# Competitive inhibition

**Competitive inhibition** is a form of enzyme inhibition where binding of the inhibitor to the enzyme prevents binding of the substrate and *vice versa*.

## Contents

## Mechanism

In competitive inhibition, the inhibitor binds to the same active site as the normal enzyme substrate, without undergoing a reaction. The substrate molecule cannot enter the active site while the inhibitor is there, and the inhibitor cannot enter the site when the substrate is there. In this case, the maximum speed of the reaction is unchanged, while the apparent affinity of the substrate to the binding site is decreased (it means: the K_{d} dissociation constant is apparently increased). The change in K_{m} (Michaelis-Menten constant) is parallel to the alteration in K_{d}. Any given competitive inhibitor concentration can be overcome by increasing the substrate concentration in which case the substrate will outcompete the inhibitor in binding to the enzyme.

## Equation

remains the same because the presence of the inhibitor can be overcome by higher substrate concentrations.

where is the inhibitors dissociation constant and [I] is the inhibitor concentration

## Derivation

In the simplest case of a single-substrate enzyme obeying Michaelis-Menten kinetics, the typical scheme

E + S <==> ES ---> E + P

is modified to include binding of the inhibitor to the free enzyme:

EI + S <==> E + I + S <==> ES + I --> E + P + I

Note that the inhibitor does not bind to the ES complex and the substrate does not bind to the EI complex. It is generally assumed that this behavior is indicative of both compounds binding at the same site, but that is not strictly necessary.
To derive the equation describing the kinetics, first assign microscopic rate constants to each step:

k_{1} = E + S --> ES

k_{-1} = ES --> E + S

k_{2} = ES --> E + P

k_{3} = E + I --> EI

k_{-3} = EI --> E + I

Just as with the derivation of the Michaelis-Menten equation, assume that the system is at steady-state, that is that the concentration of each of the enzyme species is not changing.

=> dE/dt = dES/dt = dEI/dt = 0

Furthermore, the known total enzyme concentration is E_{T} = E + ES + EI, the velocity is measured under conditions in which the substrate and inhibitor concentrations do not change substantially and an insignificant amount of product has accumulated.

We can therefore set up a system of equations:

eq 1: E_{T} = E + ES + EI

eq 2: dE/dt = 0 = -_{k1}*E*S + k_{-1}*ES + k_{2}*ES -k_{3}*E*I + k_{-3}*EI

eq 3: dES/dt = 0 = k_{1}*E*S - k_{-1}*ES - k_{2}*ES

eq 4: dEI/dt = 0 = k_{3}*E*I - k_{-3}*EI

where S, I and E_{T} are known. The initial velocity is defined as v = dP/dt = k_{2}*ES, so we need to define the unknown "ES" in terms of the knowns S, I and E_{T}.

From eq 3, we can define E in terms of ES by rearranging to

k_{1}*E*S=(k_{-1}+k_{2})*ES

Dividing by k_{1}*S gives E = (k_{-1}+k_{2})*ES/(k_{1}*S)

As in the derivation of the Michaelis-Menten equation, the term (k_{-1}+k_{2})/k_{1} can be replaced by the macroscopic rate constant K_{m}:

eq 5: E = K_{m}*ES/S

Substituting eq 5 into eq 4, we have 0 = k_{3}*I*K_{m}*ES/S - k_{-3}*EI

Rearranging, we find that EI = k_{3}*I*K_{m}*ES/(S*k_{-3}).

At this point, we can define the dissociation constant for the inhibitor as K_{i} = k_{-3}/k_{3}, giving

eq 6: EI = I*K_{m}*ES/(S*K_{i})

At this point, substitute eq 5 and eq 6 into eq 1:

E_{T} = K_{m}*ES/S + ES + I*K_{m}*ES/(S*K_{i})

Rearranging to solve for ES, we find E_{T} = ES*(K_{m}/S + 1 + I*K_{m}/(S*K_{i}))= (K_{m}*K_{i}+S*K_{i}+I*K_{m})/(S*K_{i})

=> eq 7: ES = E_{T}*S*K_{i}/(K_{m}*K_{i}+S*K_{i}+I*K_{m})

Returning to our expression for v, we now have v = k_{2}*ES = k_{2}*E_{T}*S*K_{i}/(K_{m}*K_{i}+S*K_{i}+I*K_{m})

Rearranging and replacing k_{2} with k_{cat}, we have v = k_{cat}*E_{T}*S/(K_{m} + S + K_{m}*(I/K_{i}))

Finally, we can replace k_{cat}*E_{T} with V_{max} and combine terms to yield the conventional form:

v = V_{max}*S/(S + K_{m}*(1 + I/K_{i}))

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## see also

- Schild regression for ligand receptor inhibition
- Non-competitive inhibition