Berg J.M., Tymoczko J.L., Stryer L. Biochemistry

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Substituting R = 1.987 × 10

-3

kcal mol

-1

deg

-1

and T = 298 K (corresponding to 25°C) gives

where ∆ G°

is here expressed in kilocalories per mole because of the choice of the units for R in equation 7. Thus, the

standard free energy and the equilibrium constant of a reaction are related by a simple expression. For example, an

equilibrium constant of 10 gives a standard free-energy change of -1.36 kcal mol

-1

(-5.69 kJ mol

-1

) at 25°C (Table 8.4).

Note that, for each 10-fold change in the equilibrium constant, the ∆ G°

changes by 1.36 kcal mol

-1

(5.69 kJ mol

-1

As an example, let us calculate ∆ G°

and ∆ G for the isomerization of dihydroxyacetone phosphate (DHAP) to

glyceraldehyde 3-phosphate (GAP). This reaction takes place in glycolysis (Section 16.1.4). At equilibrium, the ratio of

GAP to DHAP is 0.0475 at 25°C (298 K) and pH 7. Hence, K

= 0.0475. The standard free-energy change for this

reaction is then calculated from equation 6:

Under these conditions, the reaction is endergonic. DHAP will not spontaneously convert to GAP.

Now let us calculate ∆ G for this reaction when the initial concentration of DHAP is 2 × 10

-4

M and the initial

concentration of GAP is 3 × 10

-6

M. Substituting these values into equation 1 gives

This negative value for the ∆ G indicates that the isomerization of DHAP to GAP is exergonic and can occur

spontaneously when these species are present at the aforestated concentrations. Note that ∆ G for this reaction is

negative, although ∆ G

is positive. It is important to stress that whether the ∆ G for a reaction is larger, smaller, or the

same as ∆ G

depends on the concentrations of the reactants and products. The criterion of spontaneity for a reaction is

∆ G, not ∆ G

. This point is important because reactions that are not spontaneous based on ∆ G

can be made

spontaneous by adjusting the concentrations of reactants and products. This principle is the basis of the coupling of

reactions to form metabolic pathways (Chapter 14).

8.2.3. Enzymes Alter Only the Reaction Rate and Not the Reaction Equilibrium

Because enzymes are such superb catalysts, it is tempting to ascribe to them powers that they do not have. An enzyme

cannot alter the laws of thermodynamics and consequently cannot alter the equilibrium of a chemical reaction. This

inability means that an enzyme accelerates the forward and reverse reactions by precisely the same factor. Consider the

interconversion of A and B. Suppose that, in the absence of enzyme, the forward rate constant (k

) is 10

-4

-1

and the

reverse rate constant (k

) is 10

-6

-1

. The equilibrium constant K is given by the ratio of these rate constants:

The equilibrium concentration of B is 100 times that of A, whether or not enzyme is present. However, it might take

considerable time to approach this equilibrium without enzyme, whereas equilibrium would be attained rapidly in the

presence of a suitable enzyme. Enzymes accelerate the attainment of equilibria but do not shift their positions. The

equilibrium position is a function only of the free-energy difference between reactants and products.

I. The Molecular Design of Life 8. Enzymes: Basic Concepts and Kinetics 8.2. Free Energy Is a Useful Thermodynamic Function for Understanding Enzymes

Table 8.4. Relation between ∆ G° and K

(at 25°C)

DG°

kcal mol

kJ/mol

6.82 28.53

5.46 22.84

4.09 17.11

2.73 11.42

1.36 5.69

1 0 0

10 -1.36 -5.69

-2.73 -11.42

-4.09 -17.11

-5.46 -22.84

-6.82 -28.53

I. The Molecular Design of Life 8. Enzymes: Basic Concepts and Kinetics

8.3. Enzymes Accelerate Reactions by Facilitating the Formation of the Transition

State

The free-energy difference between reactants and products accounts for the equilibrium of the reaction, but enzymes

accelerate how quickly this equilibrium is attained. How can we explain the rate enhancement in terms of

thermodynamics? To do so, we have to consider not the end points of the reaction but the chemical pathway between the

end points.

A chemical reaction of substrate S to form product P goes through a transition state S

that has a higher free energy than

does either S or P. The double dagger denotes a thermodynamic property of the transition state. The transition state is the

most seldom occupied species along the reaction pathway because it is the one with the highest free energy. The

difference in free energy between the transition state and the substrate is called the Gibbs free energy of activation or

simply the activation energy, symbolized by ∆ G

, as mentioned in Section 8.2.1 (Figure 8.3).

Note that the energy of activation, or ∆ G

, does not enter into the final ∆ G calculation for the reaction, because the

energy input required to reach the transition state is returned when the transition state forms the product. The activation-

energy barrier immediately suggests how enzymes enhance reaction rate without altering ∆ G of the reaction: enzymes

function to lower the activation energy, or, in other words, enzymes facilitate the formation of the transition state.

One approach to understanding how enzymes achieve this facilitation is to assume that the transition state (S

) and the

substrate (S) are in equilibrium.

in which K

is the equilibrium constant for the formation of S

, and v is the rate of formation of product from S

The rate of the reaction is proportional to the concentration of S

because only S

can be converted into product. The concentration of S

is in turn related to the free energy difference

between S

and S, because these two chemical species are assumed to be in equilibrium; the greater the difference

between these two states, the smaller the amount of S

Because the reaction rate is proportional to the concentration of S

, and the concentration of S

depends on ∆ G

, the

rate of reaction V depends on ∆ G

. Specifically,

In this equation, k is Boltzmann's constant, and h is Planck's constant. The value of kT/h at 25°C is 6.2 × 10

-1

Suppose that the free energy of activation is 6.82 kcal mol

-1

(28.53 kJ mol

-1

). The ratio [S

]/[S] is then 10

-5

(see Table

8.4). If we assume for simplicity's sake that [S] = 1 M, then the reaction rate V is 6.2 × 10

-1

. If ∆ G

were lowered by

1.36 kcal mol

-1

(5.69 kJ mol

-1

), the ratio [S

]/[S] is then 10

-4

, and the reaction rate would be 6.2 × 10

-1

. As Table 8.4

shows, a decrease of 1.36 kcal mol

-1

in ∆ G

yields a tenfold larger V. A relatively small decrease in ∆ G

(20% in this

particular reaction) results in a much greater increase in V.

"I think that enzymes are molecules that are complementary in

structure to the activated complexes of the reactions that they

catalyze, that is, to the molecular configuration that is intermediate

between the reacting substances and the products of reaction for

these catalyzed processes. The attraction of the enzyme molecule for

the activated complex would thus lead to a decrease in its energy and

hence to a decrease in the energy of activation of the reaction and to

an increase in the rate of reaction."

- Linus Pauling

Nature 161(1948):707

Thus, we see the key to how enzymes operate: Enzymes accelerate reactions by decreasing ∆ G

, the activation energy.

The combination of substrate and enzyme creates a new reaction pathway whose transition-state energy is lower than

that of the reaction in the absence of enzyme (see Figure 8.3). The lower activation energy means that more molecules

have the required energy to reach the transition state. Decreasing the activation barrier is analogous to lowering the

height of a high-jump bar; more athletes will be able to clear the bar. The essence of catalysis is specific binding of the

transition state.

8.3.1. The Formation of an Enzyme-Substrate Complex Is the First Step in Enzymatic

Catalysis

Much of the catalytic power of enzymes comes from their bringing substrates together in favorable orientations to

promote the formation of the transition states in enzyme-substrate (ES) complexes. The substrates are bound to a specific

region of the enzyme called the active site. Most enzymes are highly selective in the substrates that they bind. Indeed, the

catalytic specificity of enzymes depends in part on the specificity of binding.

What is the evidence for the existence of an enzyme-substrate complex?

1. The first clue was the observation that, at a constant concentration of enzyme, the reaction rate increases with

increasing substrate concentration until a maximal velocity is reached (Figure 8.4). In contrast, uncatalyzed reactions do

not show this saturation effect. The fact that an enzyme-catalyzed reaction has a maximal velocity suggests the formation

of a discrete ES complex. At a sufficiently high substrate concentration, all the catalytic sites are filled and so the

reaction rate cannot increase. Although indirect, this is the most general evidence for the existence of ES complexes.

2. X-ray crystallography has provided high-resolution images of substrates and substrate analogs bound to the active

sites of many enzymes (Figure 8.5). In Chapter 9, we will take a close look at several of these complexes. X-ray studies

carried out at low temperatures (to slow reactions down) are providing revealing views of enzyme-substrate complexes

and their subsequent reactions. A new technique, time-resolved crystallography, depends on cocrystallizing a photolabile

substrate analog with the enzyme. The substrate analog can be converted to substrate light, and images of the enzyme-

substrate complex are obtained in a fraction of a second by scanning the crystal with intense, polychromatic x-rays from

a synchrotron.

3. The spectroscopic characteristics of many enzymes and substrates change on formation of an ES complex. These

changes are particularly striking if the enzyme contains a colored prosthetic group. Tryptophan synthetase, a bacterial

enzyme that contains a pyridoxal phosphate (PLP) prosthetic group, provides a nice illustration. This enzyme catalyzes

the synthesis of l-tryptophan from l-serine and indole-derivative. The addition of l-serine to the enzyme produces a

marked increase in the fluorescence of the PLP group (Figure 8.6). The subsequent addition of indole, the second

substrate, reduces this fluorescence to a level even lower than that of the enzyme alone. Thus, fluorescence spectroscopy

reveals the existence of an enzyme-serine complex and of an enzyme-serine-indole complex. Other spectroscopic

techniques, such as nuclear magnetic resonance and electron spin resonance, also are highly informative about ES

interactions.

8.3.2. The Active Sites of Enzymes Have Some Common Features

The active site of an enzyme is the region that binds the substrates (and the cofactor, if any). It also contains the residues

that directly participate in the making and breaking of bonds. These residues are called the catalytic groups. In essence,

the interaction of the enzyme and substrate at the active site promotes the formation of the transition state. The active

site is the region of the enzyme that most directly lowers the ∆ G

of the reaction, which results in the rate enhancement

characteristic of enzyme action. Although enzymes differ widely in structure, specificity, and mode of catalysis, a

number of generalizations concerning their active sites can be stated:

1. The active site is a three-dimensional cleft formed by groups that come from different parts of the amino acid

sequence indeed, residues far apart in the sequence may interact more strongly than adjacent residues in the amino

acid sequence. In lysozyme, an enzyme that degrades the cell walls of some bacteria, the important groups in the active

site are contributed by residues numbered 35, 52, 62, 63, 101, and 108 in the sequence of the 129 amino acids (Figure

8.7).

2. The active site takes up a relatively small part of the total volume of an enzyme. Most of the amino acid residues in an

enzyme are not in contact with the substrate, which raises the intriguing question of why enzymes are so big. Nearly all

enzymes are made up of more than 100 amino acid residues, which gives them a mass greater than 10 kd and a diameter

of more than 25 Å. The "extra" amino acids serve as a scaffold to create the three-dimensional active site from amino

acids that are far apart in the primary structure. Amino acids near to one another in the primary structure are often

sterically constrained from adopting the structural relations necessary to form the active site. In many proteins, the

remaining amino acids also constitute regulatory sites, sites of interaction with other proteins, or channels to bring the

substrates to the active sites.

3. Active sites are clefts or crevices. In all enzymes of known structure, substrate molecules are bound to a cleft or

crevice. Water is usually excluded unless it is a reactant. The nonpolar character of much of the cleft enhances the

binding of substrate as well as catalysis. Nevertheless, the cleft may also contain polar residues. In the nonpolar

microenvironment of the active site, certain of these polar residues acquire special properties essential for substrate

binding or catalysis. The internal positions of these polar residues are biologically crucial exceptions to the general rule

that polar residues are exposed to water.

4. Substrates are bound to enzymes by multiple weak attractions. ES complexes usually have equilibrium constants that

range from 10

-2

to 10

-8

M, corresponding to free energies of interaction ranging from about -3 to -12 kcal mol

-1

(from -

13 to -50 kJ mol

-1

). The noncovalent interactions in ES complexes are much weaker than covalent bonds, which have

energies between -50 and -110 kcal mol

-1

(between -210 and -460 kJ mol

-1

). As discussed in Chapter 1 (Section 1.3.1),

electrostatic interactions, hydrogen bonds, van der Waals forces, and hydrophobic interactions mediate reversible

interactions of biomolecules. Van der Waals forces become significant in binding only when numerous substrate atoms

simultaneously come close to many enzyme atoms. Hence, the enzyme and substrate should have complementary

shapes. The directional character of hydrogen bonds between enzyme and substrate often enforces a high degree of

specificity, as seen in the RNA-degrading enzyme ribonuclease (Figure 8.8).

5. The specificity of binding depends on the precisely defined arrangement of atoms in an active site. Because the

enzyme and the substrate interact by means of short-range forces that require close contact, a substrate must have a

matching shape to fit into the site. Emil Fischer's analogy of the lock and key (Figure 8.9), expressed in 1890, has proved

to be highly stimulating and fruitful. However, we now know that enzymes are flexible and that the shapes of the active

sites can be markedly modified by the binding of substrate, as was postulated by Daniel E. Koshland, Jr., in 1958. The

active sites of some enzymes assume a shape that is complementary to that of the transition state only after the substrate

is bound. This process of dynamic recognition is called induced fit (Figure 8.10).