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The Kappa biochemistry and the M{\O}D organic chemistry frameworks areamongst the most intensely developed applications of rewriting-based methods inthe life sciences to date. A typical feature of these types of rewritingtheories is the necessity to implement certain structural constraints on theobjects to be rewritten (a protein is empirically found to have a certainsignature of sites, a carbon atom can form at most four bonds, ...). In thispaper, we contribute a number of original developments that permit to implementa universal theory of continuous-time Markov chains (CTMCs) for stochasticrewriting systems. Our core mathematical concepts are a novel rule algebraconstruction for the relevant setting of rewriting rules with conditions, bothin Double- and in Sesqui-Pushout semantics, augmented by a suitable stochasticmechanics formalism extension that permits to derive dynamical evolutionequations for pattern-counting statistics. A second main contribution of ourpaper is a novel framework of restricted rewriting theories, which comprises arule-algebra calculus under the restriction to so-called constraint-preservingcompletions of application conditions (for rules considered to act only uponobjects of the underlying category satisfying a globally fixed set ofstructural constraints). This novel framework in turn renders a faithfulencoding of bio- and organo-chemical rewriting in the sense of Kappa and M{\O}Dpossible, which allows us to derive a rewriting-based formulation of reactionsystems including a full-fledged CTMC semantics as instances of our universalCTMC framework. While offering an interesting new perspective and conceptualsimplification of this semantics in the setting of Kappa, both the formalencoding and the CTMC semantics of organo-chemical reaction systems asmotivated by the M{\O}D framework are the first such results of their kind.

Rewriting Theory for the Life Sciences: A Unifying Theory of CTMC Semantics (Long version)