Introduction to Molecular Biology: From Cells to Atoms - Anthony Rees, Michael Sternberg 2002

Cells and Molecules at Work
Some Definitions and Physical Laws

Molecular weights and sedimentation coefficients of certain macromolecules

Class="center">Table 44.1

¹⁾ The sedimentation coefficient of molecules or particles depends on Temperature, as well as on the viscosity and density of the solvent. Therefore, it is more convenient to compare sedimentation coefficients normalized to identical conditions (i.e., measured under these conditions or recalculated). Usually, s is referenced to 20 °C and the viscosity and density of Water: s20w. — Ed. note.

The sedimentation coefficient s is a measure of the size and shape of a macromolecule; it is determined by placing macromolecules in a very strong gravitational field generated by an ultracentrifuge. The sedimentation coefficient is generally expressed in Svedberg units S:

IS = 10^-13 s.

To determine molecular weights in this manner, one of two Methods is typically used: the Sedimentation Velocity Method or equilibrium centrifugation. Let us outline the Procedure for finding molecular weight using the sedimentation velocity method. Within this approach, The rate of sedimentation of a macromolecule in a given solution under known gravitational conditions is measured. Next, the following relation is used

М_t = RTs/D (1 - vp),

where М_t is the relative molecular weight,

R is the universal gas constant,

Т is the absolute temperature,

D is the diffusion coefficient,

v is the partial specific volume of the macromolecule,

р is the density of the solvent,

s is the measured value of the sedimentation coefficient.

For macromolecules with s = 7,865S, D = 4,75 ∙ 10^-11 m² ∙ s^-1, v = 0,729 cm³ ∙ g^-1 at р = 0,9982 kg ∙ dm^-3, Т = 293 K, and taking into account that R = 8,31 J ∙ K^-1 ∙ mol^-1, we obtain

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Turning to Table 44.1, we see, however, that the measured value of s does not always yield the correct molecular weight; we must not forget that the molecular shape is also an important factor influencing the value of s. Several useful thermodynamic relations. Here, some of the concepts introduced in Chapter 7 are examined in greater detail.

First law of Thermodynamics. If a system (consisting of certain material objects and characterized by a definite pressure, volume, and temperature) is isolated from its environment, any change (such as a chemical reaction) occurring within this system can only lead to a redistribution of energy among its various parts, while the total energy of the system remains constant regardless of the type of process taking place. This is precisely the 1st law of thermodynamics, or the law of conservation of energy. However, if the system interacts with its environment, the energy of the system (internal energy U) changes such that

For most Chemical Reactions, the work performed

∆U = ∆q - ∆w,   (1)

where ∆U = U (at the end of the process) — U (before THE START OF the process),

∆q is the heat absorbed by the system from the environment,

∆w is the work done by the system on the environment (e.g., mechanical work performed by Muscles, or chemical work performed during The Biosynthesis of macromolecules).

is simply the energy that must be expended on expansion against atmospheric pressure (for instance, when a reaction results in a small volume change). If the system is maintained at a constant pressure (P), equation (1) can be rewritten as follows:

∆H = ∆U + P∆V,   (2)

where ∆H is the enthalpy change (which equals ∆q at constant pressure), and ∆U was defined above.

and P∆V is the work done during a volume change. Thus, ∆H can be viewed as The change in the internal energy of the system plus an additional term accounting for the work done by the system on its surroundings.

The Second Law of Thermodynamics. The First Law tells us nothing about the direction in which changes in a reacting system must occur. For example, when a bullet is fired from a gun and embeds itself in a piece of wood, we can describe the resulting energy changes in terms of

✵ the kinetic energy of the bullet,

✵ the energy expended in overcoming air resistance,

✵ the thermal energy released as the bullet decelerates in the wood.

If we were now to transfer all this total energy back to a stationary bullet in the form of heat, the First Law of Thermodynamics suggests that the bullet might absorb this energy and become extremely hot, or that the thermal energy might convert into kinetic energy, giving the bullet a very high velocity. Our experience tells us that the latter is unlikely to happen.

Therefore, it is desirable to have a criterion that helps predict the probable direction of a reaction or process. Such a criterion is provided by the Second Law of Thermodynamics, which states (in one of its many formulations) that

✵ a reaction will most likely proceed in the direction that is accompanied by an increase in the disorder of the system.

The measure of this disorder is a quantity called Entropy (S); accordingly, ∆S represents the change in the degree of randomness. For a spontaneous reaction, typically ∆S > 0. Thus (returning to our bullet example), the thermal energy acquired by the bullet simply manifests as an increase in the disorder of the metal atoms through an increase in the amplitude of their already chaotic motions. For the bullet to fire itself without a gun, all of its atoms would have to move simultaneously in the exact same direction after receiving the heat—an extremely improbable event, as it involves a massive increase in the system's order. Combining the First and Second Laws, we can formulate a condition which states roughly that if, for a given reaction, the difference between the enthalpy increment (∆H) and the entropy increment (in the same energy units, i.e., ∆S) is less than zero—that is, if

∆H - Т∆S< 0,

then this reaction will proceed spontaneously. The same condition can be formulated somewhat differently by introducing another quantity called Free energy (G) and using the equation

∆G = ∆Н - Т∆S.   (3)

Then, if ∆G < 0, the reaction will proceed spontaneously (the rate of this spontaneous process can only be estimated based on kinetic considerations); the condition ∆G = 0 means that the reaction has reached thermodynamic equilibrium; whereas if ∆G > 0, the reaction will not proceed in that direction.

The Standard Free Energy change (∆G^o) is the ∆G of a reaction before it begins, when all reactant substances are in their Standard States—that is, at standard temperature, pressure, and concentrations, the latter being equal and fixed at 1 mol ∙ L^-1. ∆G^o and the Equilibrium Constant of a reaction are related as follows. For a reaction of the type A ⇄ B, whose equilibrium constant is K' = [B]/[A] (where [A] and [B] are the equilibrium concentrations of substances A and B, respectively),

∆G° = -RTln К'.   (4)

When the same system is not at equilibrium, ∆G is determined by the equation

∆G = ∆G° + RTln [В]/[А]. (5)

Here, the concentrations of substances A and B are no longer equilibrium concentrations. Once they reach their equilibrium values, ∆G will become zero, and equation (5) will reduce to equation (4).

Standard Free energy of a biochemical reaction. The value of ∆G^o for any reaction corresponds to conditions where all reacting substances are in their standard states; therefore, if H⁺ ions participate in the reaction, their concentration must also be equal to 1 mol ∙ L^-1. Consequently, [H⁺] = 1 = 10⁰, meaning pH = 0. Such a pH value can hardly be considered physiological, since most biochemical reactions take place at or near pH = 7. Therefore, it is more convenient to use a modified form of ∆G°, denoted as ∆G°', in which all reactants except H⁺ ions remain in their standard states, while the concentration of H⁺ ions is set to 10^-7 mol ∙ L^-1, corresponding to pH = 7. Thus,

∆G°' = ∆G° + 2,303 RTlog[H⁺] =

= ∆G° - 2,303 RTрН =

= ∆G° - 2,303 RT ∙ 7.