Toward a mathematical theory of living systems focusing on developmental biology and evolution: A review and perspectives

This review paper is devoted to present a personal perspective, based on a critical analysis of the existing literature, about the conceptual difficulties that mathematics meets when attempting to describe the complexity of living matter focusing on the challenging goal of developing a mathematical theory for living systems. The authors propose a personal path, starting from the identification of a number of common features of living systems that can be viewed as sources of complexity, firstly in general, and subsequently focusing specifically on evolution problems. Further, three key questions are posed addressing to a mathematical theory. Finally, the tools of the kinetic theory of active particles are critically analyzed to understand how far this approach still is from the achievement of the afore said ambitious objective.

Research highlights► A personal perspective about the conceptual difficulties that mathematics meets when attempting to describe the complexity of living matter is proposed focusing on the challenging goal of developing a mathematical theory for living systems. ► A personal path is designed, starting from the identification of a number of common features of living systems that can be viewed as sources of complexity, focusing specifically on evolution problems. ► Constructive speculations on the rôle of mathematics are proposed within the ambitious project of building a comprehensive theory of developmental biology and evolution. ► The contents focus on complex systems constituted by a large number of interacting entities, whose interactions are nonlinearly additive. These are at the core of the emergence of new collective states that are not mere combinations of the states of the individual units. ► The mathematical tools of the kinetic theory of active particles are properly developed towards the modeling of evolutionary systems that include Darwinian selection.