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F. Antoneli, M. Golubitsky and I. Stewart

Homeostasis in a gene regulatory network motif

*Preprint.*
(2017)

The internal state of a cell is affected by inputs from the extra-cellular environment,
such as the expression level of an upstream transcription factor or external temperature.
If some output, such as the concentration of a target protein, remains approximately
constant as inputs vary, the system exhibits homeostasis. Special sub-networks called
motifs are unusually common in gene regulatory networks (GRNs) of organisms such as
E. coli and yeast, suggesting that they may have a significant biological function.
Potentially, one such function is homeostasis.
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.