The Emergence of Functional Organization in Molecular Systems - A Foundation for Theoretical Biology at the Intersection of Chemistry and Computer Science.

The current mathematical framework of evolutionary theory lacks a theory of functional organization. The evolutionary process is codified as a problem in the dynamics of alleles (gene variants) governed jointly by fitness and transmission rules. Yet concepts like gene and fitness depend on a functionally organized context known as phenotype. To address problems like the origin of life or the major transitions in evolution, the traditional dynamical systems approach has to be extended with a theory of functional organization. At the most basic level such a theory must apply to molecular reaction networks. The study of the emergence and evolution of molecular organization requires an adequate formalization of chemistry. In the absence of such a formalism, we started out by exploring a formal metaphor for chemistry called l -calculus, a syntactical system at the core of computation theory. Molecules are regarded as symbolic structures denoting mathematical functions. Chemical reactions appear as functional applications constructing new functions. Placing this model chemistry in the dynamical setting of a stochastic flow-reactor generates a diversity of self- maintaining organizations reminiscent of molecular reaction networks. Different boundary conditions result in different levels of organization. Despite doing violence to chemistry-as-we-know-it, the kinds of organizations generated in our model invite further investigation. In this proposal we outline several approaches to a more faithful formalization of chemistry. In particular we address issues pertaining to interaction specificity, symmetry, reversibility, mass conservation, and multiple product reactions. We also discuss modifications to the dynamical setting of our model. The short term goal is to improve the theoretical basis for our computer experiments. The overarching long-term goal is to develop a mathematical understanding of self-maintaining organizations. Self-maintaining natural systems include the global climate system, living organisms, many cognitive processes, and a diversity of social institutions. The capacity to construct artificial systems that are self-maintaining would be highly desirable. Yet, curiously, there exists no readily identifiable scientific tradition that seeks to understand what classes of such systems are possible or to discover conditions necessary to achieve them.

The motivation underlying this research project is search for a formal framework that allows us to think about organization in general. This interest in organization derives from questions that are central to evolutionary biology. How do new types of entities arise? What enables and constrains their change? These questions can be addressed at the molecular level because molecular species systematically imply further molecular species by means of the rules of chemical reactions - which in turn are determined by the molecules. This "feedback loop" amounts to an algebra of objects in which the "application" of an object on another object yields a third object. One can view this just like an equation A*B=C, read "A acts on B to produce C". The organization is then the described by the corresponding multiplication table, important properties of the organization are then for instance "laws of computing", such as e.g., the associative law (A*B)*C = A*(B*C) that is familiar from addition and multiplication of natural numbers. In this research project we have started the systematic investigation of such algebras both in mathematical terms and by means of computer experiments. One of the main insights is that so-called self-maintaining sets, that is collection of objects that collectively can renew themselves, play a central role for the origin of organizations. Furthmore we have developed a mathematical formalism that can be used to unify topological and algebraic descriptions of organizations. The use of this language is currently being explored. We have considered a number of formal models, such as lambda calculus, term rewrite systems, and graph rewrite systems, to see which one of them is most suitable for the modelling of chemical and biological organizations. We found that graph rewriting is most appealing, in essence because the representation of the objects fits best to their functions. A computer program for graph rewriting has been developed that forms the basis for future research both on chemical reaction networks and on the evolution of development in different life forms.