Inhibition Principles of
Biological Systems II
Enzyme Kinetics
Prof.Dr. İlhan Talınlı
Thank You Peter Birch
• Welcome to this course in enzyme kinetics
produced by Dr. Peter Birch of the Department
of Biological Sciences at the University of
Paisley.
• The course is designed to introduce you to the
theoretical concepts and practical techniques
associated with the use of kinetics as an
enzymological tool. It is divided into a series of
chapters which can either be followed in
sequence as a complete course, or used as a
reference book for help with individual areas.
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Copyright:
These pages are copyright University of Paisley.
The Italian translation is copyright:
prof. Giuseppe Striccoli
docente di Chimica presso
L'I.T.I.S. "Galileo Galilei"
via Parisi, Polivalente
70022 Altamura
Italy
The animated Italian flag is copyright Crames Studios
Any questions or comments about the course can be sent to:
Dr. Peter Birch peter.birch@paisley.ac.uk
Department of Biological Sciences
University of Paisley
PAISLEY
Renfrewshire
PA1 2BE
UK
Tel: +44 141-848 3123
Content of Enzyme Kinetics
Chapter 1: The effects of substrate concentrations on
reaction rate
Chapter 2: Determination of kinetic parameters
Chapter 3: Kinetics of enzyme inhibitors
Chapter 4: Kinetics of multisubstrate systems
Chapter 5: Kinetics of allosteric enzymes
The Effects of Substrate
Concentration on Reaction Rate
• Introduction
• The term enzyme kinetics implies a study of the speed, rate or
velocity of an enzyme catalysed reaction, and of the various factors
which may affect this.
• At the heart of any study of enzyme kinetics is a knowledge of the
way in which reaction velocity is altered by changes in the
concentration of the enzyme's substrate and of the simple
mathematics underlying this.
• To ease ourselves gently into this we will assume that the enzyme
that we are discussing has no special features, such as allosteric
properties, and catalyses the conversion of just one substrate to one
product. This may seem unrelated to real enzymes, very few have
just one substrate after all, but it will provide a basis which we can
expand on later when we study more complex systems.
• To follow the complete course go through the chapters and the
sections in their menu headings in sequence. For general reference
- click at will!
The Effects of Substrate
Concentration on Reaction Rate
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Description of the velocity/substrate concentration curve
The reaction carried out by an enzyme can be represented by the following
equation:
in which:
A is the substrate
P is the product
E is the enzyme
In a normal, non-catalysed chemical reaction we would expect that the
velocity of the reaction would be directly proportional to the concentration of
substrate. In other words, if you doubled the concentration of substrate the
velocity should also double. The reason for this is purely a statistical one. If
there are twice as many substrate molecules then the number which will
have sufficient energy to undergo reaction will be doubled. For a noncatalysed reaction, then a graph of velocity against substrate concentration
would be a straight line.
In an enzyme catalysed reaction the same type of experiment, measuring
reaction velocity at various different concentrations, does not give a straight
line but a curve.
Graph of velocity/[substrate]
The Michaelis Equation
• In order to make the v/[A] curve give us information
about the properties of the enzyme, and the way in
which it responds to its environment in the cell, we need
to understand a little about the underlying mathematics.
• Any graph which results from plotting two variables
against each other can be described by a mathematical
equation showing the relationship between the two
variables, together with one or more constants. For
instance the familiar equation of a straight line (y=mx+c)
shows the relationship between the variables y and x.
The constants m and c give us valuable information
about the graph (the slope of the line and the intercept
on the y-axis).
• Obviously the graph that we are looking at here is not a
straight line, it's described mathematically as a
rectangular hyperbola and its equation is going to be
slightly more complex, but nonetheless it should be
possible to derive it and hopefully gain some useful
information from it. Our knowledge of the equation of the
v/[A] curve is based on pioneering work carried out by
Michaelis and Menton and by Briggs and Haldane, and
is usually called the Michaelis equation after the first of
these workers. It can be written in a number of different
ways, and we'll be seeing some of them as we progress
through our study, but probably the commonest form is:
• The two variables in this equation are the reaction
velocity, v, and the concentration of substrate, a. V and
Km are two constants which will require further
explanation.
The Effects of Enzyme
Concentration
We've seen that the concentration of the substrate
affects the velocity of an enzyme reaction. Not
unreasonably the velocity is also dependent on the
concentration of enzyme. Lets examine how a change in
enzyme concentration might alter the values of the two
kinetic parameters which we have just been discussing.
Vmax is the reaction velocity at very high, saturating,
concentrations of substrate. Remember under these
conditions every enzyme molecule will have substrate
attached to it and will be interacting with it to convert it to
product as fast as it can. If we doubled the number of
enzyme molecules we would have twice as many with
substrate bound so you would expect the overall reaction
rate to double. This is in fact the case. Vmax is directly
proportional to the concentration of enzyme.
In this graph I have shown the effect of halving the
concentration of enzyme. As you can see the reaction
velocity has reduced from 10 to 5 units. Km tells us
about the affinity of the enzyme for its substrate.
Changing the number of enzyme molecules doesn't alter
their individual chemical characteristics and you would
therefore expect them to be able to bind the substrate no
better, and no worse, than before. Consequently Km is
independent of enzyme concentration. The graph also
demonstrates this. Reducing the enzyme concentration
by 50% has reduced Vmax in proportion but, of course,
Vmax/2, which is used for measuring Km is also reduced
by half. The graph shows that the Vmax/2 line for the
50% enzyme cuts the curve at exactly the same point,
relative to the horizontal axis, as the corresponding line
for the full strength enzyme. The Km in other words is
unchanged.
Determination of Kinetic
Parameters
• Introduction
• If you've studied and understood Chapter 1 of this
course it will be apparent that being able to determine
Km and Vmax accurately is important for the
enzymologist. There are a number of ways of doing this.
They are all based on the same experimental method
(that of measuring the reaction velocity at a variety of
different substrate concentrations) but differ in the way in
which the experimental data are manipulated. The aim of
this chapter is to explain some of the more important
methods, together with their advantages and
disadvantages, and give some suggestions as to the
best method to use in particular circumstances.
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To follow the complete course click on the
following headings in sequence. For general
reference - click at will!
Direct use of the v/[A] curve
The Lineweaver-Burk plot
The Eadie-Hofstee plot
The Hanes plot
The direct linear plot
Horses for courses - which method to choose
A hypothetical enzyme
• In order to study the methods of determining Vmax and Km which
this chapter covers we need a sample enzyme to work with. We
could use some real experimental results, but there are some
advantages in working with an invented enzyme which has known
values of the kinetic parameters. We can then study the techniques
using "experimental results" which are guaranteed to contain no
experimental error, but also simulate experimental error when we
wish to find out how the different methods are able to cope with it.
• The "hypothetical enzyme" which is going to be studied here is
actually the same one that was used to plot the graphs in Chapter 1
of this course. If you look back at the graph in Chapter 1 where we
discussed Vmax and Km you'll see that the enzyme has the
following values for these parameters:
Vmax
10
Km
4
Kinetic parameters for
the hypothetical enzyme
Both constants are expressed in arbitrary units.
In plotting the graphs in this chapter I've used a set of substrate
concentrations up to 10 units and carried out four "experiments". In the
first of these there is no experimental error introduced while in the other
three there is a random amount of error calculated by computer. If you
want to try plotting the graphs yourself (highly recommended!) you can
find the data I used by clicking here.
The use of the v/[A] curve for
determining kinetic parameters
As we saw in the introductory page to this chapter all
methods of determing kinetic parameters use the same
basic experiment of measuring velocity at different
substrate concentrations. We already know one way of
using this data to determine Vmax and Km - just use the
v/[A] graph that we saw in Chapter 1. It's simply a
question of plotting the graph, reading the Vmax directly
from it, and finding Km from the plot as the substrate
concentration equal to half of the Vmax value. This is
easy to understand - but there are real problems
involved that we should consider.
Measurement of Vmax
As we saw in Chapter 1 the maximal velocity is never
actually achieved. The velocity will keep rising as higher
concentrations of substrate are introduced but it never
really stops increasing however much substrate you add.
The slope of the graph just keeps getting shallower and
shallower. The velocity would only stop rising if you could
reach infinite substrate concentration. This means that
Vmax can never be directly measured. The best you can
hope for is a good estimate of its value. You can see
from the graph that we've already studied that the
highest velocity measured is about 9.5 units while the
true Vmax is 10.
Is a good estimate sufficient?
Well, sometimes it is, it depends on what you're
using your data for, but it has to be a good
estimate. Sometimes the highest velocity you
can achieve is far below the Vmax value. This
might occur because the substrate has a low
solubility so that sufficiently high concentrations
can't be used or, as happens with some
enzymes, large amounts of substrate actually
inhibit the enzyme, reducing the velocity before
Vmax can be reached. We'll be looking at this
idea of substrate inhibition in Chapter 3.
Drawing curves
Another problem with the v/[A] graph is
that it is a curved plot. Curves are
notoriously difficult to draw accurately by
eye, and this is particularly so if there is
any noticeable error in the data so that the
points are displaced from their true
positions. Under these circumstances it's
relatively easy to draw a best fit line with a
linear (straight line) plot, but much harder
with a curved plot.
Can these problems be solved
They can be solved by using one of the other techniques
discussed in this chapter but, using computer methods,
the v/[A] curve can be used itself. All you need is a
computer and software which is able to calculate a best
fit curve from your data. You simply need to give the
software your data and the equation which you expect it
to fit to (the Michaelis equation in this case). The
software will calculate the pair of Vmax and Km values
which will fit the line closest to the experimental data. Of
course, once it's done this it has automatically worked
out the parameters that you want. Software to do this is
now readily available. It doesn't need any full blown
statistical or mathematical package - a decent
spreadsheet can do it for you. Click here for some
instructions in how to do it in Microsoft Excel.
The use of the Lineweaver-Burk plot for
determining kinetic parameters
Undoubtedly the most popular, and familiar, manual
technique for calculating kinetic parameters is that
known as the Lineweaver-Burk,or double reciprocal, plot.
It is also probably the worst method to use for reasons
that I will attempt to explain and demonstrate. Firstly
though let's have a look at how the method works.
A simple algebraic conversion of the Michaelis-Menten
equation gives the following expression:
In this equation 1/v and 1/a are both variables,they are
simply the reciprocals of the velocity and substrate
concentration values, while Km/V and 1/V are constants
as they are derived from the constants Km and V. If you
compare this to the equation:
which is the equation of a straight line, youwill see that
both equations have the same structure - a variable
equals a constant times a variable plus a constant. It
follows that if you plot the two variables, 1/v and 1/a,
against each other your points will lie on a straight line.
This has removed one of the objections to the use of the
v/[A] curve. We now have the much simpler task of
drawing a straight line graph which can be extrapolated
to cut both axes.
Click on the shaded areas for explanations of the individual sections of the graph
• This seems to answer many of the problems involved in
calculating kinetic parameters from the velocity against
substrate concentration curve:
• it's a straight line which is much easier to draw
• it doesn't require a direct measurement of Vmax
• both Vmax and Km are read easily from the graph
• Unfortunately it does have some real drawbacks in
dealing with data containing significant experimental
error and is not usually recommended for determination
of kinetic parameters. You can see an explanation of
these problems or look at the next technique which is the
Eadie-Hofstee plot.
The use of the Eadie-Hofstee plot for
determining kinetic parameters
Like the Lineweaver-Burk plot this technique is
based on the conversion of the MichaelisMenten equation to give the equation of a
straight line:
This time the variables are v and v/a, so a plot of
these two would give a straight line of slope -Km
and intercept V. The intercept on the horizontal
axis would be V/Km.
The graph gives a direct
readout of the maximal
velocity. Km is probably best
determined from the V/Km
value as it's a bit of a
nuisance to calculate the
slope of the line. Like the
Lineweaver-Burk plot this is
a simple technique, but it
also has its problems. Click
here for a discussion of
these or move on to the
Hanes plot.
The use of the Hanes plot for
determining kinetic parameters
• This is another conversion of the MichaelisMenten equation to give the equation of a
straight line:
• This time a plot of substrate concentration
divided by velocity against substrate
concentration will give a straight line with an
intercept on the horizontal axis of -Km and on
the vertical axis of Km/V. The slope is I/V
This graph gives a direct
readout of Km. To avoid
having to measure the slope it
is probably best to calculate
Vmax from the Km/V
intercept. Click here to see
how the Hanes plot deals with
the problems found with the
other linear plots, or move on
to the direct linear plot.
The use of the direct linear plot for
determining kinetic parameters
• The Lineweaver-Burk, Eadie-Hofstee and Hanes plots are all
rather similar to each other, in that they are based on an algebraic
conversion of the Michaelis equation to give a straight line
equation. The direct linear plot is a very different use of a straight
line technique. It's one of those things which is much easier to do
than to explain! Consequently I'll banish the explanation to a
separate page, that you can study if you want to, and limit this
page to a description of how to carry out the technique.
• The graph below was drawn using just two of the data from the
hypothetical enzyme:
[Substrate]
velocity
(no error)
4.00
5.0000
10.00
7.1429
• Remember the Km and Vmax for this enzyme are 4.00
and 10.00 respectively (both in arbitrary units).
• Firstly the axes are drawn as shown on the graph. Notice
that the substrate axis is extended to the left (negative)
side of the origin to a value which is equal to the largest
substrate concentration to be plotted. The velocity axis
should be drawn to a value rather greater than the
expected Vmax value. The data pairs are now plotted on
the graph. The substrate value is marked as a point on
the negative side of the substrate axis and the velocity
value on the velocity axis. Notice that there is no
calculation involved here. You simply need to plot the
raw data. The pairs of data points are now joined with a
straight line which is extrapolated into the positive
substrate area of the graph. The other pair of points are
now plotted in the same way. A look at the graph should
clarify this.
As you can see the lines
intersect each other. A vertical
line dropped from the
intersection to the substrate
axis gives the value of Km, and
horizontal line to the velocity
axis gives Vmax.
This can be repeated for all of
the data. The result is the next
graph.
As you can see all of the lines
intersect at the same point,
indicating the true values for the
kinetic parameters. Remember,
though, that this graph is drawn
using the perfect data, with no
experimental error. How does this
technique cope with the real world
when error is always present? This
is dealt with on the next page.
How to Choose a Kinetic Method
Uses of kinetic plots
Before we can consider the best plot to use we need to
think about the reason for using it in the first place.
Kinetic plots tend to be used in two ways:
• For carrying out calculations of kinetic parameters, which
is really what we've been discussing so far in this
course.
• For display purposes, to demonstrate the way in which
an enzyme's activity changes with changes in its
environment. The presence of an inhibitor for instance or
a change in pH.
The best calculation plot is not necessarily the best
display plot so we need to think about both uses
separately.
Calculation plots
My own preference for calculation purposes, if the facilities are
available, is the use of a computer to determine a best fit curve to
the Michaelis equation. It avoids all of the problems of exaggeration
of error associated with the linear plots and it involves no manual
calculation. It certainly takes time to set up the spreadsheet in the
first place, but if you design it in a flexible way you can use it again
and again with different data.
If appropriate computer facilities are not available my preference
would be for the direct linear plot. It involves no calculation at all and
deals with error very well. It's the ideal plot for carrying out in the
laboratory while the assay is in process as it requires only graph
paper, pencil and ruler, and the lines are very quick to plot once the
axes have been drawn.
Of the straightforward linear plots I would probably recommend the
Hanes plot. It copes with error much better than the more commonly
used Lineweaver-Burk and avoids the difficulties of velocity being
included in the independent axis which the Eadie-Hofstee suffers
from.
Display plots
Your choice of display plot will obviously depend on what you are
trying to demonstrate. Usually however one of the linear plots is
appropriate and my own opinion is that the Lineweaver-Burk plot has
much to recommend it. It has the great advantage of familiarity. Just
about every biochemist, and many people in other disciplines, will
understand its meaning without a second thought. In addition the
substrate concentration and velocity are plotted on separate axes. I
think this makes understanding of the plot more intuitive, and more
obviously related to the v/[A] curve from which it is derived.
If you are using the Lineweaver-Burk plot for display purposes you
should calculate Km and Vmax using one of the preferred methods
and mark the intercepts on the axes. You can then draw the line
through these points rather than attempting a best fit line through the
points on the LB graph. The latter would be inaccurate for the reasons
that we've discussed elsewhere.
Kinetics of Enzyme Inhibitors
• One of the important uses of kinetic analysis is the study of
enzyme inhibitors.
• Inhibitors are compounds which interact with an enzyme to
slow down its rate of reaction.
• They may occur naturally in cells, where they might be used for
controlling metabolic reaction rates, or artificially, where they
might be used as experimental tools in the study of enzyme
reactions.
• Many toxic compounds are enzyme inhibitors, being toxic
because they inhibit enzymes responsible for vital reactions.
Some of these toxic inhibitors are specific for individual
organisms, or groups of organisms, and can be used as
antibiotics, pesticides, herbicides, and so on.
• Inhibitors can interact with an enzyme in different ways and
enzyme kinetics is a major tool in distinguishing between these
mechanisms.
• This chapter will explain the mechanisms of the different
inhibitory types and their effect on the kinetics of an enzyme,
and also examine methods of determining the inhibitor
constant.
Reversible and Irreversible Inhibitors
• 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:
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:
• 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.
• Most of this chapter will be spent discussing the kinetics of
reversible inhibitors but there is a brief discussion of the
kinetics of irreversible ones.
• The first reversible type to be considered is the competitive
inhibitor.
Competitive Inhibitors
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.
The animated graphic demonstrates this method of
inhibition
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.
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.
Noncompetitive/Mixed Inhibitors
• 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.
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.
Uncompetitive Inhibitors
• 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.
The graphic displays
the conformational
change mechanism.
Again kinetic studies
are used to
distinguish
uncompetitive
inhibitors from other
inhibitor types.
Product Inhibition
• The final reaction in the enzyme's reaction
sequence, the release of product:
is often neglected, but is of course an
essential component of the process.
• The product is bound to the active site by the
same bonds which bind the substrate.
• This reaction is therefore very similar to the
substrate binding reaction, and like that
reaction it is very fast and readily reversible.
• As a result of that, product molecules are capable of binding to
free enzyme to form enzyme-product complex. Since, in a
simple, single-substrate enzyme, the product and substrate
both attempt to occupy the active site, they cannot both bind at
the same time.
• Therefore prior binding of product prevents the enzyme from
binding substrate and the product effectively acts as an
inhibitor. In a single-substrate enzyme this is really a
specialised form of competitive inhibition as substrate and
product are mutually exclusive.
• In enzyme kinetic studies we are nearly always studying initial
rates, that is the velocity immediately following the addition of
enzyme to substrate. At this time, of course, no product exists
as it hasn't been generated yet, so we can normally ignore
product inhbition in our kinetic work.
• Product inhibition exists for all enzymes but usually doesn't
interfere with our kinetic results as just explained. Sometimes
the substrate itself can act as an inhibitor when present in large
enough amounts. It is known as excess substrate inhibition.
Excess substrate inhibition
• Normally an increase in substrate concentration increases the
velocity of the enzyme reaction. Some enzymes, however,
display the phenomenon of excess substrate inhibition.
• This means that large amounts of substrate can have the
opposite effect and actually slow the reaction down. A familiar
example of this is the enzyme invertase, or betafructofuranosidase (EC3.2.1.26), which is responsible for
hydrolysing the disaccharide sucrose to its component
monosaccharides, glucose and fructose.
• It's thought that substrate inhibition happens with this enzyme
when two substrate molecules bind to the active site at the
same time.
• They can only do this by approaching the active site in an end
on fashion which prevents either of them from positioning itself
in such a way that the enzyme can attack it.
As long as both substrate molecules are attached to the active site the
enzyme is effectively inactive, and therefore inhibited. For this process to
occur the second substrate must approach the active site very rapidly after
the first, otherwise the first substrate would quickly attain the correct
catalytic placement. As collisions between enzyme and substrate are
completely random this is only likely to occur at high substrate
concentrations when the frequency of random collisions is greatly increased,
so inhibition is only seen in the presence of excess substrate.
Now you should have a look at the effects of substrate inhibition on the
kinetics of the enzyme.
The Inhibitor Constant
Introduction
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:
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:
1) Alway possible
2) Not possible with uncompetitive inhibitors
3) Not possible with competitive inhibitors
4) Not 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.
I'll look at three ways to determine these constants:
» calculation
» use of secondary plots
» the Dixon plot
Kinetics of Multisubstrate Systems
Introduction
• Up to now we've been studying kinetics on the assumption that an
enzymic reaction involves just one substrate and one product. Such
enzymes are very rare. A typical enzyme is involved in a reaction to
convert two substrates to two products, three substrates or products
are quite common and four occur occasionally.
• That doesn't mean that the kind of kinetic analysis that we've been
studying so far is a waste of time. In studying an enzyme with more
than one substrate the normal approach is to keep all of the
substrates but one at a constant concentration. The remaining
substrate is varied in the kinetic assays and valuable information
can be gained about the kinetic relationship between that substrate,
the enzyme, inhibitors, and so on. By this method we are effectively
treating the enzyme as an apparent single substrate enzyme.
• However, if we ignore the effects of the other substrates on the
reaction we are missing a lot of information about the enzyme which
kinetic studies can give us. This chapter is intended to introduce you
to the methods used to study multisubstrate enzymes.
• To follow the complete course click on the menu headings in
sequence. For general reference - click at will!
Terminology
Multisubstrate kinetics is a little more complex than single substrate kinetics and
requires some additional basic terminology which will be used throughout this section
of the course.
•
Symbols
Obviously we'll be dealing with more substrates, more products and so on, and we
need appropriate symbols to represent them in equations. I'll be using the following to
represent the various elements of the enzymic reaction:
•
Substrates
–
•
Products
–
•
P, Q, R, S
Enzymes We'll only be dealing with one enzyme at a time but in a multisubstrate
enzyme different forms of the enzyme can be used. These are represented by the
following symbols. E is always used to represent the free enzyme - the form prior to
binding with any substrate
–
•
A, B, C, D
E, F, G
Inhibitors I won't be saying very much about inhibitors (apart from product and
substrate inhibition) but, for the sake of completeness, they can be represented by:
–
I, J
Transitory complexes
We're used to the idea of substrates and products
binding to the enzyme to form complexes. These are
called transitory complexes because they will normally
exist for only a very short period of time. They will either
break down again, as the bonds holding them together
are weak, or they will undergo a covalent change to turn
substrate into product (or vice versa).
In a multisubstrate enzyme there are more types of
transitory complex that can be generated. For an
enzyme that has two substrates (A, B) and two products
(P, Q) there are three possible enzyme-substrate
complexes (EA, EB, EAB) and three possible enzymeproduct complexes (EP, EQ, EPQ). Not all of these can
be produced by all enzymes, but we'll examine that later.
Also, in a multisubstrate enzyme there is the possibility
of forming combined enzyme-substrate-product
complexes, such as EAP and EBQ.
Central complexes
In a single substrate system the active site is effectively
full once one substrate has bound. There's no room for
another substrate, aproduct or an inhibitor to enter it. In a
multisubstrate enzyme that can bind more than one
substrate at the same time, binding of one substrate will
not completely fill the site. A central complex is a
transitory complex which is full. It may be represented by
enclosing it in parentheses. So, for instance, in an
enzyme which could bind two substrates and two
products at the same time EA, EB, EP, EQ would not be
central complexes, while (EAB) and (EPQ) would be.
Reactancy
The reactancy of a reaction is defined as the number of
reactants taking part in a reaction in a particular
direction. For instance a reaction which had two
substrates and three products would have a reactancy of
two in the forward direction and three in the reverse
direction. The numeric prefixes uni, bi, ter and quad are
frequently used to indicate reactancies of one, two three
and four respectively. So the example above could be
referred to as a bi ter reaction.
Now that we're familiar with some basic terminology we
can begin to look at the idea of a kinetic mechanism.
Kinetic mechanisms
The kinetic mechanism of an enzyme is simply
the sequence in which substrates bind to, and
products are released from the enzyme. It
should not be confused with the chemical
mechanism which is concerned with the
chemical interactions between the enzyme and
its substrate(s) which results in the creation of
product(s). Kinetic mechanisms can be broadly
divided into two main types, sequential and pingpong (also known as double displacement)
mechanisms.
Sequential mechanisms
Sequential reactions are ones in which all
reactants bind to the enzyme before the
first product is released. They can be
further subdivided into ordered reactions,
in which the reactants and products are
bound and released in an obligatory
sequence, and random reactions, in which
there is no obligatory binding sequence.
Ping-pong mechanisms
Ping-pong reactions are ones in which at least one
product is released before all the substrates have bound.
To clarify these distinctions we'll look at each of these
mechanisms in turn using a typical bi bi enzyme:
A+B
P+Q
We'll start by looking at an ordered sequential reaction,
which is perhaps the simplest in kinetic terms.
The ordered sequential mechanism
• In these enzymes the substrates bind to
the enzyme, and the products are
released in a defined sequence. Firstly the
two substrates bind to give an enzyme
substrate complex:
• E+A
EA
• EA + B
(EAB)
• Notice that the EAB complex is a central
complex and enclosed in parentheses.
The enzyme substrate complex is now
converted to enzyme product complex:
• (EAB) (EPQ)
Again this is a central complex.
The products are now released:
• (EPQ) EQ + P
• EQ
E+Q
This is of course just an extension of the usual
sequence of reactions which we are used to with
single substrate systems.
The Cleland plot
This is actually quite a simple mechanism, but the
sequence of reactions can get quite complex particularly
in enzymes with more than two substrates and products.
The Cleland plot is a diagramatic summary of the
reaction sequence in which the enzyme is represented
as a horizontal line and arrows are used to represent the
arrival and departure of substrates and products.
Transitory complexes are written below the enzyme line.
The above diagram is the Cleland plot for the ordered
sequential enzyme that we've been discussing and
should be fairly self-explanatory.
We can now look at random sequential mechanisms.
The random sequential mechanism
•
•
•
•
These enzymes work in a similar way to ordered
sequential ones with the exception that there is
no specified order in which substrates must bind
or products be released. This makes the
reaction sequence a little more complex. Initially
the substrates bind, but they can do so in either
order:
E+A
EA
EA + B
(EAB)
or
E+B
EB
EB + A
(EAB)
The central complex is identical in both cases, of
course, and the next step is the same as with an
ordered mechanism:
• (EAB)
(EPQ)
We now have a choice of sequences of product
release:
• (EPQ)
EQ + P
• EQ
E+Q
or
• (EPQ)
EP + Q
• EP
E+P
The Cleland plot gives a clear demonstration of these processes:
Random or ordered mechanisms are very similar to each other. The
important difference between them, in contributing to our
understanding of the way in which enzymes work, is that if an
enzyme is found to be ordered then there must be some reason for
its being fussy about the binding sequence, perhaps the first
substrate causes a conformational change in the enzyme for
instance.
Finally we'll take a look at ping-pong mechanisms.
The ping-pong (double
displacement) mechanism
The distinguishing feature of these enzymes is that at least one
product is released from the enzyme before all of the substrates
have bound. This might seem slightly unlikely at a first glance, but
it's actually easily explained and quite a common mechanism. Some
very familiar enzymes, for instance the serine proteases (trypsin,
chymotrypsin, etc.) and the amino transferases work in this way.
The process starts by binding of the enzyme to the first substrate in
the usual way:
• E+A
(EA)
Notice that I've used parentheses around the EA complex to indicate
that it is a central complex. The active site is full as this substrate will
be converted to product before the second substrate can bind. The
active site has no room for the second substrate so it is full.
The next reaction is the key to the whole
process:
• (EA)
(FP)
In this reaction a part of the substrate has been
removed from substrate A, converting it to
product P. The removed section has become
covalently bound to the enzyme to create a new
form of the enzyme, enzyme F.
The first product of the reaction is now released
and the second substrate binds:
• (FP) F + P
• F+B
(FB)
Now the stored section of the first substrate is transferred
to the second substrate to create the second product,
which is then released:
• (FB) (EQ)
• (EQ) E + Q)
The animation should help to clarify this.
The Cleland plot for a ping-pong enzyme is quite
distinctive:
as the first upward arrow (product P) is to the left of the
first down arrow (substrate B).
Now that we're familiar with the principal types of kinetic
mechanism we need to think about experimental
techniques to distinguish between them. To start with
we'll examine the effects of changes in substrate
concentration on multisubstrate reactions.
Effects of Substrate Concentration in
Multisubstrate Systems
You should be familiar with the typical kinetic experiment
for single substrate systems. It simply involves
measuring reaction velocity at a variety of substrate
concentrations. We can then use the data for any of the
kinetic plots that we discussed in Chapter 2.
With a multisubstrate enzyme, let's say a typical bi bi
enzyme:
• A+B
P+Q
we can carry out exactly the same experiment.
We simply need to keep one substrate (the fixed substrate) at
a constant concentration in all our assays, while we vary the
concentration of the other substrate (the variable substrate).
The results would be exactly the same as you'd expect in a
single substrate system and you could use any of the methods
that we've studied to calculate the kinetic parameters. What
would happen though if you repeated the experiment with an
increased concentration of the fixed substrate. Since you're
increasing the concentration of a substrate you would expect
the velocity to rise and in fact the reaction would be faster at
any given concentration of the variable substrate than it was in
the previous experiment. So you'd end up with a second set of
data which you could use in your chosen plot. The kinetic
parameters would change to reflect the change in velocity. If
you repeated this at a variety of concentrations of the fixed
substrate you would get a series of lines. A typical
Lineweaver-Burk plot obtained as a result of this type of
experiment can be seen below. Substrate A was used as the
variable substrate and substrate B as the fixed substrate.
The actual pattern of lines obtained will vary according to the way in
which the enzyme interacts with the two substrates and, as we'll see
in the next couple of pages, enables us to distinguish between
sequential and ping-pong enzymes.
In discussing graphs of this type we'll be considering changes in the
Vmax and slope of the line. A change in Vmax indicates the effect
that a change in the concentration of the fixed substrate on the
reaction speed at very high concentrations of the variable substrate.
Remember that Vmax is the velocity of the reaction under those
conditions. A change in slope indicates the effect of a change in
concentration of the fixed substrate on the speed at very low
concentrations of the variable substrate. Remember in our
discussion of enzyme inhibition we found that the slope was the rate
constant at low substrate concentrations.
We'll start by considering the expected results of this experiment
when carried out with a sequential enzyme.
Product Inhibition in Multisubstrate Systems
Why study product inhibition?
The experiments on the effects of substrate that we've
already examined will tell us whether an enzyme uses a
sequential or ping-pong mechanism. What it can't tell us
is whether a sequential mechanism is random or ordered
or, if it is ordered, what the order of binding of substrates
and release of products is. Product inhibition studies can
give us that information, and also sometimes give us
useful information about the mechanism of ping-pong
enzymes. Product inhibition is therefore a vital tool in the
study of multisubstrate systems.
Background to product inhibition studies
• We looked at the concept of product inhibition in single substrate systems
in Chapter 4 of this course, seeing there that this arises when product
binds to free enzyme blocking access to the substrate. This type of
inhibition will always be competitive, as the subtsrate and enzyme are
incapable of binding to the active site at the same time. It will also not be
experienced in normal kinetic studies as we are usually studying initial
velocities which occur at the start of the reaction when no product has
been generated and can't therefore inhibit the enzyme.
• Product inhibition is more complex in multisubstrate systems but also more
rewarding if studied correctly. As in a single substrate system product
inhibition wil not occur if you measure initial rates - unless you persuade it
to! You can do this by adding product to the substrate solution before
adding enzyme to start the reaction. Product hasn't yet been generated by
the reaction you're studying but it's there because you've added it.
• The complexity of product inhibition arises because we're now dealing with
more than one substrate and product and may be possible for one of the
substrates and one of the products to bind to the active site at the same
time. A substrate and product are not necessarily therefore in direct
competition with each other and the inhibition caused by the product may
not be competitive. In fact all of the reversible inhibitor types that we
studied in the last chapter may occur as product inhibitors in multisubstrate
systems and it is the pattern of product inhibitor types which give us the
information that we need in deciding on the kinetic mechanism.
Inhibitor types
If you are unclear about the different types of reversible inhibitor and how
they may be distinguished by their kinetic properties it might help you to
refer back to the chapter on enzyme inhibitors. For convenience the
types of inhibitor that we'll be considering in this section are summarised
below together with their effects on the Lineweaver-Burk plot.
Inhibitor types and their kinetic properties
Inhibitor type
Inhibits at
Lineweaver-Burk plot
Competitive
Low substrate
concentrations
Changes slope not
Vmax
Noncompetitive
Low and high
substrate
concentrations
Changes slope and
Vmax
Uncompetitve
High substrate
concentrations
Changes Vmax not
slope
Experimental approach to product
inhibition studies
• The type of experiment that we'll be discussing here is a
simple extension to that used in studying the effects of
substrate concentration on multisubstrate systems. For
any given assay one substrate will be chosen as the
variable substrate with the other kept at a fixed
concentration. This time though the enzyme will be
assayed in the presence and absence of a fixed amount
of a chosen product. With a typical bi-bi enzyme four
experiments of this kind can be carried out. Both
substrates can be used as the variable substrate and the
inhibitor effects of both products can tested against
them. The results can be plotted using any of the kinetic
plots but, as usual, I'll use the Lineweaver-Burk plot
because of its familiarity.
• We'll start by looking at the results that would be
expected with an ordered sequential enzyme.
Substrate Inhibition in Multisubstrate Systems
Introduction
• Product inhibition is used as an experimental tool in multisubstrate
studies and is found in all enzymes. Substrate inhibition (strictly
speaking excess substrate inhibition) is not a property of all enzymes
and, rather than being an experimental tool, is simply something that
you should be aware of as it can lead to some rather curious kinetic
graphs and its important that you can recognise them if you come
across them.
• We've already discussed substrate inhibition in single substrate
systems and to some extent this will be an extension of this but, like
product inhibition, will be a bit more complex. This is because we have
two, or more, substrates to consider any, or all, of which may be
inhibitory, and the type of inhibition will vary according to the kinetic
mechanism of the enzyme. The type of graph obtained will also depend
on whether it's the variable or the fixed substrate (or both!) that are
inhibitory.
• As usual we'll look at the different kinetic mechanisms in turn and
consider the effects of substrate inhibition on them. Excess substrate
inhibition doesn't occur in random sequential enzymes, for reasons that
will be explained, so we'll start by looking at ordered sequential
enzymes.
Determination of Kinetic Parameters in
Multisubstrate Systems
Introduction and definitions
In Chapter 2 of this course we looked at some length at
methods of determining Km and Vmax values for a single
substrate enzyme. Of course, true single substrate enzymes
are rather uncommon and if you've ever carried out these
procedures in practice you almost certainly used an enzyme
with more than one substrate, using one substrate as the
variable substrate and keeping the other(s) at a fixed
concentration - the same kind of experiment that we've been
using in this chapter. This will certainly give results but what
do they mean? As we've also seen in this chapter, a change
in the concentration of the fixed substrate will bring about a
change in the Km and Vmax values. By using different
concentrations of fixed substrate we can get an infinite
variety of values for the kinetic parameters.
These different values for kinetic parameters are
apparent values, the true values are defined by an
extension of the definitions used in single substrate
systems.
Maximal velocity is defined as the reaction velocity
which occurs when all substrates are at saturation
levels.
Each substrate will have its own Michaelis constant
which is defined as the concentration of that substrate
which gives a velocity of half the maximal velocity when
all other substrates are present at saturation levels.
Just as the definitions of kinetic parameters are an
extension of the single substrate ones, so the methods
of determination of these parameters in multisubstrate
systems are an extension of those used in single
substrate systems.
We'll study this by considering the way in which a
Lineweaver-Burk plot can be used in a multisubstrate
system.
Kinetics of Allosteric Enzymes
Introduction
The whole of this course so far has worked on the assumption that
enzyme kinetics is based on the Michaelis equation, indicated by the fact
that the enzyme produces a hyperbolic graph in the velocity against
substrate concentration plot. The only real deviation from this has been
in those enzymes which show excess substrate inhibition.
A large family of enzymes which deviate from Michaelis (hyperbolic)
kinetics are the allosteric enzymes. We should perhaps give honorary
membership of this family to non-enzymic allosteric proteins, such as the
blood pigment haemoglobin, as a lot of the pioneering work on allostery
was done on these.
In this chapter we'll examine some of the basic properties of allosteric
enzymes, the mechanism of allostery, the way in which allostery alters
kinetic plots and the information that can be gleaned from these plots.
To follow the complete course click on the menu headings in sequence.
For general reference - click at will!
Allosteric Enzymes - Definitions
Before we can discuss the features of allosteric enzymes we 'll need
to learn some basic definitions. To start with we need to know what
the word "allosteric" means!
An allosteric protein
An allosteric protein is defined as a protein containing two or more
topologically distinct binding sites which interact functionally
with each other.
This means that there are at least two sites in different
positions capable of binding ligands (substrates, inhibitors etc). An
event, such as the binding of a ligand, at one site alters the properties
of the other(s).
The majority of allosteric proteins are allosteric enzymes, in that they
are capable of catalysing reactions but some, such as haemoglobin,
are simply binding proteins.
Cooperativity
Cooperativity is the modification of the binding constant of the
protein for a small molecule by the prior binding of another small
molecule.
The binding constant is something like the Ks for a substrate or Ki for an
inhibitor, which is basically the dissociation constant for the protein and
ligand and indicates the binding strength, or affinity, of the protein for
that ligand. Km is usually taken as the binding constant for substrates as
it's easier to measure than Ks. The definition of cooperativity means that
binding of one ligand to the protein will either increase or decrease the
ability of the protein to bind a second ligand molecule.
If the modification increases the binding ability (or affinity) this is known
as positive cooperativity. If the binding ability is decreased it's called
negative cooperativity.
The two ligands which influence one anothers binding may be
chemically identical, for instance one substrate changing the binding of
another substrate molecule, which is known as a homotropic effect.
Alternatively they may be chemically different, for instance the influence
of an inhibitor on substrate binding. This is a heterotropic effect.
Now we can look at some of the properties of allosterics enzymes.
Properties of Allosteric Enzymes
Allosteric enzymes show a number of properties which
distinguish them from non-allosteric ones. These will be
described and explained here. It must be stressed that
not all allosteric enzymes will show all of these
properties. They are just a collection of features, at least
some of which would be expected in an individual
allosteric enzyme. They are:
•
•
•
•
•
Sigmoidal kinetics in the velocity/substrate curve
Existence of effectors
Biphasic response to competitive inhibitors
Loss of allostery by mild denaturation
Polymeric structure
Mechanisms of Allostery
• Two mechanisms have been suggested to explain the
properties of allosteric proteins and enzymes. They are
known as the concerted, or symmetry, hypothesis and
the sequential hypothesis. The concerted hypothesis
was introduced by Monod, Wyman and Changeux and is
sometimes also referred to as the MWC hypothesis,
while the sequential hypothesis is the product of
Koshland, Nemethy and Filmer and is often called the
KNF hypothesis.
• Since it's probably a gross breach of Web etiquette to
use the word "hypothesis" so many times in one page I'll
make sure I'm hung for a sheep rather than a lamb and
suggest that we now examine the first of these - the
concerted hypothesis.
Aspartate transcarbamoylase - a
sample allosteric enzyme
This page requires
the Chemscape
Chime plugin for
Netscape which can
be downloaded free
from the MDL site in
the US or the UK.
Aspartate transcarbamoylase, or carbamoyltransferase,
is an important enzyme in the biosynthesis of pyrimidine
nucleotides. The structure and allosteric properties of the
enzyme from the bacterium Escherischia coli have been
very extensively studied. It catalyses a reaction between
aspartic acid and carbamoyl phosphate to generate Ncarbamoyl aspartate with the release of inorganic
phosphate.
The complete, active enzyme (often abbreviated to ATCase) consist of no
less than 12 different protein chains or subunits. It contains two catalytic
components each made up of three identical subunits. One of these
components is shown, with the different subunits in different shades of
blue, in the diagram. These components are arranged next to each other
effectively forming the two main faces of the whole enzyme. The rest of
the enzyme is composed of three regulatory components each
comprising two subunits. These components form the corners of the
enzyme which has a roughly triangular shape. One of them is highlighted
in the diagram with the two subunits in colours of green.
The catalytic and regulatory components can be separated. If they are the
catalytic components are found to be capable of catalysing the reaction in
the usual way except that they have no allosteric properties. They show
no sign of substrate cooperativity and no reaction with allosteric effectors.
The regulatory components are unable to carry out catalysis, but can bind
the usual effectors of ATCase, CTP (an inhibitor) and ATP (an activator).
Chime images
If the Chime plugin is installed on your machine you can
examine the dynamic images on this page. There are two of
them. One of them is of the enzyme with the inhibitor CTP
bound to it. This maintains it in the tense, low substrate
affinity state. The other one has a substrate analogue bound
which maintains it in the relaxed, high affinity state. If you
click on the buttons below the images you can highlight the
different components and ligands by changing their colours.
Clicking on the "Highlight allosteric changes" buttons will
rotate the enzymes to a position where the change in shape
caused by allostery is most obvious. You'll see that the
relaxed enzyme is "swollen" compared with the tense one,
with the two catalytic components pushed further apart.
You'll see that the substrate is deeply buried within the
overall enzyme structure, it's almost invisible, presumable
this swelling makes the active site more accessible to it.
This is no doubt one reason why affinity is increased.
ATCase with inhibitor (CTP)
bound
ATCase with substrate
analogue bound
Highlight regulatory subunits
Highlight regulatory subunits
Highlight catalytic subunits
Highlight catalytic subunits
Highlight CTP
Highlight substrate analogue
Highlight allosteric changes
Highlight allosteric changes
In addition to using the buttons the images may be changed manually. If
you place the mouse cursor on one and move it around while holding
down the mouse button you'll see that you can rotate it. Doing the same
while holding down the Shift key will zoom the image. You can move the
image around by holding the Control key and the Right mouse button
down while moving the mouse. Finally, simply clicking on the image with
the Right mouse button will produce a menu which lets you change many
aspects of the image. Far too many to discuss here. Play with it and have
some fun!
The sigmoid plot and the Hill equation
One of the common characteristics of an allosteric enzyme is that it
shows a sigmoid plot when velocity is plotted against substrate
concentration. This is evidence of positive substrate cooperativity and
is in contrast to the hyperbolic plot which is expected of non-allosteric
enzymes. As we've seen the equation of the hyperbolic plot is the
Michaelis equation:
Since this equation gives a hyperbolic curve when v is plotted against
a, it clearly can't be the equation of the sigmoidal plot. Its equation is
the, closely related, Hill equation:
As you can see the two equations are very similar. The principal difference
is that in the Hill equation the substrate, a, is raised to a power, h, which is
called the Hill coefficient. The other difference is the constant on the
bottom line. Instead of Km, we have K0.5, also raised to power h. This
constant is very similar to the Michaelis constant in that it represents the
substrate concentration which gives a velocity equal to half the maximal
velocity but, since it isn't part of the Michaelis equation it shouldn't really
be refrred to as the Michaelis constant or Km.
The value of the Hill coefficient gives us a measure of the degree of
substrate cooperativity. If it were equal to one, of course, it would
effectively disappear from the equation which would then be the same as
the Michaelis equation with K0.5 equal to Km. There a value of h equal to
one means that there is no cooperativity and the graph is hyperbolic. An
increasing value of h, however, will show an increasingly sigmoidal curve
showing positive cooperativity for the substrate. A value less than one
shows negative cooperativity.
The following graph shows the effects on the v/a plot of values of h from
0.5 to 4. In each case the enzyme has a Vmax of 10 units and a K0.5 of 4
units.
You'll notice that the h=1 curve is a typical hyperbolic
plot, while the h=2 and 4 lines are clearly sigmoidal,
more so in the case of the h=4 line. In other words these
are typical plots showing positive substrate cooperativity,
with cooperativity increasing as the Hill coefficient rises.
The h=0.5 curve, showing negative cooperativity, is less
easy to distinguish from a normal hyperbolic plot but, by
comparing the two, you can see that it has a faster initial
rise and tails off much more sharply. All lines are heading
towards the same maximal velocity and, as they share
the same K0.5, they all cross at the same point.
Effects of substrate cooperativity on
linear plots
The linear plots, such as the Lineweaver-Burk
plot, commonly used in kinetic analysis are
based on an algebraic conversion of the
Michaelis equation. Since enzymes showing
substrate cooperativity do not obey the Michaelis
equation it should come as no surprise that they
do not show straight lines with these "linear"
plots. Examples of two of the popular straight
line plots with enzymes showing different values
of the Hill coefficient are shown below.
The Hill plot
The Hill equation:
can be rearranged to give:
If we now take the logarithms of this:
we have the equation of a straight line. A
plot of log[v/(V-v)] against log a should
give a straight line of slope h and intercept
on the vertical axis of -h.logK0.5. A bit of
simple algebra will show you that the
intercept on the horizontal axis is equal to
logK0.5. An example is shown in the
following graph.
In fact this graph will
usually deviate from a
straight line at very low
and very high substrate
concentrations. The
central portion of the plot,
which should be linear, is
used for the calculation.
Calculation of Vmax
One of the difficulties of the Hill plot is that you need a
value for the maximal velocity in order to draw it. There is
no obvious way in which this can be calculated using the
usual methods as all the linear plots will appear as curves
as we've seen in the previous page. In fact these plots
would give straight lines if instead of plotting substrate
concentration, wherever it appears in the plot, we used the
substrate concentration raised to the power of h. Since h
is what we're trying to calculate this isn't possible!
The way to proceed is to use any preferred linear plot to
get a rough estimate of Vmax. You can do this by using
only higher concentrations of substrate in your plot which
will give you something approaching a straight line. Now
use this value of Vmax to draw a Hill plot. This will give
you a rough estimate of h. Now use this value to replot
your original linear plot but replacing the substrate
concentration, a, with ah. This should now give you a
better straight line. You can now recalculate Vmax from
this and redraw the Hill plot with the new value. This
should give you a reasonable measure of h.
Another technique is to use the curve fit procedure
discussed earlier adapting the formula to echo the Hill
equation rather than the Michaelis equation.
Advantages of cooperativity
As we've seen already, substrate cooperativity is
not found in all allosteric enzymes, but it's found
in most of them so there must presumably be
some metabolic advantage in it. While the
reasons for allosteric inhibitors and activators
are usually fairly obvious once you realise their
role in, and the importance of, metabolic control,
the reasons for substrate cooperativity may not
be immediately apparent. To examine these
reasons have alook at the following graph:
This is the same plot that we've looked at already except that I've extended
it by using higher substrate concentrations and expressed the substrate
concentration in terms of a percentage of the Km. I've also added a couple
of horizontal lines to mark the points at which the velocity is equal to 10%
and 90% of Vmax.
Positive cooperativity
If you look at the hyperbolic (n=1) line you'll see that it takes
a large increase in substrate concentration to move the
velocity from the 10% to the 90% marker. It's quite simple
algebra to calculate the substrate concentration required to
bring the velocity to the 10% and 90% lines. It needs a
substrate concentration equal to one ninth of the Km to a
velocity of 10% of the Vmax and nine times Km to reach a
velocity of 90% of Vmax. That means that an 81x increase in
substrate concentration is needed to bring about a velocity
change of 9x. By comparison the highest positive
cooperativity plot (n=4) on the graph needs only a 3x
increase in substrate to bring about the same change. The
n=2 line shows a need of a 9x substrate change.
These effects of positive cooperativity mean that the
enzyme is much more sensitive to changes in the
amount of substrate available to it. If you consider a
situation such as glucose breakdown you'll realise that
the amount of glucose being degraded for energy will
vary greatly with the physiological situatuion of the cell. If
it needs a great deal of energy it will be degrading a lot
of glucose, if not it will be degrading very little, or none at
all. The enzymes in the breakdown pathway will
therefore have to cope with big differences in the amount
of substrate that they have to convert and the greater
sensitivity to substrate concentration produced by
positive substrate cooperativity will be of great
assistance to this.
Negative cooperativity
Negative cooperativity is much less common. You'll see
from the graph that the point at which the n=0.5 line will
cross the 90% line is way off the scale of the graph. In
fact it needs a 6561x change in substrate to bring about
the same 9x velocity change that we were discussing
above. This means that an enzyme is much less
sensitive to substrate change and almost becomes
independent of the concentration of substrate. This may
be useful with a substrate, such as a coenzyme perhaps,
whose concentration may change due to reactions in the
cell which are not directly related to the pathway in which
our enzyme is involved. It may well be an advantage in
these circumstances if the enzyme does not react to
changes in that substrate.