Formal autopoiesis: Solutions of the classical and extended functional closure equations

Formalization of autopoiesis is an ongoing effort among theoretical biologists. In this field, Letelier and co-authors proposed that Robert Rosen's (M,R)-systems theory be used as a formalism for autopoiesis. In (M,R)-systems theory, Rosen proposes that one solve a set of functional closure equations (FCEs) which account for all of the components of the system as coming from within the system itself. A key part of the functional closure equations is the repair of the metabolism component of the system. Rosen's theory gives the organizational closure of the components as well as their products, as found in autopoiesis. However, according to Razeto-Barry (M,R)-systems leaves out some of the messiness and approximation that we find in autopoiesis as he reformulates it. A related problem is that though FCEs have a long history, they are difficult in practice to solve due to their mathematical formulation. In this paper we give a novel exact solution for the FCEs for continuous real vector-valued functions which is nevertheless difficult to compute. In addition we propose an extended form of FCEs which both captures more of the messiness of autopoiesis and also helps to make the FCEs more solvable. Finally, we use our solution for the extended FCEs to give an extended repair function for a metabolism taken from a representative class of biological dynamics for gene expression (the repressilator). More generally we show that one can use our solution for the extended FCEs to get an extended repair function for continuous real vector-valued functions.