Biology 403/504, Second Lecture
Thursday 24 January 2008

Biochemical Thermodynamics

Outline

Why we care about thermodynamics in biochemistry

    Much of what we will study in this course involves reaction pathways—the conversion of one compound to another and another, onward toward some final product that gets used in a structural or functional way in an organism. In order to understand whether the reactions that produce the intermediate and final products will proceed, we need to know whether the reactions give off free energy or require free energy under physiological conditions. If they give off free energy, they will proceed without prompting; if they require energy, we will need to understand where the energy to drive the reaction comes from. This is the stuff of thermodynamics.

    We observed on Tuesday that thermodynamics alone will not tell us whether a reaction will proceed in a reasonable time frame. If the activation energy that separates reactants from products is high enough, the time required for a system to come to equilibrium will be long enough that the reaction will not play out within the lifetime of an organism. It is the job of biological catalysts—enzymes—to reduce this activation energy enough to make the kinetics of a reaction practical. Kinetics involves an understanding of energy, just as thermodynamics does. Thus the application of energetic considerations in biochemistry involves more than thermodynamics—it involves kinetics, and the ways that enzymes modify kinetics. But for today we'll focus on the energetics of equilibrium, i.e. thermodynamics.

The laws of thermodynamics

    There are four fundamental laws of thermodynamics, which for historical reasons are known as the zeroth, first, second, and third laws. The ones of immediate relevance to biochemistry are the first and second laws. These can be articulated in a variety of ways, but for our purposes:
    These laws presuppose some understanding of what energy and entropy are. The definition of energy is something you have encountered in some detail in other courses, but for this purpose we will think of it as the capacity to perform work. Entropy can be defined as the amount of disorder in a system, such that a system that is free to assume many states has more entropy than a system that is restricted to a smaller number of states.
    Note the emphasis on the notion of a "closed system" in the thermodynamic laws. An organism is not a closed system: it interacts with its surroundings. Therefore the energy associated with a cell or a larger biological entity may increase, provided that the energy comes from the environment. Ultimately the source for biological energy is the sun. Similarly, the entropy of a cell or organism may decrease, provided that, at the same time, the entropy of the surroundings increases by a larger amount than the decrease in the entropy of the biological entity. We can think of biology as the study of entities that decrease their local entropy while leaving behind a trail of increased entropy.

Enthalpy

    Enthalpy H is a concept closely related to total energy E:
    H = E + PV
where P and V are pressure and volume respectively. In biochemical systems the volume within which reactions occur is usually (but not always) constant, and the pressure within aqueous solutions is locally fixed. There are circumstances where pressure varies in biological systems. Certainly pressure varies while gases are passing in and out of permeable membranes like those of the human lung, and within a 100m long strand of aquatic seaweed there will be an appreciable difference in pressure between the top and the bottom. But in most biochemical systems both pressure and volume are effectively constant, so the difference between entropy and energy is insignificant.

Thermodynamic properties

    Energy and enthalpy are examples of extensive properties, i.e. their values are proportional to the amount of material under consideration. Thus if N molecules have, in aggregate, an energy E and an enthalpy H, then 2N molecules with the same properties will have energy 2E and enthalpy 2H. By contrast, temperature and pressure are not proportional to the number of molecules present, and as such are described as intensive properties.
    In practice we tend to work with an intensive version of energy, enthalpy, and similar properties, by measuring, for example, the energy per molecule or the energy per mole of a substance. Thus we will characterize the enthalpy of a substance in units of kJ/mole, i.e. the number of kilojoules of enthalpy per mole of substance. This has the obvious advantage of being an inherent property of the molecular species under consideration, rather than something we have to measure separately for every bundle of that substance.
    Energy, enthalpy, and entropy are state variables; that is, they do not depend on how a system was created. The path in getting from one state to another does not change the ending values of these properties, whereas other properties (like work and heat) do depend on the path. Work and heat, then, are not state variables.

Units

Recall from physics courses that the Joule is the MKS unit of energy, and is 1 kg-m2/s2. Two convenient units in biochemistry are the kilojoule/mole (kJ/mol) and the kilocalorie/mole (kcal/mol). A kilojoule is 103 Joules. A kilocalorie is the amount of energy required to increase the temperature of one kg of water at 4 deg C (277.16 K) by one degree K. This turns out to be 4.184 kJ, so 1 kcal/mol = 4.184 kJ/mol = 4184 J/mol.

In almost any thermodynamic discussion, the appropriate unit of temperature is the Kelvin, which used to be called a degree Kelvin. The abbreviation for a Kelvin is K. Occasionally I will lapse into the old-fashioned nomenclature and mention "deg-K" or "degree Kelvin" when I mean "Kelvin." You're free to jeer at me if I do that. I may say something about a "degree," by which I mean a Kelvin. This isn't actually wrong, so don't jeer if I do that. Recognize that the size of the Kelvin is the same as that of the degree Celsius; the only difference is the starting point of the scale. Zero Kelvin is the theoretical minimum of the temperature scale. Zero Celsius is defined as 273.16 Kelvin; it is the temperature (approximately) at which water freezes at 1 atmosphere of pressure.

Occasionally we may work with the electron-volt as a unit of energy. Since a volt is a Joule per Coulomb of charge, and one electron carries a charge of 1.602*10-19 Coulombs, an electron-volt is 1.602*10-19 Coulomb * 1 J/Coulomb = 1.602*10-19 J = 1.602*10-22 kJ. A mole is 6.022*1023 molecules: it is really not a unit at all, but rather just a convenient way of counting large numbers of atoms (or molecules, or electrons, or graduate students). Therefore a kJ/mol is a kJ divided by this large number of objects, so 1 kJ/mol = (1/6.022*1023) kJ = 1.661*10-24 kJ. Thus 1 kJ/mol = 1.661*10-24 kJ / (1.602*10-22 kJ/eV) = 1.037 * 10-24+22 = 0.01037 eV. Conversely, 1 eV = 96.4 kJ/mol. We will soon discuss the fact that the hydrolysis of the high-energy phosphate bond in adenosine triphosphate has a ΔGo of about 33 kJ/mol; we can see that this is about 0.34 eV.

Entropy

    We've already said that entropy is a measure of the disorder in a system. Entropy turns out to be proportional to the logarithm of the number of degrees of freedom Ω in a system:
    S = k ln Ω
where k is Boltzmann's constant, 3.4*10-24 cal/K, or 1.38*10-23 joule/K. We often measure entropy in entropy units eu = 1 cal/K. The gas constant R is the product of Avogadro's number N and k, so if the entropy of a single molecule is S, then the entropy of a mole of the same kind of molecules will be     NS = R ln Ω.
    The second law of thermodynamics says that in general the entropy of a closed system will increase, i.e. that for the most part the universe tends toward a larger number of degrees of freedom or a larger amount of disorder.
    The entropy of single molecule can be characterized by statistical-mechanical methods if the molecule is simple enough. The following table, adapted from table 2.1 in Zubay's Principles of Biochemistry, breaks the entropy of liquid propane into translational, rotational, vibrational, and electronic components:
type of entropy
kcal/(K-mol)
translational
36.04
rotational
23.38
vibrational
1.05
electronic
0.00
Total
60.47
This pattern, in which most of the entropy is translational and rotational, is typical of biomolecules. By contrast, the enthalpy in a biomolecule is usually dominated by electronic properties. Translational entropy depends primarily on (3/2)RlnMr, where Mr is the molecular weight. In a dimerization reaction the total entropy decreases, because Mr doubles, but the logarithm of it does not double—it only increases by ln 2. Thus the translational entropy goes down.

    Rigidity decreases entropy, because rigid structures cannot rotate as freely and often cannot vibrate as freely.

Entropy in solvation and binding to surfaces

What happens when molecules go into solution? The solute molecules usually undergo an increase in entropy, because they become free to dissociate from one another, and in the case of ionic solutes the cations can separate from the anions. On the other hand, the solvent molecules frequently become more organized in the vicinity of the solute molecules than they had been before the introduction of the solute, so their contribution to total change in entropy is frequently negative. The net effect is often slightly negative, i.e. the solution has a slightly lower entropy than the separated components.

When an apolar molecule is added to water, the water molecules often form a micelle around the foreign molecule. This micelle is highly ordered, so the entropy of the system decreases.

Many biochemical reactions involve binding of small molecules to a surface, e.g. the surface of a protein. In inorganic chemistry the binding of small molecules to surfaces often involves a decrease in entropy because the molecules binding to the surface lose rotational degrees of freedom. But in biochemistry the loss in rotational freedom is more than compensated for by the increase in entropy associated with the release of water molecules from the protein surface. Thus the binding of metabolites to a protein is often entropically favored.

Free Energy

Josiah Gibbs articulated the concept of free energy (sometimes called Gibbs free energy), which is related to entropy and enthalpy by
    G = H - TS
The change in free energy when a reaction occurs is
    ΔG =  ΔH - TΔS
assuming the temperature does not change. Temperature in a biochemical system in general changes very slowly, so this is a reasonable assumption. Gibbs was able to show that a chemical reaction will occur spontaneously if and only if the change in free energy is negative:
    ΔG < 0
For the most part we will analyze biochemical reactions in terms of their spontaneity and therefore in terms of whether ΔG < 0.

We can compute ΔG per mole for a wide variety of compounds. A useful formulation is that of the standard free energy of formation of a compound ΔGof, which is the difference between the free energy of the compound in its standard state and the total free energies of the elements of which the compound is composed. This table (again adapted from Zubay) contains some examples of ΔGof values for metabolites:

Substance
ΔGof, kcal/mol
ΔGof, kJ/mol
lactate, 1M
-123.76
-516
pyruvate, 1M
-113.44
-474
succinate, 1M
-164.97
-690
glycerol, 1M
-116.76
-488
water
-56.69
-280
acetate, 1M
-88.99
-369
oxaloacetate, 1M
-190.62
-797
hydrogen ions, 10-7M
-9.87
-41
carbon dioxide
-94.45
-394
bicarbonate, 1M
-140.49
-587
We can use these values to calculate the overall change in standard free energy ΔGo associated with a biochemical reaction. There are some tricks and special cases to consider. But the concept is straightforward: given the known values of ΔGof for the reactants and products in a reaction, we can calculate the overall change in standard free energy in a reaction by adding up the ΔGof values for the products and subtracting the ΔGof   values for the reactants. The ΔGof values are generally negative, so we'll be subtracting a negative number from another negative number. If the total comes out negative, the reaction is spontaneous; if it comes out positive, the reaction is not spontaneous.

Free energy and equilibrium

Gibbs established the relationship between ΔGo and the equilibrium contant of a reaction:
    ΔGo = -RTlnKeq
where Keq is the equilibrium constant of the reaction. In a bimolecular reaction
    aA + bB -> cC + dD
this equilibrium constant is
    Keq = ([C]c[D]d )/([A]a[B]b)
Thus if a reaction is just barely spontaneous, i.e. ΔGo = 0, then Keq = 1. If ΔGo < 0 then Keq > 1, i.e. there will be more products than reactants at equilibrium. If ΔGo > 0 then Keq < 1, i.e. there will be more reactants than products at equilibrium. Reactions in which ΔG o < 0 are called exergonic; reactions in which ΔGo > 0 are called endergonic.

Free energy as a source of work

The change in free energy tells us the maximum amount of useful work that can be derived from a biochemical reaction. If ΔGo is negative, then the largest amount of useful work that could be extracted from the reaction is -ΔGo. Some of that energy will go into heat, though, so the actual amount of work we can get will always be less than -ΔGo.

Organisms use this work in at least three ways:
This last case is crucial to many biochemical pathways and will be considered in greater detail.

Coupled reactions

In some cases, a single enzyme catalyzes two successive reactions, the first of which is exergonic and the second of which is endergonic. In that case, in effect, the overall reaction happens in one shot, with the energy from the exergonic part of the sequence driving the energonic part. If the overall ΔGo < 0 for the pair of reactions, the products will be produced.

In other cases, two reactions may not be spatially coupled. Instead, the fact that the first reaction produces a high concentration of its product(s) results in a high concentration of the reactant(s) for the second reaction. Based on the definition of Keq, this imbalance in concentration changes the value of ΔG enough to render the second reaction possible.

ATP as an energy currency

Some reactions encountered in biology are exergonic (have an overall negative ΔG), whereas some are endergonic (positive ΔG). The endergonic reactions in general are coupled with exergonic reactions so that they can proceed. In order for this approach to work, the cell needs a ready supply of high-energy compounds, the modifications of which can be used to drive other reactions. In general the reactions involve coupling the hydrolysis of a high-energy bond with some endergonic process. Most of the high-energy bonds are between phosphorus and oxygen atoms, and the reactions involve hydrolyzing this phosphorus-oxygen bond:
         O-
R-O~P=O + HOH -> R-O-H + PO4-3
         O-

The most common compound involved in this process is adenosine triphosphate. It can be hydrolyzed at either the gamma phosphate (the one farthest from the ribose ring) or at the beta phophate (the one in the middle). In the former case, about 7.8 kcal/mol (32.6 kJ/mol) is released by the hydrolysis:
ATP + H2O -> ADP + Pi
where Pi is a standard abbreviation for inorganic phosphate, i.e. PO4-3, HPO4-2, or H2PO4-. A similar amount is released in hydrolysis at the beta phosphate:
ATP + H2O -> AMP + PPi,
where PPi is a standard abbreviation for inorganic pyrophosphate, i.e. P2O7-4, HP2O7-3, or H2PO7-2, or H3PO7-. However, pyrophosphate hydrolyzes into two molecules of ordinary phosphate, with the release of a similar amount of energy. Appropriately coupled, the hydrolysis of ATP to AMP and two equivalents of Pi can therefore yield more than 15 kcal/mol of energy—enough to drive almost all conventionally-encountered biochemical reactions.

ATP thus acts as a kind of energy currency: a means of storing energy that can be tapped for driving endergonic reactions to completion. The energy has to come from somewhere: it comes from the creation of ATP, with its high-energy phosphate bonds, from lower-energy substituents, using various exergonic reactions as drivers. We can think of the resting concentration of ATP in a cell as the equivalent of a roll of quarters that the cell can spend when it needs energy. Each ATP molecule acts as a single quarter when it's hydrolyzed to ADP; it acts as a pair of quarters when it's hydrolyzed to AMP. Most of the purchases the cell needs to make are for prices either just below $0.50 (ATP -> AMP) or just below $0.25 (ATP -> ADP), so it's useful currency for the cell to carry around. None of the cellular vendors gives change, so if we use our quarters to buy $0.03 worth of merchandise at a time, it's not very cost-effective, but in buying items that cost $0.24 or $0.48, they're pretty efficient. When the cell runs out of quarters, it needs to go to the metabolic bank and get some more.

Other high-energy compounds

There are other compounds employed as energy-storage entities in cells. None of the others is as plentiful or as widely-used as ATP, but they play significant roles in certain pathways. Each of these compounds contains at least one high-energy phosphorus-oxygen bond, just as ATP does, so the mechanisms are similar to those found in ATP hydrolysis. But the specific ΔG values for each of these phosphate compounds differs from that of ATP, and as such they turn out to be more efficient in driving particular classes of reactions. So the cell may be carrying around several rolls of quarters (ATP molecules), but it also carries around one roll of 40-cent pieces (creatine phosphate), one roll of 35-cent pieces (phosphoenol pyruvate), and the like. Since the vendors don't make change, creatine phosphate is a useful compound to carry when making 38-cent purchases.

Dependence on Concentration

Is this bookkeeper's perspective altogether meaningful? It is if we recognize the limitations to it. A fundamental principle of chemical thermodynamics is that the free energy difference in a reaction depends on the concentrations (or, more precisely, the activities) of the products and reactants. Therefore a reaction that would have a negative ΔG if all the products and reactants had equal concentrations may have a positive ΔG if there are high concentrations of products and low concentrations of reactants when we begin to examine the system. Specifically, we write this dependence on concentration as
ΔG = ΔGo + RTln [products]/[reactants]
where, as we have discussxd, ΔGo is the standard free energy of the reaction, viz. the free energy associated with a condition in which the concentrations of products and reactants are all initially 1M. Thus if [products]/[reactants] is greater than one, the term on the right will be positive and will make ΔG more positive, or less negative, than the standard free energy ΔGo. Note that if the concentrations of products are all 1M, then [products]/[reactants] = 1 and ln[products]/[reactants] = 0, so ΔG = ΔGo. Thus it makes sense to define the standard free energy in these terms. It is often impractical to make measurements in systems where the starting concentrations are all 1M, but we generally extrapolate to those conditions without difficulty.

The way this concentration dependence affects our notion of energy currency is that the "value" of a high-energy compound depends on its concentration and the concentrations of the other molecules participating in a reaction. Thus the reaction
ATP + X → ADP + Pi + Y
in which the free energy derived from ATP hydrolysis is used to drive the conversion of X to Y, may have a ΔGo that is moderately positive in a particular cellular situation. But if the concentrations of ADP, X, Y and ATP are such that ln[products]/[reactants] is negative, then we still may find that ΔG is negative and the reaction will proceed to the right. Often [ATP] > [ADP]. This tends to increase the sponteneity of these ATP-driven reactions: [ADP]/[ATP] < 1, so the logarithm of that ratio will be negative.