Computational Properties of Cell Regulatory Pathways Through Petri Nets

Abstract

The paper develops a Petri net model of a negative feedback oscillator, Case 2a from Tyson et al. (Curr. Opin. Cell Biol. 15, 221?231, 2003), in order to be able to perform the holonomy decomposition of the automaton derived from its token markings and allowed transitions for a given initial state. The objective is to investigate the algebraic structure of the cascade product obtained from its holonomy components and to relate it to the behaviour of the physical system, in particular to the oscillations. The analysis is performed in two steps, first focusing on one of its component systems, the Goldbeter-Koshland ultrasensitive switch (Case 1c from Tyson et al. (2003), in order to verify the validity of its differential model and, from this, to validate the corresponding Petri net through a stochastic simulation. The paper does not present new original results but, rather, discusses and critiques existing results from the different points of view of continuous and discrete mathematics and stochasticity. The style is one of a review paper or tutorial, specifically to make the material and the concepts accessible to a wide interdisciplinary audience. We find that the Case 2a model widely reported in the literature violates the assumptions of the Michaelis-Menten quasi-steady-state approximation. However, we are still able to show oscillations of the full rate equations and of the corresponding Petri net for a different set of parameters and initial conditions. We find that even the automata derived from very coarse Petri nets of Case 1c and Case 2a, with places of capacity 1, are able to capture meaningful biochemical information in the form of algebraic groups, in particular the reversibility of the phosphorylation reactions. Significantly, it appears that the algebraic structures uncovered by holonomy decomposition are a larger set than what may be relevant to a specific physical problem with specific initial conditions, although they are all physically possible. This highlights the role of physical context in helping select which algebraic structures to focus on when analysing particular problems. Finally, the interpretation of Petri nets as positional number systems provides an additional perspective on the computational properties of biological systems.