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 describewith a good approximation (with the exception of the smallest avalanches) the distribution of perturbations in gene expression levelsinduced by the knock-out of single genes in Saccharomyces cerevisiae. Here we address the reason why such a simple model actuallyworks: we present a theoretical study of the distribution of avalanches and show that, in the case of a Poissonian distribution of outgoinglinks, their distribution is determined by the value of the Derrida exponent. This explains why the simulations based on the simple modelhave been effective, in spite of the unrealistic hypothesis about the number of input connections per node. Moreover, we consider herethe problem of the choice of an optimal threshold for binarizing continuous data, and we show that tuning its value provides an evenbetter agreement between model and data, valuable also in the important case of the smallest avalanches. Finally, we also discuss thechoice of an optimal value of the Derrida parameter in order to match the experimental distributions: our results indicate a value slightlybelow the critical value 1.