Experimental and theoretical analysis of the rate of solvent equilibration in the hanging drop method of protein crystal growth

Abstract

In the hanging drop method of protein crystal growth, a water droplet containing protein, buffer and a precipitating agent, such as ammonium sulfate, is suspended from a glass coverslip above a well containing an aqueous solution of the precipitating agent at a concentration double that in the drop. We present a comprehensive theoretical study of the rate of water evaporation in the hanging drop method. We find that in earth's gravity the rate controlling step in the evaporation is the rate of diffusion of water vapor across the air space separating the drop from the well. Using ammonium sulfate as the precipitating agent, we have made careful measurements at both 4°C and 25°C of the evaporation times for some 25 μL droplets at various concentrations. These results are in good agreement with our theory. As determined by the theory, the parameters affecting the rate of evaporation include the temperature, the vapor pressure of water, the ionization constant of the salt, the volume of the droplet, the contact angle between the droplet and the coverslip, the number of moles of salt in the droplet, the number of moles of water and salt in the well, the molar volumes of water and salt, the distance from the droplet to the well, and the coefficient of diffusion of water vapor through air. These parameters do not act independently; rather, they combine to form three dimensionless groups upon which the rate of evaporation depends. We evaluate numerically 18 different drop and well arrangements commonly encountered in the laboratory. In all cases considered at 25°C, the number of moles of water in the droplet achieves 95% of its final value in 3—30 h, after which further evaporation is quite slow. Our experiments confirm this. We consider qualitatively the effect of weightlessness (spaceflight) on the rate of evaporation and find it to be most likely controlled by the rate of the interdiffusion of salt and water in the droplet.