Analyzing connectivity patterns and their application to genetic networks

The connectivity underlying a complex system determines its global dynamics and its observable functional patterns. Examples are found in a variety of disciplines such as social networks, the Internet, the central nervous system including the cortex, as well as electronic circuits. Novel computational methods from fractal mathematics and "small world" networks provide an entry point to the understanding of the connectiyity and the interaction of its microscopic components from the study of the observable variables on the macroscopic system level. As an example of such an approach, we try to understand the underlying connectivity of the genome by analyzing the observable patterns of gene expression profiles made available by cDNA microarrays technology. We start by formulating different models of genetic interactions on a genomic scale and then we compute the statistics of gene expression levels produced from each model. By these means tire obtain a dictionary relating different connection topologies on the microscopic level to corresponding gene expression profiles on the macroscopic system level. To allow for comparison between theory and experiment, we compute the equivalent statistics of experimental cDNA microarrays data obtained from the public domain. Reading the theoretical dictionary backwards and applying it to the statistics of the experimental data, we are able to rule out improbable genetic connectivity patterns and identify the most promising candidates of genetic networks. Our results show that the most promising candidate of genetic network is the "small world" heterogeneous network where the value of the scaling exponent in g(k) = Ak-a is between three halves and six, 3/2 < a < 6. This conclusion is quantitatively supported by the measures of goodness of fit of the models to the experimental data. This would imply that some genes are regulated by the input from a few other genes, while some genes are regulated by the input from many other genes. However, all the genes have a similar pattern of regulatory output onto other genes. We also find that in our genetic interaction models the clustering of the input pattern of the structural connectivity matrices is reflected in the correlation pattern of the functional connectivity matrices. Hence, the model predicts a direct connection between the regulatory links among genes and the co-expression of these genes.