Design principles for the analysis and construction of robustly homeostatic biological networks

Homeostatic biological systems resist external disturbances, allowing cells and organisms to maintain a constant internal state despite perturbations from their surroundings. Many biological regulatory networks are known to act homeostatically, with examples including thermal adaptation, osmoregulation, and chemotaxis. Understanding the network topologies (sets of regulatory interactions) and biological parameter regimes that can yield homeostasis in a biological system is of interest both for the study of natural biological system, and in the context of designing new biological control schemes for use in synthetic biology. Here, we examine the mathematical properties of a function that maps a biological system's inputs to its outputs, we have formulated a novel criterion (the "cofactor condition") that compactly describes the conditions for homeostasis. We further analyze the problem of robust homeostasis, wherein the system is required to maintain homeostatic behavior when its parameter values are slightly altered. We use the cofactor condition to examine previously reported examples of robust homeostasis, showing that it is a useful way to unify a number of seemingly different analyses into a single framework. Based on the observation that all previous robustly homeostatic examples fall into one of three classes, we propose a "strong cofactor condition" and use it to provide an algorithm for designing new robustly homeostatic biological networks, giving both their topologies and constraints on their parameter values. Applying the design algorithm to a three-node biological network, we construct several robustly homeostatic genetic networks, uncovering network topologies not previously identified as candidates for exhibiting robust homeostasis.