This model shows the case of an enzyme facilitating the formation of a product from a substrate but an inhibitor exists which also can bind to the enzyme rendering it ineffective.

Description

This model represents the enzymatic conversion of a single substrate, S, to a single product, P, but there is an inhibitor, I, which also can bind to the enzyme, E, preventing it from forming the product. Initially the enzyme can either bind with the substrate forming the product yielding complex, ES, or it can bind to the inhibitor forming the non reacting complex, EI. If the ES complex is formed a reaction-release event follows, which yields the product and the enzyme. The entire binding- inhibition-reaction-release sequence may be represented symbolically as:

                        k1 -->             k2 -->
         S + I + E  <----------->  ES  <----------->  P + E
                       <-- k_1            <-- k_2
                 ^                                    
               | | ^
            k3 | | | k_3
               v | |
                 v

                EI

where k₁ is the forward binding rate of S to E, k_-1 is the backwards reaction rate of ES dissociating to E and S, k₂ is the forward reaction rate of ES forming E and P, k_-2 is the reverse reaction rate of E and P producing ES, k₃ is the forward reaction rate of E binding to I and k_-3 is the reaction rate of EI dissociating to form E and I.

Equations

This reaction is governed by a system of five ODEs which describe the concentrations of the substrate, inhibitor, enzyme complex ES, inhibition complex EI and product. The sixth equation to close the system specifies the total amount of enzyme present, E_tot which must be conserved. All of the substrate and no complex or product are present at time t=0. The system of equations are:

The backward reaction rates in this model are determined from the equilibrium dissociation rates of S binding to E, I binding to E and P binding to E. The expressions for the equilibrium dissociation rates are given by:

where K_s is the equilibrium dissociation rate of S binding to E, K_i of I binding to E and K_p of P binding to E. The reaction velocity is the current velocity of the reaction in forming the product, v, divided by the maximal reaction velocity, V_max. If we assume Michaelis-Menten kinetics the reaction is rate limited by the dissociation of ES into E and P. Therefore we have: