Diffusion-limited kinetics of the solution–solid phase transition of molecular substances

Abstract

For critical tests of whether diffusion-limited kinetics is an option for the solution-solid phase transition of molecular substances or whether they are determined exclusively by a transition state, we performed crystallization experiments with ferritin and apoferritin, a unique pair of proteins with identical shells but different molecular masses. We find that the kinetic coefficient for crystallization is identical (accuracy <7%) for the pair, indicating diffusion-limited kinetics of crystallization. Data on the kinetics of this phase transition in systems ranging from small-molecule ionic to protein and viri suggest that the kinetics of solution-phase transitions for broad classes of small-molecule and protein materials are diffusion-limited.T he kinetics of the reactions in solutions are either limited only by the rate of diffusion of the species (1) or additionally slowed down by a transition state (2). For the kinetics of the phase transitions in solutions, it is generally accepted that colloid particles follow the diffusion-limited model, whereas the growth rates of new phases of small molecules are thought to be governed by a transition state (3-7). For the intermediate case of protein solid phases, the growth kinetics largely resemble those of small molecules, and it was assumed that transition-state laws apply (8). Although rate laws reminiscent of diffusionlimited mechanisms have been postulated for small-molecule phase transitions (9-11), they were viewed as equivalent to respective transition-state expressions (10, 11), and no critical tests to discriminate between the two mechanisms were suggested or performed.In the transition-state kinetics, the rate coefficients are (i) mass-dependent (12, 13), (ii) independent on the diffusivity (2), and (iii) faster for high-symmetry molecules because of the transition-state entropy (7). Hence, for tests of the mechanism of growth of the new phase during a solution-solid phase transition, we selected the crystallization of ferritin͞ apoferritin, a unique pair of proteins that share a nearspherical protein shell, consisting of 24 identical subunits (14, 15). The crystals of ferritin and apoferritin are typically faceted by octahedral [111] faces and have been shown to grow by the lateral spreading of layers generated by two-dimensional nucleation (16, 17). For apoferritin, it was shown that the edges of the unfinished layers, called growth steps, propagate by the incorporation of single protein molecules at the growth sites, kinks (16,18). In the face-centered cubic lattice that ferritin and apoferritin share, kinks are defined as the termination points of three independent ͗011͘ rows of molecules belonging to the unfinished layer (19). Thus, the step velocity equals the product of the molecular diameter a, mean kink density n k Ϫ1 , and net molecular f lux into a kink (j ϩ Ϫ j Ϫ ) (16, 18). Methods Characterization of the Molecular Masses and Pair Interactions ofFerritin and Apoferritin. We performed static light scattering (20) in 0.2 M NaOOCCH 3 solutions....

Role of molecular charge and hydrophilicity in regulating the kinetics of crystal growth

“…. '' (29). The same diffusion-limited kinetics and potential energy barriers were also shown to control incorporation of small inorganic molecules into solids such as CaCO 3 crystals (29).…”

“…4 shows that increasing positive values of molecular hydrophilicity correlate well with step propagation rates. A possible explanation for this effect comes from recent experimental studies of step dynamics during growth from solutions (29). There it was concluded that the rate-limiting step during crystallization was desolvation of the solute, where the barrier to desolvation was that associated with bulk diffusion limited by the restructuring of water.…”

“…(29). The same diffusion-limited kinetics and potential energy barriers were also shown to control incorporation of small inorganic molecules into solids such as CaCO 3 crystals (29). Therefore, from measurements of activation energies of step incorporation of small inorganic solutes, the crossing of an energy barrier is required for the solute-to-solid phase transformation during crystallization, including that of calcite (29), for which the barrier was experimentally determined to be Ϸ33 kJ/mol (0.34 eV per molecule) (19).…”

A classical view on nonclassical nucleation

“…CNT describes the formation of nuclei from the dynamic and stochastic association of monomeric units (e.g., ions, atoms, or molecules) that overcome a free-energy barrier at a critical nucleus size and grow out to a mature bulk phase. Calcium carbonate (CaCO 3 ) is a frequently used model system to study nucleation (3)(4)(5); however, despite the many years of effort, there are still phenomena associated with CaCO 3 crystal formation where the applicability of classical nucleation concepts have been questioned (6). These include certain microstructures and habits of biominerals formed by organisms (7), or geological mineral deposits with unusual mineralogical and textural patterns (8).…”

“…We propose that a dense liquid phase (containing 4-7 H 2 O per CaCO 3 unit) forms in supersaturated solutions through the association of ions and ion pairs without significant participation of larger ion clusters. This liquid acts as the precursor for the formation of solid CaCO 3 in the form of vaterite, which grows via a net transfer of ions from solution according to z Ca 2+ + z CO 3 2− → z CaCO 3 . The results show that all steps in this process can be explained according to classical concepts of crystal nucleation and growth, and that long-standing physical concepts of nucleation can describe multistep, multiphase growth mechanisms.…”