Enzyme kinetics is the study of factors that determine the speed of enzyme-catalysed reactions. It utilizes some mathematical equations that can be confusing to students when they first encounter them. However, the theory of kinetics is both logical and simple, and it is essential to develop an understanding of this subject in order to be able to appreciate the role of enzymes both in metabolism and in biotechnology.
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Assays (measurements) of enzyme activity can be performed in either a discontinuous or continuous fashion. Discontinuous methods involve mixing the substrate and enzyme together and measuring the product formed after a set period of time, so these methods are generally easy and quick to perform. In general we would use such discontinuous assays when we know little about the system (and are making preliminary investigations), or alternatively when we know a great deal about the system and are certain that the time interval we are choosing is appropriate.
In continuous enzyme assays we would generally study the rate of an enzyme-catalysed reaction by mixing the enzyme with the substrate and continuously measuring the appearance of product over time. Of course we could equally well measure the rate of the reaction by measuring the disappearance of substrate over time. Apart from the actual direction (one increasing and one decreasing), the two values would be identical. In enzyme kinetics experiments, for convenience we very often use an artificial substrate called a chromogen that yields a brightly coloured product, making the reaction easy to follow using a colorimeter or a spectrophotometer. However, we could in fact use any available analytical equipment that has the capacity to measure the concentration of either the product or the substrate.
In almost all cases we would also add a buffer solution to the mixture. As we shall see, enzyme activity is strongly influenced by pH, so it is important to set the pH at a specific value and keep it constant throughout the experiment.
Our first enzyme kinetics experiment may therefore involve mixing a substrate solution (chromogen) with a buffer solution and adding the enzyme. This mixture would then be placed in a spectrophotometer and the appearance of the coloured product would be measured. This would enable us to follow a rapid reaction which, after a few seconds or minutes, might start to slow down, as shown in .
A common reason for this slowing down of the speed (rate) of the reaction is that the substrate within the mixture is being used up and thus becoming limiting. Alternatively, it may be that the enzyme is unstable and is denaturing over the course of the experiment, or it could be that the pH of the mixture is changing, as many reactions either consume or release protons. For these reasons, when we are asked to specify the rate of a reaction we do so early on, as soon as the enzyme has been added, and when none of the above-mentioned limitations apply. We refer to this initial rapid rate as the initial velocity (v0). Measurement of the reaction rate at this early stage is also quite straightforward, as the rate is effectively linear, so we can simply draw a straight line and measure the gradient (by dividing the concentration change by the time interval) in order to evaluate the reaction rate over this period.
We may now perform a range of similar enzyme assays to evaluate how the initial velocity changes when the substrate or enzyme concentration is altered, or when the pH is changed. These studies will help us to characterize the properties of the enzyme under study.
The relationship between enzyme concentration and the rate of the reaction is usually a simple one. If we repeat the experiment just described, but add 10% more enzyme, the reaction will be 10% faster, and if we double the enzyme concentration the reaction will proceed twice as fast. Thus there is a simple linear relationship between the reaction rate and the amount of enzyme available to catalyse the reaction ( ).
This relationship applies both to enzymes in vivo and to those used in biotechnological applications, where regulation of the amount of enzyme present may control reaction rates.
When we perform a series of enzyme assays using the same enzyme concentration, but with a range of different substrate concentrations, a slightly more complex relationship emerges, as shown in . Initially, when the substrate concentration is increased, the rate of reaction increases considerably. However, as the substrate concentration is increased further the effects on the reaction rate start to decline, until a stage is reached where increasing the substrate concentration has little further effect on the reaction rate. At this point the enzyme is considered to be coming close to saturation with substrate, and demonstrating its maximal velocity (Vmax). Note that this maximal velocity is in fact a theoretical limit that will not be truly achieved in any experiment, although we might come very close to it.
The relationship described here is a fairly common one, which a mathematician would immediately identify as a rectangular hyperbola. The equation that describes such a relationship is as follows:
y=a×xx+b
The two constants a and b thus allow us to describe this hyperbolic relationship, just as with a linear relationship (y = mx + c), which can be expressed by the two constants m (the slope) and c (the intercept).
We have in fact already defined the constant a &#; it is Vmax. The constant b is a little more complex, as it is the value on the x-axis that gives half of the maximal value of y. In enzymology we refer to this as the Michaelis constant (Km), which is defined as the substrate concentration that gives half-maximal velocity.
Our final equation, usually called the Michaelis&#;Menten equation, therefore becomes:
Initial rate of reaction(v0)=Vmax×Substrate concentrationSubstrate concentration+Km
In , Leonor Michaelis and Maud Menten first showed that it was in fact possible to derive this equation mathematically from first principles, with some simple assumptions about the way in which an enzyme reacts with a substrate to form a product. Central to their derivation is the concept that the reaction takes place via the formation of an ES complex which, once formed, can either dissociate (productively) to release product, or else dissociate in the reverse direction without any formation of product. Thus the reaction can be represented as follows, with k1, k&#;1 and k2 being the rate constants of the three individual reaction steps:
E+S&#;k&#;1k1ES&#;k2E+P
The Michaelis&#;Menten derivation requires two important assumptions. The first assumption is that we are considering the initial velocity of the reaction (v0), when the product concentration will be negligibly small (i.e. [S] &#; [P]), such that we can ignore the possibility of any product reverting to substrate. The second assumption is that the concentration of substrate greatly exceeds the concentration of enzyme (i.e. [S]&#;[E]).
The derivation begins with an equation for the expression of the initial rate, the rate of formation of product, as the rate at which the ES complex dissociates to form product. This is based upon the rate constant k2 and the concentration of the ES complex, as follows:
v0=d[P]dt=k2×[ES]
1
Since ES is an intermediate, its concentration is unknown, but we can express it in terms of known values. In a steady-state approximation we can assume that although the concentration of substrate and product changes, the concentration of the ES complex itself remains constant. The rate of formation of the ES complex and the rate of its breakdown must therefore balance, where:
Rate of ES complex formation = k1[E][S]
and
Rate of ES complex breakdown = (k&#;1 + k2)[ES]
Hence, at steady state:
k1[E][S] = k&#;1 + k2[ES]
This equation can be rearranged to yield [ES] as follows:
[ES]=k1[E][S]k&#;1+k2
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2
The Michaelis constant Km can be defined as follows:
Km=k&#;1+k2k1
Equation 2 may thus be simplified to:
[ES]=[E][S]Km
3
Since the concentration of substrate greatly exceeds the concentration of enzyme (i.e. [S] &#; [E]), the concentration of uncombined substrate [S] is almost equal to the total concentration of substrate. The concentration of uncombined enzyme [E] is equal to the total enzyme concentration [E]T minus that combined with substrate [ES]. Introducing these terms to Equation 3 and solving for ES gives us the following:
[ES]=[E]T[S][S]+Km
4
We can then introduce this term into Equation 1 to give:
v0=k2[E]T[S][S]+Km
5
The term k2[E]T in fact represents Vmax, the maximal velocity. Thus Michaelis and Menten were able to derive their final equation as:
v0=Vmax[S][S]+Km
A more detailed derivation of the Michaelis&#;Menten equation can be found in many biochemistry textbooks (see section 4 of Recommended Reading section). There are also some very helpful web-based tutorials available on the subject.
Michaelis constants have been determined for many commonly used enzymes, and are typically in the lower millimolar range ( ).
It should be noted that enzymes which catalyse the same reaction, but which are derived from different organisms, can have widely differing Km values. Furthermore, an enzyme with multiple substrates can have quite different Km values for each substrate.
A low Km value indicates that the enzyme requires only a small amount of substrate in order to become saturated. Therefore the maximum velocity is reached at relatively low substrate concentrations. A high Km value indicates the need for high substrate concentrations in order to achieve maximum reaction velocity. Thus we generally refer to Km as a measure of the affinity of the enzyme for its substrate&#;in fact it is an inverse measure, where a high Km indicates a low affinity, and vice versa.
The Km value tells us several important things about a particular enzyme.
An enzyme with a low Km value relative to the physiological concentration of substrate will probably always be saturated with substrate, and will therefore act at a constant rate, regardless of variations in the concentration of substrate within the physiological range.
An enzyme with a high Km value relative to the physiological concentration of substrate will not be saturated with substrate, and its activity will therefore vary according to the concentration of substrate, so the rate of formation of product will depend on the availability of substrate.
If an enzyme acts on several substrates, the substrate with the lowest Km value is frequently assumed to be that enzyme's &#;natural&#; substrate, although this may not be true in all cases.
If two enzymes (with similar Vmax) in different metabolic pathways compete for the same substrate, then if we know the Km values for the two enzymes we can predict the relative activity of the two pathways. Essentially the pathway that has the enzyme with the lower Km value is likely to be the &#;preferred pathway&#;, and more substrate will flow through that pathway under most conditions. For example, phosphofructokinase (PFK) is the enzyme that catalyses the first committed step in the glycolytic pathway, which generates energy in the form of ATP for the cell, whereas glucose-1-phosphate uridylyltransferase (GUT) is an enzyme early in the pathway leading to the synthesis of glycogen (an energy storage molecule). Both enzymes use hexose monophosphates as substrates, but the Km of PFK for its substrate is lower than that of GUT for its substrate. Thus at lower cellular hexose phosphate concentrations, PFK will be active and GUT will be largely inactive. At higher hexose phosphate concentrations both pathways will be active. This means that the cells only store glycogen in times of plenty, and always give preference to the pathway of ATP production, which is the more essential function.
Very often it is not possible to estimate Km values from a direct plot of velocity against substrate concentration (as shown in ) because we have not used high enough substrate concentrations to come even close to estimating maximal velocity, and therefore we cannot evaluate half-maximal velocity and thus Km. Fortunately, we can plot our experimental data in a slightly different way in order to obtain these values. The most commonly used alternative is the Lineweaver&#;Burk plot (often called the double-reciprocal plot). This plot linearizes the hyperbolic curved relationship, and the line produced is easy to extrapolate, allowing evaluation of Vmax and Km. For example, if we obtained only the first seven data points in , we would have difficulty estimating Vmax from a direct plot as shown in a.
However, as shown in b, if these seven points are plotted on a graph of 1/velocity against 1/substrate concentration (i.e. a double-reciprocal plot), the data are linearized, and the line can be easily extrapolated to the left to provide intercepts on both the y-axis and the x-axis, from which Vmax and Km, respectively, can be evaluated.
One significant practical drawback of using the Lineweaver&#;Burk plot is the excessive influence that it gives to measurements made at the lowest substrate concentrations. These concentrations might well be the most prone to error (due to difficulties in making multiple dilutions), and result in reaction rates that, because they are slow, might also be most prone to measurement error. Often, as shown in , such points when transformed on the Lineweaver&#;Burk plot have a significant impact on the line of best fit estimated from the data, and therefore on the extrapolated values of both Vmax and Km. The two sets of points shown in are identical except for the single point at the top right, which reflects (because of the plot's double-reciprocal nature) a single point derived from a very low substrate concentration and a low reaction rate. However, this single point can have an enormous impact on the line of best fit and the accompanying estimates of kinetic constants.
In fact there are other kinetic plots that can be used, including the Eadie&#;Hofstee plot, the Hanes plot and the Eisenthal&#;Cornish-Bowden plot, which are less prone to such problems. However, the Lineweaver&#;Burk plot is still the most commonly described kinetic plot in the majority of enzymology textbooks, and thus retains its influence in undergraduate education.
By Keith Michael Krise
All living things depend on millions of chemical reactions that happen constantly. Chemical reactions that keep you alive happen fast! When you eat food, breathe, play, and grow, all of these are chemical reactions, and they must take place quickly.
How does your body speed up these important reactions? The answer is enzymes. Enzymes in our bodies are catalysts that speed up reactions by helping to lower the activation energy needed to start a reaction. Each enzyme molecule has a special place called the active site where another molecule, called the substrate, fits. The substrate goes through a chemical reaction and changes into a new molecule called the product &#; sort of like when a key goes into a lock and the lock opens.
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