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An interesting problem for computational biology is the analysis of time-series expression data. Here, the application of modern methods from dynamical systems, optimization theory, numerical algorithms and the utilization of implicit discrete information lead to a deeper understanding. In [1], we suggested to represent the behavior of time-series gene expression patterns by a system of ordinary differential equations, which we analytically and algorithmically investigated under the parametrical aspect of stability or instability. Our algorithm strongly exploited combinatorial information. In this paper, we deepen, extend and exemplify this study from the viewpoint of underlying mathematical modelling. This modelling consists in evaluating DNA-microarray measurements as the basis of anticipatory prediction, in the choice of a smooth model given by differential equations, in an approach of the right-hand side with parametric matrices, and in a discrete approximation which is a least squares optimization problem. We give a mathematical and biological discussion, and pay attention to the special case of a linear system, where the matrices do not depend on the state of expressions. Here, we present first numerical examples.

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Motivation:In order to understand the behavior of a gene regulatory network, it is essential to know the genes that belong to it. Identifying the correct members (e.g. in order to build a model) is a difficult task even for small subnetworks. Usually only few members of a network are known and one needs to guess the missing members based on experience or informed speculation. It is beneficial if one can additionally rely on experimental data to support this guess. In this work we present a new method based on formal concept analysis to detect unknown members of a gene regulatory network from gene expression time series data. Results:We show that formal concept analysis is able to find a list of candidate genes for inclusion into a partially known basic network. This list can then be reduced by a statistical analysis so that the resulting genes interact strongly with the basic network and therefore should be included when modeling the network. The method has been applied to the DNA repair system of Mycobacterium tuberculosis. In this application our method produces comparable results to an already existing method of component selection while it is applicable to a broader range of problems.

We have developed a method that uses gene expression data as well as interaction data between cell components to define a set of genes that we use for our modeling. In a subsequent step, we estimate the parameters of our model of piecewise linear differential equations and evaluate the results simulating the behavior of the system with our model. We have applied our method to the DNA repair system of Mycobacterium tuberculosis. Our analysis predicts that the gene Rv2719c plays an important role in this system.

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