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

Journal of Crystal Growth 232 (2001) 63–76

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

Oleg Galkin, Peter G. Vekilov*

Centre for Microgravity and Materials Research, Department of Chemistry, University of Alabama in Huntsville,

Huntsville, AL 35899, USA

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 12.68C we varied the thermodynamicsupersaturation by changing the concentrations of protein and precipitant. We found a broken dependence of thehomogeneous 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 increasingsupersaturation as (10 or 11)! (4 or 5)! (1 or 2). Furthermore, we observe that the existence of a second liquid phaseat 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 hiddenbelow the liquidus (solubility) line in the phase diagram of the protein solution. We found that glycerol andpolyethylene glycol (PEG), which do not specifically bind to proteins, shift this phase boundary and significantlysuppress 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 theprotein concentration, or the acidity and ionicity of the solution. The effects of the two additives on the phase diagramstrongly depend on their concentration and this provides opportunities for further tuning of nucleation rates. # 2001

Elsevier Science B.V. All rights reserved.

Keywords: A1. Biocrystallization; A1. Nucleation; A1. Phase diagrams; A1. Phase equilibria; A1. Solubility; B1. Biological

macromolecules

1. Introduction

Insight into the first stage of protein crystal-lization, crystal nucleation, is still very limited. Insome cases, abundant experimental data have beenfitted to engineering or other preconceived models[1–4]. In other studies [5,6] and references therein,

select nucleation concepts were applied andthe conclusion was drawn that crystal nucleationof the studied proteins follows the classicalnucleation theories, i.e., protein crystal nucleation,similarly to crystal growth, follows the mechan-isms established for the nucleation of simpleliquids or inorganic salts from vapor orsolution. A conflicting point of view is thatmechanisms entirely different from those foundfor inorganic molecules should apply to proteincrystals [7].

*Corresponding author. Tel.: +1-256-890-6892; fax: +1-

256-890-6944.

E-mail address: [email protected] (P.G. Vekilov).

0022-0248/01/$ - see front matter # 2001 Elsevier Science B.V. All rights reserved.

PII: S 0 0 2 2 - 0 2 4 8 ( 0 1 ) 0 1 0 5 2 - 1

Furthermore, recent results on the molecularinteractions and phase behavior in proteinsolutions (that typically also contain buffer,another electrolyte and often a small-moleculeorganic additive [8]) have shown a rich variety ofphenomena that do not occur in molecularsolutions of inorganic substances [9–12] andreferences therein. Particularly intriguing isthe existence of another condensed phase, ahigh-concentration liquid, at low temperatures[9,10]. The thermodynamics of these phenomenaare more akin to the thermodynamics ofphase transitions in colloid solutions [13,14] thanto those of small-molecule inorganic solutions.Molecular dynamics simulations and analyticaltheory predict that the processes related to phaseseparation in the solution will affect the kineticsand the mechanisms of protein crystal nucleation[15].The liquid–liquid (L–L) boundary can be shifted

to higher or lower temperatures by (a) varying theacidity and ionicity of the solution [10,16], or (b)other modifiers of the protein intermolecularinteractions, e.g., non-adsorbing polymers[17,18]. If an area of enhanced nucleation existsaround the L–L boundary, such shifts provide fora nucleation control mechanism in this area.However, changing the pH and the electrolyteconcentration is not possible under physiologicalconditions, or may take laboratory or technologi-cal systems out of the crystallization conditions[19] due to changed distribution of the attractivesites on the molecular surface [20].Hence, the aims of the investigations discussed

here are to: (i) study the kinetics of crystalnucleation in a model protein system, (ii) obtaininsight into relevant mechanisms with particularattention to possible concurrent processes, e.g., theappearance of the second liquid phase that mayaffect the nucleation pathways and kinetics, (iii)test the applicability to protein crystal nucleationof the classical nucleation theory [21,22], (iv) test ifnucleation can be controlled by shifting the L–Lphase boundary, and to find suitable additives oftype (b) above.For these studies, we used lysozyme from hen

egg white (Mw �14 500Da). This enzyme was oneof the first proteins studied by X-ray diffraction

[23] and is still broadly used in studies of proteinfolding dynamics [24]. Lysozyme is a particularlyattractive crystallization model because its ther-mophysical properties are well known and it hasbeen used in numerous prior investigations (for areview see Ref. [25]). Numerous recent crystal-lization mechanism investigations using a widerange of other proteins have validated the resultsobtained with this material and justified lysozymeas a useful model system for protein crystallizationstudies.

2. Experimental section

2.1. Solution preparation

Hen egg white lysozyme, from Seikagaku, 6�crystallized, was used without additional purifica-tion. A stock solution was prepared by dissolvingthe protein powder in 0.05M acetate buffer,pH=4.5. It was then filtered through a 0.22 mmfilter (Millipore Millex-GV) and stored at 48C forfurther experiments. A 20% solution of NaCl inthe same buffer was used to add precipitant in thechosen concentration. Before each experiment asolution with desired composition was prepared bymixing the protein stock, buffer and NaCl solu-tions. The final protein concentration was deter-mined spectrophotometrically by measuringabsorbance at 280 nm using extinction coefficienta280 nm=2.64mlmg�1 cm�1 [26]. The protein solu-tion volume used in one run was about 1.5ml.

2.2. Technique for nucleation rates determinations

The existing experimental methods for determi-nations of homogeneous nucleation rates would beinapplicable or produce ambiguous results ifapplied to protein systems. For instance, differentvariants of cloud chamber [27] and supersonicnozzle expansion techniques [28] are specific forvapor–liquid nucleation. Techniques that uselevitating droplets [29,30] are prone to evaporationof solution from the liquid–air interface. Lightscattering [6,31], although a powerful technique, isheavily dependent on assumptions about the

O. Galkin, P.G. Vekilov / Journal of Crystal Growth 232 (2001) 63–7664

interactions between the molecules for data inter-pretation.Hence, we developed a novel technique that

allows direct determinations of the steady-staterate of homogeneous nucleation. In the beginningof a run, the protein solution is loaded at atemperature chosen to prevent nucleation ofcrystals or liquid–liquid demixing, see Fig. 1. Thenthe temperature is lowered to a selected T1 atwhich nucleation occurs. After a time period of Dt1temperature is raised from the nucleation tem-perature T1 to the growth temperature T2,analogous to a technique developed for studiesof electrochemical nucleation on a substrate[32]. At T2, supersaturation is at levels wherenucleation rate is practically zero, but the crystalsalready formed can grow to detectable dimensions[33]. This allows separation of the nucleationfrom the ensuing growth. After the growth stage,the crystals nucleated at T1 during Dt1 arecounted.To obtain reproducible statistical characteristics

of the random nucleation process, 400 simulta-neous trials take place under identical conditions,in solution droplets of volume 0.7 ml. To suppressthe undesired nucleation at the solution air inter-face, the droplets were suspended in inert siliconeoil, used in optimizations of the crystallizationconditions of a variety of proteins [34]. To extract

the nucleation rate from the time dependence ofthe number of nucleated crystals, five arrays of 400droplets are subjected to the nucleation super-saturation at increasing time intervals Dt1. TheseDt1’s ranged from 12min to 8 h. Thus, thedetermination of one nucleation rate data pointis based upon statistics over 2000 protein solutiondroplets. The experiment setup is schematicallyrepresented in Fig. 2. The experimental proce-dures, data collection and reproducibility, and theextensive tests to validate this technique aredescribed in detail elsewhere [35].Although the number of crystals in a droplet is a

random variable (Fig. 3), the mean number ofnucleated crystals increases linearly with time(Fig. 4) indicating steady-state nucleation at thechosen experimental conditions. Although the useof oil to cover the droplets significantly reducesheterogeneous nucleation by preventing nucleationon the droplet surface, apparently it still occurs }note the non-zero intercept of the data in Fig. 4

Fig. 1. Temperature regimes and corresponding supersatura-

tion levels used to induce nucleation during the time Dt1 and todevelop the crystals to detectable dimensions for a system with:

normal temperature dependence of the solubility, such as

lysozyme used here. Fig. 2. Schematic of experiment setup.

O. Galkin, P.G. Vekilov / Journal of Crystal Growth 232 (2001) 63–76 65

at t=0. Likely centers for this process may bethe small (50.22 mm) particles remaining in thesolution after filtration. The good linearity ofthe NðtÞ at t > 0 indicates that the heterogeneousnucleation is limited to times shorter than thoseaccessible to our technique. This mode of fastcrystal nucleation leads to a constant additive tothe number of nucleated crystals at all times. Thus,the intercept of the dependence with Y-axis in

Fig. 4 can be used to characterize the rate ofheterogeneous nucleation and the slope of thisdependence yields the homogeneous nucleation rate.

2.3. Determination of binodal and spinodal lines forliquid–liquid separation

The same set up was used to determine thebinodal of the L–L separation, TL2LðCÞ. For this,

Fig. 3. Variations in the number of crystals nucleated in a droplet between none in (a) and four in (e) under identical conditions.


arrays of 5–7 identical solution droplets weremonitored under the microscope as the tempera-ture was lowered in 0.58C increments, with 15minbetween the steps. At a certain setting, immedi-ately after the temperature was lowered, thesolutions became cloudy. Typically, this happenedsimultaneously for all droplets in an array. FurtherT-lowering by a few steps brought about nodetectable changes in the solution. After that, Twas raised in steps and typically, the dropletsbecame clear at temperatures within 0.58C fromthe T at which they had become cloudy. Theaverage of the T (cloud) and T (clarify) was takenas TL2L. Where applicable, these determinationsagree well with previous data [10]. For a real timemovie of the liquid–liquid separation, induced bysolution cooling by 18C to a value below the TL2L,see Ref. [36].First determinations of the location of the

spinodal utilized static light scattering, as de-scribed in Refs. [37,38]. We extrapolated of thetemperature dependence of the reciprocal intensityI�1 of the light scattered at 908 to the point whereI�1 reaches zero.

3. Nucleation rates

3.1. Homogeneous nucleation rates

The variations of the homogeneous nucleationrate with protein concentration at three differentconcentrations of the precipitant, NaCl, arepresented in Fig. 5a. In agreement with generalexpectations, the nucleation rate increases expo-nentially with protein concentration at eachprecipitant concentration, and, overall, is higherat higher precipitant concentrations. However,

Fig. 4. Time Dt1 dependence of the mean number of crystals in

one droplet N. Each N value is determined from the Poisson

distributions for simultaneous experiments in 400 droplets with

volume V=0.7ml, lysozyme concentration C=55.5mg/ml,

precipitant NaCl concentration CNaCl=2.5%. Error bars

correspond toffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiN=Ntrial

p. The slope of the straight line is used

to calculate the nucleation rate J in cm�3 s�1, the intercept

Nhetero characterizes heterogeneous nucleation.

Fig. 5. (a) Dependencies homogeneous nucleation rates J of

lysozyme crystals on protein concentration C at T=12.68C and

the three precipitant concentrations CNaCl indicated in the plots.

Solid lines, fits with exponential functions; dashed line for data

points at CNaCl=4% is just a guide for the eye. Datum point at

CNaCl=4% and lysozyme C=68mg/ml was obtained in a

cloudy solution and was not used in fitting procedures. Vertical

dotted line at C=66mg/ml indicates the liquid–liquid demixing

boundary at this T and CNaCl=4%. (b) Dependencies of

number of heterogeneously nucleated crystals Nhetero deter-

mined from plots similar to Fig. 4, on protein concentration at

T=12.68C and the three precipitant concentrations indicated

on the plots. Solid straight lines are just guides for the eye.


at the highest precipitant concentration,CNaCl=4%, the JðCÞ dependence contains threepeculiarities.

(i) The dependence breaks at C�=33.5mg/ml,with the sections at C5C� and C > C�following different exponents.

(ii) The data point at the highest lysozymeconcentration, C=68mg/ml and CNaCl=4%is lower than the data point determined atlower C and CNaCl, 64mg/ml and 3%respectively. We noticed that during thedetermination of the nucleation rate atCNaCl=4% and C=68mg/ml the crystalliza-tion solutions in all 2000 droplets becamecloudy immediately after temperature waslowered to the nucleation temperature ofT1=12.68C. The solutions were cloudythroughout the nucleation period Dt1 andbecame clear again when T was raised to T2

(no crystals or any other formations weredetected in the solution droplets at that time).This indicates that this set of conditions(C,CNaCl,T) is below the liquid–liquidcoexistence boundary for the lysozyme}water}NaCl system. After the growth stage,most of the crystals found in the dropletsappeared like regular tetragonal lysozymecrystals. However, in a few (�20 out of2000) droplets spherulitic crystals with thinneedles growing radially outward from acenter (‘‘sea urchin’’ morphology) weredetected, see Fig. 6. No correlation betweenthe nucleation times and the number ofdroplets with ‘‘sea urchin’’ morphology crys-tals was found. The presence of crystals withsuch unusual shape has been related tocrystallization starting below the liquid–liquid co-existence curve [10].

(iii) At C>48mg/ml the dependence bifurcateswith the data points belonging to either oftwo branches. We noticed that in the runsleading to J’s from the upper branch, in a fewdroplets (�2–3 out of 2000) spheruliticcrystals, similar to those in Fig. 6, appeared.This suggests that the closeness of the liquid–liquid coexistence boundary may be affectingcrystallization in those runs.

3.2. Heterogeneous nucleation

The mean numbers of heterogeneously nu-cleated crystals Nhetero are plotted for all studiedconditions in Fig. 5b. These Nhetero values areextracted from the intercepts of the straight linesof the time dependence of the mean number ofnucleated crystals per droplet, similar to the one inFig. 4. The number Nhetero increases with bothprotein and precipitant concentration. Note thatalthough the homogeneous nucleation rates inFig. 5a are exponential functions of the proteinconcentration, the increase in Nhetero is close tolinear.In terms of the classical nucleation theory,

heterogeneous nucleation, i.e., nucleation on aforeign surface, is enhanced if the new phase wetsthe available surface. Then, the thermodynamicwork of formation of a nucleus of the new phasecan be reduced up to several times, with thereduction factor determined by the wetting angle[39]. This is the reason for the significantly fasterrates and shorter times of this nucleation mode.The observations in Fig. 5b of more intenseheterogeneous nucleation at higher precipitant

Fig. 6. ‘‘Sea urchin’’ morphology of crystals observed in a few

of the droplets during runs at CNaCl=4% and T=12.68Cyielding higher J’s, as well as the run at CNaCl=4% and

lysozyme C=68mg/ml in which the crystals nucleated in a

cloudy solution.


and protein concentrations roughly agree with thistrend and with previous experimental observations[40]. However, the weak, linear increase in thenumber of heterogeneously nucleated crystals withprotein concentration does not fit this simplifiedscenario.

4. How many molecules are in the nucleus?

The nucleus or critical cluster of the new phaseis a cluster that has equal probability to grow or todissolve and is in a labile equilibrium with thesupersaturated solution [41– 44]. The number ofmolecules in the nucleus is the most importantcharacteristic of the nucleation process. Thenucleus size n� largely determines the height ofthe free-energy barrier for nucleation (i.e., thereversible work for nucleation), DG, and hence thenucleation rate J. The relation between DG andthe number difference between the molecules in thenucleus and in an equal solution volume, n� �n0,is treated by the nucleation theorem of Kashchievand Oxtoby [45,46], a universal, model-indepen-dent nucleation law. Since the nucleation work DGcan be estimated from the logarithm of thenucleation rate J, in terms of J and n� thenucleation theorem becomes

n* � n0 ¼ kBTq ln JqDm

þ a; ð1Þ

where a is a correction that takes values between 0and 1 [46].To present the dependencies of the nucleation

rates on the concentrations of the protein andprecipitant in Fig. 5a in the variables of thenucleation theorem, we re-plotted in Fig. 7 thedata in terms of functions of thermodynamicsupersaturation s Dm=kBT . We calculate s aslnðC=CeqÞ, where C is the protein concentration ofthe solution and Ceq is the concentration atequilibrium with large crystals at the temperatureT and precipitant concentration of the experiment[47,48]. This definition of supersaturation does notaccount for solution non-ideality [49,50]. Follow-ing the logic of Ref. [50], it is easy to show thatnon-ideality leads to a correction in the aboveexpression for the supersaturation Dm=kBT ¼

lnðC=CeqÞ þ 2B2MðC � CeqÞ þOðB3C2Þ, where

Bi are the virial coefficients and M is theprotein molecular mass. At NaCl concentrationof 2.5%, B2=�2.1� 10�4mol cm3/g, and theterms containing the higher order virial coefficientsare negligible [51]. At 75mg/ml the correctionterm 2 B2MðC � CeqÞ=�0.4, i.e., �15% oflnðC=CeqÞ=2.7. Similar estimates at the higherprecipitant concentrations require data on thesolution virial coefficients that are not available.Hence, for consistency, we employ the simplifiedexpression for supersaturation for all data inFig. 7.

Fig. 7. Dependencies of homogeneous nucleation rate J on

supersaturation s Dm=kBT at T=12.68C and at the three

precipitant concentrations indicated on the plots. Solid lines,

fits with exponential functions; dashed lines fits with the

classical nucleation theory expression, Eq. (2). Data points

with higher J’s at CNaCl=4%, and datum point obtained in a

cloudy solution at CNaCl=4% and lysozyme C=68mg/ml were

not used in fitting procedures, see text for details. Vertical

dotted lines at s=3.9 indicate the liquid–liquid demixing

boundary at this T and CNaCl=4%. (a) Linear coordinates;

(b) semi-logarithmic coordinates.


Fig. 7b indicates that at CNaCl=2.5 and 3% n�does not change throughout the respective super-saturation ranges, while at CNaCl=4% the nucleussize changes abruptly at s = 3.1, corresponding toC=33.5mg/ml. The values of n� �n0 extractedfrom the four linear segments in Fig. 7b are shownin Table 1. Since supersaturation is defined as thelogarithm of the ratio of C to Ceq, the exact valueof Ceq does not affect the slope of the straight linesin Fig. 7b. Hence, the values of n� �n0 areindependent of possible experimental errors ofthe solubility measurements.To roughly evaluate n0, we compare the

molecular diameter, �30 (AA, to the distancebetween the molecular centers n

�1=31 , where n1 is

the protein molecular concentration. AtC� 70mg/ml, n1=2.9� 1018 cm�3 and the dis-tance is � 70 (AA. At 2.5% NaCl and a proteinconcentrations close to 70mg/ml, in the volumeoccupied by a crystal consisting of �10molecules,there may be n0� 1 solute molecule. At the otherprecipitant concentrations, the n� �n0 values inTable 1 and the volume occupied by the nuclei aresmaller, hence, n0 is smaller and the correction itintroduces in the nucleus size is comparable orsmaller than a. Keeping this in mind, we canextract the n�’s corresponding to the n� �n0values in Table 1: for CNaCl=2.5% n� is 10 or11, at CNaCl=3%, n�=4 or 5, at CNaCl=4%,n�=4 or 5 and then one or two molecules. Criticalclusters consisting of one molecule have beenencountered before in investigations of electro-chemically driven nucleation of new phases underhigh overvoltages /supersaturations [32]. Underthose conditions, the nucleation rate is determinedby the kinetics of attachment of molecules to thiscritical cluster.

5. Comparison with predictions of classical

nucleation theory

One of the basic assumptions of classicalnucleation theory (CNT) is the assumption ofcontinuous cluster size changes. This is a goodapproximation to reality only for large nuclei.Thus, the size of the nuclei determined aboveprecludes application of CNT to our data. Wecompared our experimental results to the predic-tions of this theory only for the sake of complete-ness. Within the framework of this theory, thedependence of the nucleation rate on supersatura-tion s and protein molecular concentration n1 is[21,52]

J ¼ An1 exp �B=s2� �

: ð2Þ

The coefficient A is a complicated function of themolecular-level attachment-kinetics parameters.There have been attempts to analytically derivean expression for this coefficient for nucleationfrom solution [53–56]. In all cases, the finalformulae for A contain variables that are oftenimpossible to determine independently.The parameter B is related to the thermody-

namic barrier for the creation of the critical clusterDG� and for a spherical cluster can be written as

B ¼16p3

O2g3

kBTð Þ3; ð3Þ

where O is the protein molecular volume in thecrystal and g is the surface free energy of thecritical cluster. Two-parameter fits of our datawith Eq. (2) show reasonably good correspon-dence (Fig. 7a). The best-fit values of A and B areshown in Table 1. From the values of B weestimate the surface free energy g. The values of g

Table 1

Characteristics of nucleation process determined from fits of Eqs. (2)–(4) to data. For definitions and details, see text

CNaCl ð%Þ n� �n0 A (s�1) B g (mJ/m2) n�CNT

2.5 9.6� 0.2 (9� 5)� 10�16 65� 4 0.64 11–7

3 4.2� 0.2 (4� 2) � 10�18 33� 4 0.51 4–2

4 4.7� 0.3 (1.4� 0.7)� 10�17 44� 4 0.56 5–3

4 0.2� 0.3 } } } }


shown in Table 1 are lower by about half than aprevious determination in Ref. [52]. That workused expressions similar to Eqs. (2) and (3) to fitcrystallization kinetics data in which the nuclea-tion stage was not separated from growth. Hence,the estimate for g given there is of even morelimited reliability.Classical nucleation theory also allows determi-

nations of the nucleus size as

n*CNT ¼2B

s3: ð4Þ

This size continuously changes in the supersatura-tion ranges of the experiments. The nucleus sizesdetermined using Eq. (4) straddle the more accu-rate determinations based on the nucleationtheorem, Eq. (1). This correspondence seems tosupport the general belief that CNT provides a fairapproximation for the nucleation barrier [57–59],while failing to predict the pre-exponential factors.

6. Liquid–liquid separation

6.1. Effects on crystal nucleation rate

The liquid–liquid (L–L) coexistence boundariesin the system lysozyme}NaCl}water at pH=4.5maintained by acetate buffer have been determinedin our laboratory for a few concentrations of NaClusing the same materials as those used in thepresent study [10]. In the (T ,C) plane the phaseseparation curve TL2LðCÞ is best described by

TL2L ¼ T crit 1� AL2LCcrit � C

Ccrit

��1=b

( ); ð5Þ

where Ccrit=255� 30mg/ml is the critical lyso-zyme concentration, b is the critical exponent,AL�L and Tcrit are adjustable parameters with T crit

being a roughly linear function of CNaCl. Using theparameter values from Ref. [10], we calculated theliquid–liquid separation boundary for CNaCl=4%.The result is plotted in Fig. 8 together with thesolubility curve at these conditions [47] and therespective L–L and solubility curves forCNaCl=3% [47,48].The line A–A in Fig. 8 shows the locations in the

(T ,C) plane of all experiments discussed in the

previous sections. We see that the experimentalconditions are far from the L–L separation curvefor CNaCl=3% and should be even further awayfor CNaCl=2.5%. Thus, we do not expect anyeffects of the L–L demixing on the nucleation ofcrystals at these two precipitant concentrations.Correspondingly, no solution clouding or spheru-litic crystals were found in these series of experi-ments. However, the line A–A crosses the L–Lseparation curve for CNaCl=4%. The data point atthis precipitant concentration and C=68mg/ml inFig. 5a was recorded below this curve in the regionof liquid–liquid demixing, and, during nucleation,the solution was cloudy. Crystal nucleationoccurring below the L–L phase separation bound-ary is predicted to have higher-energy barriersthan at the boundary, or slightly above it [60]. Weattribute the slow nucleation at these experimentalcondition (slower that the nucleation at a lowerprotein and precipitant concentration) to thiseffect.Note that since three phases, dilute solution,

concentrated liquid, and crystals are present,the nucleation theorem does not apply to nuclea-tion.

Fig. 8. A section of the phase diagram of lysozyme solutions in

the presence of 3% and 4% NaCl. The 3% liquid–liquid

coexistence curve is according to experimental determinations

in Ref. [10], the 4% curve is calculated using Eq. (5) and

parameters determined in the same study [10]. Solubility curves

for tetragonal crystals (upper pair) at both NaCl concentrations

calculated using empirical formulae from [47]. Line A–A, B–B

and C–C mark conditions used in the studies reported here.


As shown below, the vicinity of the L–Lseparation boundary is a location of enhancednucleation of crystals. Minor variations of thesolution composition that cause small changes inthe locations of the L–L curve may have a strongeffect on the nucleation rate. We attribute thebifurcation in JðCÞ for C>48mg/ml to suchvariations [61].

6.2. Is crystal nucleation enhanced around the L–Lseparation boundary?

Fig. 9 presents the dependencies of the homo-geneous nucleation rate J on the temperature Trecorded at four solution compositions. Theconditions of these experiments are representedwith the line B–B in the phase diagram in Fig. 8. Inall cases, we see that as T is lowered, J increases,reaches a maximum near the respective TL�L andthen decreases as the system enters the L–Lseparation region.Factors that may underlie the J increase as T is

lowered before the maximum are: (i) increase ofsupersaturation for crystallization as the systemmoves away from the solubility line, (ii) theapproach to the spinodal lying below the binodalTL�LðCÞ [37] similar to the theoretical predictionsfor approaches to (Ccrit,Tcrit) [15,60,62]. At thespinodal, ðq2DG=qC2ÞT=0(DG being the free en-ergy of the solution) ([63], p. 202) and the density

fluctuations amplitudes are higher than at pointsoff of it; the critical point is a location on thespinodal where the fluctuation amplitudes are onlylimited by the container, (iii) lower free energybarrier for nucleation that could be occurring onapproach to the binodal, as suggested in Ref. [15].Note that factors (ii) and (iii) have not beenseparately considered by theory. The theoreticaltreatments of nucleation enhancement due to L–Lseparation [15,60,62] concentrate on the regionaround the critical point, where the binodal andspinodal coincide.To evaluate the effect of supersaturation in-

crease, (i) above, we note that at 208C theconcentration of a solution without additives andin equilibrium with crystals is Ceq=3.4mg/ml,while for T=148C, Ceq=2.3mg/ml [47]. For asolution containing 50mg/ml lysozyme, thesecorrespond to s=2.7 and 3.1, respectively. Fig. 7indicates that in this supersaturation range, the sincrease should lead to at most a four-fold increaseof the nucleation rate. The actual increase betweenthese two temperatures is �25-fold. Furthermore,it is unlikely that this conclusion could be affectedby possible changes in the surface free energy ofthe nuclei with temperature. Surface free energyaffects the nucleation kinetics through the nucleussize, while we found above that in this super-saturation range, this size is fixed at four or fivemolecules. We attribute the residual�six-foldincrease to the L–L separation related factors (ii)or (iii) above.To distinguish between lower nucleation bar-

riers at the binodal for L–L separation, anddensity fluctuations enhancement around thespinodal, we determined the location of thespinodal for the studied solutions (D. Petsev, workin progress). We found that the spinodal is belowthe L–L separation line by 2–88C, depending onthe protein concentration. At the concentration of50mg/ml of the experiments in Fig. 9, the differ-ence is �68C. Hence, the maximum in J is likelydue to lower nucleation barriers caused by theliquid phase.It is tempting to use the illustrative concept of

wetting to describe the enhancement of thenucleation of crystals in the area around the L–Lseparation boundary [64]. It seems to us that this

Fig. 9. Dependencies of the rate of homogeneous nucleation of

lysozyme crystals J on temperature T at pH=4.5 by 50mM

sodium acetate buffer and 4% (w/v) NaCl.


may be misleading: if the liquid wets the crystalnucleus, lowers its surface energy, and thusdecreases the nucleation barrier, it is hard tounderstand why this does not occur at tempera-tures below TL�L. Although less intuitive, theconcept of ‘‘liquid’’ structure of the crystal nucleusput forth in Ref. [15] seems more accurate. Thisinterpretation does not contradict the abovefinding of a critical cluster consisting of as few as10, or 4 or 1–2 molecules} with so few molecules,it would be difficult to assign some of them to thecrystal, and others to the liquid that wets it.For further tests of the conclusion about the

lower nucleation barriers around TL�L we carriedout JðTÞ determinations at a protein concentrationof 80mg/ml. The conditions of these experimentsare represented with the line C–C in the phasediagram in Fig. 8. The results in Fig. 9 show that Jincreases by a factor of �17� as T decreasesfrom 20 to 158C. The corresponding supersatura-tions are 3.15–3.5. As shown above, in this srange, the critical cluster contains one or twomolecules, J is a very weak function of s [61] andincrease in J under these conditions is almostentirely attributable to the liquid–liquid phaseseparation. This higher protein concentration iscloser to the critical point, and correspondingly,the effect on J is stronger.Experiments at protein concentrations of 150

and 200mg/ml revealed the expected [15,60]extremely high crystal nucleation rates close to(Ccrit,Tcrit). However, a few minutes after T waslowered to the chosen value, solution gelation[10,65] occurred and, accordingly [66], nucleationwas arrested. These observations show that gela-tion indeed plagues nucleation around the criticalpoint. A recent theory suggests that nucleationaround (Ccrit,T crit) can be thermodynamicallyuncoupled from gelation by shifting gelation tohigher C’s by finetuning the range of proteinintermolecular interactions [67]. The above ob-servations indicate that nucleation and gelationcould also be separated because of differentcharacteristic times.Upon further lowering of T in the L–L demixing

region, the nucleation rate decreases with lower T ,see Fig. 9, despite the higher crystallization super-saturation. This trend seems to correspond to the

slower nucleation at T5T crit noted in Refs. [15]and attributed the competition between thenucleation of crystals and liquid droplets [60].We cannot explain the observation that the

maximum value of J in Fig. 9 is consistentlyreached at T > TL�L by 1–1.58C. Analogousdeviation was obtained in terms DG� for the(Ccrit,T crit) region in Ref. [60] but was notaddressed in the discussion.

7. Can the rate of nucleation of protein crystals be

controlled?

The finding of a maximum in crystal nucleationrate near the L–L separation boundary, althoughnot fully understood at this point, suggests thatshifting this boundary to lower temperatures canreduce J. One may attempt this by increasing therepulsion between the protein molecules [17,18,66].Recent work with a trypsin inhibitor suggests thatglycerol increases such repulsion [68]. Glycerol ispreferentially rejected from the surroundings ofthe protein molecules [69], stabilizes their nativestructures [70], enhances their folding in aqueousmedia [24]. The JðTÞ curve in the presence ofglycerol in Fig. 9 shows that, indeed, TL�L islowered by 58C, and, at temperatures above theTL�L for solutions sans glycerol, the nucleationrate is lowered by a factor of about three. Notethat this suppression cannot be attributed toviscosity increase due to the glycerol: 5% glycerolcauses a viscosity higher by only �13% [71],which should affect the nucleation rate by the samefactor [56].In other applications, enhancement of crystal

nucleation is sought. Polyethylene glycol (PEG) isexpected to increase the attraction between colloidparticles solely due to the system’s drive tominimize the excluded volume inaccessible to thepolymer between two particles [17,18,72,73]. It hasbeen suggested that such non-adsorbing, non-bridging polymers should enhance nucleation[15]. Our results on JðTÞ in the presence of 0.2%(w/v) PEG with molecular mass 5000 Da (PEG5000) are also presented in Fig. 9. This lowconcentration of PEG does not measurably affectthe solubility or the L–L separation points [74].


Despite that, the rate of nucleation is increasedthree fold at TL�L and less than that at higher orlower T ’s. Using PEG of various molecular massesin concentrations higher than 0.5% resulted invery fast nucleation often accompanied by solu-tion gelation or amorphous precipitation of theprotein, indicating strong isotropic intermolecularattraction [75].The effect of PEG on J is considerably stronger

than that of glycerol. Investigations of the effectsof glycerol on the phase diagram of the proteinsolutions indicate that the action of PEG onTL�LðCÞ cannot be understood simply in terms ofenhanced attraction due to restricted volumeaccessible to non-adsorbing non-interacting poly-mers. Recent models and experiments [17,76,77],allowing for interactions between the polymermolecules, predict that, as the PEG concentrationincreases into the semidilute regime (where thepolymer coils overlap), the attraction between theprotein molecules should gradually taper off. Theemerging complex picture of the interactions in aPEG}protein system agrees with recent directforce measurements [78]. They show strong, non-entropic attraction at high polymer–protein se-parations, corresponding to low PEG concentra-tions, and strong repulsion at short separations.The conclusions of these works may at leastpartially explain the strong action of PEG on thenucleation kinetics.

8. Conclusions and perspectives for further work

We found that for the chosen model proteinsystem, crystal nucleation is an intrinsicallystochastic process. In this respect, protein nuclea-tion is similar to nucleation of simple liquids orwater-soluble inorganic materials. Multiple repeti-tions of a nucleation experiment under identicalconditions allow reproducible determinations ofits characteristic rates. Variations of the timeallocated for nucleation allow differentiationbetween homogeneous and heterogeneous nuclea-tion. Although the dependencies of the homo-geneous nucleation rate on protein and precipitantconcentration are similar to those found in small-molecule systems, the nuclei consist of only a few

molecules. Strictly speaking, this precludes directapplications of classical nucleation theory to thestudied system.With increasing supersaturation, imposed by

increasing protein or precipitant concentrations,the nucleus size takes discrete values of 10 or 11,then 4 or 5, then 1 or 2molecules. This leads to abroken dependence of the nucleation rate onsupersaturation that is beyond the predictingcapabilities of classical nucleation theories.Furthermore, if crystals nucleate under conditionsthat are close to the liquid–liquid separationboundary in the phase diagram of the proteinsolution, crystal nucleation kinetics are affected. Inthe region of liquid–liquid separation, nucleationrate is lower than expected for the given proteinconcentrations and temperature. On the otherhand, in the area immediately above this phaseboundary the rate of crystal nucleation is higher byup to a factor of 20 than the expected rate.Addition of glycerol shifts the L–L boundary to

lower temperatures and suppresses the rate ofnucleation above the L–L boundary, while addi-tion of PEG strongly enhances nucleation. Thisnucleation control is not confined to the twostudied additives: other substances that shift theliquid–liquid phase boundary and should helpenhance or suppress the nucleation rate of orderedsolid phases are discussed in Refs. [79,80]. Withsuitable additives, this mechanism may help toachieve a few important objectives. This mechan-ism may be applied to limit the number of crystalsto a desired few in crystallographic studies ofprotein structure [19,81], or to suppress crystal-lization and aggregation in the human bodyunderlying the protein condensation diseases[9,82]. In other cases, it may prompt strategiesfor enhancement of nucleation of hard to crystal-lize proteins, or help to achieve fast simultaneousnucleation of multiple crystallites that then cangrow to a uniform size needed for sustained drugrelease [83].Many aspects are still not well understood.

Furthermore, near the critical point for L–Lseparation, we found interplay between fastnucleation of crystals, gelation, and amorphousprecipitation. There exists no unified theory of thethermodynamics of the four or five phases and the


ensuing kinetics of the various phase transforma-tions. Such theory will have to account for thedifferent ranges of interactions that may lead togelation and the two types of phase separation[67], as well as for the anisotropy of the proteininteractions [20,75]. If polymer additives areconsidered, concentration-dependent polymer so-lution behavior and specific interactions betweenpolymer and protein molecules should be ad-dressed [76–78].

Acknowledgements

We thank D.N. Petsev, D.W. Oxtoby, and A.A.Chernov for helpful discussions and suggestionson the manuscript, S. Fraden for importantreferences, J.M. Harris, B. Hovanes and Shear-water Polymers for pure PEG, and L. Carver forexpert graphics work. Support by the NationalHeart, Lung, and Blood Institute (Grant # RO1HL58038), the Life and Microgravity Sciences andApplications Division of NASA (Grant ##NAG8-1354 and 97 HEDS 02 50), and the Stateof Alabama through the Center for Microgravityand Materials Research at the University ofAlabama in Huntsville is gratefully acknowledged.

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

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