Cellular Automata Approaches to Enzymatic Reaction Networks

Cellular automata simulations for enzymatic reaction networks differ from other models for reaction-diffusion systems, since enzymes and metabolites have very different properties. This paper presents a model where each lattice site can can contain at most one enzyme molecule, but many metabolite molecules. The rules are constructed to conform to the Michaelis-Menten kinetics by modeling the underlying mechanism of enzymatic conversion. Different possible approaches to rule construction are presented and analyzed, and simulations are shown for single reactions and simple enzyme networks. 1 Enzymatic Reaction Networks Most reactions in biological systems are catalyzed by enzymes. These enzymes are complex molecules (they are proteins) which are not consumed in the reaction, but simply facilitate the reaction of smaller molecules called metabolites (such as sugar). In a biological cell, thousands of different enzymes are active. Each enzymatic reaction takes molecules of one or more metabolite species, the substrates of this reaction, and converts them into molecules of one or more other species, the products. The products are again substrates to other reactions, and thus the reactions form complex networks. Enzymatic reactions can be described and modeled on different levels of detail: Static Interaction Networks: A first level is the static interaction network, such as the pathways collected in the KEGG database [9, 10, 11], or in the BoehringerMannheim map [13]. Such interaction networks can be constructed from purely qualitative information without relying on any quantitative information, such as reaction rates (except possibly for stoichiometric coefficients). Some analyses can extract additional information from these networks, such as inferring elementary metabolic flux modes [14, 15] (although these rely on the information whether a given reaction is reversible or not, which in turn is a semi-quantitative information on the order of magnitude of the equilibrium constant). Simply on the basis of such interaction networks, no quantitative time-dependent simulation is possible. Reactive Networks: Once quantitative data is available, reaction rates for reactions such as the conversion of a substrate S to a product P by an enzyme E can be described by some rate law, usually in the form of a Michaelis-Menten (MM) law: d[S] dt = Vmax[E][S] Km + [S] (1) with the maximum conversion rate Vmax and the Michaelis-Menten coefficient Km. A more detailed description would contain rates for the elementary reactions leading to such an overall MM rate [1]: E + S k1 ⇀↽ k−1 ES k2 →E + P. (2) Here, several rate constants need to be measured, which is usually not done or not possible. Most reactions in a realistic reaction network are more complicated, since they involve two or more substrates and two or more products, such as energy-providing ATP. In these cases, more coefficients need to be specified, and in addition the mechanism must be specified, such as BiBiRandom (in which case the two substrates may bind in any order) or Ordered BiBi (where the substrates must bind in a specific order). Other effects, such as competitive inhibition complicate the situation further. Given a network of enzymatic reactions and the corresponding reaction rates, one can simulate the network by solving the ODEs numerically, or using stochastic simulation methods [7, 8]. A static approach is to analyze steady states, parameter dependences or sensitivities, etc.. Space and Transport Phenomena. The space and diffusion or other transport phenomena are usually only taken into account by compartmentalizing the system where necessary, but they can also be included explicitly (as reactiondiffusion equations, or probabilistically in the stochastic simulation approach [6, 17]). The most detailed simulation would be a full molecular dynamics simulation of the cell, but this is by far not feasible yet. In this paper I present an approach based on cellular automata for simulating enzymatic reaction networks including diffusive transport.