The other facet of adaptation, immutability or homeostasis, is discussed. Dynamical system models that buffer external changes in a few variables to suppress changes in other variables are presented. In this case, some variable makes a transient change depending on the environmental change before returning to the original state. This transient response is shown to obey fold-change detection (or Weber–Fechner law), in which the response rate by environmental changes depends only on how many times the environmental change is to the original value. As for the multicomponent cell model, a critical state in which the abundances of each component are inversely proportional to its rank is maintained as a homeostatic state even when the environmental condition is changed. In biological circadian clocks, the period of oscillation remains almost unchanged against changes in temperature (temperature compensation) or other environmental conditions. When several reactions involved in the cyclic change use a common enzyme, enzyme-limited competition results. This competition among substrates explains the temperature compensation mentioned above. In this case, the reciprocity between the period and the plasticity of biological clocks results.

Cells regulate their proliferation, differentiation, and motility in response to external stimuli. Often, these responses involve a complex interplay of association, dissociation, and catalytic reactions, characterized by highly specific intermolecular interactions. This chapter examines cellular responses arising from such chemical reactions from a mathematical standpoint. As examples of input–output relationships, we introduce the Hill equation, Adair equation, and the MWC model concerning allosteric regulation, which describe cooperative behaviors. We discuss the Michaelis–Menten equation in enzyme reactions, covering activation, inactivation, push–pull reactions, zero-order ultrasensitivity, and positive feedback switches. Furthermore, we present the formation of a bell-shaped input–output curve by feed-forward loops, and the mechanisms of adaptation and fold-change detection utilizing feed-forward loops, or negative feedback. We explore bacterial chemotaxis mechanisms through models such as the Asakura–Honda model and the Barkai–Leibler model.