood surface is typically multi-modal, making global optimisation difficult)

13 Markov chain Monte Carlo (MCMC) for diffusions Additionally, likelihood ignores any existing information known about likely parameter values a priori, which can be very useful for regularising the inference problem better to base inference on the posterior distribution MCMC algorithms can be used to explore plausible regions of parameter space in accordance with the posterior distribution these provide rich information eg. rather than simple point estimates for parameter values, can get plausible ranges of values, together with information on parameter identifiability and confounding MCMC algorithms are computationally intensive, but given that evaluation of the likelihood is typically computationally intensive anyway, nothing to lose and everything to gain by doing a Bayesian analysis

14 Fully for diffusions In principle it is possible to carry out rigorous Bayesian statistical inference for the parameters of stochastic kinetic models 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 Techniques for exact inference for the true discrete model (Boys, W, Kirkwood 2008) do not scale well to problems of realistic size and complexity, due to the difficulty of efficiently exploring large complex integer lattice state spaces

15 for diffusions Rate parameter inference from complete data Observe process x = {x(t) : t [0, T ]} Time and type of ith reaction event is (t i, ν i ), i = 1,... n. Also define t 0 = 0, t n+1 = T Let r i be the number of type i events (so n = n i=1 r i) The complete-data likelihood for the observed sample path is { n } { T } L(c; x) = h νi (x(t i 1 ), c i ) exp h 0 (x(t), c) dt 0 i=1 (the integral is just a finite sum)

16 for diffusions Factorisation of the complete-data likelihood For rate laws of the form h i (x, c i ) = c i g(x) (true for mass-action stochastic kinetic models), the complete-data likelihood factorises as where L j (c j ; x) = c r j j L(c; x) = v L j (c j ; x) j=1 T } exp { c j g j (x(t)) dt, j = 1,..., v 0

17 Parameter estimation Introduction for diffusions Factorisation leads to MLEs of the form / T ĉ j = r j g j (x(t))dt or can be combined with priors of the form c j Γ(a j, b j ) to get full-conditionals 0 ( T ) c j x Γ a j + r j, b j + g j (x(t))dt 0

18 Discrete-time observation for diffusions Given discrete-time observations, the process breaks up into a collection of independent bridge processes that appear not to be analytically tractable Can use MCMC to explore sample paths consistent with the end-points Need to explore r t consistent with x t+1 x t = Sr t Both reversible jump and block-updating strategies are possible tedious details omitted!

19 for diffusions Inference for the Lotka-Volterra model Data: z = {x(t), y(t) : t = 0, 1, 2,..., 49} Trace plot for th1 ACF plot for th1 Density for th1 Value ACF Density Iteration Lag Value Trace plot for th2 ACF plot for th2 Density for th2 Value ACF Density Iteration Lag Value Trace plot for th3 ACF plot for th3 Density for th3 Value ACF Density Iteration Lag Value Model remains identifiable when only the prey are observed...

20 General reaction systems for diffusions In the case of perfect observation of the state of all species at discrete times, there is free software, stochinf, which does this (approximately) Also possible to extend to problems in which not all (bio-)chemical species involved in the reaction system are observed and species are observed with error (idea update intervals in pairs, relaxing constraints appropriately at the centre-point) Computational difficulties with large systems, large numbers of molecules, partial observation and measurement error...

21 for diffusions (CLE) The CLE is just a diffusion approximation for the Markov Jump Process (MJP) derived from stochastic chemical kinetic theory It is the Itô stochastic differential equation (SDE) model which most closely matches the dynamics of the MJP Formally, it is constructed by first considering a second-order approximation to the Kolmogorov forward equations for the process known in this context as the Chemical Master Equation (CME) This second-order approximation is known as the Fokker-Planck equation, and is the Kolmogorov forward equation associated with the CLE

22 Constructing the CLE Introduction for diffusions Informally, we can easily construct the CLE as the SDE with the same infinitesimal mean and variance as the MJP In a time increment, dt, the change in state, dx t, is given by dx t = SdR t, where the ith element of dr t is a Po(h i (X t, c i )dt) random quantity (independently of the other elements) Matching the mean and variance we put { } dr t h(x t, c)dt + diag h(xt, c) dw t where dw t is the increment of a v-dimensional Wiener process. Then { } dx t = Sh(X t, c)dt + S diag h(xt, c) dw t

23 Constructing the CLE (ctd.) for diffusions It is unnecessary (and sometimes inconvenient) to have the SDE being driven by a Brownian motion of higher dimension than the system state For this reason, the CLE is often written differently Since it is clear that Var(dX t ) = S diag{h(x t, c)}s dt, the CLE can be written The CLE dx t = Sh(X t, c)dt + S diag{h(x t, c)}s dw t where dw t is now the increment of a u-dimensional Wiener process

24 Good properties of the CLE for diffusions The CLE inherits many of the desirable properties of the MJP it approximates CLE properties Realisations of the CLE preserve conservation laws in the reaction network (associated with rank-degeneracy of the stoichiometry matrix, S) The CLE conserves matter (for models of a closed system) no random creation or destruction of matter The CLE describes a non-negative stochastic process

25 Inference for the CLE Introduction for diffusions Inference for a fairly general non-linear multivariate diffusion process, observed partially and discretely (and with error) Idea: Use an MCMC algorithm which fills-in the missing diffusion bridges between successive observations Use an Euler approximation to the true diffusion, but on a much finer scale than the data There are pathological mixing/convergence problems for regular MCMC schemes as the discretisation gets finer (essentially, there is an infinite amount of information about the parameters in the augmentation) It is nevertheless possible to develop effective MCMC algorithms...

26 Likelihood concepts Introduction for diffusions exp Putting µ(x, c) = Sh(x, c) and β(x, c) = S diag{h(x, c)}s : dx t = µ(x t, c)dt + β(x t, c)dw t If we choose a small enough t, we get the Euler-Maruyama approximation X t+ t X t, c N(X t + µ(x t, c) t, β(x t, c) t) Perfect observation of the system state on this time grid leads to the complete -data likelihood L(c; x) { 1 2 { n 1 } β(x i t, c) 1/2 i=0 n 1 ( xi t i=0 t ) ( ) } µ(x i t, c) β(x i t, c) 1 xi t µ(x i t, c) t t

27 Likelihood problems Introduction for diffusions Unfortunately the likelihood has no limit as t 0 If the diffusion term β(x, c) were independent of c, then it would be possible to discard some terms and then get a nice limit (exponential of the sum of a regular integral and an Itô stochastic integral), but it isn t... This is at the root of all of the computational problems concerning inference for diffusions More formally, the issue is whether or not the relevant Radon-Nikodym derivatives exist...

28 Global MCMC algorithm for diffusions Basic block-updating algorithm (perfect observation) 1 Initialise parameters, c, and sample path, x 2 For t = 2,..., n, propose a new sample path for x t using the MDB and accept/reject with a M-H step 3 Conditional on x, propose a new c and accept/reject with a M-H step 4 Output state and return to step 2 In the case of partial and/or noisy measurements, step 2 can be replaced with 2 For t = 2,..., n 1, propose a new sample path for the interval pair (x t, x t+1 ) using the MDB, and accept/reject with a M-H step together with special updating steps for the first and last intervals

29 Modified diffusion bridge (MDB) for diffusions Need a tractable process q(x t+1 c, x t, d t+1 ) that is locally equivalent to π(x t+1 c, x t, d t+1 ) Diffusion dx t = µ(x t )dt + β(x t ) 1 2 dw t The nonlinear diffusion bridge dx t = x 1 X t 1 t dt + β(x t) 1 2 dwt hits x 1 at t = 1, yet is locally equivalent to the true diffusion as it has the same diffusion coefficient This forms the basis of an efficient proposal; see Durham & Gallant (2002), Chib, Pitt & Shephard (2004), Delyon & Hu (2006), and Stramer & Yan (2007) for technical details

30 Irreducibility Introduction for diffusions The above algorithm will give an effective irreducible MCMC algorithm provided that the diffusion term of the SDE does not depend on any model parameters On the other hand, if the diffusion term does depend on model parameters, then the algorithm will be reducible, due to the fact that there is an infinite amount of information in the augmented sample path x In practice we work with finite discretisations, so the information isn t infinite, but sufficient to make the algorithm intolerably bad There is a solution to this based on a reparametrisation of the process technical details in Golightly & Wilkinson (2008)

31 Comparison of results Introduction for diffusions a b c c c c

32 Calibration of large simulators for diffusions CaliBayes Integration of GRID-based post-genomic data resources through Bayesian calibration of biological simulators Bayesian model calibration is concerned with the problem of parameter estimation, model validation, design and analysis based only on the ability to forward simulate from the model It is particularly appropriate for slow and/or complex models and/or data, where likelihood-based methods are computationally infeasible

33 for diffusions MCMC-based fully 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 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

34 Generic problem Introduction for diffusions Model parameters: c (Stochastic) model output: x (Noisy and/or partial) data: D For simplicity suppose that c D x (but can be relaxed) We wish to treat the model as a black box, which can can only be forward-simulated We are thinking about data relating to a single realisation of the model (so no need to explicitly treat initial conditions), but replicate runs and multiple conditions can be handled sequentially (as will become clear)

35 MCMC-based for diffusions Target: π(c D) Specify a measurement error model, π(d x) eg. just a product of Gaussian or t densities Generic MCMC scheme: Propose c f (c c) Accept with probability min{1, A}, where A = π(c ) π(c) f (c c ) f (c c) π(d c ) π(d c) π(d c) is the marginal likelihood (or observed data likelihood, or...)

36 Special case: deterministic model for diffusions Deterministic function g( ) such that x = g(c) Then π(d c) = π(d c, g(c)) = π(d c, x) = π(d x) Here π(d x) is just the measurement error model eg. simple product of Gaussian or t densities This setup is somewhat simplistic for the deterministic case, but we are really more concerned with the stochastic case...

37 Stochastic model Introduction for diffusions Can t get at the marginal likelihood directly, so make the target π(c, x D), where x is the true simulator output which led to the observed data... Clear that we can marginalise out x if necessary, but typically of inferential interest anyway Use ideas from likelihood-free MCMC (Marjoram et al, 2003) Propose (c, x ) f (c c)π(x c ), so that x is a forward simulation from the (stochastic) model based on the proposed new c A = π(c ) π(c) f (c c ) f (c c) π(d x ) π(d x)

38 Likelihood-free MCMC for diffusions Again π(d x) is a simple measurement error model... Crucially, because the proposal exploits a forward simulation, the acceptance probability does not depend on the likelihood of the simulator output important for complex stochastic models This scheme is completely general, and works very well provided that D is small Problem: If D is large, the MCMC scheme will mix very poorly (very low acceptance rates) Solution: Exploit the Markovian structure of the process, and adopt a sequential approach, updating one observation at a time...

39 Sequential likelihood-free algorithm for diffusions Data D t = {d 1,..., d t }, D D n. Sample paths x t {x s t 1 < s t}, t = 2, 3,..., n, so that x {x 2,..., x n }. 1 Assume at time t we have a (large) sample from π(c, x t D t ) (for time 0, initialise with sample from prior) 2 Run an MCMC algorithm which constructs a proposal in two stages: 1 First sample (c, x t ) π(c, x t D t ) by picking at random and perturbing slightly (sampling from the kernel density estimate) 2 Next sample x t+1 by forward simulation from π(x t+1 c, x t ) 3 Accept/reject (c, x t+1 ) with A = π(d t+1 x t+1 ) π(d t+1 x t+1 ) 3 Output state, put t : = t + 1, return to step 2.

40 for diffusions Advantages of the sequential algorithm 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 For slow stochastic models, there is considerable interest in developing fast emulators and embedding these into MCMC algorithms

41 for diffusions 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

42 High throughput data Introduction High-throughput data Top-down models Sparse VAR(1) models Linking VAR(1) models to the CLE Although we would prefer to use high-resolution single-cell time course data for all of our statistical modelling, such data is difficult to obtain in a high throughput (HTP) fashion for large numbers of proteins We therefore wish to integrate HTP data into our modelling approach. Such data is usually of lower resolution and possessing relative poor dynamic range, but provides (simultaneous) measurement of very large numbers of biological features Time course HTP data is most useful for making inferences about dynamics Time course microarray data is a typical example

43 High-throughput data Top-down models Sparse VAR(1) models Linking VAR(1) models to the CLE Time course microarray data w0 w1 w2 w3 w4 m0 m1 m2 m3 m4 YIL127C IPI1 LUG1 KRE33 RPC31 IMP4 TRM3 NOP1 NOP9 NOG1 ALB1 UTP21 YTM1 URB2 IKI1 BRX1 RLP7 RRP8 CIC1 ECM1 FYV7 RRP5 HCA4 BMS1 ENP2 NOP15 MPP10 DBP9 NOC3 RIX1 RPF2 TSR1 DUS3 RRP43 HAS1 TRM5 SDA1 TAD3 TRM2 REX4 PWP1 NMD3 NOP7 ATO3 MRT4 YDR161W UTP23 RRP45 LSG1 RSA3 EMG1 RTT106 GAR1 NOP58 RRP7 APT1 PAP2 TPK3 NOP12 NOP2 CGR1 RRP17 FOL1 LRP1 NIP7 SPB1 DBP10 NSA2 NUG1 KRR1 SMB1 RPO26 YCR016W SPC1 NSG1 HEM3 REI1 YNL313C IMP3 PXR1 TRM8 DAS2 UTP18 ESF2 ARX1 RRP12 RRP1 URB1 RIO1 SSF1 NOC2 DRS1 IPI3 UTP9 FAL1 SUA5 LHP1 DHR2 DUS1 DBP7 NOP53 BUD22 RNT1 YOR051C NOP4 ENA5 EBP2 SPT6 NCS2 DBP3 MTR3 ELP4 NAR1 CIA1 YOR021C DIS3 NHP2 HMT1 YLR363W A NOP10 YOL092W TIF6 RLP24 YNL108C NOP16 YML082W CFD1 YNL234W RRP15 PUF6 NSR1 ATC1 RMT2 SAS10 LOC1 RRP14 NOP8 NOP14 MRD1 RPA34 FEN1 MAK11 LTV1 PPT1 PMT5 BUD23 RRS1 DPH1 MAK21 FAF1 NOP13 SOF1 PWP2 UTP14 URA7 DBP6 RPC17 DBP8 GCD10 UTP5 URK1 UTP6 TPA1 RPA135 ERB1 RTT10 TAF6 LCP5 YMR310C TRM1 KRI1 DUS4 IOC4 MTR4 RPA190 MAK5 UTP20 RRP9 AFFX YER148w5 EDC3 BRE1 RAP1 YJL193W RKI1 PRS4 RPB3 RPS27A MAK16 TIF4631 ARB1 POL5 GIR2 UTP15 YMR259C RPC40 MES1 SPE2 PHO90 RER2 NAM7 YLR036C RPO31 NNT1 ROK1 YDL063C YIL096C CNS1 YIL091C BUD21 RSA4 ENP1 BFR2 DIP2 ELP3 UTP10 NSA1 YML096W RPB5 EFG1 UTP13 YGR272C RPA49 YGR283C NOC4 SHQ1 NAN1 RPA43 UTP8 KTI12 DRN1 RRB1 TRM7 MNI1 AAH1 NOG2 PNO1 GCD14 YLR063W VTS1 YOR304C A MPS2 LIA1 YHR032W YBL054W FOB1 UTP30 NPL6 SRP72 MSH1 NUP85 RPL22B AFFX YER148w3 GCD2 ARP4 SIS2 CTR9 ASP1 YPL279C YOR390W HIS6 POP8 NUP57 SNU56 YBL028C RHO3 SME1 FCF1 MCK1 SMX2 GPI11 GRC3 YEL048C DEG1 NUP120 HGH1 TAF5 MNP1 RPA12 RCL1 TSR2 FAP1 TAH18 NUP192 SGD1 YOL022C YML108W HTL1 LAG1 SIK1 SVF1 UME1 FLC3 BDF2 COG7 STB6 MSS116 OST6 FUN30 YJR124C NUP100 SPC19 ARP7 MNN10 MIS1 YLR042C AAT1 STH1 NUD1 RPB8 DPH2 SPC110 SFH1 RSC8 RSC9 ABF1 SWI6 AFFX r2 Sc EAF5 3 RSC58 PAM18 RSC4 HEK2 REB1 SET2 AFFX r2 Sc EAF5 M GCN5 RPB9 AFFX r2 Sc EAF5 5 SNL1 VPS9 THI80 DCP1 YBR028C YML018C TBF1 SPB4 AAC3 ANT1 SAS2 GOT1 TRM44 ORC2 SPT15 HTA2 HTB2 SYG1 DSN1 IXR1 HTZ1 YHP1 SRL1 AFFX Sc J BOI2 PSA1 TUP1 YGL101W YIP3 DAD3 TOS1 EMP70 KAR1 BSP1 GIN4 YAR066W YAR068W SCH9 STR3 RRM3 RPC19 DUT1 ARG80 DCD1 PML1 MTW1 RSC2 TPT1 PTC2 YKL061W HOM2 NHP6B YAH1 TOM20 RSM26 MCM10 OPT1 GLN3 LYS4 YJR111C ACO2 DAD1 NNF1 TFC6 SMX3 TRS20 NCS6 KIN4 TIM18 FAP7 SRM1 BUD6 SWC3 STE12 SLM2 SUN4 YML081W PTP3 HMS2 TEA1 BCP1 NCL1 MKC7 PPH3 CBF5 CDC31 YMR230W A DLT1 ARP9 YTA6 YHR020W LEU9 KEL1 YMC2 TOA2 PUS1 PAT1 MED2 HFI1 ECM22 SAP190 IKI3 GGC1 ARO2 MET22 PSF2 ARO4 TIM44 MRPL20 HIS1 RLI1 HSH155 GPM3 HIR2 RRN10 PRP19 AFFX YER148wM DOT1 TAF14 YCK2 YBR016W RPB7 TSC10 FUR1 PRP9 SAS3 MLP2 YBR271W NIP1 YGR251W TRL1 HIT1 LSM4 TGS1 DED1 YNL022C PMU1 SRP102 MET31 TRM9 SPT8 THO1 MED6 GAL11 SAM3 SVL3 CHD1 URN1 HEM2 TCB2 KAP123 ZRC1 LSM2 GDT1 AUR1 SSO2 POM152 BIM1 SPC98 KRE27 ORC1 RSN1 SMD1 ANB1 YBL081W FCY21 POP5 EAF5 SYS1 NHP6A YLR455W YPL105C YLR091W CLB3 MET16 DAD2 TYW1 ATR1 YJL047C A SCT1 UPC2 LRG1 YDR444W ISY1 RTG3 SKP2 BRL1 YLR253W CSR1 LYP1 STB4 ILV1 MIP1 HEM1 BBP1 OPY1 BRR1 SGF29 KAR5 FSH1 SKI8 DEP1 SLS1 TIF3 RPS26B LEO1 MDN1 APA1 VTC2 SMC3 BIK1 YOL014W TOF1 RAS1 PRP43 ELP2 NOB1 RIX7 NUC1 RRP6 SCY1 YBR141C AIR1 GFD2 SAP185 BAS1 SNM1 ADE6 LOS1 RBG1 NMA1 GCD7 ILS1 SPA2 TIM21 YNL097C A UBC9 RSC3 SWI3 SAC7 YDR179W A CIN8 NUP145 FMP23 EST3 ASG1 SET1 NUP53 GCR2 GON7 YBR197C NCB2 THR1 UBP12 MDE1 YOL086W A BPL1 CEF1 GCS1 CTR1 SAP155 BUD16 FCY2 VAS1 ROG1 TYW3 CDC43 YJR011C YDR336W CKA1 YOR342C PSR1 PRO1 SPT4 AIM22 TRP2 SAT4 BDF1 PCL10 PRP42 COG3 RRN9 PPM2 MET8 JJJ3 YER140W ALR1 CTF19 DFR1 YPL245W SPC42 SPS4 THP2 FET4 YNR061C ERD1 NMD4 ESF1 YDR034C A PHM6 YVH1 VTC3 UTP11 VCX1 TMA16 AIM20 HST3 RAI1 CCT3 RNA14 YHL039W DLD3 EFR3 IMD3 MMP1 RPP1 VID27 SRP40 PMP2 MEU1 FCP1 NRP1 YNL162W A DAT1 SER2 AFT2 RPL18B GEA1 MET2 MET6 SAM1 MXR1 HPT1 MNT2 YPL162C MCM6 LCB3 SLX9 TRM82 UTP4 TRM11 YOR006C NUP82 ADE4 TIM9 RPS28B SDO1 YLR287C FIT1 RCK1 YLR126C PRM6 MRS4 YDR222W MOT1 YKR041W GTT3 AGA1 SPP41 FIR1 YGL230C FKH1 GAS3 POS5 IRC8 ADE8 YPR022C DAL81 SYC1 HOL1 RPS9A YDR210W PLB2 CRP1 SUR7 HHO1 CLB1 MET14 RPS14B MET10 FTR1 MUP1 ECM17 PCL1 CSI2 AXL2 SVS1 YOX1 CLN1 KAR9 SUR2 CLN2 MET13 NAM8 HTA1 HHT2 HHF1 YOR246C AIM34 HTB1 HHT1 DSE2 DSE1 PRY3 LAC1 CTS1 COG4 BET1 SBE22 AFFX Sc U YGR146C A YNL300W PRY2 CRR1 TOF2 FAA3 SFG1 YPR013C PHO4 ELM1 VHT1 SWI1 WHI3 MDM36 NBA1 LEU3 SFL1 YCR051W FHL1 INO2 CWC21 YML007C A MNN1 YNL217W PEX2 MRM1 YAP7 YPL264C SGF73 BDS1 YIL169C HPF1 FIG2 ATF1 DMA2 EGT2 YNL046W DSE4 AMN1 SCW11 BUD9 ISR1 HO PET122 MCD1 PIR1 PCL9 DSE3 NIS1 SIC1 CYK3 PCL2 CDC6 SPO16 SWE1 YOL019W CLB6 SCW10 ORT1 ECM40 TRP4 CPA2 HOM3 RHB1 ARG8 SAM4 ICY2 APQ12 YMC1 YGL117W CAF16 ARG4 YMR321C TRP3 ARG1 ADH5 ARO3 HMO1 PYC2 HXT2 YHK8 UGA3 YHR122W DBP2 LYS1 SDT1 STR2 ZAP1 YCR023C QDR3 SOL1 IGO2 PSD1 ADK2 SNZ1 CUP1 1 MDG1 SNO1 MRPL19 MEH1 ILV3 ILV6 AIM5 HIS3 HIS7 NRK1 TMT1 YGR109W A YGR017W IMG2 HAP3 CCR4 FYV4 HIS4 YHI9 IRC7 ARG2 LYS21 HIS5 YAT2 PEX21 YGL059W SSU1 ZRT2 NPL3 CAF40 HLJ1 PUS7 MEP1 UBC11 SAS5 YER156C BRE5 ZPR1 AIM33 PHD1 ZRT1 ODC2 YNL024C GRX4 GLT1 YGR035C FRM2 TOS2 NAT4 APJ1 FES1 FUS1 PRO2 AML1 HDA1 PUS9 RAX1 TOP1 MDM20 NUP170 ALT1 KTI11 YLR364W FRE7 MMS21 YLR407W YBR184W YPL067C CWP2 ZTA1 YGR109W B FRE4 MET1 MET3 QDR2 FAR1 BIO2 YGL039W ASH1 AIM44 YLR049C YBR071W YLR194C RME1 PCL5 PST1 CRH1 ICY1 AGP1 YLR152C YIL165C SHM2 ARO10 YMR182W A YJL217W YJL213W RTS3 HXT4 CWP1 GSY2 WSC4 YBR147W HAP4 SPO74 SRX1 YOL163W YPT53 YOL162W TSA2 YNR034W A SGA1 EDS1 POG1 GUP2 YPL088W YTP1 VMR1 HMX1 HXT11 GRE1 CAR2 YOR338W MTL1 RAD16 UBI4 HOR2 ICL2 COX5B YHR140W PDH1 MAG1 CTT1 SYM1 YJL133C A YOR052C ADR1 YPL014W SPI1 YOR062C PIC2 SDH2 YPR157W OPI3 YML131W YCL049C YOR192C C SPO1 YOR289W YSW1 YCL021W A FMO1 YBL039W A AIM41 ECM38 YNL134C PET10 SDP1 ATG22 TMA17 YOL114C YLR177W YJL016W FRT2 YBR284W MCR1 YOR227W TPS1 COS12 SSP1 FMP48 MTH1 SPO11 SPS19 PES4 YEL057C GSC2 YGR067C SNZ3 YFR012W A HSP31 OSW2 USV1 UIP4 GIP2 TDH1 AFFX r2 Sc TDH1 3 AFFX r2 Sc TDH1 M AFFX r2 Sc TDH1 5 GDH3 PUT1 YOR152C MSN4 SOM1 SSE2 CAT8 PRB1 CIT3 EDC2 OM45 CYT1 NDE1 GOR1 IDP2 YFL054C RCN2 YER158C STF2 YGR153W DIT1 THI13 MIP6 YNR014W YKL161C NQM1 AGX1 YDR018C GND2 BNA2 PRM5 YNL058C GTT1 MMR1 PTP2 YDL124W YIL055C PPE1 PLM2 RNR3 DUN1 YOR114W RAD51 RNR2 YNL092W YOL048C UBP11 FSH3 YBR204C YGR053C ARA1 ALG13 FIS1 SPO12 FBP26 UGX2 YLR345W GGA1 YJL163C SDS24 PFK26 HFD1 GLK1 YPL119C A YPK2 NDE2 DMC1 SRL3 GAL3 VPS73 YPS6 PYK2 CRG1 ECM4 YOL083W ATG19 TPS2 YDR379C A DCS1 YNL200C ROM1 YBL101W A PEX27 UBC5 PAU3 PRY1 YKR011C YHR138C YJR008W EMI2 NGL3 YKL133C YOR020W A YNL115C FYV10 UGA2 EMP46 YDL027C ATG8 ALD2 YGR250C SIA1 YKL151C GSM1 MPM1 KIN82 YJR096W YJL132W COS111 PIG1 PLB1 FUN14 PRX1 GPD1 TPK1 YBR085C A YBR056W YGR146C ALG14 YJL070C LAP4 PNC1 RNY1 AFR1 YNL305C XKS1 HOR7 ATH1 YBR053C UBC8 NTH1 STF Value Color Key

44 Time course microarray data High-throughput data Top-down models Sparse VAR(1) models Linking VAR(1) models to the CLE Small number of very high-dimensional short time courses

45 Introduction High-throughput data Top-down models Sparse VAR(1) models Linking VAR(1) models to the CLE Using high-throughput omics data to fit statistical models and uncover networks of interacting bio-molecules The models are often static, but there is increasing interest in dynamic models, fitted to time course data Many approaches, including Dynamic Bayesian Networks (DBNs) for discretised data and sparse Dynamic Linear Models (DLMs) for (normalised) continuous data A special case of the DLM is the sparse vector auto-regressive model of order 1, known as the sparse VAR(1) model, and this appears to be a particularly effective model for uncovering dynamic network interactions (Opgen-Rhein & Strimmer, 2007)

46 Sparse VAR(1) model Introduction High-throughput data Top-down models Sparse VAR(1) models Linking VAR(1) models to the CLE Observe a p-dimensional vector X t, at each of n time points, t = 1,..., n (with p >> n) X t+1 = µ + A(X t µ) + ɛ t, ɛ t N(0, V ) The p p matrix A is assumed to be sparse (ie. most elements are expected to be exactly zero) Sparsity can be modelled in many ways. Simplest: Pr(a ij 0) = π, i, j, a ij a ij 0 N(0, σ 2 ), i, j The non-zero structure of A can be associated with a graph (network) of dynamic interactions (non-zero a ij implies arc from node j to node i)

47 High-throughput data Top-down models Sparse VAR(1) models Linking VAR(1) models to the CLE Inference for model parameters and structure from data Can get a point estimate for the network structure by computing a shrinkage estimate of A and then thresholding (Opgen-Rhein & Strimmer, 2007) Can also use Bayesian MCMC methods to explore the space of plausible interaction graphs MCMC methods allow computation of useful quantities such as Pr(a ij 0 D) Inference for graphs is a hard problem...

48 MCMC for sparse VAR(1) models High-throughput data Top-down models Sparse VAR(1) models Linking VAR(1) models to the CLE RJ-MCMC algorithm to explore both graphical structure and model parameters (auto-regressive coefficients, mean vector, variance components) routine to develop and implement, but exhibits very poor mixing in high-dimensional settings Conditional on the graphical structure, possible (but messy) to develop a variational algorithm which gives an approximate marginal log-likelihood for the model after a few iterations can embed this in a very simple MCMC algorithm to explore just the graphical structure Even this algorithm mixes poorly for large p (say, p > 200), but there are 2 p2 graphs, after all... Could probably get reasonable speed-up by using (parallel) sparse matrix algorithms

49 Sparse VAR(1) sparsity structure High-throughput data Top-down models Sparse VAR(1) models Linking VAR(1) models to the CLE Image plot Col Row

50 High-throughput data Top-down models Sparse VAR(1) models Linking VAR(1) models to the CLE Sparse VAR(1) autoregressive parameter posteriors (a) (b) (c) Density Probability Density Probability Density Probability Value Value Value (d) (e) (f) Density Probability Density Probability Density Probability Value Value Value (g) (h) (i) Density Probability Density Probability Density Probability Value Value Value

51 Application to some EEG data High-throughput data Top-down models Sparse VAR(1) models Linking VAR(1) models to the CLE Image plot Col Row

52 Linking VAR(1) models to the CLE High-throughput data Top-down models Sparse VAR(1) models Linking VAR(1) models to the CLE In the CLE, expand h(x t, c) about a typical value, x h(x t, c) h( x, c) + h (x t x) x x Plug into CLE (first-order for the drift and zeroth-order for diffusion) [ dx t = S h( x, c) + h ] (X t x) dt+ S diag{h( x t, c)}s x x dw t Gives a multivariate Gaussian Ornstein-Uhlenbeck (OU) process

53 Linear Gaussian OU process High-throughput data Top-down models Sparse VAR(1) models Linking VAR(1) models to the CLE Linearisation of the CLE gives a Gaussian OU process dx t = H(X t µ)dt + Λ dw t where H = S h x x µ = x H 1 Sh( x, c) Λ = S diag{h( x t, c)}s the auto-regressive matrix H is sparse (due the the sparsity of S and the Jacobian)

54 Time discretisation Introduction High-throughput data Top-down models Sparse VAR(1) models Linking VAR(1) models to the CLE The Gaussian OU process is analytically tractable In particular, it can be time-discretised exactly, giving a VAR(1) model with auto-regressive matrix A = exp{ H t} Although H will be very sparse, A typically won t be as sparse, due to fill-in Two possibilities: 1 Although A = exp{ H t} isn t so sparse, the first order approx A I H t is (could get same result by discretising time first with an Euler approximation and then linearising) 2 Don t fit VAR(1) models with sparse A, but models with sparse log(a)

55 Issues Introduction Discussion Acknowledgements References H doesn t have the same sparsity structure as S in general, so there s still work to do to get at sparsity of S, even if we can infer the sparsity of H Population averaged versus single-cell data complicates argument slightly (the mean of the CLE is typically not the RRE), but doesn t change the essential conclusions Consistency of sparsity structures under a log transformation Consistency of sparsity structures under marginalisation can t measure everything of interest to a bottom-up modeller in a top-down high-throughput experiment (rules from graphical modelling theory can be applied)

56 Introduction Discussion Acknowledgements References The CLE sits at the interface between many different but related bottom-up modelling paradigms The Gaussian OU process formed by linearising the CLE links naturally to a class of sparse discrete time linear models used as part of a top-down modelling approach for high-dimensional HTP data Direct fully for the CLE based on discrete time course data is also possible, and is especially useful for inferring quantitative dynamics Likelihood-free methods for inference can sometimes work well in this context, and are very simple and general in terms of their applicability

57 Acknowledgements Introduction Discussion Acknowledgements References Stochastic kinetic models: Richard Boys, Tom Kirkwood, Colin Gillespie, Conor Lawless, Pete Milner, Daryl Shanley, Carole Proctor, Wan Ng (funding from BBSRC, EPSRC, MRC, DTI, Unilever) Likelihood-free modelling: Richard Boys, Daniel Henderson, Eryk Wolski, Jake Wu (funding from BBSRC) Inference for diffusions: Andrew Golightly (studentship funding from EPSRC) Sparse VAR(1) modelling: Richard Boys, Colin Gillespie, Adrian Houghton, Guiyuan Lei (funding from BBSRC and studentship funding from EPSRC)

58 Discussion Acknowledgements References Boys, R. J., D. J. Wilkinson and T. B. L. Kirkwood (2008). Bayesian inference for a discretely observed stochastic kinetic model. Statistics and Computing, 18(2), Golightly, A. and D. J. Wilkinson (2006). Bayesian sequential inference for stochastic kinetic biochemical network models. Journal of Computational Biology. 13(3), Golightly, A. and D. J. Wilkinson (2008). for nonlinear multivariate diffusion models observed with error. Computational Statistics and Data Analysis. 52(3), Henderson, D. A., Boys, R. J., Krishnan, K. J., Lawless, C., Wilkinson, D. J. (2008). Bayesian emulation and calibration of a stochastic computer model of mitochondrial DNA deletions in substantia nigra neurons. Journal of the American Statistical Association, to appear. Wilkinson, D. J. (2006). Stochastic Modelling for Systems Biology. Chapman & Hall/CRC Press.