Algebraic Analysis of the Computation in the Belousov-Zhabotinksy Reaction

We analyse two very simple Petri nets inspired by the Oregonator model of the Belousov-Zhabotinsky reaction using our stochastic Petri net simulator. We then perform the Krohn-Rhodes holonomy decomposition of the automata derived from the Petri nets. The simplest case shows that the automaton can be expressed as a cascade of permutation-reset cyclic groups, with only 2 out of the 12 levels having only trivial permutations. The second case leads to a 35-level decomposition with 5 different simple non-abelian groups (SNAGs), the largest of which is A9. Although the precise computational significance of these algebraic structures is not clear, the results suggest a correspondence between simple oscillations and cyclic groups, and the presence of SNAGs indicates that even extremely simple chemical systems may contain functionally complete algebras. Introduction In self-organising systems, the “self” or autonomous aspect is provided by the fall towards equilibrium, which serves as the driver or energy source. As a consequence, a system that needs to maintain self-organising behaviour indefinitely must be open since, if it were closed, once it had reached equilibrium it would stop functioning. Therefore, in order to keep going it must be open and connected to a source of (free) energy that can keep it ‘far from equilibrium’, to use Prigogine’s famous phrase [19], even whilst it is continually falling towards it. The Belousov-Zhabotinsky (BZ) reaction has been studied extensively [20] because it was the first reaction to exhibit sustained oscillations even in an isolated system, although they do die down eventually. Before Belousov’s discovery in the 1930s and Zhabotinsky’s confirmation of the phenomenon in the 1960s, 2 P Dini, C L Nehaniv, A Egri-Nagy and M J Schilstra species concentrations were believed to vary monotonically unless driven by a periodic forcing function. In a constant-flow reactor the oscillations are periodic and can be sustained indefinitely, as long as the inflow and outflow are kept constant. This qualifies the BZ reaction as a system far from equilibrium. In this paper we analyse a simplified ordinary differential equation (ODE) model of the BZ reaction, the almost equally famous “Oregonator” model, developed by Field and Noyes at the University of Oregon [11]. We compare the structure and behaviour of a very simple system inspired by the Oregonator model of the BZ reaction from the different viewpoints of systems biology and algebraic automata theory. In particular, we focus on its oscillatory behaviour. Although the computational properties of chemical oscillations are not clear, the fact that we are very familiar with them at both an intuitive and a mathematical level makes them a useful reference system when attempting to decipher the computational significance of the algebraic structures uncovered in the corresponding finite state automata, as we discuss below. This was, in fact, the main motivation for selecting the BZ reaction as an object of study. Thus, this work aims to merge two research traditions: dynamical systems theory rooted in physics and informing much of modern-day systems biology, and theoretical computer science rooted in algebraic automata theory [16].