Analysis and generic properties of gene regulatory networks with graded response functions

The mass of new genomic data has lead to a growing interest in gene regulatory models describing the regulatory aspects of gene activity. In these models, the rates of change of gene product concentrations are expressed as a sum of regulatory switches in terms of sums of products of sigmoid functions, mimicking complex logical on-off functions turning gene activity on and off. The customary way to analyse such equations is to replace the sigmoid functions by step functions and disregard models with autoregulation. This leads to discontinuous equations of motion, but in models without autoregulation, continuous solutions can easily be obtained. With effective autoregulation this simple approach breaks down. l regulators have finite gain, and autoregulation is ubiquitous in biological systems, we propose a generalised gene regulatory model framework admitting autoregulation and graded sigmoid functions with different steepnesses. Using singular perturbation analysis methods we derive a set of reformulated, continuous equations for the limit solution when the sigmoids approach step functions, and show that the solution for steep sigmoids approaches this limit solution uniformly in a finite time interval. A conspicuous feature of the limit solution is so-called sliding motion during which the solution slides along a threshold hyperplane or the intersection of such hyperplanes. A particular mapping leads to a dual picture of the flow, where the fast parts are magnified at the expense of the slow parts. Combining the two dual pictures, a simple and powerful method to analyse the flow is obtained. w that for steep sigmoid functions the flow may be highly sensitive to the relative steepnesses of the sigmoids. This has important consequences when stochastic effects are taken into account. Also, it implies that the observation by Glass and Kauffman in 1973 that the qualitative features of the solution do not change provided the underlying “logical” structure of the model is preserved, is not a generic property of gene regulatory models.