F. Antoneli, M. Golubitsky and I. Stewart
Homeostasis in a gene regulatory network motif
In support of this hypothesis, we use the new technique of infinitesimal homeostasis to show that a specific feed-forward loop GRN produces homeostasis. Here the inputs are subsumed into a single parameter that affects only the first node in the motif, and the output is the concentration of a target protein. The biochemical dynamics of the motif lead to homeostasis over a range of the input parameter. The analysis uses the mathematical notion of infinitesimal homeostasis, which occurs when the input-output map has a critical point (zero derivative). Such points can be located using calculus. If the second derivative also vanishes, the input-output map has a `chair': as the input increases (or in some cases decreases), the output rises roughly linearly, then flattens out (the homeostasis region), and then starts to rise again. Chair points are a common cause of homeostasis.
We apply this method to a standard family of differential equations modeling the feed-forward loop GRN, and deduce that chair points occur when a sigmoid function in the model has an inflection point. This function determines the production of a particular mRNA, and its inflection points, and the resulting chair points, are found analytically. The same method can potentially be used to find homeostasis regions in other GRNs. We also discuss why homeostasis in the motif may persist even when the rest of the network is taken into account, and in the presence of low-level stochastic noise.