Modelling Transcriptional Regulation with Gaussian Processes

Transcription

1 Modelling Transcriptional Regulation with Gaussian Processes Neil Lawrence School of Computer Science University of Manchester Joint work with Magnus Rattray and Guido Sanguinetti 8th March 7

2 Outline Application Application Methodology Overview 3 Linear Response with MLP Kernel Non-linear Responses

3 Online Resources Application 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.

4 Framework Application Methodology Overview Latent functions Many interaction networks have latent functions. Assume a Gaussian process (GP) prior distribution for the latent function. Gaussian processes (GPs) are probabilistic models for functions. O Hagan [978, 99], Rasmussen and Williams [6] Our Approach Take a differential equation model for the system. Derive GP covariance jointly for observed and latent functions. 3 Maximise likelihood with respect to parameters (mostly physically meaningful).

5 Methodology Overview This Talk Transcription Network Introduce Gaussian Processes for dealing with latent functions in transcription networks. Show how in a linear response model the latent function can be dealt with analytically. Discuss extensions to systems with non-linear responses.

6 p53 Inference [Barenco et al., 6] 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). () 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.

7 Linear Response Solution Solve for TFC The equation given in () can be solved to recover x j (t) = B t j + S j exp ( D j t) f (u) exp (D j u) du. () D j If we model f (t) as a GP then as () only involves linear operations x j (t) is also a GP.

8 Gaussian Processes Application GP Advantages GPs allow for inference of continuous profies, accounting naturally for temporal structure. GPs allow joint estimation of a mrna concentration and production rates (derivative observations). GPs deal consistently with the uncertainty inherent in the measurements. GPs outstrip MCMC for computational efficiency. Note: GPs have previously been proposed for solving differential equations [Graepel, 3] and dynamical systems [Murray-Smith and Pearlmutter].

9 Gaussian Processes Application Gaussian process is governed by a mean function and a covariance function p (f) = N ( f f, K ) k ij = k (t i, t j ), f i = f (t i ) Mean function often taken to be zero. Covariance function is any positive definite function, e.g. RBF Covariance.

10 Covariance Functions Visualisation of RBF Covariance RBF Kernel Function k ( t, t ) = α exp ( (t t ) l ) Covariance matrix is built using the time inputs to the function, t. For the example above it was based on Euclidean distance. The covariance function is also known as a kernel. m n

11 Covariance Samples Application demcovfuncsample sample from the prior Figure: RBF kernel with γ =, α =

12 Covariance Samples Application demcovfuncsample sample from the prior Figure: RBF kernel with l =, α =

13 Covariance Samples Application demcovfuncsample sample from the prior Figure: RBF kernel with l =.3, α =

14 Different Covariance Functions MLP Kernel Function k ( t, t ) ( = αsin wtt ) + b wt + b + wt + b + A non-stationary covariance matrix Williams [997]. Derived from a multi-layer perceptron (MLP). m n

15 Covariance Samples Application demcovfuncsample samples from the prior Figure: MLP kernel with α = 8, w = and b =

16 Covariance Samples Application demcovfuncsample samples from the prior Figure: MLP kernel with α = 8, b = and w =

17 Prior to Posterior Application Prediction with GPs GPs provide a probabilistic prior over functions. By combining with data we get a posterior over functions. This is obtained through combining a covariance function with data. Toy Example: regression with GPs.

18 Gaussian Process Regression demregression 3 3 Figure: Going from prior to posterior with data.

19 Gaussian Process Regression demregression 3 3 Figure: Going from prior to posterior with data.

20 Gaussian Process Regression demregression 3 3 Figure: Going from prior to posterior with data.

21 Gaussian Process Regression demregression 3 3 Figure: Going from prior to posterior with data.

22 Gaussian Process Regression demregression 3 3 Figure: Going from prior to posterior with data.

23 Gaussian Process Regression demregression 3 3 Figure: Going from prior to posterior with data.

24 Gaussian Process Regression demregression 3 3 Figure: Going from prior to posterior with data.

25 Gaussian Process Regression demregression 3 3 Figure: Going from prior to posterior with data.

26 Covariance of Latent Function Prior Distribution for TFC We assume that the TF concentration is a Gaussian Process. We will assume an RBF covariance function p (f) = N (f, K) k ( t, t ) ( = exp (t ) t ) l.

27 Computation of Joint Covariance Covariance Function Computation We rewrite solution of differential equation 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.

28 Induced Covariance Application 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.

29 Covariance for Transcription Model 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 (t), x (t) and f (t). Here: D S D S f(t) x (t) x (t) f(t) x (t) x (t)

30 Joint Sampling of x (t) and f (t) gpsimtest Figure: Left: joint samples from the transcription covariance, blue: f (t), cyan: x (t) and red: x (t). Right: numerical solution for f (t) of the differential equation from x (t) and x (t) (blue and cyan). True f (t) included for comparison.

31 Joint Sampling of x (t) and f (t) gpsimtest Figure: Left: joint samples from the transcription covariance, blue: f (t), cyan: x (t) and red: x (t). Right: numerical solution for f (t) of the differential equation from x (t) and x (t) (blue and cyan). True f (t) included for comparison.

32 Joint Sampling of x (t) and f (t) gpsimtest Figure: Left: joint samples from the transcription covariance, blue: f (t), cyan: x (t) and red: x (t). Right: numerical solution for f (t) of the differential equation from x (t) and x (t) (blue and cyan). True f (t) included for comparison.

33 Noise Corruption Application 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 =,..., T, y j (t i ) = x j (t i ) + ɛ j (t i ) () with ɛ j (t i ) N (, σ ji ). Estimate noise level using probe-level processing techniques of Affymetrix microarrays (e.g. mmgmos, [Liu et al., 5]). The covariance of the noisy process is then K yy = Σ + K xx, with Σ = diag ( σ,..., σ T,..., σ N,..., σ NT ).

34 Artificial Data Application 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: Learn decays, sensitivities and basal transcription rates. Infer a posterior distribution over the missing TFC.

35 Artificial Data Results 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= (lines) along with noise corrupted data. Right: The inferred TFC (with error bars). demtoyproblem

36 Artificial Data Results 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= (lines) along with noise corrupted data. Right: The inferred TFC (with error bars). demtoyproblem

37 Artificial Data Results 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= (lines) along with noise corrupted data. Right: The inferred TFC (with error bars). demtoyproblem

38 Artificial Data Results 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= (lines) along with noise corrupted data. Right: The inferred TFC (with error bars). demtoyproblem

39 Artificial Data Results 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= (lines) along with noise corrupted data. Right: The inferred TFC (with error bars). demtoyproblem

40 Artificial Data Results 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= (lines) along with noise corrupted data. Right: The inferred TFC (with error bars). demtoyproblem

41 Artificial Data Results 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= (lines) along with noise corrupted data. Right: The inferred TFC (with error bars). demtoyproblem

42 Artificial Data Results 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= (lines) along with noise corrupted data. Right: The inferred TFC (with error bars). demtoyproblem

43 Artificial Data Results 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= (lines) along with noise corrupted data. Right: The inferred TFC (with error bars). demtoyproblem

44 Artificial Data Results 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= (lines) along with noise corrupted data. Right: The inferred TFC (with error bars). demtoyproblem

45 Artificial Data Results 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= (lines) along with noise corrupted data. Right: The inferred TFC (with error bars). demtoyproblem

46 Artificial Data Results 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= (lines) along with noise corrupted data. Right: The inferred TFC (with error bars). demtoyproblem

47 Artificial Data Results 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= (lines) along with noise corrupted data. Right: The inferred TFC (with error bars). demtoyproblem

48 Artificial Data Results 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= (lines) along with noise corrupted data. Right: The inferred TFC (with error bars). demtoyproblem

49 Artificial Data Results 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= (lines) along with noise corrupted data. Right: The inferred TFC (with error bars). demtoyproblem

50 Artificial Data Results 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= (lines) along with noise corrupted data. Right: The inferred TFC (with error bars). demtoyproblem

51 Artificial Data Results 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= (lines) along with noise corrupted data. Right: The inferred TFC (with error bars). demtoyproblem

52 Results Application Linear System Recently published biological data set studied using linear response model by Barenco et al. [6]. Study focused on the tumour suppressor protein p53. mrna abundance measured for five targets: DDB, p, SESN/hPA6, BIK and TNFRSFb. 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).

53 Linear response analysis Experimental Setup We analysed data using the linear response model. Raw data was processed using the mmgmos model of Liu et al. [5] 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, 996).

54 Linear Response Results dembarenco 3 5 Figure: Predicted protein concentration for p53. Solid line is mean, dashed lines 95% credibility intervals. The prediction of [Barenco et al., 6] was pointwise and is shown as crosses.

55 Results Transcription Rates Estimation of Equation Parameters dembarenco DDB hpa6 TNFRSFb p BIK Figure: Basal transcription rates. Our results (black) compared with Barenco et al. [6] (white).

56 Results Transcription Rates Estimation of Equation Parameters dembarenco DDB hpa6 TNFRSFb p BIK Figure: Sensitivities. Our results (black) compared with Barenco et al. [6] (white).

57 Results Transcription Rates Estimation of Equation Parameters dembarenco DDB hpa6 TNFRSFb p BIK Figure: Decays. Our results (black) compared with Barenco et al. [6] (white).

58 Linear Response Discussion GP Results Note oscillatory behaviour, possible artifact of RBF covariance Rasmussen and Williams [see page 3 in 6]. 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 3 minutes to produce Barenco et al. required million iterations of Monte Carlo.

59 Linear Response with MLP Kernel Non-linear Responses More Realistic Response All the quantities in equation () 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.

60 Formalism Application 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.

61 Example: linear response Linear Response with MLP Kernel Non-linear Responses Using non-rbf kernels Start by taking g( ) to be linear. Provides sanity check and allows arbitrary covariance functions. Avoids double numerical integral that would normally be required.

62 Response Results Application Linear Response with MLP Kernel Non-linear Responses dembarencomap, dembarencomap Figure: Left: RBF prior on f (log likelihood -.); Right: MLP prior on f (log likelihood -5.6).

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

64 exp ( ) Response Results Linear Response with MLP Kernel Non-linear Responses dembarencomap3, dembarencomap Figure: Left: shows results of using a squared exponential prior covariance on f (log likelihood -.6); Right: shows results of using an MLP prior covariance on f (log likelihood -6.).

65 log ( + exp (f )) Response Results Linear Response with MLP Kernel Non-linear Responses dembarencomap5, dembarencomap Figure: Left: shows results of using a squared exponential prior covariance on f (log likelihood -.9); Right: shows results of using an MLP prior covariance on f (log likelihood -.).

66 3 +exp( f ) Application Response Results Linear Response with MLP Kernel Non-linear Responses dembarencomap7, dembarencomap Figure: Left: shows results of using a squared exponential prior covariance on f (log likelihood -.); Right: shows results of using an MLP prior covariance on f (log likelihood -.).

67 Discussion Application 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

68 Acknowledgements References Future Directions What Next? 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.

69 Acknowledgements Acknowledgements References Covariance Computation Posterior for f Non-linear Response Implementation 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.

70 Covariance Result Acknowledgements References Covariance Computation Posterior for f Non-linear Response Implementation Covariance Result where k xj x k ( t, t ) = S j S k π h `t kj, t = exp (γ k) D j + D k jexp ˆ D `t k t» t t erf l Here γ k = D kl. [ 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 «

71 Cross Covariance Acknowledgements References Covariance Computation Posterior for f Non-linear Response Implementation 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. (5) For RBF we have k `t πlsj e xj f, t γ j = exp ˆ D `t t j t»erf ««t t γ j + erf + γ j. l l

72 Posterior for f Acknowledgements References Covariance Computation Posterior for f Non-linear Response Implementation Prediction for TFC Standard Gaussian process regression techniques [see e.g. Rasmussen and Williams, 6] yield f post = K f x K xx x K post ff = K ff K f x K 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 )

73 Implementation Acknowledgements References Covariance Computation Posterior for f Non-linear Response Implementation 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= so that = t p t p is constant. The vector f = {f p } M p= 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=

74 Acknowledgements References Covariance Computation Posterior for f Non-linear Response Implementation Log Likelihood Functional Gradient Given noise-corrupted data y j (t i ) the log-likelihood is TX log p(y f, θ j ) = i= j= " NX `xj (t i ) y j (t i ) σ ji # log σji NT log(π) 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 g (f (t))e D j (t i t) i= j= ji Θ(x) Heaviside step function.

75 Log Likelihood Acknowledgements References Covariance Computation Posterior for f Non-linear Response Implementation Functional Hessian Given noise-corrupted data y j (t i ) the log-likelihood is TX log p(y f, θ j ) = i= j= " NX `xj (t i ) y j (t i ) σ ji # log σji NT log(π) 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 ji g (f (t))e D j (t i t) i= j= TX + Θ(t i t)θ `t NX i t σ ji g (f (t)) g `f (t ) e D j (t i t t ) i= j= g (f ) = g/ f and g (f ) = g/ f.

76 Implementation II Acknowledgements References Covariance Computation Posterior for f Non-linear Response Implementation 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, 6, chapter 3] Ψ(f) log p(y f) = K f f f Ψ(f) f = (W + K ) 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) ˆf T K ˆf log I + KW. (7) (6)

77 Acknowledgements References References M. Barenco, D. Tomescu, D. Brewer, R. Callard, J. Stark, and M. Hubank. Ranked prediction of p53 targets using hidden variable dynamic modeling. Genome Biology, 7(3):R5, 6. T. Graepel. Solving noisy linear operator equations by Gaussian processes: Application to ordinary and partial differential equations. In T. Fawcett and N. Mishra, editors, Proceedings of the International Conference in Machine Learning, volume, pages 3. AAAI Press, 3. ISBN X. Liu, M. Milo, N. D. Lawrence, and M. Rattray. A tractable probabilistic model for Affymetrix probe-level analysis across multiple chips. Bioinformatics, (8): , 5. R. Murray-Smith and B. A. Pearlmutter. Transformations of Gaussian process priors. R. M. Neal. Bayesian Learning for Neural Networks. Springer, 996. Lecture Notes in Statistics 8. A. O Hagan. Curve fitting and optimal design for prediction. Journal of the Royal Statistical Society, B, :, 978. A. O Hagan. Some Bayesian numerical analysis. In J. M. Bernardo, J. O. Berger, A. P. Dawid, and A. F. M. Smith, editors, Bayesian Statistics, pages , Valencia, 99. Oxford University Press. C. E. Rasmussen and C. K. I. Williams. Gaussian Processes for Machine Learning. MIT Press, Cambridge, MA, 6. ISBN 6853X. C. K. I. Williams. Computing with infinite networks. In M. C. Mozer, M. I. Jordan, and T. Petsche, editors, Advances in Neural Information Processing Systems, volume 9, Cambridge, MA, 997. MIT Press.