Past exam of the mathematics and the environment and mathematical physiology course of the University of Oxford / 2007 / B / 5 / ii

Write concentrations as $E,S,X,Y,E_i,P$ in this calculation. The law of mass action assigns the forward bimolecular rate $k_{1}ES$ and the first-order rates $k_{-1}X,k_{2}X,k_{3}Y,k_{4}Y$. Each species gains the rates of reactions producing it and loses those consuming it:Adding the equations for all enzyme-bearing species cancels every reaction rate, so $\boxed{E+X+Y+E_i=e_0}$. This is conservation of the total enzyme-derived material, including the inactivated form. It does not assert conservation of active enzyme: mechanism-based inhibition transfers some of that conserved material irreversibly into $E_i$.