Hessian Geometric Structure of Equilibrium and Nonequilibrium Chemical Reaction Newtworks

Cells are the basic units of all living things, and their functions are realized by circuits and networks of chemical reactions. Thus, understanding the mechanism how various cellular functions are implemented by chemical reaction networks (CRN) is the central challenge in biophysics and quantitive biology. Among various aspects of CRN, its thermodynamic property is particularly important because most of biological functions are energy-consuming nonequilibrium phenomena. However, even though the equilibrium chemical thermodynamics and kinetics of chemical reactions were founded more than one century ago, the nonequilibrium theory of CRN is still immature. One reason is the nonlinearity in the constitutive equation between chemical force and flux, which prevents us from associating the tangent and cotangent spaces of the dynamics by the usual inner product structure. In this work, we show that the nonlinear relation between chemical force and flux can be captured by Legendre transformation and the geometric aspects of CRN dynamics can be characterized by Hessian geometry. Hessian geometry is the geometry generated by Legendre dual pairs of convex functions and is the basis of dually flat structure of information geometry and also equilibrium thermodynamics. Thus, we have dually flat structures in CRN dynamics, one on the state-potential space where equilibrium and energetic aspect is formulated (1,2), and the other on the force-flux space where nonequilibrium and kinetics aspect is characterized(3). Two of them are consistently connected by topological property of the underlying hypergraph structure of CRN. We discuss potential applications of this structure not only for CRN but also for other phenomena and problems(4,5).