The Working, Equation, and Examples of Non-competitive Inhibition

Non-competitive Inhibition: Working, Equation, and Examples
The activity of any enzyme depends on its interaction with two effector molecules - the substrate/activator and the inhibitor. Binding of the substrate brings about enzymatic activity, whereas binding of the inhibitor ceases the activity. There are various mechanisms employed for the inhibition of an enzyme by the inhibitor. This Buzzle post explains non-competitive inhibition along with examples.
Heavy metals such as lead, mercury, copper, and silver are poisonous since they act as non-competitive inhibitors of cysteine residues present in various enzymes and proteins in the human body.
Enzyme inhibition by an inhibitor can be either irreversible or reversible. Irreversible inhibitors covalently bind to the enzyme and modify its structure and function. This modification, hence, renders the enzyme permanently inactive. On the other hand, reversible inhibitors only cause the cessation of enzyme activity temporarily, i.e. the enzyme is inactive only till it is bound to the inhibitor, once the inhibitor is unbound, the enzyme may resume its activity. These reversible inhibitions are brought about by three different mechanisms of inhibitor binding. They are explained as follows.
Competitive Inhibition: The inhibitor possesses an affinity to the active binding site of the enzyme, and hence, competes with the substrate to bind to the enzyme. Therefore, if the inhibitor has bound itself to the active site, the substrate is unable to bind, and thus is not converted to the product.
Uncompetitive Inhibition: The inhibitor has an affinity towards the bound enzyme-substrate complex, i.e, it does not bind to the enzyme if the substrate is not bound to the enzyme. It therefore achieves inhibition by forming the enzyme-substrate-inhibitor (ESI) complex, which cannot form a product.
Non-competitive Inhibition: The inhibitor molecule shows an affinity to the enzyme itself as well as the enzyme-substrate complex. It binds to a site different from the active site of the enzyme such that it alters the active site.
It involves the binding of the inhibitor at an allosteric site (alternate binding site that is not the active site). This binding brings about an alteration in the active site such that, either the substrate is not able to bind at all or the substrate binds but fails to activate the enzyme. Here, if E is the enzyme, I is the inhibitor, S is the substrate, and P is the product, the following equation can be used to explain the working of this type of inhibition.
noncompetitive inhibition general equation
Given above is a set of two reactions presented one below the other, with the corresponding constituents being reversible reactions themselves.

Let us consider the first equation.

It shows that in the presence of the enzyme, inhibitor, and substrate, if the substrate binds to the enzyme and activates it, the enzyme-substrate (ES) complex that is formed catalyzes the substrate and forms a product (enzyme is recovered).

Now, consider the second equation.

It shows that in case the inhibitor has already bound to the enzyme forming a enzyme-inhibitor (EI) complex, the subsequent binding of the substrate merely forms a trimeric enzyme-substrate-inhibitor (ESI) complex, but does not result in the activation of the enzyme. Thus the substrate is not subjected to enzymatic activity, and hence no product is obtained.

These two reactions are inter-connected due to their reversible nature, i.e., the formation of the EI and ESI complexes are reversible, and hence have a dissociation constant Ki in a reaction with the presence of an inhibitor.
Michaelis-Menten Equation
The Michaelis-Menten equation provides a relation between the rate of reaction and the concentration of the substrate.

Since the inhibitor does not compete with the substrate to bind to the enzyme, the concentration of the substrate has no apparent effect on the activity of the enzyme, and in turn, the rate of reaction (Vm). Instead, since the inhibitor binds to not only the pure enzyme, but also to the enzyme-substrate complex, the rate of reaction depends on the concentration of the inhibitor in the given system.

Therefore, Vm, that gives the enzyme activity or rate of reaction in ideal conditions without the presence of inhibitor, is replaced by Vm,app (apparent value of Vm), which changes the equation to accommodate the reduction of enzyme activity. This reduction takes into account the concentration of the inhibitor [I] and its dissociation constant KI. Therefore, the Michaelis-Menten equation for such a reaction is given by,
noncompetitive inhibition kinetic equation
Lineweaver-Burk Plot
It is a graphical method used to conduct analysis of the Michaelis-Menten equation of a given system, and it emphasizes the relationship between substrate concentration [S] and the rate of reaction (Vm). Hence, it can be used to compare the changes in the kinetics of a system in the presence and absence of an inhibitor.

The inhibition constant (Km) refers to the required concentration of the inhibitor in a reaction so as to reduce the rate of reaction (Vm) by half in a Michaelis-Menten enzymatic model. In the presence of a non-competitive inhibitor, the substrate and the inhibitor do not have competing affinities, and hence the observed inhibition constant (Km,app) is equivalent to the Km exhibited by the system in ideal conditions. The inhibitor binds to the enzyme as well as the ES complex with equal affinity, thus a state of equilibrium is achieved and maintained. However, as the ESI complex yields no product, the effectivity of the enzyme is lowered.

In such a system, the inhibitor does not determine the availability of the active site, and hence does not have any significant effect on the inhibition constant. So, the Km (X-intercept) is the same whether an inhibitor is present or not, i.e., it remains unchanged. However, since the enzyme activity is affected by the inhibitor, the rate or velocity of the reaction (Y-intercept) is affected, i.e., it is reduced (as shown and explained below). The resultant graph is shown below where the rate of reaction is compared between two systems, one with an inhibitor and one without.
lineweaver burk noncompetitive inhibition
Hence, in the Lineweaver-Burk plot, since Km is unchanged, all plotted curves pass through the same X-intercept. In case of the reaction velocity, the Y-axis plots 1/Vm, where the decrease in the Vm corresponds to the increase in the 1/Vm (since it is an inverse of itself).
Monoamine oxidase (MAO) inhibitors, used as antidepressants, act as non-competitive inhibitors of the enzyme MAO and functions to prevent the breakdown of monoamine neurotransmitters (e.g. serotonin, dopamine, epinephrine, etc.), thereby increasing their availability.
Cytochrome P450 2C9 (abbrev. CYP2C9) is an enzyme that oxidizes various endogenous and xenobiotic compounds. Its non-competitive inhibitors are nifedipine, phenethyl isothiocyanate, medroxyprogesterone acetate, and 6-hydroxyflavone.
Carbon dioxide (CO2) is a non-competitive ligand, as it binds to the amino (i.e. NH2) groups present on the surface of hemoglobin molecules.
Digitalis, a compound extracted from the foxglove plants, inhibits ATPase, and results in an increase in the contraction of heart muscles.
Foxglove Plants
Strychnine, a toxic alkaloid, binds to glycine receptors and inhibits the binding of glycine (inhibitory neurotransmitter), therefore inducing the motor neurons to remain active. This leads to rapid and repeated contraction and relaxation of the body muscles.
Penicillin binds to a bacterial enzyme DD-transpeptidase, which polymerizes the peptidoglycan units to form the cell wall. The inhibition of this enzyme causes the bacteria to have a weak cell wall which no longer protects it from exogenous entities.
Non-competitive inhibition is also referred to as mixed inhibition since it exhibits affinities of both competitive (enzyme binding only) as well as uncompetitive inhibitors (ES complex binding only).