## Estimation of Multiple Transcription Factor Activities using ODEs and Gaussian Processes

published: April 16, 2009, recorded: April 2009, views: 338

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# Description

Recently, ordinary differential equations (ODEs) have been used to infer the concentration of a single transcription factor (TF) protein from time series expression data of a set of target genes. For instance, this has been applied to uncover the concentration of the p53 protein; see Barenco et al. (2006). In the present work, we propose a framework to estimate multiple TFs from a set of observed gene expressions that are co-regulated by these TFs. We assume that the connectivity network (that describes which TFs regulate each of the genes) is partially and probabilistically observed. For example, such side information can be available through a technique such as Chromatine Immunoprecipitation (ChIP). The objective of inference is to estimate the structure of the sub-network, the concentration of the transcription factor proteins continuously in time as well as to infer the type of regulation in each network link (i.e. activation, repression or non-regulation). This multiple-TF framework uses Gaussian process priors to model the unobserved TF activities continuously in time, as considered in Lawrence, et al. (2007) for the single-TF case. The ODE model of transcriptional regulation using multiple TFs is based on the following linear differential equation, dy_j(t)/dt = B_j+ S_j*g(f_1(t),...,f_R (t);w_j)− D_jy_j(t), where y_j(t) denotes the gene expression of jth gene at time t, (B_j,S_j,D_j) are the kinetic parameters of the equation, each f_r(t) is a TF concentration function, w_j are the connectivity weights between the gene and the TFs and g is a sigmoid (e.g. Michaelis-Menten) type of function. Given a set of observations of the gene expression at discrete time points, the parameters {B_j,S_j,w_j, D_j} and the protein concentration functions {f_r(t)} are estimated by using a full Bayesian methodology that employs a Markov chain Monte Carlo algorithm. Gaussian process priors are placed on the functions {f_r(t)}, while the connectivity weights {w_j} are given sparse priors so that the side prior information about the network connectivity is taken into account. The whole framework is currently applied to sub-networks in yeast cell-cycle gene expression data in Spellman et al. (1998) and Orlando et al. (2008) by using the connectivity ChiP information provided in Lee et al. (2002). This is a joint work with Magnus Rattray and Neil Lawrence.

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