CHAPTER 4 Biophysical Principles^{*}

**T**he concepts in this chapter form the basis for understanding all the molecular interactions in chemistry and biology. To illustrate some of these concepts with a practical example, the chapter concludes with a section on an exceptionally important family of enzymes that bind and hydrolyze the nucleotide GTP. This example provides the background knowledge to understand how GTPases participate in numerous processes covered in later chapters.

The extent of chemical reactions is characterized by the **equilibrium constant;** the rates of these reactions are described by **rate constants.** This chapter reviews the physical basis for rate constants and how they are related to the thermodynamic parameter, the equilibrium constant. These simple but powerful principles permit a deeper appreciation of molecular interactions in cells. On the basis of many examples presented in this book, it will become clear to the reader that rate constants are at least as important as equilibrium constants, since the rates of reactions govern the dynamics of the cell. The chapter includes discussion of the chemical bonds important in biochemistry. Box 4-1 lists key terms used in this chapter.

Reaction rates are expressed as follows:

At equilibrium, the forward rate equals the reverse rate:

and concentrations of reactants R_{eq} and products P_{eq} do not change with time.

# First-Order Reactions

First-order reactions have one reactant (R) and produce a product (P). The general case is simply

Some common examples of first-order reactions (Fig. 4-1) include conformational changes, such as a change in shape of protein A to shape A^{*}:

Figure 4-1 **first-order reactions.** In first-order reactions, a single reactant undergoes a change. In these examples, molecule A changes conformation to ^{*} and the bimolecular complex AB dissociates to A and B. The rate constant for a first-order reaction (arrows) is a simple probability.

and the dissociation of complexes, such as

A first-order rate constant can be viewed as a **probability** per unit of time. For a conformational change, it is the probability that any A will change to ^{*} in a unit of time. For dissociation of complex AB, the first-order rate constant is determined by the strength of the bonds holding the complex together. This “dissociation rate constant” can be viewed as the probability that the complex will fall apart in a unit of time. The probability of the conformational change of any particular A to ^{*} or of the dissociation of any particular AB is independent of its concentration. ** The concentra-tions of A and AB are important only in determining the rate of the reaction observed in a bulk sample** (Box 4-2).

BOX 4-2 Relationship of the Half-Time to a First-Order Rate Constant

To review, the rate of a first-order reaction is simply the product of a constant that is characteristic of the reaction and the concentration of the single reactant. The constant can be calculated from the half-time of a reaction (Box 4-2).

# Second-Order Reactions

Second-order reactions have two reactants (Fig. 4-2). The general case is

Figure 4-2 **second-order reactions**. In second-order reactions, two molecules must collide with each other. The rate of these collisions is determined by their concentrations and by a collision rate constant (arrows). The collision rate constant depends on the sum of the diffusion coefficients of the reactants and the size of their interaction sites. The rate of diffusion in a given medium depends on the size and shape of the molecule. Large molecules, such as proteins, move more slowly than small molecules, such as adenosine triphosphate (ATP). A protein with a diffusion coefficient of 10^{−11} m^{2} s^{−1} diffuses about 10 mm in a second in water, while a small molecule such as ATP diffuses 100 times faster. The rate constants (arrows) are about the same for A + B and C + ** D** because the large diffusion coefficient of

**offsets the small size of its interaction site on C. Despite the small interaction size,**

*D***+**

*D***is faster because both reactants diffuse rapidly.**

*D*A common example in biology is a bimolecular association reaction, such as

the same as a first-order reaction.

The value of a second-order “association” rate constant, *k*_{+}, is determined mainly by the rate at which the molecules collide. This collision rate depends on the rate of diffusion of the molecules (Fig. 4-2), which is determined by the size and shape of the molecule, the viscosity of the medium, and the temperature. These factors are summarized in a parameter called the **diffusion coefficient, D**, with units of m

^{2}s

^{−1}.

**is a measure of how fast a molecule moves in a given medium. The rate constant for collisions is described by the Debye-Smoluchowski equation, a relationship that depends only on the diffusion coefficients and the area of interaction between the molecules:**

*D*