DeepAI AI Chat
Log In Sign Up

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

by   Nicolas Behr, et al.

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.


Rewriting Theory for the Life Sciences: A Unifying Theory of CTMC Semantics

The Kappa biochemistry and the MØD organo-chemistry frameworks are among...

Combinatorial Conversion and Moment Bisimulation for Stochastic Rewriting Systems

We develop a novel method to analyze the dynamics of stochastic rewritin...

Sesqui-Pushout Rewriting: Concurrency, Associativity and Rule Algebra Framework

Sesqui-pushout (SqPO) rewriting provides a variant of transformations of...

Commutators for Stochastic Rewriting Systems: Theory and Implementation in Z3

In the semantics of stochastic rewriting systems (SRSs) based on rule al...

On Stochastic Rewriting and Combinatorics via Rule-Algebraic Methods

Building upon the rule-algebraic stochastic mechanics framework, we pres...

Concurrency Theorems for Non-linear Rewriting Theories

Sesqui-pushout (SqPO) rewriting along non-linear rules and for monic mat...

Combinatorics of chemical reaction systems

We propose a concise stochastic mechanics framework for chemical reactio...