Enzyme kinetics are just a special case of the kinetics that go on in any chemical system. Enzymes are biological catalysts, which are chemicals that increase the rate at which equilibrium is achieved without themselves being permanently transformed by the reaction. A reactant in an enzymatic reaction is usally called a substrate.
Substrate(s) + Enzyme → Product(s) + Enzyme
Note that although they alter the rate of equilibrium achievement, they do not the actual equilibrium point.
Enzyme catalysis rates are affected by three important factors...
- Temperature. The rate of any reaction increases with temperature. However, enzymes require a specific shape to work, and as the temperature increases, vibrations in the enzyme molecule will mess with its catalytic ability: at very high temperatures, the enzyme will completely denature (like egg white), becoming catalytically inactive. These competing processes lead to a temperature 'optimum' for the enzyme. This is not a true optimum, since the length of the assay will affect its value: a short assay will benefit from the increased rate, and not suffer too much from the exponential decay (over time) of enzyme activity at raised temperature. A longer assay will have more time for the enzyme to denature, so the observed 'optimum' will be lower.
- pH. Enzymes bristle with ionisable side groups that can lose and gain protons. If the active site contains a basic and an acidic amino acid that are required for catalysis, and both need to be ionised to interact with the substrate, then there will be a (true) optimum pH somewhere between the pKa of the acid and the pKb of the base.
- Substrate concentration. We will cover this in a minute.
Enzymes catalyse reactions by reducing the activation energy and by altering the steric constant in the Arrhenius equation. The former they do by decreasing the energy of the transition state, often by increasing the number of transition states, so dividing the big mountain of the uncatalysed activation energy with a few smaller foot-hills between several intermediate states. The latter they do by ensuring that substrates interact with one another in the correct orientation by binding them in a specific orientation to the active site.
The maximum rate an enzyme at STP can possibly run is 7 x 109 mol s-1 L-1. This situation is called diffusion limitation: essentially all collisions between substrate and enzyme are productive (lead to product formation). Few enzymes are this perfect, but a few, such as catalase, do manage these dizzying heights of enzymatic perfection.
The most important equation in enzymology is that derived by Michaelis and Menten in 1913, which relates the enzyme rate of reaction (or velocity, V) to the rate constants for the reactions...
E + S ↔ ES → E + P
When measuring the rate of reaction of an enzyme, we usually measure the initial rate, since as enzymes 'work' they will use up reactant, and this loss of reactant (and possible negative product feedback) will slow the reaction down. The initial velocity of a reaction , V, is...
V = Vm [S] / ( [S] + Km )
Vm is the maximum rate (measured in moles of product per litre degraded per second, or similar units: often V will actually be scaled to the amount of enzyme present, and so have units moles per second per milligram of protein) that can be attained at a given pH and T: this rate is achieved only when the substrate concentration is very high. Km is the Michaelis constant (the concentration of substrate giving a velocity of ½Vm), and [S] is the concentration of substrate. Here is a graph for an enzyme with Vm 10 and Km 4...
At low [S], V is first order w.r.t. [S]: nearly a straight line [S] < 4. That is to say, it is pseudo-first order w.r.t. [S]. At saturating [S] > 6, it becomes zeroeth order (rate not dependent on [S]), as active sites run out. (In fact, it becomes pseudo-first order w.r.t. enzyme concentration if the substrate is present in massive excess).
The specific activity of a protein is the rate at which a given mass of protein catalyses reactant degradation, and has units mol s-1 mg-1 or similar. It is much used to measure the purity of an enzyme in protein purification. The turnover number kcat(catalytic capacity) of an enzyme is the number of moles of substrate degraded per mole of enzyme per second. It is also termed the zeroeth order rate constant, since it is the velocity at saturating [S], and is quite literally the number of molecules of substrate a single enzyme molecule can degrade in one second, which is rather nice.
Enzymes are specific for their substrates: even for certain chiralities of a molecule, because they are themselves chiral (being composed of L-amino acids). Some enzymes (DNA polymerase, aminoacyl tRNA synthetases) have error checking mechanisms, and some multicomponent enzymes (haemoglobin, not really an enzyme though!) become more specific as they bind more molecules of substrate. The specificity constant for an enzyme is defined as kcat / Km, i.e. a specific enzyme is one that both 'finds' its substrate easily (a low Km indicates that the enzyme will run at an appreciable fraction kcat at even very low [S]), and rapidly breaks it down (high kcat).
Enzymes can be readily inhibited by certain compounds. Irreversible inhibitors bind to the enzyme and destroy the active site. Suicide inhibitors, a special class of such inhibitors, are activated by the normal catalytic activity of the enzyme, but form an intermediate that binds to and destroys the active site.
Reversible inhibitors come in several varieties: they do not permanently damage the enzyme, rather they form an enzyme-inhibitor complex, EI, with an inhibitor constant of Ki...
E + I ↔ EI → no products
Ki = [E] [I] / [EI]
We will look at competitive and noncompetitive inhibitors.
Competitive inhibitors are substrate analogues that bind reversibly to the active site. They reduce the apparent Km, because binding to the inhibitor is unproductive: since the enzyme accidentally picks up inhibitor, this means that a higher concentration of substrate is required to get the reaction to run at ½Vm. This Kmapp is...
Kmapp = Km( 1 + [I]/Ki )
Vm is not affected, as at infinite [S], all bindings will be enzyme to substrate.
Noncompetitive inhibitors may block the active site, change the protein conformation allosterically, etc. They have no effect on Km as the uninhibited enzyme molecules will only encounter substrate, and no unproductive binding will occur. They do however reduce the apparent Vm, as they effectively 'kill' a proportion of the enzyme, and consequently the apparent Vm will be reduced, since some protein will no longer be enzymatically competent.
Vmapp = Vm / ( 1 + [I]/Ki )
The Lineweaver-Burke double reciprocal plot can be used to measure the Km and Vm of an enzyme catalysed reaction. By taking the reciprocal of the Michaelis-Menten equation, and rearranging it, we obtain...
1/V = ( Km/Vm )( 1/[S] ) + 1/Vm
which is the equation of a straight line, y = mx + c. The Lineweaver Burke plot (of 1/V vs. 1/[S]) allows rapid estimation of Vm and Km for both inhibited and uninhibited enzymes. The intercept on the y axis is 1/Vm, and the intercept on the x-axis is -1/Kmm. The slope is also Km/Vm. It does have the problem that it gives undue weighting to small values of [S] (since 1/[S] is large): these are unfortunately the very values of [S] most likely to be inaccurate. Other plots and algorithms are available, but the double reciprocal plot is still widely used due to its simplicity.
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Different modes of inhibition can be distinguished using such a Lineweaver-Burke double reciprocal plot.
The intercept on the y-axis of this plot is 1/Vm. Note that the competitive inhibitor crosses the control at this point (showing they both have the same Vm, as mentioned earlier), but the noncompetitive inhibitor has a higher intercept (hence a lower Vm, i.e. Vmapp).
The intercept on the x-axis is -1/Km: note that the noncompetitive inhibitor and the control cross here (showing they both have the same Km as mentioned earlier), but that the competitive inhibitor has a smaller negative intercept (hence a higher positive Km, i.e. Kmapp).
