The work is devoted to the theory of nucleation and growth of crystals in a supersaturated (supercooled) liquid. An integro-differential model of the kinetic and balance equations for the crystal-size distribution function and supersaturation (supercooling) of the liquid has been formulated and analytically solved, taking into account the following factors: (i) fluctuations in crystal growth rates leading to diffusion of the distribution function in the particle size space, (ii) nonstationary growth rates of individual crystals, and (iii) arbitrary initial crystal size distribution. The problem is solved for arbitrary crystal nucleation kinetics, and the Weber-Volmer-Frenkel-Zel'dovich and Meirs kinetic mechanisms are considered as special cases for calculations. Analytical solutions of the nonstationary problem are derived: the crystal-size distribution function and the supersaturation (supercooling) of the liquid. Some biomedical applications of the developed theory for crystal growth from supersaturated solutions are discussed. The theory is compared with experimental data on protein crystallization of lysozyme and canavalin, as well as bovine and porcine insulin. The time-dependent dynamics of solution supersaturation and a bell-shaped particle-size distribution function are studied for these substances.

The theory we have developed is important for describing the bulk crystallization of insulin, proteins and other vital chemicals. For example, the study of the protein lysozyme, most commonly released from chicken egg whites, is important because this enzyme hydrolyses polysaccharides on bacterial cell walls. It is used as an antiseptic and also as a food additive. It should be noted that the rate of decomposition of protein supersaturation in crystallizing solutions of lysozyme was investigated in Ref. [1] when the crystallized protein is more stable than the dissolved one. The growth dynamics of another important protein, canavalin, was studied in Ref. [2]. In this tudy, we have compared our theory and experimental data on crystallization of lysozyme and canavalin proteins. For a more precise description of bulk crystallzation it is necessary to take the crystal anisotropy into account. The simplest way to do this is to use an ellipsoidal coordinate system. To generalise the present theory to the case of anisotropic particle growth, an approach recently developed in Refs. [3,4] can be used.