Inferring Latent Functions with Gaussian Processes in Differential Equations

Transcription

1 Inferring Latent Functions with in Differential Equations Neil Lawrence School of Computer Science University of Manchester Joint work with Magnus Rattray and Guido Sanguinetti 9th December 26

2 Outline 1 Transcription Factor Concentrations 2 Exact Inference in Linear System Toy Problem Biological Problem 3 Riemann Quadrature Linear Response with MLP Kernel Non-linear Responses

3 Advert! PUMA Project Work is part of PUMA project for Microarray analysis Will be appointing a new Post-doc in the new year!

4 Online Resources All source code and slides are available online This talk available from home page (see talks link on side). Scripts available in the gpsim toolbox MATLAB commands used for examples given in typewriter font.

5 Inference about functions Differential equation models of systems often have functional unknowns. We will focus on a protein concentration example. Gaussian processes (GPs) are probabilistic models for functions. O Hagan [1978, 1992], Rasmussen and Williams [26] GPs provide a framework for performing inference about these functions in the presence of uncertainty.

6 Overview Transcription Factor Concentrations Transcription Factor Inference Model the dynamics of gene transcription. Infer transcription factor concentration (TFC). Infer parameters of the transcription model (decay rates, sensitivities etc.) Infer these quantities using a set of known target genes.

7 Methodology Transcription Factor Concentrations Latent function Treat TFC as a latent function in a differential equation model. Assume a Gaussian process (GP) prior distribution for the latent function. Derive GP covariance jointly for genes and transcription factor. Maximise likelihood with respect to parameters (mostly physically meaningful).

8 Overview Transcription Factor Concentrations Background Understanding of cellular processes is improving through microarrays, chromatin immunoprecipitation etc.. Quantitative description of regulatory mechanisms requires: transcription factor (TF) concentrations, gene-specific constants such as the baseline expression, mrna decay rate and sensitivity to TF concentrations (TFCs).

9 Overview Transcription Factor Concentrations Background II These quantities are hard to measure directly. We show how the They can be inferred using a systems biology model and Gaussian processes (GPs). Our work provides a extensible, principled framework for attacking this problem.

10 Transcription Factor Concentrations GP Advantages GPs allow for inference of continuous profies, accounting naturally for temporal structure. GPs avoid cumbersome interpolation to estimate mrna production rates. GPs deal consistently with the uncertainty inherent in the measurements. GPs outstrip MCMC for computational efficiency. GPs have previously been proposed for solving differential equations [Graepel, 23] and dynamical systems [Murray-Smith and Pearlmutter].

11 Exact Inference in Linear System Toy Problem Biological Problem p53 Inference [Barenco et al., 26] Data consists of T measurements of mrna expression level for N different genes. We relate gene expression, x j (t), to TFC, f (t), by dx j (t) dt = B j + S j f (t) D j x j (t). (1) B j basal transcription rate of gene j, S j is sensitivity of gene j D j is the decay rate of the mrna. Dependence of mrna transcription rate on TF is linear.

12 Linear Response Solution Exact Inference in Linear System Toy Problem Biological Problem Solve for TFC The equation given in (1) can be solved to recover x j (t) = B t j + S j exp ( D j t) f (u) exp (D j u) du. (2) D j If we model f (t) as a GP then as (2) only involves linear operations x j (t) is also a GP. If we use the RBF covariance everything is analytic.

13 Covariance Functions Visualisation of RBF Covariance Exact Inference in Linear System Toy Problem Biological Problem RBF Kernel Function k ( t, t ) = exp ( (t t ) 2 l 2 ) Covariance matrix is built using the time inputs to the function. For the example above it was based on Euclidean distance. The covariance function is also known as a kernel. m n

14 Covariance Samples Exact Inference in Linear System Toy Problem Biological Problem demcovfuncsample (oxford toolbox) Figure: RBF kernel with l =.25, α = 1

15 Covariance Samples Exact Inference in Linear System Toy Problem Biological Problem demcovfuncsample (oxford toolbox) Figure: RBF kernel with l = 1, α = 1

16 Computation of Joint Covariance Exact Inference in Linear System Toy Problem Biological Problem Covariance Function Computation We rewrite equation (2) as where x j (t) = B j D j + L j [f ] (t) t L j [f ] (t) = S j exp ( D j t) f (u) exp (D j u) du (3) is a linear operator.

17 Induced Covariance Exact Inference in Linear System Toy Problem Biological Problem Gene s Covariance The new covariance function is then given by cov ( L j [f ] (t), L k [f ] ( t )) = L j L k [k ff ] ( t, t ). more explicitly k xj x k `t, t = S j S k exp ` D j t D k t Z t exp (D j u) Z t exp `D k u k ff `u, u du du. With RBF covariance these integrals are tractable.

18 Covariance Result Exact Inference in Linear System Toy Problem Biological Problem Covariance Result where k xj x k ( t, t ) = S j S k π 2 h `t kj, t = exp (γ k) 2 D j + D k jexp ˆ D `t k t» t t erf l Here γ k = D kl 2. [ hkj ( t, t ) + h jk ( t, t )] «t γ k + erf l exp ˆ `D k t»erf t «+ D j γ k + erf (γ k ) l ff. + γ k «

19 Cross Covariance Exact Inference in Linear System Toy Problem Biological Problem Correlation of x j (t) and f (t ) Need the cross-covariance terms between x j (t) and f (t ), which is obtained as k xj f ( t, t ) t ( = S j exp ( D j t) exp (D j u) k ff u, t ) du. (4) For RBF we have k `t πlsj e xj f, t 2γ j = 2 exp ˆ D `t t j t»erf ««t t γ j + erf + γ j. l l

20 Posterior for f Exact Inference in Linear System Toy Problem Biological Problem Prediction for TFC Standard Gaussian process regression techniques [see e.g. Rasmussen and Williams, 26] yield f post = K f x K 1 xx x K post ff = K ff K f x K 1 xx K xf Model parameters B j, D j and S j estimated by type II maximum likelihood, log p (x) = N (x, K xx )

21 Covariance for Transcription Model Exact Inference in Linear System Toy Problem Biological Problem RBF Kernel function for f (t) x i (t) = B t i + S i exp ( D i t) f (u) exp (D i u) du. D i Joint distribution for x 1 (t), x 2 (t) and f (t). Here: D 1 S 1 D 2 S f(t) x 1 (t) x 2 (t) f(t) x 1 (t) x 2 (t)

22 Joint Sampling of x (t) and f (t) Exact Inference in Linear System Toy Problem Biological Problem gpsimtest Figure: Left: joint samples from the transcription covariance, blue: f (t), cyan: x 1 (t) and red: x 2 (t). Right: numerical solution for f (t) of the differential equation from x 1 (t) and x 2 (t) (blue and cyan). True f (t) included for comparison.

23 Joint Sampling of x (t) and f (t) Exact Inference in Linear System Toy Problem Biological Problem gpsimtest Figure: Left: joint samples from the transcription covariance, blue: f (t), cyan: x 1 (t) and red: x 2 (t). Right: numerical solution for f (t) of the differential equation from x 1 (t) and x 2 (t) (blue and cyan). True f (t) included for comparison.

24 Joint Sampling of x (t) and f (t) Exact Inference in Linear System Toy Problem Biological Problem gpsimtest Figure: Left: joint samples from the transcription covariance, blue: f (t), cyan: x 1 (t) and red: x 2 (t). Right: numerical solution for f (t) of the differential equation from x 1 (t) and x 2 (t) (blue and cyan). True f (t) included for comparison.

25 Covariance for Transcription Model Exact Inference in Linear System Toy Problem Biological Problem RBF Kernel function for f (t) x i (t) = B t i + S i exp ( D i t) f (u) exp (D i u) du. D i Joint distribution for x 1 (t), x 2 (t) and f (t). Here: D 1 S 1 D 2 S f(t) x 1 (t) x 2 (t) f(t) x 1 (t) x 2 (t)

26 Noise Corruption Exact Inference in Linear System Toy Problem Biological Problem Estimate Underlying Noise Allow the mrna abundance of each gene at each time point to be corrupted by noise, for observations at t i for i = 1,..., T, y j (t i ) = x j (t i ) + ɛ j (t i ) (5) with ɛ j (t i ) N (, σ 2 ji ). Estimate noise level using probe-level processing techniques of Affymetrix microarrays (e.g. mmgmos, [Liu et al., 25]). The covariance of the noisy process is then K yy = Σ + K xx, with Σ = diag ( σ 2 11,..., σ2 1T,..., σ2 N1,..., σ2 NT ).

27 Artificial Data Exact Inference in Linear System Toy Problem Biological Problem Toy Problem Results from an artificial data set. We used a known TFC and derived six mrna profiles. Known TFC composed of three Gaussian basis functions. mrna profiles derived analytically. Fourteen subsamples were taken and corrupted by noise. This data was then used to infer a distribution over plausible TFCs.

28 Artificial Data Results Exact Inference in Linear System Toy Problem Biological Problem demtoyproblem Figure: Left: The TFC, f (t), which drives the system. Middle: Five gene mrna concentration profiles each obtained by using different parameter sets {B i, S i, D i } 5 i=1 (lines) along with noise corrupted data. Right: The inferred TFC (with error bars).

29 Results Exact Inference in Linear System Toy Problem Biological Problem Linear System Recently published biological data set studied using linear response model by Barenco et al. [26]. Study focused on the tumour suppressor protein p53. mrna abundance measured for five targets: DDB2, p21, SESN1/hPA26, BIK and TNFRSF1b. Quadratic interpolation for the mrna production rates to obtain gradients. They used MCMC sampling to obtain estimates of the model parameters B j, S j, D j and f (t).

30 Linear response analysis Exact Inference in Linear System Toy Problem Biological Problem Experimental Setup We analysed data using the linear response model Raw data was processed using the mmgmos model of Liu et al. [25] which provides variance as well as expression level. We present posterior distribution over TFCs. Results of inference on the values of the hyperparameters B j, S j and D j. Samples from the posterior distribution were obtained using Hybrid Monte Carlo (see e.g. Neal, 1996).

31 Linear Response Results Exact Inference in Linear System Toy Problem Biological Problem dembarenco Figure: Predicted protein concentration for p53. Solid line is mean, dashed lines 95% credibility intervals. The prediction of [Barenco et al., 26] was pointwise and is shown as crosses.

32 Results Transcription Rates Exact Inference in Linear System Toy Problem Biological Problem Estimation of Equation Parameters dembarenco DDB2 hpa26 TNFRSF2b p21 BIK Figure: Basal transcription rates. Our results (black) compared with Barenco et al. [26] (white).

33 Results Transcription Rates Exact Inference in Linear System Toy Problem Biological Problem Estimation of Equation Parameters dembarenco DDB2 hpa26 TNFRSF2b p21 BIK Figure: Sensitivities. Our results (black) compared with Barenco et al. [26] (white).

34 Results Transcription Rates Exact Inference in Linear System Toy Problem Biological Problem Estimation of Equation Parameters dembarenco DDB2 hpa26 TNFRSF2b p21 BIK Figure: Decays. Our results (black) compared with Barenco et al. [26] (white).

35 Linear Response Discussion Exact Inference in Linear System Toy Problem Biological Problem GP Results Note oscillatory behaviour, possible artifact of RBF covariance Rasmussen and Williams [see page 123 in 26]. Results are in good accordance with the results obtained by Barenco et al.. Differences in estimates of the basal transcription rates probably due to: different methods used for probe-level processing of the microarray data. Our failure to constrain f () =. Our results take about 13 minutes to produce Barenco et al. required 1 million iterations of Monte Carlo.

36 Riemann Quadrature Linear Response with MLP Kernel Non-linear Responses More Realistic Response All the quantities in equation (1) are positive, but direct samples from a GP will not be. Linear models don t account for saturation. Solution: model response using a positive nonlinear function.

37 Formalism Riemann Quadrature Linear Response with MLP Kernel Non-linear Responses Non-linear Response Introduce a non-linearity g ( ) parameterised by θ j dx j dt = B j + g(f (t), θ j ) D j x j x j (t) = B j + exp ` D Z t j t du g(f (u), θ j ) exp `D j u. D j The induced distribution of x j (t) is no longer a GP. Derive the functional gradient and learn a MAP solution for f (t). Also compute Hessian so we can approximate the marginal likelihood.

38 Implementation Riemann Quadrature Linear Response with MLP Kernel Non-linear Responses Riemann quadrature Implementation requires a discretised time. Compute the gradient and Hessian on a grid. Integrate them by approximate Riemann quadrature. We choose a uniform grid {t p } M p=1 so that = t p t p 1 is constant. The vector f = {f p } M p=1 is the function f at the grid points. Z t I (t) = f (u) exp `D j u du I (t) MX f (t p) exp `D j t p p=1

39 Log Likelihood Riemann Quadrature Linear Response with MLP Kernel Non-linear Responses Functional Gradient Given noise-corrupted data y j (t i ) the log-likelihood is TX log p(y f, θ j ) = 1 2 i=1 j=1 " NX `xj (t i ) y j (t i ) 2 σ 2 ji # log σji 2 NT 2 log(2π) The functional derivative of the log-likelihood wrt f is δ log p(y f ) δf (t) TX NX (x j (t i ) y j (t i )) = Θ(t i t) σ 2 g (f (t))e D j (t i t) i=1 j=1 ji Θ(x) Heaviside step function.

40 Log Likelihood Riemann Quadrature Linear Response with MLP Kernel Non-linear Responses Functional Hessian Given noise-corrupted data y j (t i ) the log-likelihood is TX log p(y f, θ j ) = 1 2 i=1 j=1 " NX `xj (t i ) y j (t i ) 2 σ 2 ji # log σji 2 NT 2 log(2π) The negative Hessian of the log-likelihood wrt f is w(t, t ) = TX Θ(t i t)δ `t NX t `xj (t i ) y j (t i ) σ 2 ji g (f (t))e D j (t i t) i=1 j=1 TX + Θ(t i t)θ `t NX i t σ 2 ji g (f (t)) g `f (t ) e D j (2t i t t ) i=1 j=1 g (f ) = g/ f and g (f ) = 2 g/ f 2.

41 Implementation II Riemann Quadrature Linear Response with MLP Kernel Non-linear Responses Combine with Prior Combine these with prior to compute gradient and Hessian of log posterior Ψ(f) = log p(y f) + log p(f) [see Rasmussen and Williams, 26, chapter 3] Ψ(f) log p(y f) = K 1 f f f 2 Ψ(f) f 2 = (W + K 1 ) K prior covariance evaluated at the grid points. Use to find a MAP solution via, ˆf, using Newton s algorithm. The Laplace approximation is then log p(y ) log p(y ˆf) 1 2ˆf T K 1ˆf 1 log I + KW. (7) 2 (6)

42 Example: linear response Riemann Quadrature Linear Response with MLP Kernel Non-linear Responses Using non-rbf kernels Start by taking g( ) to be linear. Provides sanity check and smooth (i.e. non-stochastic) covariance function. Avoids double numerical integral that would normally be required.

43 Oscillatory Behaviour Fix with MLP Covariance Riemann Quadrature Linear Response with MLP Kernel Non-linear Responses MLP Kernel Function k `t, t = αarcsin! wtt + b p (wt 2 + b + 1) (wt 2 + b + 1) A non-stationary covariance matrix [Williams, 1997]. Derived from a multi-layer perceptron (MLP). m n

44 Covariance Samples Riemann Quadrature Linear Response with MLP Kernel Non-linear Responses demcovfuncsample (oxford toolbox) Figure: RBF kernel with l =.25, α = 1

45 Covariance Samples Riemann Quadrature Linear Response with MLP Kernel Non-linear Responses demcovfuncsample (oxford toolbox) Figure: RBF kernel with l = 1, α = 1

46 Covariance Samples Riemann Quadrature Linear Response with MLP Kernel Non-linear Responses demcovfuncsample (oxford toolbox) Figure: MLP kernel with α = 8, w = 1 and b = 1

47 Covariance Samples Riemann Quadrature Linear Response with MLP Kernel Non-linear Responses demcovfuncsample (oxford toolbox) Figure: MLP kernel with α = 8, b = and w = 1

48 Response Results Riemann Quadrature Linear Response with MLP Kernel Non-linear Responses dembarencomap1, dembarencomap Figure: Predicted protein concentration for p53 using a MAP estimation with linear response model: Left: RBF prior on f (log likelihood -11.4); Right: MLP prior on f (log likelihood -15.6). Solid line is mean prediction, dashed lines are 95% credibility intervals. The prediction of Barenco et al. was pointwise and is shown as crosses.

49 Non-linear response analysis Riemann Quadrature Linear Response with MLP Kernel Non-linear Responses Non-linear responses Exponential response model (constrains protein concentrations positive). log (1 + exp (f )) response model. 3 1+exp( f ) Inferred MAP solutions for the latent function f are plotted below.

50 exp ( ) Response Results Riemann Quadrature Linear Response with MLP Kernel Non-linear Responses dembarencomap3, dembarencomap Figure: Predicted protein concentration for p53 using an exponential response: Left: shows results of using a squared exponential prior covariance on f (log likelihood -1.6); Right: shows results of using an MLP prior covariance on f (log likelihood -16.4). Solid line is mean prediction, dashed lines show 95% credibility intervals. The results shown are for exp(f ), hence the asymmetry of

51 log (1 + exp (f )) Response Results dembarencomap5, dembarencomap6 Riemann Quadrature Linear Response with MLP Kernel Non-linear Responses Figure: Predicted protein concentration for p53 using a log (1 + exp (x)) response: Left: shows results of using a squared exponential prior covariance on f (log likelihood -1.9); Right: shows results of using an MLP prior covariance on f (log likelihood -11.). Solid line is mean prediction, dashed lines show 95% credibility intervals. The results shown are for exp(f ), hence the

52 3 1+exp( f ) Response Results Riemann Quadrature Linear Response with MLP Kernel Non-linear Responses dembarencomap7, dembarencomap Figure: Predicted protein concentration for p53 using a sigmoid response: Left: shows results of using a squared exponential prior covariance on f (log likelihood -14.1); Right: shows results of using an MLP prior covariance on f (log likelihood ). Solid line is mean prediction, dashed lines show 95% credibility intervals. The results shown are for exp(f ), hence the asymmetry of Neil thelawrence, credibility Guidointervals. Sanguinetti and The Magnus prediction Rattray of Gaussian Barenco Processes et al. was pointwise and is

53 Discussion Riemann Quadrature Linear Response with MLP Kernel Non-linear Responses We have described how GPs can be used in modelling dynamics of a simple regulatory network motif. Our approach has advantages over standard parametric approaches: there is no need to restrict the inference to the observed time points, the temporal continuity of the inferred functions is accounted for naturally. GPs allow us to handle uncertainty in a natural way. MCMC parameter estimation in a discretised model can be computationally expensive. Parameter estimation can be achieved easily in our framework by type II maximum likelihood or by using efficient hybrid Monte Carlo sampling techniques All code on-line

54 Future Directions Riemann Quadrature Linear Response with MLP Kernel Non-linear Responses This is still a very simple modelling situation. We are ignoring transcriptional delays. Here we have single transcription factor: our ultimate goal is to describe regulatory pathways with more genes. All these issues can be dealt with in the general framework we have described. Need to overcome the greater computational difficulties.

55 Other Relevant Work Riemann Quadrature Linear Response with MLP Kernel Non-linear Responses Simpler genome wide models [Sabatti and James, 26, Sanguinetti et al., 26b,a]. Better MCMC models [Rogers et al., 26]. Sorry for anyone missed!

56 Open Question Riemann Quadrature Linear Response with MLP Kernel Non-linear Responses Functional Approximations Without sampling, how can we produce a better posterior approximation?

57 Acknowledgements Riemann Quadrature Linear Response with MLP Kernel Non-linear Responses Data and Support We thank Martino Barenco for useful discussions and for providing the data. We gratefully acknowledge support from BBSRC Grant No BBS/B/76X Improved processing of microarray data with probabilistic models.

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