Genetic network structure and dynamics: identifying simple negative feedback loops

A broad array of experimental techniques have been used to determine the interactions between genes that regulate key cellular processes such as differentiation, metabolism and the cell cycle. The experimental studies are often complemented by development of models of varying degrees of complexity. We consider the 'inverse problem': to determine the underlying interactions based solely on the observed dynamics. In earlier work, we considered a specific class of ordinary differential equations that are continuous analogues of a Boolean switching network. We developed techniques to analyse and classify the dynamics based on their logical structure. We also developed techniques to solve the inverse problem. In the current work, we extend these earlier methods to analyse a model equation for a genetic network proposed by Cummins and colleagues. For a simple negative feedback system in which there is a cyclic interaction diagram with an odd number of inhibitory links, if the data is sampled at a sufficiently fine time scale with sufficient accuracy that maxima and minima can be determined, the structure can be deduced by considering sequences of maxima and minima. Alternatively, one can use the sequence of logical states found by discretizing the dynamics based on the first derivative of the variables as a function of time. The most useful technique for determining the interactions involves assessing the dependence of the rate of change of each variable as a function of the other variables, taken one at a time.