Detection and Analysis of Unexpected State Components in Biological Systems

This contribution presents a methodology aimed at detecting and analysing unexpected states of biological systems. We suggest using: (a) Petri nets in order to represent the causal structure of a biological system and its prescribed functional behaviour; and (b) algebraic coding theory (Hamming codes) to detect unexpected system states (mutations, unwanted situations, errors).Recent research aimed at modelling biological processes has successfully applied Petri nets to represent the causal structure and processes of biological systems in a way that makes the formal verification of the model easy. The formal language of Petri Nets is a powerful tool for representing biological systems together with their ”regular” behaviour. However, in every living system sooner or later something will ”go wrong” - a disease, or a mutation will occur - bringing the system into a state which our model didn’t take into account, a state that we will call an ”error state” for short. This paper is about introducing algebraic error-detection into Petri-net models. The basic idea is to turn reachable markings into ”legal” words of a linear error-correcting code. This is achieved by adding some control places to the Petri net, so that: (a) its incidence matrix becomes the generatrix of a linear code, and (b) reachable markings can be characterised as solutions of the code’s linear homogeneous system.”Illegal” markings - or, unexpected system states - may then be detected via linear algebra, without having to construct, and search, the net’s (always quite complex) reachability graph. Using suitable linear codes - that is, codes for which fast error-correction algorithms are available - ”mutant” components of ”illegal” markings are instantaneously identified, and can be analysed with regard to other structural properties of the net, such as boundedness and liveness. This contribution considers the case of single errors (point mutations): at most one unexpected component per system state. Single errors are detected via Hamming codes, for which a very fast error detection algorithm is known. We show that the number of control places to be added to the Petri net decreases exponentially with the net’s size. Coding theory offers a wide range of algorithms for the detection of terrors in transmission systems. Applying such algorithms to the detection of unexpected states of biological systems seems a very promising approach.