A model-based optimization framework for the inference of regulatory interactions using time-course DNA microarray expression data

Abstract

Background: Proteins are the primary regulatory agents of transcription even though mRNA expression data alone, from systems like DNA microarrays, are widely used. In addition, the regulation process in genetic systems is inherently non-linear in nature, and most studies employ a time-course analysis of mRNA expression. These considerations should be taken into account in the development of methods for the inference of regulatory interactions in genetic networks.

Results: We use an S-system based model for the transcription and translation process. We propose an optimization-based regulatory network inference approach that uses time-varying data from DNA microarray analysis. Currently, this seems to be the only model-based method that can be used for the analysis of time-course "relative" expressions (expression ratios). We perform an analysis of the dynamic behavior of the system when the number of experimental samples available is varied, when there are different levels of noise in the data and when there are genes that are not considered by the experimenter. Our studies show that the principal factor affecting the ability of a method to infer interactions correctly is the similarity in the time profiles of some or all the genes. The less similar the profiles are to each other the easier it is to infer the interactions. We propose a heuristic method for resolving networks and show that it displays reasonable performance on a synthetic network. Finally, we validate our approach using real experimental data for a chosen subset of genes involved in the sporulation cascade of Bacillus anthracis. We show that the method captures most of the important known interactions between the chosen genes.

Conclusion: The performance of any inference method for regulatory interactions between genes depends on the noise in the data, the existence of unknown genes affecting the network genes, and the similarity in the time profiles of some or all genes. Though subject to these issues, the inference method proposed in this paper would be useful because of its ability to infer important interactions, the fact that it can be used with time-course DNA microarray data and because it is based on a non-linear model of the process that explicitly accounts for the regulatory role of proteins.

Figure 1

Time profiles of log mRNA expression ratios for three representative synthetic networks. Logarithm of mRNA expression ratios as a function of time for three networks. The network in (a), "Low" results in a relatively lower degree of similarity between the different gene-expression patterns in the system, the network in (b), "Medium" in a medium degree of similarity, while the network in (c), "High" results in a relatively high degree of similarity. The units of time are arbitrary but are consistent with the units of the parameters of the system.

Figure 2

Variation of correct identifications and identification errors with experimental samples and discretizations for "Low" network. Variation of the percentage of correctly identified interactions among 30 known interactions and the error as a percentage of the error obtained with the smallest number of samples. The variations are with respect to the number of experimental samples chosen and the number of discretizations, N_t. The "experimental" data are obtained by simulation using the "Low" synthetic network (see Figure 1).

Figure 3

Variation of correct identifications and identification errors with experimental samples for "Medium" network. Variation of the percentage of correctly identified interactions among 30 known interactionsand the error as a percentage of the error obtained with the smallest number of samples. The variations are with respect to the number of experimental samples chosen. The "experimental" data are obtained from simulations using the "Medium" synthetic network of Figure 1. 10 time points were used in the optimization method.

Figure 4

Time-course variation of subset of important genes in the sporulation cascade of B. Anthracis. The time-course (in hours) variation of the logarithm of expression ratios in color-coded format (green indicates up-regulation, red indicates down-regulation, grey indicates missing data and the intensity of the color indicates the level of regulation) of 24 important genes in the sporulation cascade of B. anthracis ([29]). An approximate measure of the intensity of the color to the magnitude of the log₂ ratio (fold change) is also given. Note that the ratio refers to that of the actual expression value to the expression value at the reference state.