Peter G. Vekilov

Principles of Crystal Nucleation and Growth

In the most general sense, biomineralization is a process by which organisms produce materials solutions for their own functional requirements. Because so many biomineral products are derived from an initial solution phase and are either completely crystalline or include crystalline components, an understanding of the physical principles of crystallization from solutions is an important tool for students of biomineralization. However, crystal growth is a science of great breadth and depth, about which many extensive texts have been written. In addition, there are already other thorough reviews that specifically address the crystal growth field of study as it relates to biomineral formation. Consequently, the goals of this chapter are both modest and specific. It is intended to provide: 1) a simple narrative explaining the physical principles behind crystallization for those who are completely new to the topic, 2) a few basic equations governing nucleation and growth for those who wish to apply those principles—at least in a semi-quantitative fashion—to experimental observations of mineralization, and 3) an overview of some recent molecular-scale studies that have revealed new insights into the control of crystal growth by small molecules, both organic and inorganic. This last topic gets to the heart of what makes crystallization in biological systems unique. Every day, many tons of crystals are produced synthetically in non-biological processes, but by-and-large, the degree of control over nucleation and growth achieved by deterministic additions of growth modifiers or the presence of a controlling matrix is very minor. More commonly, crystal growers view modifying agents as unwanted impurities and work extremely hard to eliminate them from the starting materials. Indeed, the degree to which living organisms are able to control the crystallization process is most striking when contrasted to the products of such synthetic crystallization processes. This contrast applies to both the compositional differences that …

Entropy and surface engineering in protein crystallization.

Protein crystallization remains a key limiting step in the characterization of the atomic structures of proteins and their complexes by X-ray diffraction methods. Current data indicate that standard screening procedures applied to soluble well folded prokaryotic proteins yield X-ray diffraction crystals with an approximately 20% success rate and for eukaryotic proteins this figure may be significantly lower. Protein crystallization is predominantly dependent on entropic effects and the driving force appears to be the release of ordered water from the sites of crystal contacts. This is countered by the entropic cost of ordering of protein molecules and by the loss of conformational freedom of side chains involved in the crystal contacts. Mutational surface engineering designed to create patches with low conformational entropy and thereby conducive to formation of crystal contacts promises to be an effective tool allowing direct enhancement of the success rate of macromolecular crystallization.

Quasi-planar nucleus structure in apoferritin crystallization

First-order phase transitions of matter, such as condensation and crystallization, proceed through the formation and subsequent growth of ‘critical nuclei’ of the new phase. The thermodynamics and kinetics of the formation of these critical nuclei depend on their structure, which is often assumed to be a compact, three-dimensional arrangement of the constituent molecules or atoms. Recent molecular dynamics simulations have predicted compact nucleus structures for matter made up of building blocks with a spherical interaction field, whereas strongly anisotropic, dipolar molecules may form nuclei consisting of single chains of molecules. Here we show, using direct atomic force microscopy observations, that the near-critical-size clusters formed during the crystallization of apoferritin, a quasi-spherical protein, and which are representative of the critical nucleus of this system, consist of planar arrays of one or two monomolecular layers that contain 5–10 rods of up to 7 molecules each. We find that these clusters contain between 20 and 50 molecules each, and that the arrangement of the constituent molecules is identical to that found in apoferritin crystals. We anticipate that similarly unexpected critical nucleus structures may be quite common, particularly with anisotropic molecules, suggesting that advanced nucleation theories should treat the critical nucleus structure as a variable.

Nucleation and Crystallization of Globular Proteins: What we Know and What is Missing

Recently, much progress has been made in understanding the nucleation and crystallization of globular proteins, including the formation of compositional and structural crystal defects. Insight into the interactions of (screened) protein macro-ions in solution, obtained from light scattering, small angle X-ray scattering and osmotic pressure studies, can guide the search for crystallization conditions. These studies show that the nucleation of globular proteins is governed by the same principles as that of small molecules. However, failure to account for direct and indirect (hydrodynamic) protein interactions in the solutions results in unrealistic aggregation scenarios. Microscopic studies of numerous proteins reveal that crystals grow by the attachment of growth units through the same layer-spreading mechanisms as inorganic crystals. Investigations of the growth kinetics of hen-egg-white lysozyme (HEWL) reveal non-steady behavior under steady external conditions. Long-term variations in growth rates are due to changes in step-originating dislocation groups. Fluctuations on a shorter timescale reflect the non-linear dynamics of layer growth that results from the interplay between interfacial kinetics and bulk transport. Systematic gel electrophoretic analyses suggest that most HEWL crystallization studies have been performed with material containing other proteins at percent levels. Yet, sub-percent levels of protein impurities impede growth step propagation and play a role in the formation of structural/compositional inhomogeneities. In crystal growth from highly purified HEWL solutions, however, such inhomogeneities are much weaker and form only in response to unusually large changes in growth conditions. Equally important for connecting growth conditions to crystal perfection and diffraction resolution are recent advances in structural characterization through high-resolution Bragg reflection profiling and X-ray topography.

Liquid–liquid separation in solutions of normal and sickle cell hemoglobin

We show that in solutions of human hemoglobin (Hb)—oxy- and deoxy-Hb A or S—of near-physiological pH, ionic strength, and Hb concentration, liquid–liquid phase separation occurs reversibly and reproducibly at temperatures between 35 and 40°C. In solutions of deoxy-HbS, we demonstrate that the dense liquid droplets facilitate the nucleation of HbS polymers, whose formation is the primary pathogenic event for sickle cell anemia. In view of recent results that shifts of the liquid–liquid separation phase boundary can be achieved by nontoxic additives at molar concentrations up to 30 times lower than the protein concentrations, these findings open new avenues for the inhibition of the HbS polymerization.

Interactions and aggregation of apoferritin molecules in solution: effects of added electrolytes.

We have studied the structure of the protein species and the protein-protein interactions in solutions containing two apoferritin molecular forms, monomers and dimers, in the presence of Na(+) and Cd(2+) ions. We used chromatographic, and static and dynamic light scattering techniques, and atomic force microscopy (AFM). Size-exclusion chromatography was used to isolate these two protein fractions. The sizes and shapes of the monomers and dimers were determined by dynamic light scattering and AFM. Although the monomer is an apparent sphere with a diameter corresponding to previous x-ray crystallography determinations, the dimer shape corresponds to two, bound monomer spheres. Static light scattering was applied to characterize the interactions between solute molecules of monomers and dimers in terms of the second osmotic virial coefficients. The results for the monomers indicate that Na(+) ions cause strong intermolecular repulsion even at concentrations higher than 0.15 M, contrary to the predictions of the commonly applied Derjaguin-Landau-Verwey-Overbeek theory. We argue that the reason for such behavior is hydration force due to the formation of a water shell around the protein molecules with the help of the sodium ions. The addition of even small amounts of Cd(2+) changes the repulsive interactions to attractive but does not lead to oligomer formation, at least at the protein concentrations used. Thus, the two ions provide examples of strong specificity of their interactions with the protein molecules. In solutions of the apoferritin dimer, the molecules attract even in the presence of Na(+) only, indicating a change in the surface of the apoferritin molecule. In view of the strong repulsion between the monomers, this indicates that the dimers and higher oligomers form only after partial denaturation of some of the apoferritin monomers. These observations suggest that aggregation and self-assembly of protein molecules or molecular subunits may be driven by forces other than those responsible for crystallization and other phase transitions in the protein solution.

Dependence of lysozyme growth kinetics on step sources and impurities

Interferometric microscopy was used to investigate the growth morphology and kinetics of {110} and {101} faces of tetragonal lysozyme crystals. Solutions were prepared from as-received Sigma and Seikagaku material, and Seikagaku lysozyme further purified by cation exchange liquid chromatography under salt-free conditions. The protein composition of the solutions was characterized by sodium dodecyl sulphate (SDS) electrophoresis with silver staining. We found that on crystals smaller than about 150 μm, 2D nucleation sites were randomly distributed over the faces. With increasing crystal size, surface nucleation became restricted to facet edges and, eventually, to facet corners. This reflects the higher interfacial supersaturation at these locations. However, on some crystals, we observed 2D nucleation at preferred non-corner sites presumably associated with defects. Upon abrupt temperature decreases, dislocation step sources formed on faces that previously had none. Within groups of dislocations, the dominating step source changed frequently. Depending on the activity of the dislocation groups, growth rates of different crystals differed by up to a factor of five during the same experiment. On facets with dislocation step sources, step generation by 2D nucleation became dominant above a critical supersaturation σ∗. In the absence of dislocations, nucleation-induced growth set in at σ < σ∗. In solutions with higher impurity concentrations, the density of the steps generated by 2D nucleation was higher and σ∗ was lower. Hence, it appears that impurity adspecies are active in surface nucleation. The presence of less than 1% of protein impurities with molecular weight (MW) ≥ 30 kD had significant effects on the crystallization kinetics. Step motion was impeded even at high σ, presumably through blocking of kink sites. In solutions without these high MW impurities, facets containing step sources did not grow below σ = ln(C / Csat) < 0.5. In the less pure solutions such a “dead zone” was not observed. Hence, it appears that in lysozyme dead zones are caused by non-protein impurities. In growth from the highly purified material no growth sector boundaries were visible, in contrast to the as-received lysozyme, and striae formation on growth temperature changes appeared drastically reduced.

Heterogeneity determination and purification of commercial hen egg-white lysozyme.

Hen egg-white lysozyme (HEWL) is widely used as a model protein, although its purity has not been adequately characterized by modern biochemical techniques. We have identified and quantified the protein heterogeneities in three commercial HEWL preparations by sodium dodecyl sulfate polyacrylamide gel electrophoresis with enhanced silver staining, reversed-phase fast protein liquid chromatography (FPLC) and immunoblotting with comparison to authentic protein standards. Depending on the source, the contaminating proteins totalled 1-6%(w/w) and consisted of ovotransferrin, ovalbumin, HEWL dimers, and polypeptides with approximate M(r) of 39 and 18 kDa. Furthermore, we have obtained gram quantities of electrophoretically homogeneous [> 99.9%(w/w)] HEWL by single-step semi-preparative scale cation-exchange FPLC with a yield of about 50%. Parallel studies of crystal growth kinetics, salt repartitioning and crystal perfection with this highly purified material showed fourfold increases in the growth-step velocities and significant enhancement in the structural homogeneity of HEWL crystals.

Nucleation of protein crystals: critical nuclei, phase behavior, and control pathways

Abstract We have studied the nucleation of crystals of the model protein lysozyme using a novel technique that allows direct determinations of homogeneous nucleation rates. At constant temperature of 12.6°C we varied the thermodynamic supersaturation by changing the concentrations of protein and precipitant. We found a broken dependence of the homogeneous nucleation rate on supersaturation that is beyond the predictions of the classical nucleation theory. The nucleation theorem allows us to relate this to discrete changes of the size of the crystal nuclei with increasing supersaturation as (10 or 11)→(4 or 5)→(1 or 2). Furthermore, we observe that the existence of a second liquid phase at high protein concentrations strongly affects crystal nucleation kinetics. We show that the rate of homogeneous nucleation of lysozyme crystals passes through a maximum in the vicinity of the liquid–liquid phase boundary hidden below the liquidus (solubility) line in the phase diagram of the protein solution. We found that glycerol and polyethylene glycol (PEG), which do not specifically bind to proteins, shift this phase boundary and significantly suppress or enhance the crystal nucleation rates, although no simple correlation exists between the action of PEG on the phase diagram and the nucleation kinetics. This provides for a control mechanism which does not require changes in the protein concentration, or the acidity and ionicity of the solution. The effects of the two additives on the phase diagram strongly depend on their concentration and this provides opportunities for further tuning of nucleation rates.

Nucleation of ordered solid phases of proteins via a disordered high-density state: phenomenological approach.

Nucleation of ordered solid phases of proteins triggers numerous phenomena in laboratory, industry, and in healthy and sick organisms. Recent simulations and experiments with protein crystals suggest that the formation of an ordered crystalline nucleus is preceded by a disordered high-density cluster, akin to a droplet of high-density liquid that has been observed with some proteins; this mechanism allowed a qualitative explanation of recorded complex nucleation kinetics curves. Here, we present a simple phenomenological theory that takes into account intermediate high-density metastable states in the nucleation process. Nucleation rate data at varying temperature and protein concentration are reproduced with high fidelity using literature values of the thermodynamic and kinetic parameters of the system. Our calculations show that the growth rate of the near-critical and supercritical ordered clusters within the dense intermediate is a major factor for the overall nucleation rate. This highlights the role of viscosity within the dense intermediate for the formation of the ordered nucleus. The model provides an understanding of the action of additives that delay or accelerate nucleation and presents a framework within which the nucleation of other ordered protein solid phases, e.g., the sickle cell hemoglobin polymers, can be analyzed.