A Probabilistic Dynamical Model for Quantitative Inference of

A Probabilistic Dynamical Model for Quantitative Inference of the Regulatory Mechanism of Transcription Guido Sanguinetti, Magnus Rattray and Neil D. Lawrence

Talk plan • • • Overview of the problem Extending regression Introducing dynamics Modelling separately concentrations What next?

The problem • The Central Dogma Genes Transcription Easy to measure m. RNA Translation Hard to measure COMPLEX! Proteins Protein interactions Life

Specific problem • Transcription factors produce proteins that promote or repress transcription of other genes; they play a fundamental role in gene networking • Deduce the activity of the transcription factors’ proteins (in an experimental condition) from the m. RNA expression data.

Why not use the TFs expressions? TFs are often low expressed, noisy TFs are post-transcriptionally regulated TFs interact non-trivially with each other

Current approaches • Integrate with Ch. IP-on-chip data • Ch. IP-on-chip gives a binary matrix X of transcription factors binding genes (connectivity matrix) • Regress microarray expression data on X bmt is the transcription factor activity (TFA) of TF m at time t, monotonically linked to protein concentrations (Liao et al, Boulesteix and Strimmer, Gao et al, . . . )

Problems • All genes bound by the TF contribute equally to the estimate of the TFA, regardless of the regulation type. • TFAs are gene-independent, but the influence of a transcription factor varies from gene to gene (and according to condition) • The model is linear (inevitable)

Extending Regression Modify the regression model to allow different TFAs for different genes and experiments Reduce the number of parameters by placing a prior distribution over the gene-specific TFAs. The choice of the prior distribution depends on the situation we model. E. g. , for independent samples we may assume TFAs at different time points to be independent

Introducing dynamics • To model time series data, we choose a Kalman filter prior on the rows of B where This is equivalent to assuming TFAs vary smoothly

Likelihood function • Given the model and the prior, we can obtain a likelihood The likelihood can be estimated efficiently using the sparsity of the covariance and recursion relations.

Estimating the TFAs can be estimated a posteriori using Bayes’s Theorem and moment matching Error bars associated with each TFA are given by the squared root of the diagonal entries in the posterior covariance. Mean TFAs can be obtained by averaging genespecific TFAs over the target genes.

Testing the model • We compared our averaged TFAs with the ones obtained by regression for the Spellman dataset (Mol. Biol. Cell, 1998), Ch. IP data from Lee et al. (Science 2002). The diagrams show the TFA for ACE 2 p.

. . . but we also get. . . TFA for SCW 11 TFA for YER 124 C TFA for CTS 1 TFA for YKL 151 C

. . . and we can do more! Gene Name Maximum TFA with error YER 124 C YHR 143 W ACE 2=1. 1± 0. 2, FKH 2=0. 03± 0. 04 PHO 3 AGA 1 NDD 1=1. 6± 0. 2, FKH 2=0. 06± 0. 02 ACE 2=1. 4± 0. 2, FKH 1=0. 011± 0. 009, FKH 2=0. 03± 0. 04 MBP 1=1. 5± 0. 4, SWI 4=1. 0± 0. 4, MCM 1=0± 0. 003 • Error bars allow to determine which regulations are significant • Correlations among TFs can be obtained from Σ

Decoupling action and concentration • It is not clear in the model whether a high gene-specific TFA is the result of a high affinity or of a high protein concentration • We modify the model to distinguish the effects of protein concentration and affinity • Specifically, we model

Estimating the parameters • The model is no longer exact. • Approximate inference is performed using a variational EM algorithm • This exploits Jensen’s inequality to get a bound on the log likelihood Under a factorization assumption on the approximating distribution q, the E-step becomes exactly solvable via fixed point equations.

Results The left hand picture shows the expression level of ACE 2 in the yeast cell cycle, the middle shows the inferred protein concentration and right shows the significance of the activities.

Problems • Ch. IP data is notoriously noisy; for example the same transcription factor (MSN 4) in the same conditions (rich medium) is found to bind 32 genes in Lee et al. and 57 genes in Harbison et al. (the intersection is 20 genes). • Posterior estimation helps with false positives, not with false negatives. • The model is additive (in log space) and doesn’t model combinatorial effects.

What next? • Collaborate with biologists to validate our predictions on novel data • Microarray and Ch. IP data from same lab should be more consistent • Use the model results as a starting point for systems biology modeling • Introduce combinatorial effects