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.

Diffusion plays crucial roles in cells and tissues, and the purpose of this chapter is to theoretically examine it. First, we describe the diffusion equation and confirm that its solution becomes a Gaussian distribution. Then, we discuss concentration gradients under fixed boundary conditions and the three-color flag problem to address positional information in multicellular organism morphogenesis. We introduce the possibility of pattern formation by feed-forward loops, which can transform one gradient into another or convert a chemical gradient into a stripe pattern. Next, we introduce Turing patterns as self-organizing pattern formation, outlining the conditions for Turing instability through linear stability analysis and demonstrating the existence of characteristic length scales for Turing patterns. We provide specific examples in one-dimensional and two-dimensional systems. Additionally, we present instances of traveling waves, such as the cable equation, Fisher equation, FitzHugh–Nagumo equation, and examples of their generation from limit cycles. Finally, we introduce the transformation of temporal oscillations into spatial patterns, exemplified by models like the clock-and-wavefront model.