Evaldo Araújo de Oliveira

Inference of gene regulatory networks from time series by Tsallis entropy

BackgroundThe inference of gene regulatory networks (GRNs) from large-scale expression profiles is one of the most challenging problems of Systems Biology nowadays. Many techniques and models have been proposed for this task. However, it is not generally possible to recover the original topology with great accuracy, mainly due to the short time series data in face of the high complexity of the networks and the intrinsic noise of the expression measurements. In order to improve the accuracy of GRNs inference methods based on entropy (mutual information), a new criterion function is here proposed.ResultsIn this paper we introduce the use of generalized entropy proposed by Tsallis, for the inference of GRNs from time series expression profiles. The inference process is based on a feature selection approach and the conditional entropy is applied as criterion function. In order to assess the proposed methodology, the algorithm is applied to recover the network topology from temporal expressions generated by an artificial gene network (AGN) model as well as from the DREAM challenge. The adopted AGN is based on theoretical models of complex networks and its gene transference function is obtained from random drawing on the set of possible Boolean functions, thus creating its dynamics. On the other hand, DREAM time series data presents variation of network size and its topologies are based on real networks. The dynamics are generated by continuous differential equations with noise and perturbation. By adopting both data sources, it is possible to estimate the average quality of the inference with respect to different network topologies, transfer functions and network sizes.ConclusionsA remarkable improvement of accuracy was observed in the experimental results by reducing the number of false connections in the inferred topology by the non-Shannon entropy. The obtained best free parameter of the Tsallis entropy was on average in the range 2.5 ≤ q ≤ 3.5 (hence, subextensive entropy), which opens new perspectives for GRNs inference methods based on information theory and for investigation of the nonextensivity of such networks. The inference algorithm and criterion function proposed here were implemented and included in the DimReduction software, which is freely available at http://sourceforge.net/projects/dimreduction and http://code.google.com/p/dimreduction/.

Gradient descent learning in and out of equilibrium

Relations between the off thermal equilibrium dynamical process of on-line learning and the thermally equilibrated off-line learning are studied for potential gradient descent learning. The approach of Opper to study on-line Bayesian algorithms is used for potential based or maximum likelihood learning. We look at the on-line learning algorithm that best approximates the off-line algorithm in the sense of least Kullback-Leibler information loss. The closest on-line algorithm works by updating the weights along the gradient of an effective potential, which is different from the parent off-line potential. A few examples are analyzed and the origin of the potential annealing is discussed.

Inference From Aging Information

For many learning tasks the duration of the data collection can be greater than the time scale for changes of the underlying data distribution. The question we ask is how to include the information that data are aging. Ad hoc methods to achieve this include the use of validity windows that prevent the learning machine from making inferences based on old data. This introduces the problem of how to define the size of validity windows. In this brief, a new adaptive Bayesian inspired algorithm is presented for learning drifting concepts. It uses the analogy of validity windows in an adaptive Bayesian way to incorporate changes in the data distribution over time. We apply a theoretical approach based on information geometry to the classification problem and measure its performance in simulations. The uncertainty about the appropriate size of the memory windows is dealt with in a Bayesian manner by integrating over the distribution of the adaptive window size. Thus, the posterior distribution of the weights may develop algebraic tails. The learning algorithm results from tracking the mean and variance of the posterior distribution of the weights. It was found that the algebraic tails of this posterior distribution give the learning algorithm the ability to cope with an evolving environment by permitting the escape from local traps.

Analysis of the GRNs Inference by Using Tsallis Entropy and a Feature Selection Approach

An important problem in the bioinformatics field is to understand how genes are regulated and interact through gene networks. This knowledge can be helpful for many applications, such as disease treatment design and drugs creation purposes. For this reason, it is very important to uncover the functional relationship among genes and then to construct the gene regulatory network (GRN) from temporal expression data. However, this task usually involves data with a large number of variables and small number of observations. In this way, there is a strong motivation to use pattern recognition and dimensionality reduction approaches. In particular, feature selection is specially important in order to select the most important predictor genes that can explain some phenomena associated with the target genes. This work presents a first study about the sensibility of entropy methods regarding the entropy functional form, applied to the problem of topology recovery of GRNs. The generalized entropy proposed by Tsallis is used to study this sensibility. The inference process is based on a feature selection approach, which is applied to simulated temporal expression data generated by an artificial gene network (AGN) model. The inferred GRNs are validated in terms of global network measures. Some interesting conclusions can be drawn from the experimental results, as reported for the first time in the present paper.

Solving the Levins' paradox in the logistic model to the population growth

We introduce a new method to improve Markov maps by means of a Bayesian approach. The method starts from an initial map model, wherefrom a likelihood function is defined which is regulated by a temperature-like parameter. Then, the new constraints are added by the use of Bayes rule in the prior distribution. We applied the method to the logistic map of population growth of a single species. We show that the population size is limited for all ranges of parameters, allowing thus to overcome difficulties in interpretation of the concept of carrying capacity known as the Levins paradox.

Inference of restricted stochastic boolean GRN's by Bayesian error and entropy based criteria

This work compares two frequently used criterion functions in inference of gene regulatory networks (GRN), one based on Bayesian error and another based on conditional entropy. The network model utilized was the stochastic restricted Boolean network model; the tests were realized in the well studied yeast cell-cycle and in randomly generated networks. The experimental results support the use of entropy in relation to the use of Bayesian error and indicate that the application of a fast greedy feature selection algorithm combined with an entropy-based criterion function can be used to infer accurate GRNs, allowing to accurately infer networks with thousands of genes in a feasible computational time cost, even though some genes are influenced by many other genes.

Revealing temporal genetic regulatory networks from steady-state distributions

The design of dynamical networks from steady-state distributions usually presents some inherent limitations. The dynamical behavior of the system can not be determined from the steady-state, it can only be constrained by it. In general, there is a huge number of dynamical systems that can produce the same steady-state. Nevertheless, it is possible to further constrain the possibilities by adopting the Probabilistic Genetic Networks model which is based on axioms that usually make sense in biological systems. In this work we introduce a new method for the inference of dynamical systems, and their underlying logical structures, from a steady-state distribution. Our method is based on the assumption that biological systems are quasi-deterministic. The technique is based on an integer programming model that selects stochastic matrices with a known limit distribution. These transition matrices reveal how the dynamical system evolves, allowing the application of standard inference methods to discover dependencies among elements of the system.