Understanding gene regulatory networks is key in Systems Biology. Reverse engineering such networks is thus paramount and a plethora of literature dealing with the problem has developed in recent years. These approaches often assume one observed time series for each gene. However, gene expression data is normally obtained by destructive sampling and the idea of a longitudinal time series becomes problematic. This is because a single individual is not followed throughout the experiment, but rather a population of cells or individuals maintained in controlled, homogeneous conditions are sampled and their gene expression measured. Thus, rather than "real'' gene expression measurements, we are faced with a set of surrogate measures. In addition to the uncertainty involved in the sampling process, it is well known that gene expression measurement technologies, such as microarrays, render noisy data and frequently exhibit outliers. When repeated measurements are available, time series used for reverse engineering gene interaction networks are commonly obtained as a (weighted) average of these replicates and therefore these sources of uncertainty are ignored.

By acting as though the averaged series was the sole source of information, the additional information and uncertainty present in the repeated measurements is neglected and some biases can be passed on to the estimated network. Neglecting the variability within the replicates can have severe effects when fitting a linear model, with perhaps the most important being the so-called attenuation of the coefficient estimates. This attenuation can in turn render a underestimation of the interaction coefficients within the network, with a spurious sense of security given the concomitant underestimation of the variability of these estimates.  

In the sequel we present a model that takes into account repeated measurements of time course gene expression data for estimating the topology of a gene interaction network. To ease the presentation, we assume an underlying first-order linear autoregressive process, AR(1), for the interaction network. As gene expression measurements frequently exhibit heavier-than-normal tail behaviour, we model the measurement process with Student-t errors. The model fitting is illustrated with data from the circadian clock in Arabidopsis thaliana.

E.R. Morrissey, M.A. Juarez, K. Denby and N.J. Burroughs.  "Inferring the topology of a non-linear sparse gene regulatory network using fully Bayesian spline regression"

We propose a semi-parametric Bayesian model, based on penalised splines, for the recovery of an interaction network topology from longitudinal data. Our motivation is inference of gene regulatory networks from low resolution microarray time series (10-50 time points). Such biological networks are known to be sparse, imposed in the network by augmenting the model with parent indicators and providing these with either an overall or gene-wise hierarchical structure. We give conditions for posterior propriety under a broad class of frequently used improper priors. Appropriate specification of the prior is crucial to control the flexibility of the splines, especially under circumstances of scarce data; thus we provide a more informative, proper prior and analyse sensitivity. The posterior is analytically intractable and numerical methods are needed. A Metropolis-within-Gibbs sampler is proposed, with a novel Metropolis-Hastings step for sampling the topology and the spline coefficients simultaneously. We also construct a linear model for comparison purposes. The models are illustrated using syntheticand gene expression data drawn from ODE models and an experimental data set for the circadian rhythm.