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Kinetics: Inhibition of an Enzymes Activity - Background

There is a lot of information provided in this background, use the following table of contents to quickly jump to the subsection you need:

Kinetics of Enzyme Inhibitors
 

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We can make a broad division of enzyme inhibitors into reversible and irreversible types.

Reversible inhibitors bind to the enzyme using weak bonds, similar to those used in binding the substrate. These bonds are formed rapidly, but also break easily. In consequence reversible inhibitors are effectively instantaneous in their action, but do not permanently disable the enzyme. The inhibitor comes to an equilibrium with the enzyme, to form an enzyme-inhibitor complex:

Reversible Inhibition

the amount of inhibition depending on the amount of enzyme which has inhibitor bound, in other words, the position of the equilibrium.

Irreversible inhibitors are also known as enzyme inactivators. They combine with the enzyme by forming a strong, usually covalent bond:

Irreversible Inhibition

Since the reaction is more or less irreversible, the enzyme is effectively permanently disabled. Unlike reversible inhibitors these inactivators take some time to react with the enzyme as covalent bonds are slower to form. Consequently irreversible inhibitors usually display time dependency, the degree of inhibition increasing with the time with which the enzyme is in contact with the enzyme.

Competitive inhibition by active site binding

Classically, a competitive inhibitor is a compound which bears a close structural and chemical similarity to the substrate of the enzyme. Because of this similarity the inhibitor binds to the active site in place of the substrate - a sort of molecular mistake. However, because the substrate and inhibitor are not identical the enzyme is unable to convert the inhibitor into product. The inhibitor simply blocks the active site. While it's there the substrate can't enter and consequently the enzyme can't convert it to product. Similarly, though, if the substrate binds to the active site before the inhibitor, the inhibitor is incapable of binding. The two are said to be mutually exclusive - it is impossible for both of them to bind to the active site at the same time.

animation 1
The animated graphic demonstrates this method of inhibition.

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Competitive inhibition by conformational change

This is the obvious, and commonest, way for competitive inhibitors to work but it isn't the only way. Another possibility is that the inhibitor binds not to the active site but to an inhibitor binding site which is remote from the active site. On binding, however, the inhibitor causes a change in the three-dimensional shape - a conformation change - in the enzyme. This has the effect of altering the active site such that the substrate can no longer bind to it. Similarly, prior binding of the substrate to the active site causes a change in the inhibitor site which prevents the inhibitor from binding.

Once again it is impossible for both inhibitor and substrate to bind to the enzyme at the same time. They are mutually exclusive.

In this kind of competitive inhibition there is no need for the inhibitor to have any chemical similarity to the substrate, as they are both binding to separate enzyme sites.

animation 2

The animated graphic shows this mechanism

Kinetics of competitive inhibitors

Since any kind of inhibitor slows down an enzymic reaction it must clearly have an effect on the kinetics. The nature of that effect may be used to distinguish between inhibitor types. In the presence of a competitive inhibitor the enzyme can bind to the substrate:

eq2

to form an enzyme-substrate complex, or the inhibitor:

graph

to form an enzyme-inhibitor complex. Formation of a ternary complex, in which both substrate and inhibitor are bound to the same enzyme:

eq5
eq6

is not possible as substrate and inhibitor are mutually exclusive.

Both of these binding reactions are very rapid and reversible so they exist in equilibrium. The positions of these equilibria will, of course, depend on the concentrations of the reactants. The equilibria are also linked as the enzyme is a reactant in both of them. This means that at very high levels of inhibitor virtually all of the enzyme molecules will be converted to EI complex and the enzyme will be almost completely inhibited as there is no free enzyme to react with the substrate. On the other hand, at very high substrate concentrations nearly all of the enzyme will be converted into EA complex and there will be no free enzyme to react with the inhibitor. For that reason competitive inhibitors lose their inhibitory power at high substrate concentrations. They are said to be competed out of the system. What effects will this have on the kinetics?

Effects on Km

Km is an indication of enzyme-substrate affinity. In the presence of a competitive inhibitor some enzyme molecules will exist as free enzymes, others as enzyme-inhibitor complexes. The former will have normal affinity and the latter zero affinity as they are totally incapable of binding substrate.Kmis measuring the overall affinity of the enzyme in the reaction mixture which will be an average value between the normal and zero affinity enzymes which will clearly be less than the normal value. So a competitive inhibitor reduces enzyme-substrate affinity, or increases Km.

Effects on Vmax

Vmax is the velocity at very high substrate concentration. As we've just seen, under these conditions the inhibitor is competed out by the substrate and does not inhibit the enzyme at all. So competitive inhibitors do not slow the reaction at high substrate concentrations and their is no change in Vmax.

Effects on Vmax/Km

Vmax/Km, which is simply a ratio between the other two constants, is a useful constant in its own right. If we look at the Michaelis equation:

eq3

we can actually simplify it at low substrate concentrations. Under these circumstances Km + a is almost equal to Km, since the value of a is so small, and the equation simplifies to:

eq4

Remember that both the maximal velocity and the Michaelis constant are constants so the ratio between them must also be a constant. The velocity of a chemical reaction at a known reactant concentration is given by:

velocity = constant x [reactant]

the constant being known as the rate constant of the reaction. It follows that Vmax/Km is the rate constant of the reaction at low substrate concentration, and anything that alters the velocity at low substrate levels will cause a change in this constant.

The Lineweaver-Burk plot

The results of these changes are displayed on a Lineweaver-Burk plot. I'll be using this throughout because of my preference for this as a display plot. Remember that the slope of the plot is equal to Km/Vmax, the inverse of the ratio discussed above, so an increase in the slope in the presence of the inhibitor indicates a reduction in the reaction speed at low substrate levels.

eq1

Summary

Competitive inhibitors prevent the substrate from binding to the enzyme and thereby prevent the enzyme from converting it to product. They are mutually exclusive with the substrate so prior binding of the substrate prevents the inhibitor from binding. Consequently competitive inhibitors are inactive at very high substrate concentrations and do not therefore alter the maximal velocity. They are active at low substrate concentrations which is seen as an increase in the slope of the Lineweaver-Burk plot. They reduce the affinity of the enzyme for its substrate; seen as an increase in the Michaelis constant.

Competitive inhibitors may work by direct competition with the substrate by binding to the active site, or by binding to a remote site and causing a conformational change in the enzyme. Both mechanisms give identical kinetic results.

A noncompetitive inhibitor binds to an inhibitor site on the enzyme which is remote from the active site and brings about a conformational change in the active site. In this sense it's very similar to one of the competitive inhibitor types. The difference is that this time the change in the active site is such that it does not prevent substrate binding but, rather, prevents the enzyme from converting the bound substrate to product.

noncomp
Again this is demonstrated by the graphic.

A classical noncompetitive inhibitor has absolutely no effect on substrate binding. In fact a change to the shape of the active site is almost certain to alter the ability of the substrate to bind. It won't stop it altogether but the affinity will be reduced. Inhibitors like this are often called mixed inhibitors as they appear to have some of the properties of competitive and noncompetitive types. In fact classical noncompetitive inhibitors are very rare, if they exist at all, and I tend to use the words "noncompetitive" and "mixed" interchangeably.

These inhibitors can be distinguished from competitive ones by their effect on the enzyme's kinetics.


Like a competitive inhibitor, a noncompetitive one can bind to both substrate or inhibitor to form enzyme-substrate or enzyme inhibitor complexes:
eq2
revequn

This time though both substrate and inhibitor are capable of combining with the enzyme at the same time, so a ternary complex can beformed:

yeseis1
yeseis2

This complex is not capable of generating a product.

Notice that the EIA complex can be produced by two routes, but the final result is the same - an inactive, abortive, complex. The inhibitor can work equally well at low substrate concentrations, when most of the enzyme is in the form of free enzyme, and at high substrate concentrations, when it is in the form of EA complex, as it binds equally well to both of them.

Effects on Km

A classical noncompetitive inhibitor has no effect whatsoever on substrate binding so the enzyme-substrate affinity, and hence the Km, are unchanged.

A mixed inhibitor allows the substrate to bind, but reduces its affinity, so the Km is increased.

Effects on Vmax

Noncompetitive, of both the classical and mixed varieties, inhibit at high substrate concentrations so the Vmaxis decreased.

Effects on Vmax/Km

Noncompetitive inhibitors can also work at low substrate concentrations so Vmax/Km is decreased

The Lineweaver-Burk plot

The following graph is a Lineweaver-Burk plot for a typical classical noncompetitive inhibitor:

ncomplb
 

A mixed inhibitor would give a plot similar to the following:

mcomplb
 

Notice that in the second graph the two lines have different intercepts on the 1/[Substrate] axis, showing the increase in Km, but in the first the Kmvalues are identical. However both graphs show an increase in the intercept on the velocity axis and the slope, demonstrating that the inhibitor is effective at both high and low enzyme concentrations.

Summary

Non-competitive inhibitors bind to an inhibitor site which is remote from the active site. They inhibit the enzyme by causing a conformational change which prevents enzyme from converting substrate to product. The substrate and inhibitor are capable of binding to the enzyme at the same time to create a ternary complex. This means that the inhibitor is not competed out by high substrate concentrations and works equally well at low and high concentrations of the substrate. Usually, although the substrate can bind to the enzyme-inhibitor complex, its ability to bind (its affinity) is reduced. This results in what is sometimes known as mixed kinetics but there is no real mechanistic difference between a so called mixed inhibitor and a classical noncompetitive inhibitor.

The next inhibitor to discuss is the uncompetitive inhibitor.


Probably the main claim to fame of uncompetitive inhibitors is the frequent confusion of names between them and noncompetitive inhibitors! In fact they are very rare, but you should be aware of them to complete the story, and also because they have some importance in the study of multisubstrate enzymes which are covered in the next chapter.

The key feature of these inhibitors is they are incapable of binding to free enzyme. They can only bind to the enzyme-substrate complex. This could be because the substrate is itself directly involved in binding the inhibitor or because it brings about a conformational change in an inhibitor binding site which was previously incapable of binding the inhibitor. Once the inhibitor has bound it prevents the enzyme from turning the substrate into product. Again this could be some kind of direct interaction, or due to a change in conformation of the active site.

uncompin
The graphic displays the conformational change mechanism.

Again kinetic studies are used to distinguish uncompetitive inhibitors from other inhibitor types.

Kinetics of Uncompetitive Inhibitors

For an uncompetitive inhibitor to work the enzyme must first combine with substrate:
equn2

followed by combination of inhibitor with the enzyme-substrate complex:

yeseis2

Combination of the inhibitor with free enzyme:

noei

is not possible.

The consequence is that uncompetitive inhibitors are inactive at very low substrate concentrations when nearly all of the enzyme is present in the free form. At high substrate concentrations, when most enzyme molecules are present in the form of enzyme-substrate complex, the inhibitor is effective.

Effects on Km

Uncompetitive inhibitors have a rather unexpected effect on the Km. As the inhibitor binds to the enzyme-substrate complex it effectively reduces the concentration of that complex by converting some of it into the ternary, EIA, complex. By the Law of Mass Action this has the effect of pulling the equilibrium of the substrate binding reaction to the right. In effect, then, the inhibitor has increased the amount of substrate which binds to the enzyme, giving an apparent increase in enzyme-substrate affinity and a decrease in Km.

Effects on Vmax

As the inhibitor is not competed out by large amounts of substrate, quite the opposite as it needs substrate to bind to the enzyme first, it is effective at high substrate concentrations and therefore decreasesVmax.

Effects on Vmax/Km

Since these inhibitors do not work at low substrate concentrations, Vmax/Km is unchanged.

The Lineweaver-Burk plot

The following is a typical Lineweaver-Burk plot for an uncompetitive inhibitor.

ucomplb

Summary

Uncompetitive inhibitors can bind only to enzyme substrate complex, not to free enzyme. As a result they do not inhibit at very low enzyme concentrations. They show an apparent increase in affinity for the substrate as more substrate binds to the enzyme but only in the formation of an abortive ternary complex.

We've already seen the importance of the affinity of an enzyme for its substrate in determining how rapidly an enzyme carries out its reaction. Our experimental measure of this affinity is Ks, the dissociation constant of the enzyme-substrate complex or, more usually, Km, the Michaelis constant, which is easier to measure experimentally and is usually taken as a reasonable estimate of Ks. Enzyme-inhibitor affinity is also important, as an inhibitor with a high affinity for the enzyme is more likely to bind to it and will therefore be a more powerful inhibitor. The inhibitor constant ( Ki) is a measure of this affinity. It is the dissociation constant of the enzyme-inhibitor complex and is directly comparable with the Ks. So a large value of Ki indicates a low affinity and vice versa.

The inhibitor equations
In fact things are a little more complicated than that. Binding of an enzyme to its substrate and inhibitor can be represented by the following set of equations:

inhequns
 

The free enzyme is capable of reacting with the substrate, to give an EA complex, or with the inhibitor, to give an EI complex. In addition either of these can be converted to the ternary enzyme-substrate-inhibitor complex (EIA) by binding with the second component. There are therefore four reactions to consider, each of which has its own dissociation constant. These are referred to as Ks, Ks', Ki, and Ki' on the diagram. The reactions are also numbered for easy reference.

As we've seen in our study of the different inhibitor types, not all inhibitors are capable of taking part in all of these reactions:

        Reaction 1Alway possible

        Reaction 2Not possible with uncompetitive inhibitors

        Reaction 3Not possible with competitive inhibitors

        Reaction 4Not possible with competitive or uncompetitive inhibitors

          A noncompetitive inhibitor is capable of all four reactions, but the classical noncompetitive inhibitor, as opposed to a mixed one, is a special case. With these inhibitors Ks and Ks' are equal to each other, as are Ki and Ki'.
Determination Ki of and Ki'

It's clear from the above that mixed inhibitors are the most complex type kinetically, in that they can carry out all four reactions, each of which has a different constant. I'll therefore use this type of an inhibitor as an example in discussing methods of determining inhibitor constants. The others would simply use a simplified version of the same method.


The presence of an inhibitor brings about a change in the Km and/or V of the reaction. We can call these altered values Km' and V'. The amount of change will depend on the concentration of inhibitor used (i) and also on the affinity of the enzyme for the inhibitor, which can be measured by the inhibitor constants. Clearly there must be some kind of mathematical relationship between these. This is shown by the following equations:

kmpequn     vpequn

Notice that the change in Km is linked to the Kivalue and the change in V is linked to the Ki' value, so both inhibitor constants may be calculated using this pair of equations.

With a noncompetitive (mixed) inhibitor these two constants would be different from each other.

With a noncompetitive (classical) inhibitor they would be equal to each other.

With a competitive inhibitor the second equation can't be used as there would be no change in maximal velocity. This is to be expected as Ki'is not relevant to a competitive inhibitor.

With an uncompetitive inhibitor the first equation can be used, as there is a change in Km, but a negative value for Ki would result. This is because the Kmchange with these inhibitors is only an apparent increase, as explained in the uncompetitive inhibition page. In fact, of course, Ki is not relevant to an uncompetitive inhibitor so this result can be ignored.

Of course, if the experiment is repeated at a number of different inhibitor concentrations a series of values for Km and V will be obtained and a number of estimates of the inhibitor constant(s) can be calculated. Under these circumstances a mean value of the results can be calculated, but it makes better use of this data to work out the kinetic parameters by using a secondary plot.

The primary plot
The following graph is a Lineweaver-Burk plot of the results of a series of kinetic assays carried out at a variety of different inhibitor concentrations. The inhibitor is a noncompetitive (mixed) type causing changes in the slope and intercepts of the graph. The inhibitor constants could be calculated using the equations discussed in the previous section but we can also use the data from this graph to draw two secondary plots.

prinplot
 

Secondary plot 1 - 1/Vapp against inhibitor concentration
In the primary plot, each inhibitor concentration gives a different value for the maximal velocity. We can call these apparent maximal velocities and they are read from the graph as their reciprocals (1/Vapp). In the secondary plot these are plotted against the inhibitor concentration. The intercept on the inhibitor axis gives us the value for -Ki' . This is shown in the graph below - the Ki' value being 15 units.

secplot1
 

Secondary plot 2- slope against inhibitor concentration

The primary plot also shows a change in slope as you alter the inhibitor concentration, and this gives us another secondary plot.
secplot2

This time the intercept on the inhibitor axis indicates -Ki. As can be seen from the graph the Ki for this inhibitor is 8 units.

In plotting this graph remember that the slope of the Lineweaver-Burk plot is equal to Km/V. It's probably easier to calculate the slope from the Km and V values rather than measuring it directly.

Secondary plots with different inhibitor types
The sample plots shown here were produced using a mixed inhibitor, as this is the kinetically most complex of the types that we've studied. For the other types matters are simplified as follows:

  • Classical noncompetitive inhibitor
    • The secondary plots are made as above but the two plots should give identical results as Kiand Ki' are equal for these inhibitors.
  • Competitive inhibitor
    • The first secondary plot cannot be made as there is no change in maximal velocity. This plot is not required though as it gives Ki' which is irrelevant for a competitive inhibitor.
  • Uncompetitive inhibitor
    • This is really the opposite of the competitive inhibitor. The second secondary plot can't be made as there is no change in slope. Again this is not required as the Ki is irrelevant to an uncompetitive inhibitor.

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