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

TL;DR

This paper develops a universal theoretical framework for stochastic rewriting systems in life sciences, integrating CTMC semantics with structural constraints, and unifies existing biochemical and chemical rewriting models.

Contribution

It introduces a novel rule algebra and a framework for constrained rewriting, enabling a unified CTMC semantics for bio- and organo-chemical systems.

Findings

Unified CTMC framework for rewriting systems

Encoding of bio-chemical and organo-chemical rewriting

First formal CTMC semantics for M{\

Abstract

The Kappa biochemistry and the M{\O}D 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 M{\O}D 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 M{\O}D framework are the first such results of their kind.