Cells reproduce under nonequilibrium conditions. By noting that a cell contains enzymes that drastically increase the equilibration process, it is shown that a cell is an apparatus that reveals the nonequilibrium property of the environment and accelerates equilibration. As a consequence, the entropy generation rate per cell growth is minimized at a finite growth rate, not in the adiabatic limit as in the Carnot cycle. General statistical properties of cells are then presented, including the power law in abundances and the lognormal distribution of cell-to-cell variation. The transition from exponential growth to the dormant state (where cell growth is arrested) is shown to be a general consequence of the accumulation of waste (non-autocatalytic) components, which leads to a jamming of the reaction. Related experiments using single-cell measurements elucidate the distribution of cell-to-cell variation in protein concentrations and growth rates. How cell reproduction and molecular replication achieve consistency is also a fundamental question for constructing protocells and understanding the origin of life. The relationship between minority molecules and genetic information, the synchronization of minority molecular replication and cell division, the separation of genetic information and catalytic function, and the acquisition of evolutionary potential are discussed as universal properties that must be satisfied for all cell reproduction systems.

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.