The equilibrium constant describes the equilibrium of the reaction

In theory, all chemical reactions are reversible. The reaction equilibrium can be determined experimentally by mixing substrates (or products) with a suitable catalyst and allowing the reaction to proceed to completion. At this point, the concentrations of substrates and products can be measured to determine the equilibrium constant K_equ, which is defined as the ratio of product concentration to substrate concentration at equilibrium. For a simple reaction

[B] and [A] are the molar concentrations of product B and substrate A at equilibrium.

When more than one substrate or product participate, their concentrations have to be multiplied. For the reaction

The alcohol dehydrogenase (ADH) reaction provides an example:

   (1)

NAD⁺ (nicotinamide adenine dinucleotide) is a coenzyme that accepts hydrogen in this reaction (see Chapter 5). The equilibrium constant of the reaction is

   (2)

From Equation (2) we can calculate the relative concentrations of acetaldehyde and ethanol at equilibrium when [NADH] = [NAD⁺] and pH = 7.0:

   (3)

There is 10,000 times more ethanol than acetaldehyde at equilibrium!

Under aerobic conditions, however, NAD⁺ is far more abundant than NADH in the cell. When [NAD⁺] is 1000 times higher than [NADH], Equation (3) assumes the numerical values of

The pH also is important. At pH of 8.0 and [NAD⁺]/[NADH] ratio of 1000, for example, Equation (3) yields

This example shows that a reaction can be driven toward product formation by raising the concentration of a substrate or lowering the concentration of a product.

To adapt the equilibrium constant to physiological conditions, a “biological equilibrium constant,” K′_equ, is used. In the definition of K′_equ, a value of 1.0 is assigned to the water concentration if water participates in the reaction, and a value of 1.0 to a proton concentration of 10⁻⁷ mol/L (pH = 7.0) if protons participate in the reaction. The K′_equ of the alcohol dehydrogenase reaction, for example, is not 10⁻¹¹ mol/L but 10⁻⁴ mol/L. At a pH of 8.0 in the preceding example, the proton concentration would be given a numerical value of 10⁻¹.

The free energy change is the driving force for chemical reactions

During chemical reactions, energy is either released or absorbed. This is described as the enthalpy change ΔH:

(4)

ΔE is the heat that is released or absorbed. It is measured in either kilocalories per mole (kcal/mol) or kilojoules per mole (kJ/mol, where 1 kcal = 4.184 kJ). By convention, a negative sign of ΔE means that heat is released; a positive sign indicates that heat is absorbed. P is the pressure, and ΔV is the volume change. P × ΔV is the work done by the system. It can be substantial in a car motor when the volume in the cylinder expands against the pressure of the piston, but volume changes are negligible in the human body. Therefore ΔH ≈ ΔE. The enthalpy change describes the difference in the total chemical bond energies between the substrates and products.

Reactions are driven not only by ΔH but by the entropy change (ΔS) as well. Entropy is a measure of the randomness or disorderliness of the system. A cluttered desk is often cited as an example of a high-entropy system. A positive ΔS means that the system becomes more disordered during the reaction. Entropy change and enthalpy change are combined in the free energy change ΔG:

(5)

where T = absolute temperature measured in Kelvin.

ΔG is the driving force of the reaction. Like ΔE and ΔH in Equation (4), it is measured in kilocalories per mole (kcal/mol) or kilojoules per mole (kJ/mol). A negative sign of ΔG defines an exergonic reaction. It can proceed only in the forward direction. A positive sign of ΔG signifies an endergonic reaction. It can proceed only in the backward direction. At equilibrium, ΔG equals zero.

Equation (5) shows that a reaction can be driven by either a decrease in the chemical bond energies of the reactants (negative ΔH) or an increase in their randomness (positive T × ΔS). Low energy content and high randomness are the preferred states. Like most students, Nature tends to slip from energized order into energy-depleted chaos.

Entropy changes are small in most biochemical reactions, but diffusion is an entropy-driven process (Fig. 4.2, A). There is no making and breaking of chemical bonds during diffusion. Therefore the enthalpy change ΔH is zero. This leaves the T × ΔS part of Equation (5) as the only driving force. Thus diffusion can produce only a random distribution of the dissolved molecules.

Figure 4.2 Diffusion as an entropy-driven process. A, In a hypothetical two-compartment system, molecules diffuse until their concentrations are equal. This is the state of maximal entropy. B, The living cell maintains a gradient of sodium ions across its plasma membrane. The cell can maintain this gradient, which represents a low-entropy state, only by “pumping” sodium out of the cell. The pump is fueled by the chemical bond energy in adenosine triphosphate (ATP). ADP, Adenosine diphosphate; P_i, inorganic phosphate. C, The dead cell lacks ATP; therefore, it cannot maintain its sodium gradient. A high-entropy state develops spontaneously, with intracellular [Na⁺] = extracellular [Na⁺].

The human body is a very orderly system. To maintain this improbable and therefore thermodynamically disfavored state of affairs, biochemical reactions must antagonize the spontaneous increase in entropy.Equation (5) shows that a reaction can reduce entropy (negative T × ΔS) only when it consumes chemical bond energy (negative ΔH). In other words, the human body must consume chemical bond energy to maintain its low-entropy state.

In the example of Figure 4.2, B and C, the cell maintains a sodium gradient across the membrane by pumping sodium out of the cell (negative T × ΔS). Sodium pumping is driven by the hydrolysis of adenosine triphosphate (ATP) to adenosine diphosphate (ADP) and inorganic phosphate (P_i) (negative ΔH). Without ATP the gradient dissipates, the entropy of the system increases, and the cell dies. That is what death and dying are all about: a sharp rise in the entropy of the body.

The standard free energy change determines the equilibrium

The free energy change ΔG is affected by the relative reactant concentrations. It is not a property of the reaction as such. To describe the energy balance of a reaction, the standard free energy change, ΔG⁰′, must be defined: ΔG⁰′ is the free energy change under standard conditions. Standard conditions are defined by a concentration of 1 mol/L for all reactants (except protons and water) at a pH of 7.0. As in the definition of K′_equ, values of 1 are assigned both to the water concentration and to the proton concentration at pH 7.

For the reaction

the standard free energy change ΔG⁰′ is related to the real free energy change ΔG by Equation (6):

   (6)

where R = gas constant, and T = absolute temperature measured in Kelvin. The numerical value of R is 1.987 × 10⁻³ kcal × mol⁻¹ × K⁻¹. At a “standard temperature” of 25°C (298K), Equation (6) assumes the form of

   (7)

At equilibrium, ΔG = 0, and Equation (7) therefore yields

   (8)

The reactant concentrations under the logarithm are now the equilibrium concentrations. Their ratio defines the biological equilibrium constant K′_equ

   (9)

Substituting Equation (9) into Equation (8) yields

(10)

There is a negative logarithmic relationship between ΔG⁰′ and the equilibrium constant K′_equ (Table 4.1). When ΔG⁰′ is negative, product concentrations are higher than substrate concentrations at equilibrium; when it is positive, substrate concentrations are higher.

Table 4.1 Relationship between the Equilibrium Constant K′_equ and the Standard Free Energy Change ΔG⁰′

Enzymes are both powerful and selective

There is no compelling reason why only proteins should catalyze reactions, and catalytic ribonucleic acids (RNAs) are known to exist. By and large, however, almost all enzymes are globular proteins.

Enzymes can accelerate a chemical reaction enormously. Many reactions that proceed within minutes in the presence of an enzyme would require thousands of years to reach their equilibrium in the absence of a catalyst. The turnover number describes the catalytic power of the enzyme. It is defined as the maximal number of substrate molecules converted to product by one enzyme molecule per second.Table 4.2 lists the turnover numbers of some enzymes.

Table 4.2 Approximate Turnover Numbers of Some Enzymes

* Turnover numbers are measured at saturating substrate concentrations. They depend on the assay conditions, including temperature and pH.

Another key property of enzymes is their substrate specificity. Typically, each reaction requires its own enzyme. For example, when an enzyme is inhibited by a drug or is deficient because of a genetic defect, only one reaction is blocked.

The substrate must bind to its enzyme before the reaction can proceed

Enzymatic catalysis, like sex, requires intimate physical contact. It starts with the formation of an enzyme-substrate complex:

In the enzyme-substrate complex, the substrate is bound noncovalently to the active site on the surface of the enzyme protein. The active site contains the functional groups for substrate binding and catalysis. If a prosthetic group participates in the reaction as a coenzyme, it is present in the active site.

According to the lock-and-key model, substrate and active site bind each other because their surfaces are complementary. In many cases, however, substrate binding induces a conformational change in the active site that leads to further enzyme-substrate interactions and brings catalytically active groups to the substrate. This is called induced fit (Fig. 4.3).

Figure 4.3 Two models of enzyme-substrate binding. A, Lock-and-key model. B, Induced-fit model. S, Substrate.

The enzyme’s substrate specificity is determined by the geometry of enzyme-substrate binding. If the substrate is optically active, generally only one of the isomers is admitted. This is to be expected because the enzyme, being formed from optically active amino acids, is optically active itself. A three-point attachment (shown schematically in Fig. 4.4) is the minimal requirement for stereoselectivity.

Figure 4.4 Three-point attachment is the minimal requirement for stereoselectivity. In this hypothetical example, the enzyme-substrate complex is formed by a salt bond, a hydrogen bond, and a hydrophobic interaction. The substrate binds, whereas its enantiomer (bottom), is not able to form an enzyme-substrate complex.

Rate constants are useful for describing reaction rates

The rate (velocity) of a chemical reaction can be described by a rate constant k:

In this one-substrate reaction, the reaction rate is defined by

(11)

The rate constant has the dimension s⁻¹ (per second), and the velocity V is the change in substrate concentration per second.

For a reversible reaction, the forward and backward reactions must be considered separately:

   (12)

(13)

At equilibrium, V_forward = V_backward. Therefore the net reaction is zero:

(14)

   (15)

Equation (15) shows that the equilibrium constant K_equ, previously defined as [B]/[A] at equilibrium, is also the ratio of the two rate constants.

The forward reaction in Equation (12) is a first-order reaction. In a first-order reaction, the reaction rate is directly proportional to the substrate concentration. When the substrate concentration [A] is doubled, the reaction rate V is doubled as well. Uncatalyzed one-substrate reactions follow first-order kinetics. A classic example is the decay of a radioactive isotope.

When two substrates participate, the reaction rate is likely to depend on the concentrations of both substrates. This is called a second-order reaction. For

the following is obtained:

(16)

Doubling the concentration of one substrate doubles the reaction rate; doubling the concentrations of both raises it fourfold.

A zero-order reaction is independent of the substrate concentration. No matter how many substrate molecules are present in the test tube, only a fixed number is converted to product per second:

(17)

Zero-order kinetics are observed only in catalyzed reactions when the substrate concentration is high, and the amount and turnover number of the catalyst, rather than the substrate availability, is the limiting factor.

Enzymes decrease the free energy of activation

Reactions with a negative ΔG can

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