## Statistical Thermodynamics Arguments of Independence of Solubility on Molecular Mass

The values for ne, determined as discussed in Subheading 3.2. and in refs. 10,25 for ferritin and apoferritin are, respectively, (2.7 ± 0.5) x 1013 cm-3 and (3.0 ± 0.5) x 1013 cm-3. They are equal within the error limits, suggesting that the solubilities of these proteins do not depend on their molecular mass. Next, I provide statistical-mechanical arguments that indicate that this may be the case for protein molecules with near-spherical symmetry.

To rationalize the apparently equal solubility of ferritin and apoferritin, we consider the equilibrium between the solution and a crystal. This is equivalent to equilibrium between the states of a molecule in a kink on the crystal surface and in the solution (42-44). At constant temperature and pressure, the activity of a molecule in the crystal does not depend on the concentration of the solute and is equal to the activity of the standard crystal state (45,46). Then, the equilibrium constant Kcryst can be written as

in which ge is the protein activity coefficient at a protein concentration equal to the solubility, and Ce is the solubility. The activity coefficient depends on the protein concentration and on the intermolecular interactions. Hence, we expect equal ys in solutions of ferritin and apoferritin of equal concentration. As shown in Subheading 3.2., determinations of y for apoferritin solutions of concentrations up to 20-fold higher than the solubility have yielded y= 1 (10). We expect the same to be true for ferritin, and this is the basis of the second equality in Eq. 13 above for these two proteins.

From the point of view of statistical thermodynamics, the equilibrium constant for crystallization Kcryst can be written as (47)

Kcryst = qoexpClVkfiT) (4)

in which q0 is the partition function of a molecule in a kink (which is only a function of temperature and pressure), and lo is the standard chemical potential of a molecule in the solution.

To evaluate q0 and |0, we assume that the internal molecular vibrations in the solution are the same as in the crystal and are decoupled from the other degrees of freedom. This allows us to neglect the internal vibrational partition function for both states. Furthermore, we limit ourselves to only translational contributions to the solute partition function, neglecting the rotational contributions, and those stemming from the intermolecular interactions. This limits the validity of the considerations below to molecules similar to the ferritin-apoferritin pair: with symmetry close to spherical, and that only exhibit very weak intermolecular interactions and activity coefficients close to 1.

We do not take into account the contribution of the release or binding of the solvent molecules to the free-energy changes in the phase transition. Although arguments presented in Subheading 5.2. indicate that these contributions may be significant (10), we expect the contributions of the solvent effects to be identical for ferritin and apoferritin. This justifies neglecting them while aiming at comparisons between the two proteins. We also neglect the rotational vibrations in the crystal.

With these assumptions, we can use the expressions for the partition functions from ref. 47, and write qo = qxqyqz = (qvb)3 (5)

in which qt (i = x, y, z) are the partition functions for translational vibrations along the respective coordinate. In turn, with h the Planck constant, n the vibration frequency, and U the mean-force potential of a molecule in a kink, exp - -

hv i

kBT'

Combining, we obtain for qvib and q0

2nkBT

Fig. 6. (A) Molecular structure of growth step on apoferritin crystal at protein concentration of 70 mg/mL, corresponding to supersaturation o = AyJkBT = 1.1, (C/Ce - 1) = 2.04. Dark area: lower layer; light area: advancing upper layer. Adsorbed impurity clusters and surface vacancies are indicated. (B-D) The distribution of the number of molecules between kinks on steps located >0.5 |im apart, obtained from images similar to Fig. 1, at the three supersaturations s indicated in the plots is shown. The mean values of the distributions for each case are also shown. The protein concentrations corresponding to these o's are (B) 25 |g/mL, (C) 70 |g/mL, and (D) 1 |g/mL. (From ref. 10.)

Fig. 6. (A) Molecular structure of growth step on apoferritin crystal at protein concentration of 70 mg/mL, corresponding to supersaturation o = AyJkBT = 1.1, (C/Ce - 1) = 2.04. Dark area: lower layer; light area: advancing upper layer. Adsorbed impurity clusters and surface vacancies are indicated. (B-D) The distribution of the number of molecules between kinks on steps located >0.5 |im apart, obtained from images similar to Fig. 1, at the three supersaturations s indicated in the plots is shown. The mean values of the distributions for each case are also shown. The protein concentrations corresponding to these o's are (B) 25 |g/mL, (C) 70 |g/mL, and (D) 1 |g/mL. (From ref. 10.)

12%mkBT\

and exp

kBT\

kBT \2nmkBT I

We see that q0 contains m3/2, while exp(|0/kBT)is proportional to m 3/2; that is, their product Kcryst, and Ce do not depend on the mass of the molecule.