title: A method for writing enzyme rate equations: application to the estimation of the number and size of key proton families.
abstract: We develop a method to derive the rate equation for enzyme models that include pH-dependent activation. Our presentation is based on a kinetic model recently described for sucrase, the three-key-proton model of Vasseur and coworkers, which considers the existence, in the acid ionization reaction, of two functionally distinct prototropic groups, respectively responsible for either V-type or K-type kinetic effects. In contrast, as concerns the basic ionization reaction, the model conforms to classical concepts of pH-dependent activation, whereby a single proton participates in either V-type or K-type effects but not in both at the same time. Enzymes with more than three key protons have been described, indicating that, rather than isolated protons, groups of protons should be considered, and therefore the model can be better described as a three-proton-family model, where a proton family is defined as one or several protons that are gained or lost as a block and perform the same kinetic function. The resulting model is treated here as a useful framework upon which other models can be built. To facilitate the writing of the rate equations, we define two new entities: (1) intralevel coefficients, which describe the various combinations of the enzyme with either the substrate(s), the allosteric effector(s), or both at a given protonation level, and (2) interlevel coefficients, which describe the interplay between the various protonation levels. The resulting rate equation can be used in a global fit procedure permitting in a single computer run the estimation of (1) the entire set of dissociation and microscopic ionization constants of the model, (2) the number and kinetic function of proton families characterizing the enzyme under consideration, and (3) the number of key protons constituting each family, which is derived from the derivatives of the kinetic parameters, Vm/Km, Vm, and Km.
authors.forename: M
authors.initials: M
authors.lastname: Vasseur
authors.forename: G
authors.initials: G
authors.lastname: van Melle
authors.forename: F
authors.initials: F
authors.lastname: Alvarado
mesh_terms: D004798:Enzymes / Q000378:metabolism*; D007700:Kinetics; D008433:Mathematics; D008954:Models, Biological; D011522:Protons
chemical_list: D004798:Enzymes; D011522:Protons