Advancements (and challenges) in the study of protein crystal nucleation and growth; thermodynamic and kinetic explanations and comparison with small-molecule crystallization

This paper reviews advancements and some novel ideas (not yet covered by reviews and monographs) concerning thermodynamics and kinetics of protein crystal nucleation and growth, as well as some outcomes resulting therefrom. By accounting the role of physical and biochemical factors, the paper aims to present a comprehensive (rather than complete) review of recent studies and efforts to elucidate the protein crystallization process. Thermodynamic rules that govern both protein and small-molecule crystallization are considered firstly. The thermodynamically substantiated EBDE method (meaning equilibration between the cohesive energy which maintains the integrity of a crystalline cluster and the destructive energies tending to tear-up it) determines the supersaturation dependent size of stable nuclei (i.e., nuclei that are doomed to grow). The size of the stable nucleus is worth-considering because it is exactly related to the size of the critical crystal nucleus, and permits calculation of the latter. Besides, merely stable nuclei grow to visible crystals, and are detected experimentally. EBDE is applied for considering protein crystal nucleation in pores and hydrophobicity assisted protein crystallization. The logistic functional kinetics of nucleation (expressed as nuclei number density vs. nucleation time) explains quantitatively important aspects of the crystallization process, such as supersaturation dependence of crystal nuclei number density at fixed nucleation time and crystal size distribution (CSD) resulting from batch crystallization. It is shown that the CSD is instigated by the crystal nucleation stage, which produces an ogee-curve shaped CSD vs. crystal birth moments. Experimental results confirm both the logistic functional nucleation kinetics and the calculated CSD. And even though Ostwald ripening modifies the latter (because the smallest crystals dissolve rendering material for the growth of larger crystals), CSD during this terminal crystallization stage retains some traces of the CSD shape inherited from the nucleation stage. Another objective of this paper is to point-out some biochemical aspects of the protein crystallization, such as bond selection mechanism (BSM) of protein crystal nucleation and growth and the effect of electric fields exerted on the process. Finally, an in-silico study on crystal polymorph selection is reviewed.     

Protein crystallization in low gravity by step gradient diffusion method.

Two-step crystallization experiments were conducted in low gravity employing a liquid-liquid diffusion method in an effort to eliminate problems associated with protein crystal growth under the supersaturating conditions required for nucleation. Experiments were performed in diffusion cells formed by the sliding of blocks on orbit. Step gradient diffusion experiments consisted of first exposing protein solutions in diffusion half-wells for brief periods to initiating buffer solutions of high precipitant concentrations to induce nucleation followed by expoure of the same protein solutions to solutions of lower precepitant concentration to promote growth of induced nuclei into crystals. To avoid convective disturbances that occur when solutions of discrepant densities are interfaced at normal gravity, crystallization of hen egg-white lysozyme and rabbit skeletal muscle aldolase by step gradient diffusion was investigated in low gravity on four NASA space shuttle flights. In general, the largest ctystals of both proteins formed at the highest initiating precipitant concentration used, which is consistent with nuclei formation upon brief exposure to high precipitant concentration, and that these nuclei are competent for sustained growth at lower precipitant concentration. The two-step approach dissociates nucleation events from crystal growth allowing parameters affecting nucleation kinetics such as time, precipitant concentration and temperature of nucleation to be varied separately from conditions used for post-nucleation growth.     

Study on crystallization kinetics and the crystal internal defects of HMX

In this paper, the solubility data of HMX (1,3,5,7‐tetranitro‐1,3,5,7‐tetrazocane) in acetone from 323.15 K to 293.15 K were accurately measured by use of the laser‐monitoring observation technique. Intermittent dynamic method was utilized to study crystallization kinetics of HMX in acetone. The data of crystallization kinetics were obtained by moment analysis, and the parameters of the growth rate and nucleation rate equations were derived by using multiple linear least squares method. Subsequently, growth rate and nucleation rate at different conditions were calculated according to these equations. In addition, Optical Microscopy Images (qualitative) and Particle Apparent Density (quantitative) experiments were applied to study the crystal internal defects of HMX under different crystallization conditions. It can be found that the crystal apparent density of HMX is in the range of 1.8993 g·cm⁻³ to 1.9017 g·cm⁻³, very close to the theory density of HMX; the internal defects and the crystal size do not increase after 25 °C, from which we predict that the HMX crystal growth reaches the steady growth segment. These results suggest that the nucleation rate is a significant factor influencing the crystal internal defects, and larger nucleation kinetics can reduce crystal internal defects.     

DTA peak shift studies of primary crystallization in glasses

Following the recognition during the last decade that knowledge of the shift in the exothermic crystallization DTA peak as a function of pre-DTA isothermal heat-treatment times and temperatures, can provide quantitative information about the crystallization kinetics, there has been renewed interest in DTA investigations of crystallization of glasses. Most studies to date, however, have focussed on the kinetics of polymorphic crystallization (where the compositions of the crystal and the parent glass are the same). These studies have established that the DTA peak shifts to lower temperatures with increased pre-DTA heat-treatment times and JMAK-based formalisms have been developed to extract the steady state nucleation rate from the DTA peak shift data. In this paper, we report new results on the DTA peak shift in systems undergoing primary crystallization (where the compositions of the crystal and glass are different). The DTA results show that the exothermic peak temperature decreases initially but increases later on, becoming significantly larger than the initial value, with increase in the pre-DTA heat-treatment time at a fixed temperature. This increase at long times has not been reported previously and is qualitatively different than the monotonic decrease reported for polymorphic crystallization. To rationalize these new results, a model of primary crystallization has been developed which includes homogeneous nucleation, diffusion-controlled growth, Gibbs-Thomson effect, and a mean field soft-impingement correction during growth. Based on this model and experimental results, it is concluded that the initial shift to lower temperatures is due to an increase in the number of nuclei (as concluded previously by others for the case of polymorphic crystallization) and the later shift towards high temperatures in our experiments is due to diffusion-controlled growth during the pre-DTA heat treatment.     

Electrocrystallization of silver: Growth kinetics of screw-dislocation-free (100) crystal faces

At given conditions, especially at higher supersaturation, the growth rate of a close packed, perfect crystal face depends on the formation rate of two-dimensional nuclei and on the propagation rate of the monoatomic layers. This multinuclear multilayer growth as well as the advancement rate of growth steps have been studied experimentally on electrocrystallization of silver. The advancement rate of mono- and polyatomic growth steps has been measured on screw dislocation-free (100) crystal faces. For low overvoltages a linear dependence of the rate on overvoltage has been found. A strong influence of the surface condition of the crystal face — “fresh” or “aged” on the step advancement rate has been established. It was also found that on a “fresh” surface mono- and polyatomic steps advance with the same rate. The average monoatomic step spacing of the polyatomic step has been determined. The kinetic constants of the step growth rate are established and a conclusion regarding the mechanism of electrolytic deposition of silver is drawn. The initial current—time curves were recorded on applying potentiostatic pulses on a perfect crystal face. The shape of these curves coincides very well with those theoretically calculated for the cases of multinuclear growth. On the basis of the theoretical dependences, one can determine from these curves the formation rate J of two-dimensional nuclei at a given overvoltage η since the rate of step advancement is known. A linear dependence of log J on 1/η has been established. The values of the pre-exponential term in Volmer's equation and the specific edge energy of the two-dimensional nucleus have been determined. The surface condition of the crystal face influences strongly also the process of two-dimensional nucleation.     

Spontaneous crystallization of potassium chloride from aqueous and aqueous‐ethanol solutions; Part 3: Model of the crystallization process

A model of spontaneous crystallization process was proposed. The model describes kinetics of the crystallization process after the end of the induction period. To test the model the published earlier data on crystallization and aggregation kinetics of potassium chloride at its spontaneous crystallization from supersaturated aqueous and aqueous‐ethanol solutions were used. It was found excellent coincidence of the experimental and theoretical data on concentration of the salt and the total number of crystals in solution at crystallization. Somewhat change for the worse was at the theoretical calculations of crystal size distribution at the end of the crystallization process. It indicated that the ways of calculation of size of crystals and their weight fraction in deposit were very approximate. The model allows predicting with satisfactory accuracy kinetics of crystallization using such general parameters of potassium chloride as the specific surface energy and the height of the nucleus‐bridges between crystals at coalescence. It needs further test of the model for other salts. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The estimation of crystallization kinetics from linear in n(L) versus L steady-state data including growth rate dispersion

A discussion of modelling of nucleation and crystal growth kinetics is presented for those mixedsuspension, mixed product removal (MSMPR) crystallizer data available in the literature, for which log population density data appears to be a linear function of crystal size. The modelling crystallization kinetics is done twice – once assuming that the system is obeying McCabe's Δ law, and again when the crystals exhibit growth rate dispersion (GRD). In the latter case the gamma, inverse gamma, log-normal, beta, Fisher, sum of two gammas, sum of two inverse gammas, sum of two log-normal, and sum of two constant rates of growth rate dispersion models are used for crystal growth rate estimation. Finally, McCabe's and GRD predictions are compared with the size-dependent growth rate estimations.     

Crystallization of n-doped amorphous silicon PECVD films: comparison between SPC and RTA methods

Amorphous silicon films are deposited by radio-frequency plasma-enhanced chemical-vapor deposition (RF-PECVD) with different n-doping rates. The amorphous films are subsequently crystallized using either solid phase crystallization (SPC) or rapid thermal annealing (RTA). We compare the effect of the n-doping rate on some properties of the microcrystalline silicon films obtained with both techniques. In the SPC process, the time required for the beginning of the crystallization decreases with increasing phosphorus doping. Moreover, doped films present slightly higher crystal size than intrinsic films but the doping rate does not significantly influence the grain size. For RTA, the doping rate decreases the crystallization temperature and increases significantly the crystal size. Whatever the doping rate, the average grain sizes obtained by RTA are larger than those obtained by SPC. The electrical resistance of the crystallized films also depends on the crystallization process: RTA films present a lower dark conductivity than SPC films. These results are discussed taking into account the different kinetics of both crystallization techniques and the role played by the silicon dangling bonds and their charge states on the crystal growth.     

纳米氧化镱对熔融石英析晶动力学机制的影响

本文研究引入纳米氧化镱对熔融石英析晶机制及动力学过程的影响.通过X射线衍射仪(XRD)测试结果分析了纳米氧化镱的引入对熔融石英陶瓷析晶率的影响,采用动力学方法分析了纳米氧化镱的引入对熔融石英陶瓷析晶机制的影响,同时探讨了其等温析晶动力学过程.研究表明,熔融石英陶瓷的晶粒向二维兼有一维及三维的方式生长,引入纳米氧化镱的熔...     

Study of phosphoric acid crystallization using a focused beam reflectance measurement method

A way for restoring the crystal size distributions (CSD) from measured chord length distributions (CLD) was reported in this paper. The kinetics of phosphoric acid crystallization process was investigated in cooling mode using focused beam reflectance measurement (FBRM) and digital photo technique. In order to restore the CSD from measured CLD and verify the reliability of FBRM data, digital photo technique in real time and optical microscope were applied in large crystal size and small range, respectively. Results indicated a converting constant A existed between CLD and CSD when crystal growth follows size‐independent growth (Mcabe's ΔL law) law. It was verified by Malvern particles size analysis method. The converting constant A varied with crystal morphology. The crystal growth order increased with the stirring increasing speed during phosphoric acid crystallization process. The trend was especially notable at higher speed situations. It can illustrate that the state of phosphoric acid hemihydrate crystal growth was controlled by both diffusion and surface‐integration with the increasing stirring speed. (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Crystallization in solid solution-aqueous solution systems: Thermodynamic and kinetic approaches

A new phenomenological approach is proposed to describe the crystallization kinetics in solid solution-aqueous solution binary systems. The phase diagrams, equilibria, and quasie-quilibria are considered within this approach. The crystallization kinetics near the true equilibrium and the crystallization features at large deviations from equilibrium are discussed on this basis. Special attention is paid to possible interactions in a solution with a seed crystal placed in it. In particular, the interactions leading to the seed’s crystal growth or dissolution and to a possible exchange or metasomatic reactions are considered. In addition, the effect of the generated mismatch stress on the crystal growth rate and composition is analyzed.     

Determination of reactive crystallization kinetics of 5‐(difluromethoxy)‐2‐mercapto‐1H‐benzimidazole in a batch crystallizer

5‐(difluromethoxy)‐2‐mercapto‐1H‐benzimidazole (DMB) was crystallized via the reaction between sodium salt of DMB and acetic acid. In this work, we firstly measured the solubility data of DMB in binary ethanol‐water mixture at different temperatures. Then, the reactive crystallization kinetics of DMB was determined in the batch crystallizer. Based on the population balance and mass balance, the kinetics model of reactive crystallization process of DMB was established. And the nucleation rate, size‐independent crystal growth and agglomeration kernel were determined by method of classes. Finally, based on the population balance, numeric simulation was done using the above crystallization kinetics in order to verify its reliability.     

定位硅酸锌结晶釉析晶动力学研究

本文利用定位结晶法研究了氧化锌和硅酸锌两种晶种对硅酸锌结晶釉釉面晶花的影响,根据结晶动力学理论,对结晶釉中硅酸锌晶体的生长规律和结晶机理进行了探讨,并比较了不同晶种对硅酸锌晶体生长规律的影响.实验结果表明:采用氧化锌和硅酸锌作定位晶种的釉面晶花外观效果基本相同;釉熔体中硅酸锌晶体生长半径与保温时间成线性关系;不同的定位晶种对硅酸锌晶体的生长基本没有影响;结晶常数随着析晶保温温度的提高而增加,当析晶保温温度分别为1150 ℃,1100 ℃和1050 ℃时,所对应的结晶常数依次为1.0911,0.9996和0.7177;即使在同一析晶保温温度下,不同的保温时间段其结晶常数也不同.     

Main regularities of crystallization of calcium oxalate in the presence of amino acids

Processes of crystallization of calcium oxalate from aqueous solutions are investigated. Main regularities of the nucleation and crystal growth of calcium oxalate during mass crystallization without admixtures and in the presence of amino acids are established. The inhibiting action of the aspartic and glutamic amino acids on the growth of calcium oxalate crystals is demonstrated. Addition of amino acids does not affect the nucleation kinetics.     

Kinetics and mechanism of corundum formation in the thermal decomposition of basic aluminium chloride gels

Compositions and corundum contents in products, formed by thermal decomposition of a basic aluminium chloride gel at temperatures up to 600 °C, were determined at variable temperatures and times of decomposition. For the kinetics of α-alumina crystallization a model is proposed, valid for diffusion-controlled two-dimensional growth. A mechanistic interpretation of these results and of those from previous examinations is based on the essential influence of the gel structure and of the atmosphere in thermal decomposition. As the rate-controlling step of corundum formation the transportation of HCl and H₂O molecules is discussed.     

Sinter-crystallization of a glass obtained from basaltic tuffs

Glass-ceramic materials, obtained by sinter-crystallization of melted alkaline-olivine basaltic tuffs, were investigated. The kinetics of bulk crystallization was evaluated by differential thermal analysis (DTA) at different heating rates. The phase formation and the sintering behavior of glass powders (<75 μm) were studied in air and in nitrogen atmospheres by DTA and dilatometry, respectively. The crystalline phases formed were identified by X-ray diffraction. The DTA traces showed an unusual phase formation behavior with a higher crystallization trend for the bulk samples. The crystallization activation energy was evaluated as 590 ± 20 kJ/mol in the range 1080–1110 K. A value of about 3 of the Avrami constant, corresponding to three-dimensional growth on a fixed number of nuclei, was evaluated by Ozawa and Augis–Bennet methods. The densification at low-temperatures is reduced by the intensive crystallization process in both air and nitrogen atmospheres. The sintering starts again at 1150–1250 K. In air atmospheres, due to the FeO oxidation, the sintering temperature increases and the percentage of formed crystal phase decreases by 15–20%.     

A method for determination of size-dependent crystal growth kinetics from batch experiments (I). One-experiment estimation

The method developed for determining size-dependent crystal growth kinetics makes use of the assumption that the linear growth rate is a product of a size-dependent factor and a supersaturation-dependent one. The transformed form of the population balance equation, obtained by means of this assumption, is the basis of the evaluation utilizing measurement data from one seeded batch crystallization experiment, namely the size distributions of the seed and product crystals as well as the supersaturation course. The application of the method is illustrated by evaluating an experiment with potash alum crystals.     

Kinetic studies on the influence of temperature and growth rate history on crystal growth

Crystallization experiments of sucrose were performed in a batch crystallizer to study the effect of temperature and growth rate history on the crystal growth kinetics. In one of the growth methods adopted, the isothermal volumetric growth rate (R_V) is determined as a function of supersaturation (S) at 35, 40 and 45 ºC. In the other, crystals are allowed to grow at constant supersaturation by automatically controlling the solution temperature as the solute concentration decreased. Using the latter method R_V is calculated as the solution is cooled. The obtained results are interpreted using empirical, engineering and fundamental perspectives of crystal growth. Firstly, the overall activation energy (E_A) is determined from the empirical growth constants obtained in the isothermal method. The concept of falsified kinetics, widely used in chemical reaction engineering, is then extended to the crystal growth of sucrose in order to estimate the true activation energy (E_T) from the diffusion‐affected constant, E_A. The differences found in the isothermal and constant supersaturation methods are explained from the viewpoint of the spiral nucleation mechanism, taking into account different crystal surface properties caused by the growth rate history in each method. Finally, the crystal growth curve obtained in the batch crystallizer at 40 ºC is compared with the one obtained in a fluidized bed crystallizer at the same temperature. Apparently divergent results are explained by the effects of crystal size, hydrodynamic conditions and growth rate history on the crystallization kinetics of sucrose. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solvent effects on the growth kinetics of subtilisin crystals

The effects of salts on subtilisin crystallization were investigated. Three salts—NaCl, NaNO₃ and NaSCN—were selected to study the effects of different anions on growth kinetics of three subtilisin mutants—Properase^®, Purafect^® and Purafect^®OX. The effectiveness of salts in decreasing the solubility of Properase^® and Purafect^® subtilisin followed the reverse order of the Hofmeister series: SCN⁻>NO₃⁻>Cl⁻. The average length and diameter of crystals were measured during crystallization. The nature of salt changed the length/diameter ratio of crystals, indicating the changes in the relative growth rate of different crystal faces. The required supersaturation, (c−s)/s, for a given growth rate increased in the order of NaCl, NaNO₃ and NaSCN. The observed trend in required supersaturation indicates a kinetic effect and was counter to the trend for the solubility data. A rationale is provided based on the influence of ion binding and kinetics on the energetics of crystal growth and growth rate is correlated to the molar Gibbs free energy of hydration of the anion.