On the interplay between mathematics and biology: Hallmarks toward a new systems biology

•A critical analysis of the existing literature on mathematical tools developed toward systems biology approaches, a new approach is proposed, a variety of challenging mathematical problems are illustrated.

This paper proposes a critical analysis of the existing literature on mathematical tools developed toward systems biology approaches and, out of this overview, develops a new approach whose main features can be briefly summarized as follows: derivation of mathematical structures suitable to capture the complexity of biological, hence living, systems, modeling, by appropriate mathematical tools, Darwinian type dynamics, namely mutations followed by selection and evolution. Moreover, multiscale methods to move from genes to cells, and from cells to tissue are analyzed in view of a new systems biology approach.