Comparative evaluation of reverse engineering gene regulatory networks with relevance networks, graphical gaussian models and bayesian networks

MOTIVATION: An important problem in systems biology is the inference of biochemical pathways and regulatory networks from postgenomic data. Various reverse engineering methods have been proposed in the literature, and it is important to understand their relative merits and shortcomings. In the present paper, we compare the accuracy of reconstructing gene regulatory networks with three different modelling and inference paradigms: (1) Relevance networks (RNs): pairwise association scores independent of the remaining network; (2) graphical Gaussian models (GGMs): undirected graphical models with constraint-based inference, and (3) Bayesian networks (BNs): directed graphical models with score-based inference. The evaluation is carried out on the Raf pathway, a cellular signalling network describing the interaction of 11 phosphorylated proteins and phospholipids in human immune system cells. We use both laboratory data from cytometry experiments as well as data simulated from the gold-standard network. We also compare passive observations with active interventions. RESULTS: On Gaussian observational data, BNs and GGMs were found to outperform RNs. The difference in performance was not significant for the non-linear simulated data and the cytoflow data, though. Also, we did not observe a significant difference between BNs and GGMs on observational data in general. However, for interventional data, BNs outperform GGMs and RNs, especially when taking the edge directions rather than just the skeletons of the graphs into account. This suggests that the higher computational costs of inference with BNs over GGMs and RNs are not justified when using only passive observations, but that active interventions in the form of gene knockouts and over-expressions are required to exploit the full potential of BNs. AVAILABILITY: Data, software and supplementary material are available from http://www.bioss.sari.ac.uk/staff/adriano/research.html     

Mining functional modules in genetic networks with decomposable graphical models

In recent years, graphical models have become an increasingly important tool for the structural analysis of genome-wide expression profiles at the systems level. Here we present a new graphical modelling technique, which is based on decomposable graphical models, and apply it to a set of gene expression profiles from acute lymphoblastic leukemia (ALL). The new method explains probabilistic dependencies of expression levels in terms of the concerted action of underlying genetic functional modules, which are represented as so-called "cliques" in the graph. In addition, the method uses continuous-valued (instead of discretized) expression levels, and makes no particular assumption about their probability distribution. We show that the method successfully groups members of known functional modules to cliques. Our method allows the evaluation of the importance of genes for global cellular functions based on both link count and the clique membership count.     

Temporal graphical models for cross-species gene regulatory network discovery

Many genes and biological processes function in similar ways across different species. Cross-species gene expression analysis, as a powerful tool to characterize the dynamical properties of the cell, has found a number of applications, such as identifying a conserved core set of cell cycle genes. However, to the best of our knowledge, there is limited effort on developing appropriate techniques to capture the causality relations between genes from time-series microarray data across species. In this paper, we present hidden Markov random field regression with L(1) penalty to uncover the regulatory network structure for different species. The algorithm provides a framework for sharing information across species via hidden component graphs and is able to incorporate domain knowledge across species easily. We demonstrate our method on two synthetic datasets and apply it to discover causal graphs from innate immune response data.     

Likelihood analysis of phylogenetic networks using directed graphical models

A method for computing the likelihood of a set of sequences assuming a phylogenetic network as an evolutionary hypothesis is presented. The approach applies directed graphical models to sequence evolution on networks and is a natural generalization of earlier work by Felsenstein on evolutionary trees, including it as a special case. The likelihood computation involves several steps. First, the phylogenetic network is rooted to form a directed acyclic graph (DAG). Then, applying standard models for nucleotide/amino acid substitution, the DAG is converted into a Bayesian network from which the joint probability distribution involving all nodes of the network can be directly read. The joint probability is explicitly dependent on branch lengths and on recombination parameters (prior probability of a parent sequence). The likelihood of the data assuming no knowledge of hidden nodes is obtained by marginalization, i.e., by summing over all combinations of unknown states. As the number of terms increases exponentially with the number of hidden nodes, a Markov chain Monte Carlo procedure (Gibbs sampling) is used to accurately approximate the likelihood by summing over the most important states only. Investigating a human T-cell lymphotropic virus (HTLV) data set and optimizing both branch lengths and recombination parameters, we find that the likelihood of a corresponding phylogenetic network outperforms a set of competing evolutionary trees. In general, except for the case of a tree, the likelihood of a network will be dependent on the choice of the root, even if a reversible model of substitution is applied. Thus, the method also provides a way in which to root a phylogenetic network by choosing a node that produces a most likely network.     

Regularized estimation of large-scale gene association networks using graphical Gaussian models

Background  

Graphical Gaussian models are popular tools for the estimation of (undirected) gene association networks from microarray data. A key issue when the number of variables greatly exceeds the number of samples is the estimation of the matrix of partial correlations. Since the (Moore-Penrose) inverse of the sample covariance matrix leads to poor estimates in this scenario, standard methods are inappropriate and adequate regularization techniques are needed. Popular approaches include biased estimates of the covariance matrix and high-dimensional regression schemes, such as the Lasso and Partial Least Squares.     

Discrete time piecewise affine models of genetic regulatory networks

We introduce simple models of genetic regulatory networks and we proceed to the mathematical analysis of their dynamics. The models are discrete time dynamical systems generated by piecewise affine contracting mappings whose variables represent gene expression levels. These models reduce to boolean networks in one limiting case of a parameter, and their asymptotic dynamics approaches that of a differential equation in another limiting case of this parameter. For intermediate values, the model present an original phenomenology which is argued to be due to delay effects. This phenomenology is not limited to piecewise affine model but extends to smooth nonlinear discrete time models of regulatory networks. In a first step, our analysis concerns general properties of networks on arbitrary graphs (characterisation of the attractor, symbolic dynamics, Lyapunov stability, structural stability, symmetries, etc). In a second step, focus is made on simple circuits for which the attractor and its changes with parameters are described. In the negative circuit of 2 genes, a thorough study is presented which concern stable (quasi-)periodic oscillations governed by rotations on the unit circle – with a rotation number depending continuously and monotonically on threshold parameters. These regular oscillations exist in negative circuits with arbitrary number of genes where they are most likely to be observed in genetic systems with non-negligible delay effects.     

Transient dynamics of reduced-order models of genetic regulatory networks

In systems biology, a number of detailed genetic regulatory networks models have been proposed that are capable of modeling the fine-scale dynamics of gene expression. However, limitations on the type and sampling frequency of experimental data often prevent the parameter estimation of the detailed models. Furthermore, the high computational complexity involved in the simulation of a detailed model restricts its use. In such a scenario, reduced-order models capturing the coarse-scale behavior of the network are frequently applied. In this paper, we analyze the dynamics of a reduced-order Markov Chain model approximating a detailed Stochastic Master Equation model. Utilizing a reduction mapping that maintains the aggregated steady-state probability distribution of stochastic master equation models, we provide bounds on the deviation of the Markov Chain transient distribution from the transient aggregated distributions of the stochastic master equation model.     

Extracting protein alignment models from the sequence database.

Biologists often gain structural and functional insights into a protein sequence by constructing a multiple alignment model of the family. Here a program called Probe fully automates this process of model construction starting from a single sequence. Central to this program is a powerful new method to locate and align only those, often subtly, conserved patterns essential to the family as a whole. When applied to randomly chosen proteins, Probe found on average about four times as many relationships as a pairwise search and yielded many new discoveries. These include: an obscure subfamily of globins in the roundworm Caenorhabditis elegans ; two new superfamilies of metallohydrolases; a lipoyl/biotin swinging arm domain in bacterial membrane fusion proteins; and a DH domain in the yeast Bud3 and Fus2 proteins. By identifying distant relationships and merging families into superfamilies in this way, this analysis further confirms the notion that proteins evolved from relatively few ancient sequences. Moreover, this method automatically generates models of these ancient conserved regions for rapid and sensitive screening of sequences.     

Ultrasensitive synthetic protein regulatory networks using mixed decoys

Cellular protein interaction networks exhibit sigmoidal input-output relationships with thresholds and steep responses (i.e. ultrasensitivity). Although cooperativity can be a source of ultrasensitivity, we examined whether the presence of "decoy" binding sites that are not coupled to activation could also lead to this effect. To systematically vary key parameters of the system, we designed a synthetic regulatory system consisting of an autoinhibited PDZ domain coupled to an activating SH3 domain binding site. In the absence of a decoy binding site, this system is non-ultrasensitive, as predicted by modeling of this system. Addition of a high-affinity decoy site adds a threshold, but the response is not ultrasensitive. We found that sigmoidal activation profiles can be generated utilizing multiple decoys with mixtures of high and low affinities, where high affinity decoys act to set the threshold and low affinity decoys ensure a sigmoidal response. Placing the synthetic decoy system in a mitotic spindle orientation cell culture system thresholds this physiological activity. Thus, simple combinations of non-activating binding sites can lead to complex regulatory responses in protein interaction networks.     

Hierarchical analysis of piecewise affine models of gene regulatory networks

A key point in the analysis of dynamical models of biological systems is to handle systems of relatively high dimensions. In the present paper we propose a method to hierarchically organize a certain type of piecewise affine (PWA) differential systems. This specific class of systems has been extensively studied for the past few years, as it provides a good framework to model gene regulatory networks. The method, shown on several examples, allows a qualitative analysis of the asymptotic behavior of a PWA system, decomposing it into several smaller subsystems. This technique, based on the well-known strongly connected components decomposition, is not new. However, its adaptation to the non-smooth PWA differential equations turns out to be quite relevant because of the strong discrete structure underlying these equations. Its biological relevance is shown on a 7-dimensional PWA system modeling the gene network responsible for the carbon starvation response in Escherichia coli.

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Probabilistic inference in general graphical models through sampling in stochastic networks of spiking neurons

An important open problem of computational neuroscience is the generic organization of computations in networks of neurons in the brain. We show here through rigorous theoretical analysis that inherent stochastic features of spiking neurons, in combination with simple nonlinear computational operations in specific network motifs and dendritic arbors, enable networks of spiking neurons to carry out probabilistic inference through sampling in general graphical models. In particular, it enables them to carry out probabilistic inference in Bayesian networks with converging arrows ("explaining away") and with undirected loops, that occur in many real-world tasks. Ubiquitous stochastic features of networks of spiking neurons, such as trial-to-trial variability and spontaneous activity, are necessary ingredients of the underlying computational organization. We demonstrate through computer simulations that this approach can be scaled up to neural emulations of probabilistic inference in fairly large graphical models, yielding some of the most complex computations that have been carried out so far in networks of spiking neurons.