Exploring the scale invariance property in enzymatic networks

Fold-change detection (FCD), the (approximate) invariance of the complete output trajectory with respect to a rescaling of input magnitudes, is exhibited by many biological systems. Such a "logarithmic sensing" property of scale invariance is a genuinely nonlinear behavior, as it can never be exhibited by linear systems. This paper employs a combination of computational search and theoretical analysis to characterize 3-node enzymatic networks that have an approximate FCD property. After showing the impossibility of perfect FCD for these systems, it describes a computational screen and its results, and a novel property, "uniform linearizations with fast output" (ULFO). All tested networks which are (approximately) scale invariant also satisfy (approximate) ULFO, and the (exact) ULFO property implies approximate scale invariance. We also discuss a pair of other behaviors related to FCD, in which the shape of the response is the same up to time or output rescalings.