Nonlinear response of layer growth dynamics in the mixed kinetics-bulk-transport regime

Through Diffusion Measurements of Molecules to a Numerical Model for Protein Crystallization in Viscous Polyethylene Glycol Solution

Protein crystallography has become a popular method for biochemists, but obtaining high-quality protein crystals for precise structural analysis and larger ones for neutron analysis requires further technical progress. Many studies have noted the importance of solvent viscosity for the probability of crystal nucleation and for mass transportation; therefore, in this paper, we have reported on experimental results and simulation studies regarding the use of viscous polyethylene glycol (PEG) solvents for protein crystals. We investigated the diffusion rates of proteins, peptides, and small molecules in viscous PEG solvents using fluorescence correlation spectroscopy. In high-molecular-weight PEG solutions (molecular weights: 10,000 and 20,000), solute diffusion showed deviations, with a faster diffusion than that estimated by the Stokes–Einstein equation. We showed that the extent of the deviation depends on the difference between the molecular sizes of the solute and PEG solvent, and succeeded in creating equations to predict diffusion coefficients in viscous PEG solutions. Using these equations, we have developed a new numerical model of 1D diffusion processes of proteins and precipitants in a counter-diffusion chamber during crystallization processes. Examples of the application of anomalous diffusion in counter-diffusion crystallization are shown by the growth of lysozyme crystals.

Hydrodynamics and Mass Transfer during the Solution Growth of the K2(Co,Ni)(SO4)2•6H2O Mixed Crystals in the Shapers

Mathematical models of the hydrodynamics and mass transfer processes during the mixed crystal growth from low-temperature aqueous solutions have been analyzed. The features of these processes are caused by complex design of the crystallizer with a shaper. Two models of the solution flowing into the shaper have been considered. In the first model, the solution is fed to the central part of the crystal. The second model presents a peripheral solution supply along the shaper perimeter, which allows us to create a swirling flow. The calculation models correspond to laminar and turbulent regimes of solution flow during the growth of K2(Co,Ni)(SO4)2•6H2O mixed crystal from an aqueous solution.

Applications of the second virial coefficient: protein crystallization and solubility

This article begins by highlighting some of the ground-based studies emanating from NASA's Microgravity Protein Crystal Growth (PCG) program. This is followed by a more detailed discussion of the history of and the progress made in one of the NASA-funded PCG investigations involving the use of measured second virial coefficients (B values) as a diagnostic indicator of solution conditions conducive to protein crystallization. A second application of measured B values involves the determination of solution conditions that improve or maximize the solubility of aqueous and membrane proteins. These two important applications have led to several technological improvements that simplify the experimental expertise required, enable the measurement of membrane proteins and improve the diagnostic capability and measurement throughput.

Upscaling calcite growth rates from the mesoscale to the macroscale

Quantitative prediction of mineral reaction rates in the subsurface remains a daunting task partly because a key parameter for macroscopic models, the reactive site density, is poorly constrained. Here we report atomic force microscopy (AFM) measurements on the {1014} calcite surface of monomolecular step densities, treated as equivalent to the reactive site density, as a function of aqueous calcium-to-carbonate ratio and saturation index. Data for the obtuse step orientation are combined with existing step velocity measurements to generate a model that predicts overall macroscopic calcite growth rates. The model is quantitatively consistent with several published macroscopic rates under a range of alkaline solution conditions, particularly for two of the most comprehensive data sets, without the need for additional fit parameters. The model reproduces peak growth rates, and its functional form is simple enough to be incorporated into reactive transport or other macroscopic models designed for predictions in porous media. However, it currently cannot model equilibrium or pH effects and it may overestimate rates at high aqueous calcium-to-carbonate ratios. The discrepancies in rates at high calcium-to-carbonate ratios may be due to differences in pretreatment, such as exposing the seed material to SI ≥ 1.0 to generate/develop growth hillocks, or other factors.

Improving marginal crystals

The physical chemistry of crystal growth can help to identify directions in which to look for improved crystal properties. In this chapter, we summarize how crystal growth depends on parameters that can be controlled experimentally, and relate them to the tools available for optimizing a particular crystal form for crystal shape, volume, and diffraction quality. Our purpose is to sketch the conceptual basis of optimization and to provide sample protocols derived from those foundations. We hope to assist even those who chose not to use systematic methods by enabling them to carry out rudimentary optimization searches armed with a better understanding of how the underlying physical chemistry operates.

A Metastable Prerequisite for the Growth of Lumazine Synthase Crystals

Dense liquid phases, metastable with respect to a solid phase, form in solutions of proteins and small-molecule materials. They have been shown to serve as a prerequisite for the nucleation of crystals and other ordered solid phases. Here, using crystals of the protein lumazine synthase from Bacillus subtilis, which grow by the generation and spreading of layers, we demonstrate that within a range of supersaturations the only mechanism of generation of growth layers involves the association of submicrometer-size droplets of the dense liquid to the crystal surface. The dense liquid is metastable not only with respect to the crystals, but also with respect to the low-concentration solution: dynamic light scattering reveals that the droplets' lifetime is limited to several seconds, after which they decay into the low-concentration solution. The short lifetime does not allow growth to detectable dimensions so that liquid-liquid phase separation is not observed within a range of conditions broader than the one used for crystallization. If during their lifetime the droplets encounter a crystal surface, they lower their free energy not by decay, but by transformation into crystalline matter, ensuring perfect registry with the substrate. These observations illustrate two novel features of phase transformations in solutions: the existence of doubly metastable, short-lifetime dense phases and their crucial role for the growth of an ordered solid phase.

Crystallization of Transmembrane Proteins in cubo: Mechanisms of Crystal Growth and Defect Formation

Crystallization of membrane proteins is a major stumbling block en route to elucidating their structure and understanding their function. The novel concept of membrane protein crystallization from lipidic cubic phases, "in cubo", has yielded well-ordered crystals and high-resolution structures of several membrane proteins, yet progress has been slow due to the lack of understanding of the molecular mechanisms of protein transport, crystal nucleation, growth, and defect formation in cubo. Here, we examine at molecular and mesoscopic resolution with atomic force microscopy the morphology of in cubo grown bacteriorhodopsin crystals in inert buffers and during etching by detergent. The results reveal that crystal nucleation occurs following local rearrangement of the highly curved lipidic cubic phase into a lamellar structure, which is akin to that of the native membrane. Crystals grow within the bulk cubic phase surrounded by such lamellar structures, whereby transport towards a growing crystalline layer is constrained to within an individual lamella. This mechanism leads to lack of dislocations, generation of new crystalline layers at numerous locations, and to voids and block boundaries. The characteristic macroscopic lengthscale of these defects suggests that the crystals grow by attachment of single molecules to the nuclei. These insights into the mechanisms of nucleation, growth and transport in cubo provide guidance en route to a rational design of membrane protein crystallization, and promise to further advance the field.

Crystallization mechanisms of hemoglobin C in the R state

Crystallization of the mutated hemoglobin, HbC, which occurs inside red blood cells of patients expressing betaC-globin and exhibiting the homozygous CC and the heterozygous SC (in which two mutant beta-globins, S and C, are expressed) diseases, is a convenient model for processes underlying numerous condensation diseases. As a first step, we investigated the molecular-level mechanisms of crystallization of this protein from high-concentration phosphate buffer in its stable carbomonoxy form using high-resolution atomic force microscopy. We found that in conditions of equilibrium with the solution, the crystals' surface reconstructs into four-molecule-wide strands along the crystallographic a (or b) axis. However, the crystals do not grow by the alignment of such preformed strands. We found that the crystals grow by the attachment of single molecules to suitable sites on the surface. These sites are located along the edges of new layers generated by two-dimensional nucleation or by screw dislocations. During growth, the steps propagate with random velocities, with the mean being an increasing function of the crystallization driving force. These results show that the crystallization mechanisms of HbC are similar to those found for other proteins. Therefore, strategies developed to control protein crystallization in vitro may be applicable to pathology-related crystallization systems.

Capillarity effects on crystallization kinetics: insulin

During layerwise growth of crystals, capillarity governs the generation of new crystal layers. Theory predicts that the line tension of the layer edge determines, via the characteristic two-dimensional capillary length L(c), the rates of generation and initial growth of the new layers. To test the correlation between L(c) and the rate of layer generation, we used in situ Tapping Mode Atomic Force Microscopy (TM-AFM) to study the generation and spreading of layers during crystallization of rhombohedral, R3, porcine insulin. We show that crystallization of this insulin form is uniquely suitable for such an investigation due to the linear kinetics of step growth it exhibits. This linear kinetics reflects the abundance of the incorporation sites along the rough steps, the lack of long-range step-step interactions, and the transport control of the growth kinetics. The kinetic coefficients are 7 x 10(-)(3) and 4 x 10(-)(2) cm s(-)(1), respectively, in the absence and presence of the cosolvent acetone-somewhat high for proteins and comparable to values for inorganic systems. We show that (i). the relevant capillary length, the size of a critical quadrangular 2D nucleus L(c), is the main scaling factor for the density of growth steps, while (ii). all steps longer than L(c) grow with a rate determined only by the supersaturation and independent of their length. We explain the divergence of (ii). from theoretical predictions with the high supersaturations typical of the growth of this protein system.

The growth and the fluid dynamics of protein crystals and soft organic tissues: models and simulations, similarities and differences

The fluid-dynamic environment within typical growth reactors as well as the interaction of such flow with the intrinsic kinetics of the growth process are investigated in the frame of the new fields of protein crystal and tissue engineering. The paper uses available data to introduce a set of novel growth models. The surface conditions are coupled to the exchange mass flux at the specimen/culture-medium interface and lead to the introduction of a group of differential equations for the nutrient concentration around the sample and for the evolution of the construct mass displacement. These models take into account the sensitivity of the construct/liquid interface to the level of supersaturation in the case of macromolecular crystal growth and to the "direct" effect of the fluid-dynamic shear stress in the case of biological tissue growth. They then are used to show how the proposed surface kinetic laws can predict (through sophisticated numerical simulations) many of the known characteristics of protein crystals and biological tissues produced using well-known and widely used reactors. This procedure provides validation of the models and associated numerical method and at the same time gives insights into the mechanisms of the phenomena. The onset of morphological instabilities is discussed and investigated in detail. The interplay between the increasing size of the sample and the structure of the convective field established inside the reactor is analysed. It is shown that this interaction is essential in determining the time evolution of the specimen shape. Analogies about growing macromolecular crystals and growing biological tissues are pointed out in terms of behaviours and cause-and-effect relationships. These aspects lead to a common source (in terms of original mathematical models, ideas and results) made available for the scientific community under the optimistic idea that the contacts established between the "two fields of engineering" will develop into an ongoing, mutually beneficial dialogue.

Stable equidistant step trains during crystallization of insulin

Bunching of growth steps plagues layerwise crystallization of materials in laboratory, industrial, and geological environments, and theory predicts that equidistant step trains are unstable under a variety of conditions. Searching for an example of stable equidistant step trains, we monitored the generation and spatiotemporal evolution of step trains on length scales from 100 nm to 1 mm during the crystallization of insulin, using atomic force microscopy and phase-shifting interferometry. We show that near-equidistant step trains are generated by single and cooperating screw dislocation. The lack of step-step interaction and the overall transport-controlled growth regime further regularize the step train and ensure the stability of the obtained equidistant arrangement.

Noninvasive in situ observation of the crystallization kinetics of biological macromolecules by confocal laser scanning microscopy

High-resolution confocal laser scanning microscopy (CLSM) is a powerful tool for in situ observation and analysis of protein crystal growth kinetics. Because the resolution of CLSM is not diffraction-limited by the object, it is possible to visualize, under certain conditions, objects in molecular dimensions. A modified batch technique is applied which allows the growth kinetics of sufficiently small crystallites fixed at the lower side of a cover glass, within a hanging drop, to be studied in reflected light near the total reflection angle. A gap, or cavity, filled with solution is formed between the cover glass and the upper crystal face, which acts to fix small crystallites by hydrodynamic friction forces. The cavity height enables the propagation of molecular steps across the upper crystal face without constraint, so that the propagation velocity and geometrical parameters can be measured by CLSM. The layer growth kinetics of monoclinic crystallites of a long-acting insulin derivative (Insulin Glargine) is investigated. For a twofold supersaturation of the solution, the growth is governed by 2D nucleation at the edges of the crystallites followed by a spreading of molecular steps. The layer growth kinetics are well fitted by the simple cubic kinetic lattice model. We find that only about one of a thousand solute (protein) molecules which push a kink place due to their Brownian motion becomes really incorporated into the growing crystal.

Dissipating step bunches during crystallization under transport control

In studies of crystal formation by the generation and spreading of layers, equidistant step trains are considered unstable-bunches and other spatiotemporal patterns of the growth steps are viewed as ubiquitous. We provide an example to the opposite. We monitor the spatiotemporal dynamics of steps and the resulting step patterns during crystallization of the proteins ferritin and apoferritin using the atomic force microscope. The variations in step velocity and density are not correlated, indicating the lack of a long-range attraction between the steps. We show that (i) because of its coupling to bulk transport, nucleation of new layers is chaotic and occurs at the facet edges, where the interfacial supersaturation is higher; (ii) step bunches self-organize via the competition for supply from the solution; and, (iii) bunches of weakly interacting steps decay as they move along the face. Tests by numerical modeling support the conclusions about the mechanisms underlying our observations. The results from these systems suggest that during crystallization controlled by transport, with weakly or noninteracting growth steps, the stable kinetic state of the surface is an equidistant step train, and step bunches only arise during nucleation of new layers. Since nucleation only occurs at a few sites on the surface, the surface morphology may be controllably patterned or smoothened by locally controlling nucleation.

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

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 <or=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.

Mathematical model for diffusion of a protein and a precipitant about a growing protein crystal in microgravity

Equations are presented that model diffusion of a protein to the surface of a growing crystal in a convection-free environment. The equations apply to crystal growth solutions that contain both a protein and a protein precipitant. The solutions are assumed ternary and the equations include all four diffusion coefficients necessary for the full description of the diffusion process. The four diffusion coefficients are assumed constant. Effects of crystal/solution moving boundary and the effect of a protein adsorption barrier at the crystal interface are included. The equations were applied to the system lysozyme chloride + NaCl + H2O, which has served as the primary model system for the study of crystal growth of proteins and for which there are now published ternary diffusion coefficients. Calculated results with and without the inclusion of cross-term diffusion coefficients are compared.

The mass transfer process and the growth rate of protein crystals

In this paper, protein crystal growth is studied by a Mach-Zehnder interferometer and an image process system. The interference fringe images are recorded during the crystallization of tetragonal hen egg white lysozyme crystal. Concentration distributions of the protein solution are given from the interference fringe images recorded by the Mach-Zehnder interferometer with a real time servo system of a four-step phase shift. The mass transfer flux and the crystal growth rates are obtained from the concentration distribution. The results show that the observed rates are in accordance with those demonstrated by measurements of the experimental images; therefore the method for determining growth rate by the diffusion process is reasonable.

Analysis of protein crystal growth at molecular resolution by atomic force microscopy

High-resolution atomic force microscopy (AFM) studies have been performed to analyze the molecularity of growth steps of the (1 1 0) face of tetragonal lysozyme crystals. Besides a major population of step heights of about 5.5 nm also step heights of about half this size were observed. The latter steps always appeared pairwise. Both surfaces the 1 1 0) face and the (1 0 1) face could be imaged at molecular level. Comparison of the height pattern of the corresponding surface structure indicates that the (1 1 0) face is relatively smooth of less than 0.2 nm compared to the (1 0 1) face of about 1.5 nm. AFM linescan images of the (1 0 1) face indicate the insertion of lysozyme aggregates in solution to the crystal surface rather than lysozyme monomers. This study suggests that insertion of lysozyme aggregates in the solution yields growth steps of the (1 1 0) face of monomolecular as well as of bimolecular unit height.

Molecular-level thermodynamic and kinetic parameters for the self-assembly of apoferritin molecules into crystals

The self-assembly of apoferritin molecules into crystals is a suitable model for protein crystallization and aggregation; these processes underlie several biological and biomedical phenomena, as well as for protein and virus self-assembly. We use the atomic force microscope in situ, during the crystallization of apoferritin to visualize and quantify at the molecular level the processes responsible for crystal growth. To evaluate the governing thermodynamic parameters, we image the configuration of the incorporation sites, "kinks", on the surface of a growing crystal. We show that the kinks are due to thermal fluctuations of the molecules at the crystal-solution interface. This allows evaluation of the free energy of the intermolecular bond phi=3.0 k(B)T=7.3 kJ/mol. The crystallization free energy, extracted from the protein solubility, is -42 kJ/mol. Published determinations of the second virial coefficient and the protein solubility between 0 and 40 degrees C revealed that the enthalpy of crystallization is close to zero. Analyses based on these three values suggest that the main component in the crystallization driving force is the entropy gain of the water molecules bound to the protein molecules in solution and released upon crystallization. Furthermore, monitoring the incorporation of individual molecules in to the kinks, we determine the characteristic frequency of attachment of individual molecules at one set of conditions. This allows a correlation between the mesoscopic kinetic coefficient for growth and the molecular-level thermodynamic and kinetic parameters determined here. We found that step growth velocity, scaled by the molecular size, equals the product of the kink density and attachment frequency, i.e. the latter pair are the molecular-level parameters for self-assembly of the molecules into crystals.

Intrinsic instability of the concentration field in diffusion-limited growth and its effect on crystallization

The dynamic behavior of the concentration field in crystallization is investigated by considering the coupling of the bulk concentration field and interfacial kinetics. It is shown that the concentration field may become unstable for perturbations with certain wavelength. When instability occurs, the physical environment in front of the growing interface will fluctuate and the interfacial growth mode will be affected accordingly. We suggest that our analysis can be used to interpret some spatial-temporal instabilities observed in crystallization.