Uncovering the design principles of circadian clocks: Mathematical analysis of flexibility and evolutionary goals

In this paper, we present the mathematical details underlying both an approach to the flexibility of regulatory networks and an analytical characterization of evolutionary goals of circadian clock networks. A fundamental problem in cellular regulation is to understand the relation between the form of regulatory networks and their function. Circadian clocks present a particularly interesting instance of this. Recent work has shown that they have complex structures involving multiple interconnected feedback loops with both positive and negative feedback. We address the question of why they have such a complex structure and argue that it is to provide the flexibility necessary to simultaneously attain multiple key properties of circadian clocks such as robust entrainment and temperature compensation. To do this we address two fundamental problems: (A) to understand the relationships between the key evolutionary aims of the clock and (B) to ascertain how flexible the clock's structure is. To address the first problem we use infinitesimal response curves (IRCs), a tool that we believe will be of general utility in the analysis of regulatory networks. To understand the second problem we introduce the flexibility dimension d, show how to calculate it and then use it to analyse a range of models. We believe our results will generalize to a broad range of regulatory networks.