This model describes the kinetics of an enzymatic reaction where an inhibitor can bind to the enzyme in a non-competitive manner.

Description

This model describes the enzymatic conversion of a single substrate, S, to a single product, P, with an inhibitor, I, which can also bind to the enzyme, E, preventing it from forming the product. The difference between competitive and non-competitive inhibition is that the enzyme-inhibitor complex, EI can still bind to the substrate in the non-competitive case. The resulting enzyme-inhibitor-substrate complex, EIS, can then dissociate into ES and I. The ES complex can then yield the product through a reaction release step. The entire binding-inhibition-reaction-release sequence may be represented symbolically as:

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

where k₁ is the forward binding rate of S to E to EI, k_-1 is the backwards reaction rate of ES dissociating to E and S and EIS to EI 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 and ES binding to I and k_-3 is the reaction rate of EI dissociating to form E and I and EIS to ES 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 and EI, K_i of I binding to E and ES 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: