Rewriting theory for the life sciences: A unifying theory of CTMC semantics.

• First-of-its-kind general framework for a CTMC theory of stochastic rewriting systems for rules with conditions. • New rule-algebraic restricted rewriting theory formalism. • First-of-its-kind encoding of bio- and organo-chemical reaction systems via rule-algebraic restricted rewriting theory. The Kappa biochemistry and the MØD organic chemistry frameworks are amongst the most intensely developed applications of rewriting-based methods in the life sciences to date. A typical feature of these types of rewriting theories is the necessity to implement certain structural constraints on the objects to be rewritten (a protein is empirically found to have a certain signature of sites, a carbon atom can form at most four bonds,...). In this paper, we contribute a number of original developments that permit to implement a universal theory of continuous-time Markov chains (CTMCs) for stochastic rewriting systems. Our core mathematical concepts are a novel rule algebra construction for the relevant setting of rewriting rules with conditions, both in Double- and in Sesqui-Pushout semantics, augmented by a suitable stochastic mechanics formalism extension that permits to derive dynamical evolution equations for pattern-counting statistics. A second main contribution of our paper is a novel framework of restricted rewriting theories, which comprises a rule-algebra calculus under the restriction to so-called constraint-preserving completions of application conditions (for rules considered to act only upon objects of the underlying category satisfying a globally fixed set of structural constraints). This novel framework in turn renders a faithful encoding of bio- and organo-chemical rewriting in the sense of Kappa and MØD possible, which allows us to derive a rewriting-based formulation of reaction systems including a full-fledged CTMC semantics as instances of our universal CTMC framework. While offering an interesting new perspective and conceptual simplification of this semantics in the setting of Kappa , both the formal encoding and the CTMC semantics of organo-chemical reaction systems as motivated by the MØD framework are the first such results of their kind. [ABSTRACT FROM AUTHOR]