Why a simple model of genetic regulatory networks describes the distribution of avalanches in gene expression data

In a previous study it was shown that a simple random Boolean network model, with two input connections per node, can describe with a good approximation (with the exception of the smallest avalanches) the distribution of perturbations in gene expression levels induced by the knock-out of single genes in Saccharomyces cerevisiae. Here we address the reason why such a simple model actually works: we present a theoretical study of the distribution of avalanches and show that, in the case of a Poissonian distribution of outgoing links, their distribution is determined by the value of the Derrida exponent. This explains why the simulations based on the simple model have been effective, in spite of the unrealistic hypothesis about the number of input connections per node. Moreover, we consider here the problem of the choice of an optimal threshold for binarizing continuous data, and we show that tuning its value provides an even better agreement between model and data, valuable also in the important case of the smallest avalanches. Finally, we also discuss the choice of an optimal value of the Derrida parameter in order to match the experimental distributions: our results indicate a value slightly below the critical value 1.