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.

Large temporal fluctuations or oscillations in cellular states are widely observed in biological systems, for instance, in neural firing, circadian rhythms, and collective motion of amoebae. These phenomena arise from the interplay between positive and negative feedback mechanisms, as discussed in previous chapters. In this chapter, we focus on such dynamic changes in cellular states. Using trajectories of oscillatory dynamics in phase planes such as the Brusselator, we provide detailed explanations of conditions for oscillation through the use of nullcline and Jacobian matrix analyses. We confirm the existence of two mechanisms: the activator-inhibitor system and the substrate-depletion system. Furthermore, we extensively introduce the Hodgkin–Huxley equations concerning membrane potential and excitability, which represent a significant milestone in the fields of biophysics, theoretical biology, and electrophysiology. Through quantitative comparison with experimental data, we elucidate the mechanisms underlying its dynamics, which are explained by the reduction of variables leading to the FitzHugh–Nagumo equations.