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.Department of Biological SciencesUniversity 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.

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 Altamura Italy The animated Italian flag is copyright Crames Studios Any questions or comments about the course can be sent to: Dr. Peter Birch Department of Biological Sciences University of Paisley PAISLEY Renfrewshire PA1 2BE Tel:

Content of Enzyme Kinetics Chapter 1Chapter 1: The effects of substrate concentrations on reaction rate Chapter 2:Chapter 2: Determination of kinetic parameters Chapter 3:Chapter 3: Kinetics of enzyme inhibitors Chapter 4Chapter 4: Kinetics of multisubstrate systems Chapter 5Chapter 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 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 non- catalysed 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.v/[A] curve 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: Michaelis and MentonBriggs and Haldane 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.V Km

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.

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:graph

Vmax10 Km4 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.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. v/[A]

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.here

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:Michaelis-Menten equation

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.explanation of these problems 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 Michaelis- Menten 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. here Hanes plot

The use of the Hanes plot for determining kinetic parameters This is another conversion of the Michaelis- Menten 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.heredirect 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. separate page The graph below was drawn using just two of the data from the hypothetical enzyme: [Substrate]velocity (no error)

Remember the Km and Vmax for this enzyme are 4.00 and 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.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.best fit curve to the Michaelis equation 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.direct linear plot 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. Hanes

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 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.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.kinetic studies

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.xcess 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 beta- fructofuranosidase (EC ), 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. 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: »calculationcalculation »use of secondary plotsuse of secondary plots »the Dixon plotthe 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 –A, B, C, D 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 –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 enzyme- product 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 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 ping- pong (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.ordered sequential reaction

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.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.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 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.rate constant at low substrate concentrations We'll start by considering the expected results of this experiment when carried out with a sequential enzyme.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 in single substrate systems 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 and their kinetic properties Inhibitor typeInhibits atLineweaver-Burk plot CompetitiveLow substrate concentrations Changes slope not Vmax Non- competitive Low and high substrate concentrations Changes slope and Vmax UncompetitveHigh substrate concentrations Changes Vmax not slope 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.chapter on enzyme inhibitors

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.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.substrate inhibition in single substrate systems 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.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.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. 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.Michaelis equationhyperbolic graph in the velocity against substrate concentration plot 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 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. 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.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 N- carbamoyl 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 catalytic subunits Highlight CTP Highlight substrate analogue 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:sigmoid plot when velocity is plotted against substrate concentrationhyperbolic plot which is expected of non-allosteric enzymes 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.curve fit procedure

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:already

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.already

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.