Mathematical expressions describing enzyme velocity and inhibition at high enzyme concentration

Enzyme behaviour is characterised in the laboratory using diluted solutions of enzyme. However, in vivo processes usually occur at [ S _T ] ≈ [ E _T ] ≈  K _m . Furthermore, the study of enzyme action involves characterisation of inhibitors and their mechanisms. However, to date, there have been no reports proposing mathematical expressions that can be used to describe enzyme activity at high enzyme concentration apart from the simplest single substrate, irreversible case. Using a continued fraction approach, equations can be easily derived for the most common cases in monosubstrate reactions, such as irreversible or reversible reactions and effector (inhibitor or activator) kinetic interactions. These expressions are an extension of the classical Michaelis-Menten equations. A first analysis using these expressions permits to deduce some differences at high versus low enzyme concentration, such as the greater effectiveness of allosteric inhibitors compared to catalytic ones. Also, they can be used to understand catalyst saturation in a reaction. Although they can be linearised, these equations also show differences that need to be taken into account. For example, the different meaning of line intersection points in Dixon plots. All in all, these expressions may be useful tools for modelling in vivo and biotechnological processes.