# Generic kinetic equations for modelling multisubstrate reactions in computational systems biology

Thesis (MSc (Biochemistry))--University of Stellenbosch, 2006.

Thesis

Systems biology is a rapidly developing field, studying biological systems by methodically perturbing them either chemically, genetically or biologically. The system response is observed and incorporated into mathematical models. These computational models describe the system structure, predicting its behaviour in response to individual perturbations. Metabolic networks are examples of such systems and are modelled in silico as kinetic models. These kinetic models consist of the constituent enzyme reactions that make up the different pathways of a metabolic network. Each enzyme reaction is represented as a mathematical equation. The main focus of a kinetic model is to portray as realistically as possible a view in silico of physiological behaviour. The equations used to describe model reactions therefore need to make accurate predictions of enzyme behaviour. Numerous enzymes in metabolic networks are cooperative enzymes and many equations have been put forward to describe these reactions. Examples of equations used to model cooperative enzymes are the Adair equation, the uni-reactant Monod, Wyman and Changeux model, Hill equation, and the recently derived reversible Hill equation. Hill equations fit the majority of experimental data very well and have many advantages over their uni-substrate counterparts. In contrast to the abovementioned equations, the majority of enzyme reactions in metabolism are of a multisubstrate nature. Moreover, these multisubstrate reactions should be modelled as reversible reactions, as the contribution of the reverse reaction rate on the net conversion rate can not be ignored [1]. To date, only the bi-substrate reversible MWC equation has been formulated to describe cooperativity for a reversible reaction of more than one substrate. It is, however, difficult to use as a result of numerous parameters, not all of which have clear operational meaning. Moreover, MWC equations do not predict realistic allosteric modifier behaviour [2, 3]. Hofmeyr & Cornish-Bowden [3] showed how the uni-reactant reversible Hill equation succeeds in predicting realistic allosteric inhibitor behaviour, compared to the uni-reactant MWC equation, which does not. The aim of this study was to therefore derive a reversible Hill equation that can describe multisubstrate cooperative reactions and predicts realistic allosteric modifier behaviour. In this work, we present a generalised multisubstrate reversible Hill (GRH) equation. The bi-substrate and three substrate cases of this equation were also extended to incorporate any number of independently binding allosteric modifiers. The derived GRH equation is evaluated against the above mentioned cooperative models and shows good correlation. Moreover, the predicted behaviour of the bi-substrate reversible Hill equation with one allosteric inhibitor is compared to the MWC equation with one allosteric inhibitor in silico. This showed how the bi-substrate reversible Hill equation is able to account for substrate-modifier saturation, unlike the MWC equation, which does not. Additionally, the bi-substrate reversible Hill equation behaviour was evaluated against in vitro data from a cooperative bi-substrate enzyme which was allosterically inhibited. The experimental data confirm the validity of the behaviour predicted by the bi-substrate reversible Hill equation. Furthermore, we also present here reversible Hill equations for two substrates to one product and one substrate to two products reactions. Reactions of this nature are often found in metabolism and the need to accurately describe their behaviour is as important as reactions with equal substrates and products. The proposed reversible Hill equations are all independent of underlying enzyme mechanism, they contain parameters that have clear operational meaning and all of the newly derived equations can be transformed to non-cooperative equations by setting the Hill coefficient equal to one. These equations are of great use in computational models, enabling the modeller to accurately describe the behaviour of a vast number of cooperative and non-cooperative enzyme reactions with only a few equations.