Chapter 7 – Complementary Experimental Tools  283

as a virial expansion (see Equation 4.25) in which the parameter B is the second virial coeffi­

cient and is negative, indicative of net attractive forces between the protein molecules.

Once a highly purified biomolecule solution has been prepared, then, in principle, the

attractive forces between the biomolecules can result in crystal formation if a supersaturated

solution in generated. It is valuable to depict the dependence of protein solubility on precipi­

tant concentration as a 2D phase diagram (Figure 7.3). The undersaturation zone indicates

solubilized protein, whereas regions to the upper right of the saturation curve indicate the

supersaturation zone in which there is more protein present than can be dissolved in the

water available.

The crystallization process involves local decrease in entropy S due to an increase in the

order of the constituent molecules of the crystals, which is offset by a greater increase in

local disorder of all of the surrounding water molecules due to breakage of solvation bonds

with the molecules that undergo crystallization. Dissolving a crystal breaks strong molecular

bonds and so releases enthalpy H as heat (i.e., an exothermic process), and similarly crystal

formation is an endothermic process. Thus, we can say that

∆G

G

G

H

H

crystallization

crystal

solution

crystal

solution

=

−

=

−

(

) ^{−T S}

S

crystal

solution

−

(

)

The change in enthalpy for the local system composed of all molecules in a given crystal

is positive for the transition of disordered solution to ordered crystal (i.e., crystallization),

as is the change in entropy in this local system. Therefore, the likelihood that the crystal­

lization process occurs spontaneously, which requires ΔGcrystallization < 0, increases at lower

temperatures T. This is the same basic argument for a change of state from liquid to solid.

Optimal conditions of precipitant concentration can be determined in advance to find a

crystallization window that maximizes the likely number of crystals formed. For example,

static light scattering experiments (see Chapter 4) can be performed on protein solutions

containing different concentrations of the precipitant. Using the Zimm model as embodied

by Equation 4.30 allows the second virial coefficient to be estimated. Estimated preliminary

values of B can then be plotted against the empirical crystallization success rate (e.g., number

of small crystals observed forming in a given period of time) to determine an empirical crys­

tallization window by extrapolating back to the ideal associated range in precipitant concen­

tration, focusing on efforts for longer time scale in crystal growth experiments.

FIGURE 7.3  Generating protein crystals. Schematic of 2D phase diagram showing the

dependence of protein concentration on precipitant concentration. Water vapor loss from a

concentrated solution (point I) results in supersaturated concentrations, which can result in

crystal nucleation (point II). Crystals can grow until the point III is reached, and further vapor loss

can result in more crystal nucleation (point IV).