In this chapter, we explore the concept of information in living organisms in its broadest sense. Biological organisms perceive the external environment, alter their own state, and take action (selection among possibilities). To capture these properties intrinsic to the organisms, we begin by discussing the “information quantity” that quantifies such situations. Starting with the definition of information quantity, we introduce Shannon entropy and provide an overview of Shannon’s information theory framework. We also discuss Kullback–Leibler divergence and mutual information. Next, moving on to information in DNA sequences, we cover various aspects such as differences in the frequency of AT and GC occurrence, the structure of genetic codes, long-range correlations in DNA sequences, and recent findings in intergenic sequences. Additionally, we explain kinetic proofreading as one candidate for achieving high accuracy in molecular recognition from a combination of unreliable elements. Furthermore, we explore the relationship between entropy in statistical mechanics and information, elucidating the connection between Maxwell’s demon and information using the Szilard engine as a mediator. Finally, we introduce intriguing points from the perspective of dynamics and information, highlighting the dynamic interplay between the two.

This chapter quantitatively examines molecule numbers and reaction rates within a cell, along with thermal fluctuations and Brownian motion, from a mesoscopic perspective. Thermal fluctuations of molecules are pivotal in chemical reactions, protein folding, molecular motor systems, and so on. We introduce estimations of cell size and molecule numbers within cells, highlighting the possible significance of the minority of molecules. Describing their behaviors necessitates dealing with stochastic fluctuations, and the Gillespie algorithm, widely employed in Monte Carlo simulations for stochastic chemical reactions, is described. We elaborate on extrinsic and intrinsic noise in cells, and on why understanding how cells process fluctuations for sensing is crucial. To facilitate this comprehension, we revisit the fundamentals of statistics, including the law of large numbers and the central limit theorem. We derive the diffusion equation from random walk and confirm the dimensionality dependence of random walks, and elucidate Brownian motion as the continuous limit of random walk and explain the Einstein relation. As examples of the physiological significance of fluctuations in cell biology, we estimate the diffusion constant of proteins inside cells, diffusion-limited reactions, and introduce bacterial random walks and chemotaxis, and amoeboid movements of eukaryotic cells.

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.

In this chapter, we introduce various modeling approaches capable of addressing pattern formation by cell populations. Firstly, we discuss the Delta–Notch system as an example of pattern formation by local interaction. We then explore the Kessler–Levin model, which combines cellular automaton and continuous system approaches, illustrating the evolution of cAMP waves in cellular slime molds. Next, our attention turns to methodologies requiring active consideration of cellular arrangements and deformations, including models involving cell proliferation and movement. We present reaction–diffusion systems that explain structures formed in bacterial colonies resembling Diffusion Limited Aggregation (DLA). Additionally, we introduce the cellular Potts model to investigate pattern formation among moving cells, incorporating variations in cellular adhesion force. The cell-vertex model represents a cell population as a collection of vertices of a polygon or polyhedron. We also discuss the phase field model, employing partial differential equations to depict relatively simple morphological changes in complex structures. By employing these modeling techniques, we can capture the characteristics of various pattern formations orchestrated by cell populations.

In this chapter, we explore theoretical aspects of the origin of life problem. Firstly, we address the Chicken and Egg problem referring to the “RNA world.” We explain a mathematical model of the RNA replication system introduced by Eigen and discuss the conditions necessary for self-replication, referring “error catastrophe.” As a potential solution, we discuss the “hypercycle,” alongside its vulnerabilities and the acquisition of evolvability through compartmentalization. On another front, we examine Dyson’s catalytic reaction system as an alternative hypothesis, showing that catalytic reaction networks capable of maintaining themselves and undergoing imperfect reproduction may have appeared first. We also refer to a simple model of polymer reactions, arguing that such autocatalytic reaction networks can stochastically emerge, as proposed by Kauffman. Furthermore, we describe a cell model featuring an intracellular chemical reaction network that divides based on its state, highlighting the universal nature of reaction dynamics in replicating cells and the power-law distribution of chemical abundance (Zipf’s law), which has been verified across many organisms. Additionally, we introduce the concept of “minority control” in catalytic reaction networks, which can carry primitive genetic information. Finally, we discuss perspectives on research regarding the origin of life.

This chapter serves as an intuitive introduction to dynamical systems within the realm of biological systems, through visual representations of state space dynamics. Biological examples and experimental realizations are described to demonstrate how dynamical systems concepts are applicable in solving fundamental problems in cell biology. Differential equations are taken as typical of dynamical systems, and we explain topics such as nullcline and fixed points, linear stability analysis, and attractors, elucidating their significance using systems such as gene toggle switches. The introduction of limit cycles and the Poincaré–Bendixson theorem in two-dimensional systems is followed by examples such as the Brusselator and the repressilator system. Furthermore, we explore the basin structure in multi-attractor systems and provide detailed explanations using toggle switch systems to illustrate time-scale separation between variables and adiabatic elimination of variables. Several instances of co-dimension 1 bifurcations commonly observed in biological systems are presented, with a discussion of their biological significance in processes like cell differentiation. Finally, chaos theory is introduced.

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.

In this chapter, we discuss dynamical system approaches for cellular differentiation. We explain how intracellular reaction dynamics can give rise to various attractors using a simple discrete-time and discrete-state reaction model known as a Boolean network. Subsequently, we outline the behavior of a simple stochastic differentiation model of stem cells, where the scaling law discovered therein aligns well with that observed in the distribution of clonal cell populations generated by epidermal stem cells. To integate both approaches, we introduce a theory wherein cell–cell interactions induce transitions between attractors, and stability at the cell-population level emerges through the regulation of these dynamic transitions. Such a circular relationship satisfies the consistency between the cell and the cell population. We expound on three types of differentiation processes, that by Turing instability, transition from an oscillatory state (limit-cycle) to a fixed point, and retaining oscillatory expression dynamics. Additionally, we analyze stability at the cell population level through the regulation of differentiation ratios and the differentiation dynamics of stem cells. Finally, we engage in a discussion of unresolved issues in the field.

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.

Considers communication in dense bacterial suspensions with an emphasis on quorum sensing.

Introduces Nernst potentials for bacterial cells, simple Hodgkin–Huxley models for action potentials and describes experimental methods to measure membrane potentials.

Considers systems biology, synthetic biology, gene motifs and the central dogma of molecular biology.

Brief introduction to applications of bacteria in food, petrochemicals, photonics, water purification and ecology.

Considers the swimming of bacterial cells, sedimentation and swimming near surfaces.

Introduces mesoscopic forces including DLVO, steric potentials, depletion and bridging forces, hydrophobic interactions and adhesion.

Introduces swarming, superfluids, non-linear rheology, bacterial turbulence and biofilm viscoelasticity.

Considers bacterial electrophysiology highlighting actions potentials, electrical communication and electrical conductivity.