The intermolecular bond energy Q contains both enthalpy and entropy components. The enthalpy ones are owing to the ion-mediated, hydrogen, and other bonds between the molecules, and the entropy components stem from the net release or binding of water and other small molecules on crystallization (45). With this in mind, we can write an expression for the free energy for crystallization as follows:
Here, AH° - TAS°solvent are contributions associated with Q, and AS°protein is the loss of entropy of the protein molecules. A crude estimate of AS°protein and of the relative weights of the two entropy contributions can be obtained by comparing the standard free-energy change for apoferritin crystallization AG° determined from the solubility to the value corresponding to the intermolecular bond free energy Q.
By converting the solubility of the apoferritin (23 mg/mL) to molality, Ce = 5.2 x 10-8 mol/kg. At equilibrium between crystal and solution, for apoferritin
AG = G°(crystal) - [G°(solution) + NAkBT ln(Ce)] = 0 (14)
in which Na is the Avogadro number, and the product NAkB = R is the universal gas constant. Hence,
AG° = G°(crystal) - G°(solution) = NAkBT lnCe (15)
To get AH° - TAS°solvent from Q = 3 kBT = 7.3 kJ/mol, we have to multiply Q by Zl/2 = 6, the half-number of neighbors in the crystal lattice (two molecules partake in a bond, in an f.c.c. lattice Z1 = 12) and, accounting for the sign, we get -44 kJ/mol. The closeness of AH° - TAS°solvent to AG° indicates the insignificance of AS°protein for the free energy of crystallization.
As shown above, the enthalpy of crystallization and the related energy of pair interactions in the solution are close to zero. In combination with the insignificance of AS°protein, this allows us to conclude that crystallization is mostly driven by the maximization of the entropy of the solvent. Such disordering may stem from the release on crystallization of the water and other solvent components bound to the protein molecules in the solution. A similarity can be traced to the processes that underlie hydrophobic attraction, which governs many processes in nature (58), including some stages of protein folding (59).
The standard free energy of formation of a single intermolecular bond in apoferritin crystals is -7.8 kJ/mol (see Subheading 4.2.) and is fully attributable to the entropy gain caused by the release of water, AS°solvent = 26.6 J/(mol-K) per intermolecular bond (10).
This conclusion allows us to estimate crudely the number of water molecules nw released at the contact between two hemoglobin molecules. Following an analogy first put forth by Tanford (60), we compare the entropy effect of Hb crystallization to the entropy change for melting of ice = at 273 K, AS°ice =
22 J/(mol-K) (45,61,62). Similarly, estimates of the entropy loss owing to the tying up of hydration water in crystals have yielded 25 to 29 J/(mol-K) (62). Using these numbers, the values of AS°solvent reflect the release of one or two water molecules per intermolecular bond.
This low number of water molecules can be tentatively linked to the structure of the intermolecular bonds in the face-centered cubic apoferritin crystals. The X-ray structure reveals that each of the 12 such bonds consists of a pair of Cd2+ ions (4). Around each Cd2+ ion of the pair, two of the six coordination spots are occupied by an aspartic acid residue from the one apoferritin molecule partaking in the bond and a glutamic acid residue from the other (5). The fact that the entropy change corresponds to the release of one or two, rather than four, water molecules suggests that the Cd2+ ions may be prebound to either the incoming apoferritin molecule or the apoferritin molecules already in the crystal.
Thus, as suggested by the crystal structures of ferritin and apoferritin (3-5), the main component of the crystallization driving force stems from the strong Cd2+-mediated bond between each pair of molecules. The unexpected part of our conclusion is that this driving force is not of enthalpy origin (the likely large negative enthalpy of such a bond must have been compensated for by unfavorable enthalpy effects of other patches of the molecules), but comes from the entropy of the water released during the formation of this bond.
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