“Alice and the Queen of
Hearts,” illustrated by John Tenniel, The Nursery Alice. (Mary Evans
Picture Library,
Chapter 14
Enzyme Kinetics
Living organisms seethe with metabolic activity. Thousands of chemical reactions are proceeding very rapidly at any given instant within all living cells. Virtually all of these transformations are mediated by enzymes, proteins (and occasionally RNA) specialized to catalyze metabolic reactions. The substances transformed in these reactions are often organic compounds that show little tendency for reaction outside the cell. An excellent example is glucose, a sugar that can be stored indefinitely on the shelf with no deterioration. Most cells quickly oxidize glucose, producing carbon dioxide and water and releasing lots of energy:
C6H12O6 + 6 O2 ® 6 CO2 = 6 H2O + 2870 kJ of energy
(-2870 kJ/mol is the standard free energy change [DG°'] for the oxidation of glucose; see Chapter 3). In chemical terms, 2870 kJ is a large amount of energy, and glucose can be viewed as an energy-rich compound even though at ambient temperature it is not readily reactive with oxygen outside of cells. Stated another way, glucose represents thermodynamic potentiality: its reaction with oxygen is strongly exergonic, but it just doesn’t occur under normal conditions. On the other hand, enzymes can catalyze such thermodynamically favorable reactions so that they proceed at extraordinarily rapid rates (Figure 14.1). In glucose oxidation and countless other instances, enzymes provide cells with the ability to exert kinetic control over thermodynamic potentiality. That is, living systems use enzymes to accelerate and control the rates of vitally important biochemical reactions.
Figure
14.1 • Reaction
profile showing large D G‡ for glucose
oxidation, free energy change of -2,870 kJ/mol; catalysts lower D
G‡, thereby accelerating rate.
Enzymes Are the Agents of Metabolic Function
Acting in sequence, enzymes form metabolic pathways by which nutrient molecules are degraded, energy is released and converted into metabolically useful forms, and precursors are generated and transformed to create the literally thousands of distinctive biomolecules found in any living cell (Figure 14.2). Situated at key junctions of metabolic pathways are specialized regulatory enzymes capable of sensing the momentary metabolic needs of the cell and adjusting their catalytic rates accordingly. The responses of these enzymes ensure the harmonious integration of the diverse and often divergent metabolic activities of cells so that the living state is promoted and preserved.
Figure 14.2 • The breakdown of glucose by glycolysis provides a prime example of a metabolic pathway. Ten enzymes mediate the reactions of glycolysis. Enzyme 4, fructose 1,6, biphosphate aldolase, catalyzes the C-C bond- breaking reaction in this pathway.
14.1 · Enzymes—Catalytic Power, Specificity, and Regulation
Enzymes are characterized by three distinctive features: catalytic power, specificity, and regulation.
Catalytic Power
Enzymes display enormous catalytic power, accelerating reaction rates as much as 1016 over uncatalyzed levels, which is far greater than any synthetic catalysts can achieve, and enzymes accomplish these astounding feats in dilute aqueous solutions under mild conditions of temperature and pH. For example, the enzyme jack bean urease catalyzes the hydrolysis of urea:
At 20°C, the rate constant for the enzyme-catalyzed reaction is 3 x 104/sec; the rate constant for the uncatalyzed hydrolysis of urea is 3 x 10-10/sec. Thus, 1014 is the ratio of the catalyzed rate to the uncatalyzed rate of reaction. Such a ratio is defined as the relative catalytic power of an enzyme, so the catalytic power of urease is 1014.
Specificity
A given enzyme is very selective, both in the substances with which it interacts and in the reaction that it catalyzes. The substances upon which an enzyme acts are traditionally called substrates. In an enzyme-catalyzed reaction, none of the substrate is diverted into nonproductive side-reactions, so no wasteful by-products are produced. It follows then that the products formed by a given enzyme are also very specific. This situation can be contrasted with your own experiences in the organic chemistry laboratory, where yields of 50% or even 30% are viewed as substantial accomplishments (Figure 14.3). The selective qualities of an enzyme are collectively recognized as its specificity. Intimate interaction between an enzyme and its substrates occurs through molecular recognition based on structural complementarity; such mutual recognition is the basis of specificity. The specific site on the enzyme where substrate binds and catalysis occurs is called the active site.
Figure 14.3 • A 90% yield over 10 steps, for example, in a metabolic pathway, gives an overall yield of 35%. Therefore, yields in biological reactions must be substantially greater; otherwise, unwanted by-products would accumulate to unacceptable levels.
Regulation
Regulation of enzyme activity is achieved in a variety of ways, ranging from controls over the amount of enzyme protein produced by the cell to more rapid, reversible interactions of the enzyme with metabolic inhibitors and activators. Chapter 15 is devoted to discussions of enzyme regulation. Because most enzymes are proteins, we can anticipate that the functional attributes of enzymes are due to the remarkable versatility found in protein structures.
Enzyme Nomenclature
Traditionally, enzymes often were named by adding the suffix-ase to the name of the substrate upon which they acted, as in urease for the urea-hydrolyzing enzyme or phosphatase for enzymes hydrolyzing phosphoryl groups from organic phosphate compounds. Other enzymes acquired names bearing little resemblance to their activity, such as the peroxide-decomposing enzyme catalase or the proteolytic enzymes (proteases) of the digestive tract, trypsin and pepsin. Because of the confusion that arose from these trivial designations, an International Commission on Enzymes was established in 1956 to create a systematic basis for enzyme nomenclature. Although common names for many enzymes remain in use, all enzymes now are classified and formally named according to the reaction they catalyze. Six classes of reactions are recognized (Table 14.1).
Within each class are subclasses, and under each subclass are sub-subclasses within which individual enzymes are listed. Classes, subclasses, sub-subclasses, and individual entries are each numbered, so that a series of four numbers serves to specify a particular enzyme. A systematic name, descriptive of the reaction, is also assigned to each entry. To illustrate, consider the enzyme that catalyzes this reaction:
ATP + d-glucose ® ADP + d-glucose-6-phosphate
A phosphate group is transferred from ATP to the C-6-OH group of glucose, so the enzyme is a transferase (Class 2, Table 14.1). Subclass 7 of transferases is enzymes transferring phosphorus-containing groups, and sub-subclass 1 covers those phosphotransferases with an alcohol group as an acceptor. Entry 2 in this sub-subclass is ATP: d-glucose-6-phosphotransferase, and its classification number is 2.7.1.2. In use, this number is written preceded by the letters E.C., denoting the Enzyme Commission. For example, entry 1 in the same sub-subclass is E.C.2.7.1.1, ATP: d-hexose-6-phosphotransferase, an ATP-dependent enzyme that transfers a phosphate to the 6-OH of hexoses (that is, it is nonspecific regarding its hexose acceptor). These designations can be cumbersome, so in everyday usage, trivial names are employed frequently. The glucose-specific enzyme, E.C.2.7.1.2, is called glucokinase and the nonspecific E.C.2.7.1.1 is known as hexokinase. Kinase is a trivial term for enzymes that are ATP-dependent phosphotransferases.
Coenzymes
Many enzymes carry out their catalytic function relying solely on their protein structure. Many others require nonprotein components, called cofactors (Table 14.2). Cofactors may be metal ions or organic molecules referred to as coenzymes. Cofactors, because they are structurally less complex than proteins, tend to be stable to heat (incubation in a boiling water bath). Typically, proteins are denatured under such conditions. Many coenzymes are vitamins or contain vitamins as part of their structure. Usually coenzymes are actively involved in the catalytic reaction of the enzyme, often serving as intermediate carriers of functional groups in the conversion of substrates to products. In most cases, a coenzyme is firmly associated with its enzyme, perhaps even by covalent bonds, and it is difficult to separate the two. Such tightly bound coenzymes are referred to as prosthetic groups of the enzyme. The catalytically active complex of protein and prosthetic group is called the holoenzyme. The protein without the prosthetic group is called the apoenzyme; it is catalytically inactive.
14.2 · Introduction to Enzyme Kinetics
Kinetics is the
branch of science concerned with the rates of chemical reactions. The study
of enzyme kinetics addresses the biological roles of enzymatic catalysts
and how they accomplish their remarkable feats. In enzyme kinetics, we seek
to determine the maximum reaction velocity that the enzyme can attain and its
binding affinities for substrates and inhibitors. Coupled with studies on the
structure and chemistry of the enzyme, analysis of the enzymatic rate under
different reaction conditions yields insights regarding the enzyme’s mechanism
of catalytic action. Such information is essential to an overall understanding
of metabolism.
Significantly,
this information can be exploited to control and manipulate the course of metabolic
events. The science of pharmacology relies on such a strategy. Pharmaceuticals,
or drugs, are often special inhibitors specifically targeted at a particular
enzyme in order to overcome infection or to alleviate illness. A detailed knowledge
of the enzyme’s kinetics is indispensable to rational drug design and successful
pharmacological intervention.
Review of Chemical Kinetics
Before beginning a quantitative treatment of enzyme kinetics, it will be fruitful to review briefly some basic principles of chemical kinetics. Chemical kinetics is the study of the rates of chemical reactions. Consider a reaction of overall stoichiometry
A ® P
Although we treat this reaction as a simple, one-step conversion of A to P, it more likely occurs through a sequence of elementary reactions, each of which is a simple molecular process, as in
A ® I ® J ® P
where I and J represent intermediates in the reaction. Precise description of all of the elementary reactions in a process is necessary to define the overall reaction mechanism for A ® P.
Let us assume that A ® P is an elementary reaction and that it is spontaneous and essentially irreversible. Irreversibility is easily assumed if the rate of P conversion to A is very slow or the concentration of P (expressed as [P]) is negligible under the conditions chosen. The velocity, v, or rate, of the reaction A ® P is the amount of P formed or the amount of A consumed per unit time, t. That is,
or 
(14.1)
The mathematical relationship between reaction rate and concentration of reactant(s) is the rate law. For this simple case, the rate law is
(14.2)
From this expression, it is obvious that the rate is proportional to the concentration of A, and k is the proportionality constant, or rate constant. k has the units of (time) -1, usually sec-1. v is a function of [A] to the first power, or, in the terminology of kinetics, v is first-order with respect to A. For an elementary reaction, the order for any reactant is given by its exponent in the rate equation. The number of molecules that must simultaneously interact is defined as the molecularity of the reaction. Thus, the simple elementary reaction of A ® P is a first-order reaction. Figure 14.4 portrays the course of a first-order reaction as a function of time. The rate of decay of a radioactive isotope, like 14C or 32P, is a first-order reaction, as is an intramolecular rearrangement, such as A ® P. Both are unimolecular reactions (the molecularity equals 1).
Figure 14.4 • Plot of the course of a first-order reaction. The half-time, t1/2, is the time for one-half of the starting amount of A to disappear.
Biomolecular Reactions
Consider the more complex reaction, where two molecules must react to yield products:
A + B ® P + Q
Assuming this reaction is an elementary reaction, its molecularity is 2; that is, it is a bimolecular reaction. The velocity of this reaction can be determined from the rate of disappearance of either A or B, or the rate of appearance of P or Q:
(14.3)
The rate law is
v = k[A][B] (14.4)
The rate is proportional
to the concentrations of both A and B. Because it is proportional to the product
of two concentration terms, the reaction is second-order overall, first-order
with respect to A and first-order with respect to B. (Were the elementary reaction
2 ® P + Q, the rate law would be v =
k[A]2, second-order overall and Second-order with respect
to A.) Second-order rate constants have the units of (concentration) -1
(time) -1, as in M-1 sec-1.
Molecularities
greater than two are rarely found (and greater than three, never). When the
overall stoichiometry of a reaction is greater than two (for example, as in
A+B+Cn or 2A+Bn), the reaction almost always proceeds via uni- or bimolecular
elementary steps, and the overall rate obeys a simple first- or Second-order
rate law.
At this
point, it may be useful to remind ourselves of an important caveat that is the
first principle of kinetics: Kinetics cannot prove a hypothetical mechanism.
Kinetic experiments can only rule out various alternative hypotheses because
they don’t fit the data. However, through thoughtful kinetic studies, a process
of elimination of alternative hypotheses leads ever closer to the reality.
Free Energy of Activation and the Action of Catalysts
In a first-order chemical reaction, the conversion of A to P occurs because, at any given instant, a fraction of the A molecules has the energy necessary to achieve a reactive condition known as the transition state. In this state, the probability is very high that the particular rearrangement accompanying the A ® P transition will occur. This transition state sits at the apex of the energy profile in the energy diagram describing the energetic relationship between A and P (Figure 14.5).
Figure 14.5 • Energy diagram for a chemical reaction (A ® P) and the effects of (a) raising the temperature from T1 to T2 or (b) adding a catalyst. Raising the temperature raises the average energy of A molecules, which increases the population of A molecules having energies equal to the activation energy for the reaction, thereby increasing the reaction rate. In contrast, the average free energy of A molecules remains the same in uncatalyzed versus catalyzed reactions (conducted at the same temperature). The effect of the catalyst is to lower the free energy of activation for the reaction.
The average free energy of A molecules defines the initial state and the average free energy of P molecules is the final state along the reaction coordinate. The rate of any chemical reaction is proportional to the concentration of reactant molecules (A in this case) having this transition-state energy. Obviously, the higher this energy is above the average energy, the smaller the fraction of molecules that will have this energy, and the slower the reaction will proceed. The height of this energy barrier is called the free energy of activation, DG‡. Specifically, DG‡ is the energy required to raise the average energy of one mole of reactant (at a given temperature) to the transition-state energy. The relationship between activation energy and the rate constant of the reaction, k, is given by the Arrhenius equation:
k = Ae- DG‡/RT (14.5)
where A is a constant for a particular reaction (not to be confused with the reactant species, A, that we’re discussing). Another way of writing this is 1/k = (1/A)e DG‡/RT. That is, k is inversely proportional to e DG‡/RT. Therefore, if the energy of activation decreases, the reaction rate increases.
Decreasing DG‡ Increases Reaction Rate
We are familiar with two general ways that rates of chemical reactions may be accelerated. First, the temperature can be raised. This will increase the average energy of reactant molecules, which in effect lowers the energy needed to reach the transition state (Figure 14.5a). The rates of many chemical reactions are doubled by a 10°C rise in temperature. Second, the rates of chemical reactions can also be accelerated by catalysts. Catalysts work by lowering the energy of activation rather than by raising the average energy of the reactants (Figure 14.5b). Catalysts accomplish this remarkable feat by combining transiently with the reactants in a way that promotes their entry into the reactive, transition-state condition. Two aspects of catalysts are worth noting: (a) they are regenerated after each reaction cycle (A ® P), and so can be used over and over again; and (b) catalysts have no effect on the overall free energy change in the reaction, the free energy difference between A and P (Figure 14.5b).
14.3 · Kinetics of Enzyme-Catalyzed Reactions
Examination of the change in reaction velocity as the reactant concentration is varied is one of the primary measurements in kinetic analysis. Returning to A ® P, a plot of the reaction rate as a function of the concentration of A yields a straight line whose slope is k (Figure 14.6).
Figure
14.6 • A
plot of v versus [A] for the unimolecular chemical reaction, A ®
P, yields a straight line having a slope equal to k.
The more A that is available, the greater the rate of the reaction, v. Similar analyses of enzyme-catalyzed reactions involving only a single substrate yield remarkably different results (Figure 14.7). At low concentrations of the substrate S, v is proportional to [S], as expected for a first-order reaction. However, v does not increase proportionally as [S] increases, but instead begins to level off. At high [S], v becomes virtually independent of [S] and approaches a maximal limit. The value of v at this limit is written Vmax. Because rate is no longer dependent on [S] at these high concentrations, the enzyme-catalyzed reaction is now obeying zero-order kinetics; that is, the rate is independent of the reactant (substrate) concentration. This behavior is a saturation effect: when v shows no increase even though [S] is increased, the system is saturated with substrate. Such plots are called substrate saturation curves. The physical interpretation is that every enzyme molecule in the reaction mixture has its substrate-binding site occupied by S. Indeed, such curves were the initial clue that an enzyme interacts directly with its substrate by binding it.
Figure 14.7 • Substrate saturation curve for an enzyme-catalyzed reaction. The amount of enzyme is constant, and the velocity of the reaction is determined at various substrate concentrations. The reaction rate, v, as a function of [S] is described by a rectangular hyperbola. At very high [S], v = Vmax. That is, the velocity is limited only by conditions (temperature, pH, ionic strength) and by the amount of enzyme present; v becomes independent of [S]. Such a condition is termed zero-order kinetics. Under zero-order conditions, velocity is directly dependent on [enzyme]. The H2O molecule provides a rough guide to scale. The substrate is bound at the active site of the enzyme.
The Michaelis - Menten Equation
Lenore Michaelis and Maud L. Menten proposed a general theory of enzyme action in 1913 consistent with observed enzyme kinetics. Their theory was based on the assumption that the enzyme, E, and its substrate, S, associate reversibly to form an enzyme-substrate complex, ES:
(14.6)
This association/dissociation is assumed to be a rapid equilibrium, and Ks is the enzyme : substrate dissociation constant. At equilibrium,
k-1[ES] = k1[E][S] (14.7)
and
(14.8)
Product, P, is formed in a second step when ES breaks down to yield E + P.
(14.9)
E is then free to interact with another molecule of S.
Steady-State Assumption
The interpretations of Michaelis and Menten were refined and extended in 1925 by Briggs and Haldane, by assuming the concentration of the enzyme-substrate complex ES quickly reaches a constant value in such a dynamic system. That is, ES is formed as rapidly from E + S as it disappears by its two possible fates: dissociation to regenerate E + S, and reaction to form E + P. This assumption is termed the steady-state assumption and is expressed as
(14.10)
That is, the change in concentration of ES with time, t, is 0. Figure 14.8 illustrates the time course for formation of the ES complex and establishment of the steady-state condition.
Figure
14.8 • Time
course for the consumption of substrate, the formation of product, and the establishment
of a steady-state level of the enzyme-substrate [ES] complex for a typical enzyme
obeying the Michaelis-Menten, Briggs-Haldane models for enzyme kinetics. The
early stage of the time course is shown in greater magnification in the bottom
graph.
Initial Velocity Assumption
One other simplification will be advantageous. Because enzymes accelerate the rate of the reverse reaction as well as the forward reaction, it would be helpful to ignore any back reaction by which E1P might form ES. The velocity of this back reaction would be given by v = k-2[E][P]. However, if we observe only the initial velocity for the reaction immediately after E and S are mixed in the absence of P, the rate of any back reaction is negligible ecause its rate will be proportional to [P], and [P] is essentially 0. Given such simplification, we now analyze the system described by Equation (14.9) in order to describe the initial velocity v as a function of [S] and amount of enzyme.
The total amount of enzyme is fixed and is given by the formula
Total enzyme, [ET] = [E] + [ES] (14.11)
where [E] = free enzyme and [ES] = the amount of enzyme in the enzyme-
substrate complex. From Equation (14.9), the rate of [ES] formation is
vf = k1([ET] - [ES])[S]
where
[ET] - [ES] = [E] (14.12)
From Equation (14.9), the rate of [ES] disappearance is
vd = k-1[ES] + k2[ES] = (k-1 + k2)[ES] (14.13)
At steady state, d[ES]/dt = 0, and therefore, vf = vd.
So,
K1([ET] - [ES])[S] = (k-1 + k2)[ES] (14.14)
Rearranging gives
(14.15)
The Michaelis Constant, Km
The ratio of constants (k-1 + k2)/k1 is itself a constant and is defined as the Michaelis constant, Km
(14.16)
Note from (14.15) that Km is given by the ratio of two concentrations (([ET] - [ES]) and [S]) to one ([ES]), so Km has the units of molarity. From Equation (14.15), we can write
(14.17)
which rearranges to
(14.18)
Now, the most important parameter in the kinetics of any reaction is the rate of product formation. This rate is given by
(14.19)
and for this reaction
v = k2[ES] (14.20)
Substituting the expression for [ES] from Equation (14.18) into (14.20) gives
(14.21)
The product k2[ET] has special meaning. When [S] is high enough to saturate all of the enzyme, the velocity of the reaction, v, is maximal. At saturation, the amount of [ES] complex is equal to the total enzyme concentration, ET, its maximum possible value. From Equation (14.20), the initial velocity v then equals k2[ET] = Vmax. Written symbolically, when [S] >> [ET] (and Km), [ET] = [ES] and v = Vmax. Therefore,
Vmax = k2[ET] (14.22)
Substituting this relationship into the expression for v gives the Michaelis-
Menten equation
(14.23)
This equation says that the rate of an enzyme-catalyzed reaction, v, at any moment is determined by two constants, Km and Vmax, and the concentration of substrate at that moment.
When [S] = Km, v = Vmax/2
We can provide an operational definition for the constant Km by rearranging Equation (14.23) to give
(14.24)
Then, at v = Vmax/2, Km = [S]. That is, Km is defined by the substrate concentration that gives a velocity equal to one-half the maximal velocity. Table 14.3 gives the Km values of some enzymes for their substrates.
Relationships Between Vmax, Km, and Reaction Order
The Michaelis-Menten equation (14.23) describes a curve known from analytical geometry as a rectangular hyperbola.1 In such curves, as [S] is increased, v approaches the limiting value, Vmax, in an asymptotic fashion. Vmax can be approximated experimentally from a substrate saturation curve (Figure 14.7), and Km can be derived from Vmax /2, so the two constants of the Michaelis-Menten equation can be obtained from plots of v versus [S]. Note, however, that actual estimation of Vmax, and consequently Km, is only approximate from such graphs. That is, according to Equation (14.23), in order to get v = 0.99 Vmax, [S] must equal 99 Km, a concentration that may be difficult to achieve in practice.
From Equation (14.23), when [S] >> Km, then v = Vmax. That is, v is no longer dependent on [S], so the reaction is obeying zero-order kinetics. Also, when [S] < Km, then v » (Vmax/ Km)[S]. That is, the rate, v, approximately follows a first-order rate equation, v = k'[A], where k' = Vmax / Km.
Km and Vmax, once known explicitly, define the rate of the enzyme-catalyzed reaction, provided:
1. The reaction involves only one substrate, or if the reaction is multisubstrate, the concentration of only one substrate is varied while the concentration of all other substrates is held constant.
2. The reaction ES ® E + P is irreversible, or the experiment is limited to observing only initial velocities where [P] = 0.
3. [S]0 > [ET] and [ET] is held constant.
4. All other variables that might influence the rate of the reaction (temperature, pH, ionic strength, and so on) are constant.
Enzyme Units
In many situations, the actual molar amount of the enzyme is not known. However, its amount can be expressed in terms of the activity observed. The International Commission on Enzymes defines One International Unit of enzyme as the amount that catalyzes the formation of one micromole of product in one minute. (Because enzymes are very sensitive to factors such as pH, temperature, and ionic strength, the conditions of assay must be specified.) Another definition for units of enzyme activity is the katal. One katal is that amount of enzyme catalyzing the conversion of one mole of substrate to product in one second. Thus, one katal equals 6 x 107 international units.
Turnover Number
The turnover number of an enzyme, kcat, is a measure of its maximal catalytic activity. kcat is defined as the number of substrate molecules converted into product per enzyme molecule per unit time when the enzyme is saturated with substrate. The turnover number is also referred to as the molecular activity of the enzyme. For the simple Michaelis-Menten reaction (14.9) under conditions of initial velocity measurements, k2 = kcat. Provided the concentration of enzyme, [ET], in the reaction mixture is known, kcat can be determined from Vmax. At saturating [S], v = Vmax = k2 [ET]. Thus,
(14.25)
The term kcat represents the kinetic efficiency of the enzyme. Table 14.4 lists turnover numbers for some representative enzymes. Catalase has the highest turnover number known; each molecule of this enzyme can degrade 40 million molecules of H2O2 in one second! At the other end of the scale, lysozyme requires 2 seconds to cleave a glycosidic bond in its glycan substrate.
|
Values of kcat (Turnover Number) for Some Enzymes |
|
| Enzyme | kcat (sec-1) |
| Catalase | 40,000,000 |
| Carbonic anhydrase | 1,000,000 |
| Acetylcholinesterase | 14,000 |
| Penicillinase | 2,000 |
| Lactate dehydrogenase | 1,000 |
| Chymotrypsin | 100 |
| DNA polymerase I | 15 |
| Lysozyme | 0.5 |
kcat/Km
Under physiological conditions, [S] is seldom saturating, and kcat itself is not particularly informative. That is, the in vivo ratio of [S]/Km usually falls in the range of 0.01 to 1.0, so active sites often are not filled with substrate. Nevertheless, we can derive a meaningful index of the efficiency of Michaelis-Menten-type enzymes under these conditions by employing the following equations. As presented in Equation (14.23), if
and Vmax = kcat [ET], then
(14.26)
When [S] << Km, the concentration of free enzyme, [E], is approximately equal to [ET], so that
(14.27)
That is, kcat / Km is an apparent Second-order rate constant for the reaction of E and S to form product. Because Km is inversely proportional to the affinity of the enzyme for its substrate and kcat is directly proportional to the kinetic efficiency of the enzyme, kcat / Km provides an index of the catalytic efficiency of an enzyme operating at substrate concentrations substantially below saturation amounts.
An interesting point emerges if we restrict ourselves to the simple case where kcat = k2. Then
(14.28)
But k1 must always be greater than or equal to k1k2/(k-1 + k2). That is, the reaction can go no faster than the rate at which E and S come together. Thus, k1 sets the upper limit for kcat / Km. In other words, the catalytic efficiency of an enzyme cannot exceed the diffusion-controlled rate of combination of E and S to form ES. In H2O, the rate constant for such diffusion is approximately 109/M × sec. Those enzymes that are most efficient in their catalysis have kcat / Km ratios approaching this value. Their catalytic velocity is limited only by the rate at which they encounter S; enzymes this efficient have achieved so-called catalytic perfection. All E and S encounters lead to reaction because such “catalytically perfect” enzymes can channel S to the active site, regardless of where S hits E. Table 14.5 lists the kinetic parameters of several enzymes in this category. Note that kcat and Km both show a substantial range of variation in this table, even though their ratio falls around 108/M × sec.
| Table 14.5 | ||||
| Enzymes Whose kcat /Km Approaches the Diffusion-Controlled Rate of Association with Substrate | ||||
| Enzyme | Substrate | kcat (sec-1) |
Km (M) |
kcat
/Km (sec-1 M-1) |
| Acetylcholinesterase | Acetylcholine | 1.4 x 104 | 9 x 10-5 | 1.6 x 108 |
| Carbonic anhydrase |
CO2 HCO3- |
1
x 106 4 x 105 |
0.012 0.026 |
8.3
x 107 1.5 x 107 |
| Catalase | H2O2 | 4 x 107 | 1.1 | 4 x 107 |
| Crotonase | Crotonyl-CoA | 5.7 x 103 | 2 x 10-5 | 2.8 x 108 |
| Fumarase | Fumarate Malate |
800 900 |
5 x 10-6 2.5 x 10-5 |
1.6
x 108 3.6 x 107 |
| Triosephosphate isomerase |
Glyceraldehyde- 3-phosphate* |
4.3 x 103 | 1.8 x 10-5 | 2.4 x 108 |
| b-Lactamase | Benzylpenicillin | 2 x 103 | 2 x 10-5 | 1 x 108 |
|
* Km for glyceraldehyde-3-phosphate is calculated on the basis that only 3.8% of the substrate in solution is unhydrated and therefore reactive with the enzyme. Adapted from Fersht, A. 1985. Enzyme Structure and Mechanism, 2nd ed. New York : W.H. Freeman & Co. |
||||
Linear Plots Can Be Derived from the Michaelis - Menten Equation
Because of the hyperbolic shape of v versus [S] plots, Vmax can only be determined from an extrapolation of the asymptotic approach of v to some limiting value as [S] increases indefinitely (Figure 14.7); and Km is derived from that value of [S] giving v = Vmax/2. However, several rearrangements of the Michaelis-Menten equation transform it into a straight-line equation. The best known of these is the Lineweaver-Burk double-reciprocal plot:
Taking the reciprocal of both sides of the Michaelis-Menten equation, Equation (14.23), yields the equality
(14.29)
This conforms to y5mx1b (the equation for a straight line), where y = 1/v; m, the slope, is Km/Vmax; x = 1/[S]; and b = 1/Vmax. Plotting 1/v versus 1/[S] gives a straight line whose x-intercept is -1/ Km, whose y-intercept is 1/ Vmax, and whose slope is Km / Vmax (Figure 14.9).
Figure 14.9 • The Lineweaver-Burk double-reciprocal plot, depicting extrapolations that allow the determination of the x- and y-intercepts and slope.
The Hanes-Woolf plot is another rearrangement of the Michaelis-Menten equation that yields a straight line:
Multiplying both sides of Equation (14.29) by [S] gives
(14.30)
and
(14.31)
Graphing [S]/v versus [S] yields a straight line where the slope is 1/ Vmax, the y-intercept is Km/Vmax, and the x-intercept is -Km, as shown in Figure 14.10. The common advantage of these plots is that they allow both Km and Vmax to be accurately estimated by extrapolation of straight lines rather than asymptotes. Computer fitting of v versus [S] data to the Michaelis-Menten equation is more commonly done than graphical plotting.
Figure 14.10 • A Hanes-Wolff plot of [S]/v versus [S], another straight-line rearrangement of the Michaelis-Menten equation.
Departures from Linearity: A Hint of Regulation?
If the kinetics of the reaction disobey the Michaelis-Menten equation, the violation is revealed by a departure from linearity in these straight-line graphs. We shall see in the next chapter that such deviations from linearity are characteristic of the kinetics of regulatory enzymes known as allosteric enzymes. Such regulatory enzymes are very important in the overall control of metabolic pathways.
Effect of pH on Enzymatic Activity
Enzyme-substrate recognition and the catalytic events that ensue are greatly dependent on pH. An enzyme possesses an array of ionizable side chains and prosthetic groups that not only determine its secondary and tertiary structure but may also be intimately involved in its active site. Further, the substrate itself often has ionizing groups, and one or another of the ionic forms may preferentially interact with the enzyme. Enzymes in general are active only over a limited pH range and most have a particular pH at which their catalytic activity is optimal. These effects of pH may be due to effects on Km or Vmax or both. Figure 14.11 illustrates the relative activity of four enzymes as a function of pH. Although the pH optimum of an enzyme often reflects the pH of its normal environment, the optimum may not be precisely the same. This difference suggests that the pH-activity response of an enzyme may be a factor in the intracellular regulation of its activity.
Figure
14.11 • The
pH activity profiles of four different enzymes. Trypsin, an intestinal
protease, has a slightly alkaline pH optimum, whereas pepsin, a gastric
protease, acts in the acidic confines of the stomach and has a pH optimum near
2. Papain, a protease found in papaya, is relatively insensitive to pHs
between 4 and 8. Cholinesterase activity is pH-sensitive below pH 7 but
not between pH 7 and 10. The cholinesterase pH activity profile suggests that
an ionizable group with a pK' near 6 is essential to its activity. Might
it be a histidine residue within the active site?
Effect of Temperature on Enzymatic Activity
Like most chemical reactions, the rates of enzyme-catalyzed reactions generally increase with increasing temperature. However, at temperatures above 50° to 60°C, enzymes typically show a decline in activity (Figure 14.12). Two effects are operating here: (a) the characteristic increase in reaction rate with temperature, and (b) thermal denaturation of protein structure at higher temperatures. Most enzymatic reactions double in rate for every 10°C rise in temperature (that is, Q10 = 2, where Q10 is defined as the ratio of activities at two temperatures 10° apart) as long as the enzyme is stable and fully active. Some enzymes, those catalyzing reactions having very high activation energies, show proportionally greater Q10 values. The increasing rate with increasing temperature is ultimately offset by the instability of higher orders of protein structure at elevated temperatures, where the enzyme is inactivated. Not all enzymes are quite so thermally labile. For example, the enzymes of thermophilic bacteria (thermophilic = ”heat-loving”) found in geothermal springs retain full activity at temperatures in excess of 85°C.
Figure 14.12 • The effect of temperature on enzyme activity. The relative activity of an enzymatic reaction as a function of temperature. The decrease in the activity above 50°C is due to thermal denaturation.
If the velocity of an enzymatic reaction is decreased or inhibited, the kinetics of the reaction obviously have been perturbed. Systematic perturbations are a basic tool of experimental scientists; much can be learned about the normal workings of any system by inducing changes in it and then observing the effects of the change. The study of enzyme inhibition has contributed significantly to our understanding of enzymes.
Reversible Versus Irreversible Inhibition
Enzyme inhibitors are classified in several ways. The inhibitor may interact either reversibly or irreversibly with the enzyme. Reversible inhibitors interact with the enzyme through noncovalent association/dissociation reactions. In contrast, irreversible inhibitors usually cause stable, covalent alterations in the enzyme. That is, the consequence of irreversible inhibition is a decrease in the concentration of active enzyme. The kinetics observed are consistent with this interpretation, as we shall see later.
Reversible Inhibition
Reversible inhibitors fall into two major categories: competitive and noncompetitive (although other more unusual and rare categories are known). Competitive inhibitors are characterized by the fact that the substrate and inhibitor compete for the same binding site on the enzyme, the so-called active site or S-binding site. Thus, increasing the concentration of S favors the likelihood of S binding to the enzyme instead of the inhibitor, I. That is, high [S] can overcome the effects of I. The other major type, noncompetitive inhibition, cannot be overcome by increasing [S]. The two types can be distinguished by the particular patterns obtained when the kinetic data are analyzed in linear plots, such as double-reciprocal (Lineweaver-Burk) plots. A general formulation for common inhibitor interactions in our simple enzyme kinetic model would include
(14.32)
That is, we consider here reversible combinations of the inhibitor with E and/or ES.
Competitive Inhibition
Consider the following system
(14.33)
where an inhibitor, I,
binds reversibly to the enzyme at the same site as S. S-binding and I-binding
are mutually exclusive, competitive processes. Formation of the ternary
complex, EIS, where both S and I are bound, is physically impossible. This condition
leads us to anticipate that S and I must share a high degree of structural similarity
because they bind at the same site on the enzyme. Also notice that, in our model,
EI does not react to give rise to E + P. That is, I is not changed by interaction
with E. The rate of the product-forming reaction is v = k2[ES].
It is
revealing to compare the equation for the uninhibited case, Equation (14.23)
(the Michaelis-Menten equation) with Equation (14.43) for the rate of the enzymatic
reaction in the presence of a fixed concentration of the competitive inhibitor,
[I]
(see also Table 14.6). The Km term in the denominator in the inhibited case is increased by the factor (1 + [I]/KI); thus, v is less in the presence of the inhibitor, as expected. Clearly, in the absence of I, the two equations are identical. Figure 14.13 shows a Lineweaver-Burk plot of competitive inhibition. Several features of competitive inhibition are evident. First, at a given [I], v decreases (1/v increases). When [S] becomes infinite, v = Vmax and is unaffected by I because all of the enzyme is in the ES form. Note that the value of the -x-intercept decreases as [I] increases. This -x-intercept is often termed the apparent Km (or Kmapp) because it is the Km apparent under these conditions. The diagnostic criterion for competitive inhibition is that Vmax is unaffected by I; that is, all lines share a common y-intercept. This criterion is also the best experimental indication of binding at the same site by two substances. Competitive inhibitors resemble S structurally.
Figure 14.13 • Lineweaver-Burk plot of competitive inhibition, showing lines for no I, [I], and 2[I]. Note that when [S] is infinitely large (1/[S]50), Vmax is the same, whether I is present or not. In the presence of I, the negative x-intercept521/Km(11[I]/KI).
| A Deeper Look | |
|
The Equations of Competitive Inhibition Given the relationships between E, S, and I described previously and recalling the steady-state assumption that d[ES]/dt = 0, from Equations (14.14) and (14.16) we can write Assuming that E + I ⇌ EI reaches rapid equilibrium, the rate of EI formation, vf' = k3[E][I], and the rate of disappearance of EI, vd' = k-3[EI], are equal. So, Therefore,
If we define KI as k-3/ k3, an enzyme-inhibitor dissociation constant, then knowing [ET] = [E] + [ES] + [EI]. Then
|
Solving for [E] gives Because the rate of product formation is given by v = k2[ES], from Equation (14.34) we have So, Because Vmax = k2[ET], or |
Succinate Dehydrogenase—A Classic Example of Competitive Inhibition
The enzyme succinate dehydrogenase (SDH) is competitively inhibited by malonate. Figure 14.14 shows the structures of succinate and malonate. The structural similarity between them is obvious and is the basis of malonate’s ability to mimic succinate and bind at the active site of SDH. However, unlike succinate, which is oxidized by SDH to form fumarate, malonate cannot lose two hydrogens; consequently, it is unreactive.
Figure 14.14 • Structures of succinate, the substrate of succinate dehydrogenase (SDH), and malonate, the competitive inhibitor. Fumarate (the product of SDH action on succinate) is also shown.
Noncompetitive Inhibition
Noncompetitive inhibitors interact with both E and ES (or with S and ES, but this is a rare and specialized case). Obviously, then, the inhibitor is not binding to the same site as S, and the inhibition cannot be overcome by raising [S]. There are two types of noncompetitive inhibition: pure and mixed.
Pure Noncompetitive Inhibition
In this situation, the binding of I by E has no effect on the binding of S by E. That is, S and I bind at different sites on E, and binding of I does not affect binding of S. Consider the system
(14.44)
Pure noncompetitive inhibition occurs if KI = KI'. This situation is relatively uncommon; the Lineweaver-Burk plot for such an instance is given in Figure 14.15. Note that Km is unchanged by I (the x-intercept remains the same, with or without I). Note also that Vmax decreases. A similar pattern is seen if the amount of enzyme in the experiment is decreased. Thus, it is as if I lowered [E].
Figure 14.15 • Lineweaver-Burk plot of pure noncompetitive inhibition. Note that I does not alter Km but that it decreases Vmax. In the presence of I, the y-intercept is equal to (1/Vmax)(1 + I/KI).
Mixed Noncompetitive Inhibition
In this situation, the
binding of I by E influences the binding of S by E. Either the binding sites
for I and S are near one another or conformational changes in E caused by I
affect S binding. In this case, KI and KI', as defined
previously, are not equal. Both Km and Vmax are
altered by the presence of I, and Km / Vmax is
not constant (Figure 14.16).
Figure
14.16 • Lineweaver-Burk
plot of mixed noncompetitive inhibition. Note that both intercepts and the slope
change in the presence of
(b) when KI is greater than KI'.
This inhibitory pattern is commonly encountered. A reasonable explanation is that the inhibitor is binding at a site distinct from the active site, yet is influencing the binding of S at the active site. Presumably, these effects are transmitted via alterations in the protein’s conformation. Table 14.6 includes the rate equations and apparent Km and Vmax values for both types of noncompetitive inhibition.
| Table 14.6 | |||
| The
Effect of Various Types of Inhibitors on the Michaelis–Menten Rate Equation and on Apparent Km and Apparent Vmax |
|||
|
Inhibition Type |
Rate Equation | Apparent Km | Apparent Vmax |
| None | v = Vmax[S]/( Km + [S]) | Km | Vmax |
| Competitive | v = Vmax [S]/([S] + Km (1 + [I]/KI)) | Km (1 + [I]/ KI) | Vmax |
| Noncompetitive | v = (Vmax [S]/(1 + [I]/ KI))/( Km + [S]) | Km | Vmax /(1 + [I]/ KI) |
| Mixed | v = Vmax [S]/((1 +[I]/ KI) Km 1(1 + [I]/ KI' [S])) | Km (1 + [I]/ KI)/(1 + [I]/ KI') | Vmax /(1 + [I]/ KI') |
| KI is defined as the enzyme:inhibitor dissociation constant KI = [E][I]/[EI]; KI' is defined as the enzyme substrate complex : inhibitor dissociation constant KI' = [ES][I]/[ESI] | |||
Irreversible Inhibition
If the inhibitor combines irreversibly with the enzyme—for example, by covalent attachment—the kinetic pattern seen is like that of noncompetitive inhibition, because the net effect is a loss of active enzyme. Usually, this type of inhibition can be distinguished from the noncompetitive, reversible inhibition case since the reaction of I with E (and/or ES) is not instantaneous. Instead, there is a time-dependent decrease in enzymatic activity as E + I ® EI proceeds, and the rate of this inactivation can be followed. Also, unlike reversible inhibitions, dilution or dialysis of the enzyme: inhibitor solution does not dissociate the EI complex and restore enzyme activity.
Suicide Substrates—Mechanism-Based Enzyme Inactivators
Suicide substrates are inhibitory substrate analogs designed so that, via normal catalytic action of the enzyme, a very reactive group is generated. This reactive group then forms a covalent bond with a nearby functional group within the active site of the enzyme, thereby causing irreversible inhibition. Suicide substrates, also called Trojan horse substrates, are a type of affinity label. As substrate analogs, they bind with specificity and high affinity to the enzyme active site; in their reactive form, they become covalently bound to the enzyme. This covalent link effectively labels a particular functional group within the active site, identifying the group as a key player in the enzyme’s catalytic cycle.
Penicillin—A Suicide Substrate
Several drugs in current medical use are mechanism-based enzyme inactivators. For example, the antibiotic penicillin exerts its effects by covalently reacting with an essential serine residue in the active site of glycoprotein peptidase, an enzyme that acts to cross-link the peptidoglycan chains during synthesis of bacterial cell walls (Figure 14.17). Once cell wall synthesis is blocked, the bacterial cells are very susceptible to rupture by osmotic lysis, and bacterial growth is halted.
Figure 14.17 • Penicillin is an irreversible inhibitor of the enzyme glycoprotein peptidase, which catalyzes an essential step in bacterial cell wall synthesis. Penicillin consists of a thiazolidine ring fused to a b-lactam ring to which a variable group R is attached. A reactive peptide bond in the b-lactam ring covalently attaches to a serine residue in the active site of the glycopeptide transpeptidase. (The conformation of penicillin around its reactive peptide bond resembles the transition state of the normal glycoprotein peptidase substrate.) The penicilloyl-enzyme complex is catalytically inactive. The bond between the enzyme and penicillin is indefinitely stable; that is, penicillin binding is irreversible.
14.5 · Kinetics of Enzyme-Catalyzed Reactions
Involving Two or More Substrates
Thus far, we have considered only the simple case of enzymes that act upon a single substrate, S. This situation is not common. Usually, enzymes catalyze reactions in which two (or even more) substrates take part.
Consider the case of an enzyme catalyzing a reaction involving two substrates, A and B, and yielding the products P and Q:
(14.45)
Such a reaction is termed a bisubstrate reaction. In general, bisubstrate reactions proceed by one of two possible routes:
1. Both A and B are bound to the enzyme and then reaction occurs to give P1Q:
(14.46)
Reactions of this type are defined as sequential or single-displacement reactions. They can be either of two distinct classes:
a. random, where either A or B may bind to the enzyme first, followed by the other substrate, or
b. ordered, where A, designated the leading substrate, must bind to E first before B can be bound.
Both classes of single-displacement reactions are characterized by lines that intersect to the left of the 1/v axis in Lineweaver-Burk double-reciprocal plots (Figure 14.18).
Figure 14.18 • Single-displacement bisubstrate mechanism. Double-reciprocal plots of the rates observed with different fixed concentrations of one substrate (B here) are graphed versus a series of concentrations of A. Note that, in these Lineweaver-Burk plots for single-displacement bisubstrate mechanisms, the lines intersect to the left of the 1/v axis.
2. The other general possibility is that one substrate, A, binds to the enzyme and reacts with it to yield a chemically modified form of the enzyme (E') plus the product, P. The second substrate, B, then reacts with E',
group of donors
(14.34)
(14.35)
(14.36)
(14.37)
(14.38)
(14.39)
(14.40)
(14.41)
(14.42)
(14.43)
