Is Agarose an Impurity or an Impurity Filter? In Situ Observation of the Joint Gel/Impurity Effect on Protein Crystal Growth Kinetics Alexander E. S. Van Driessche,† Fermin Ota´lora,*,† Jose A. Gavira,† and Gen Sazaki†,‡ Laboratorio de Estudios Cristalogra´ficos, IACT, CSIC, P.T. Ciencias de la Salud, AVenida del conocimiento s/n, 18100 Armilla (Granada), Spain, and Center for Interdisciplinary Research, Tohoku UniVersity, Aramaki, Aoba-ku, Sendai 980-8578, Japan
CRYSTAL GROWTH & DESIGN 2008 VOL. 8, NO. 10 3623–3629
ReceiVed February 11, 2008; ReVised Manuscript ReceiVed July 21, 2008
ABSTRACT: The joint effect of agarose gel and impurities on hen egg white lysozyme crystal growth kinetics was investigated in situ by comparing the two-dimensional (2D) nucleation rate and the step velocity of crystals growing from free and gelled (agarose) solutions having two different levels of purity: highly purified (99.99% pure) and commercial grade (98.5% pure). The 2D nucleation rate and step velocity were measured on {110} faces of tetragonal lysozyme crystals using laser confocal microscopy combined with differential interference contrast microscopy (LCM-DIM). 2D nucleation rates are enhanced by the presence of gel fibers that act as heterogeneous nucleation sites. These results also show that the specific surface energies are similar for the gel fiber/crystal interface and for the gel fiber/solution interface. This is consistent with the observed incorporation of agarose fibers into the lysozyme crystal lattice and the small effect of gel fibers on step velocity. 2D nucleation in the presence of both gel and impurities is also enhanced but not as much as for gelled purified solutions. The presence of agarose has an almost negligible effect on the step velocity in purified solutions but significantly modifies the step velocity in crystals growing from impure solutions, shifting these values closer to the velocities measured in purified solutions. This velocity increase corresponds to a 7-fold reduction in the concentration of adsorbed impurities at the crystal surface with respect to ungelled experiments. This direct evidence of the diffusive impurity filtering concept is also consistent with the qualitative observations on 2D island morphologies.
1. Introduction Gelled solutions1–5 are, along with thin capillaries,6–8 the best methods to implement a diffusive crystallization media on earth. They have, over capillaries, the advantage of being much more effective in reducing convection, making handling of crystals easier, especially for cryocooling techniques, and not limiting crystal size. In the field of macromolecular crystal growth, several studies have shown2,4,9–11 that gel crystal growth techniques, mainly using low concentration agarose gels, produce crystals of very good quality and larger size, which has been attributed to reduced and steady supersaturation at the crystal/solution interface4,12,13 and filtering of impurities14,15 through the diffusive concentration gradient around growing crystals. A striking peculiarity of these crystals is that they “trap” the gel within the crystal volume, incorporating, apparently undistorted, the three-dimensional frameworks of cross-linked agarose fibers.16 Although incorporation of gel fibers does not seem to have any major detrimental impact on the X-ray diffraction quality of the crystals,10 chemical interaction must exist between the gel fibers and the protein molecules, thus qualifying agarose gel as a potential impurity for the crystal growth process. Despite the large number of macromolecules reported to produce good crystals in agarose gels17 and the detailed characterization of this media in crystal growth,12,16 the study of their impact on crystal growth kinetics has been restricted to three-dimensional nucleation, showing that agarose gel works as a nucleation promoter.18,19 But crystal quality is much more related to growth kinetics than to nucleation and, up to today, we do not even know if agarose gels operate as an impurity, as an impurity filter, or both. * Corresponding author. E-mail:
[email protected]. Phone: +34958 181621. Fax: +34 958 181632. † LEC, IACT, CSIC, P.T. Ciencias de la Salud. ‡ Tohoku University.
In this paper this dual character of agarose gels in crystal growth experiments is investigated from the viewpoint of growth kinetics. The use of a noninvasive technique able to image in situ the advancement of growth steps in gelled media and the orthogonal approach to the effect of both gel and impurities are the two key aspects to face this challenging problem.
2. Experimental Methods 2.1. In Situ Observation by LCM-DIM. Growing {110} faces of tetragonal crystals of the model protein hen egg white lysozyme were observed in situ by laser confocal interference contrast microscopy (LCM-DIM),20–22 which provides a significant contrast level for elementary steps of nanometer height. This advanced optical technique is implemented using a confocal system (FV300, Olympus) attached to an inverted optical microscope (IX70, Olympus) with a 20× objective lens (LUCplan FLN 20×, Olympus) and equipped with a Nomarski prism introduced into the optical path and a partially coherent superluminescent diode (Amonics Ltd., model ASLD68-050-B-FA: 680 nm) to eliminate interference fringes. The observation cell used in this work was made of two sandwiched glass plates of 0.17 mm thickness separated by 1-mm-thick polystyrene spacers glued by silicone adhesive to one of the glass plates. After the polymerization of the adhesive, the cell was carefully washed by ultrasonic cleaning in Milli-Q water. Previously grown lysozyme seed crystals (0.1-0.3 mm in height) were transferred to the observation cell with the {110} faces parallel to the bottom glass plate. Step velocities and 2D nucleation rates were measured at the free (upper) solution-crystal interface. More details about this experimental setup can be found in previous works.20–22 2.2. Lysozyme Solutions. To unravel the possible impurity effects of agarose gel on crystal growth, a highly purified lysozyme solution (99.99% purity, Maruwa food Inc.) was used. To test for the effect of protein impurities in gelled solution a commercial grade lysozyme (98.5% purity: 6× recrystallized, Seikagaku Co.) was used. The protein impurities present in Seikagaku lysozyme were previously identified and quantified23 to be 0.5% of lysozyme dimer, 1.0% of a 18 kDa polypeptide and less than 0.1% of a 39 kDa polypeptide. Gelled solutions at different concentrations (Ca), 0.025%, 0.075%, 0.125% and 0.175% (w/v) were prepared following standard protocols.16
10.1021/cg800157t CCC: $40.75 2008 American Chemical Society Published on Web 09/09/2008
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2.3. Production of Macroseeds and Seeding Procedure. Tetragonal seed crystals of lysozyme were grown at 20.0 ( 0.1 °C from a solution containing 70 mg/mL commercial grade lysozyme (98.5%, Seikagaku), 25 mg/mL NaCl and 50 mM sodium acetate (pH 4.5) buffer. After seed crystals were transferred to an observation cell (1 × 10 × 20 mm3), the solution inside the cell was replaced with a supersaturated solution of highly purified lysozyme (99.99%) or commercial grade lysozyme (98.5%) and a given concentration of agarose gel (Hispanagar D5). Replacement of the solution inside the cell with a gelled solution is possible because the gelification process has a certain induction time, especially for low agarose concentrations. Once the solution was replaced, the observation cell was incubated for 24 h at 23.0 °C until the surfaces of the seed crystals were slightly overgrown. 2.4. Supersaturation Control and Notation. To control the solubility of the protein solution, the observation cell was set on a temperature controlled stage with Peltier elements. Solubility was calculated from data reported by Sazaki and co-workers.24 Step velocities (V) are assumed to be a linear function of C - Ce:25 V ) Ωβ(C - Ce)
(1)
25
according to Chernov’s hypothesis on the direct integration of growth units in the steps. Measurements shown in Figure 3 will prove that this is a usefull approximation for step velocities in ungelled and gelled purified lysozyme solutions. Here Ω ) 3 × 10-20 cm3 is the volume of one lysozyme molecule,26 β is a kinetic coefficient, C is the bulk concentration of lysozyme and Ce is the solubility. Observations were carried out in the supersaturation range C - Ce ) 0-45 mg/mL in free solution and C Ce ) 18-35 mg/mL in gelled solution. Nucleation rates are represented versus
σ ≡ ln(C ⁄ Ce) in log/log plots because22,27
(2)
(
J ) A exp -
πsκ2 2k2T2 ln(C ⁄ Ce)
)
(3)
where s ) 1.06 × 10-17 m2 is the area of one molecule occupied inside a nucleus,28 κ a ledge free energy of a 2D cluster, and the factor 2 in the denominator arises, as reported in our previous paper,22 from the bimolecular height of steps on {110} faces.29–32 Nucleation rate measurements were done in the supersaturation range σ ) 0.0-1.4 in free solution and σ ) 0.3-1.3 in gelled solution.
3. Results 3.1. Effect of Agarose Gel. To test for the effects of agarose gel on lysozyme crystallization, a series of experiments were performed using exclusively purified lysozyme solutions (99.99% pure). All results in this subsection come from these “almost” impurity-free experiments. Lens-shaped 2D islands, pointed in the direction and having an elementary step height of 5.6 nm,27–30 develop on the {110} face of tetragonal lysozyme crystals growing from ungelled purified solutions (Figure 1a). When the same experiment is repeated using gelled solutions (Figure 1b-d), images look noisier owing to the presence of gel fibers in the confocal plane but the overall island shape is very similar to the one observed for free solution growth (Figure 1a). Consequently, the presence of gel fibers has not a major impact on the overall island morphology. Within the range of supersaturation (σ ) 0.3-1.3) and agarose concentration (Ca ) 0 - 0.175%) used in our experiments, 2D nucleation was observed to occur randomly over the entire crystal surface, as was the case in free solutions.22 Neither
Figure 1. LCM-DIM photomicrographs of the morphology of 2D islands on {110} faces of growing tetragonal lysozyme crystals. (a) 0% gel, (b) 0.075% (w/v) gel, (c) 0.125% (w/v) and (d) 0.175% (w/v). Growth conditions: 99.99% pure lysozyme 45 mg/mL (a), 44 mg/mL (b), 49 mg/mL (c), 44.5 mg/mL (d), NaCl 25 mg/mL, in 50 mM sodium acetate buffer (pH 4.5), at 24.0 °C (a), 25.0 °C (b) and (c), 24.5 °C (d).
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Figure 2. Changes in ln J as a function of 1/[T2 ln(C/Ce)], measured on {110} faces of tetragonal lysozyme crystals. J is the 2D nucleation rate, T absolute temperature, C the bulk concentration of lysozyme, and Ce the solubility. In the figure legend, Ca corresponds to the agarose concentration. Straight lines are the results of regression by eq 3. Growth conditions: 99.99% pure lysozyme, NaCl 25 mg/mL, in 50 mM sodium acetate buffer (pH 4.5).
Figure 3. Step velocities V, in directions, measured on {110} faces of tetragonal lysozyme crystals under various supersaturations and agarose gel concentrations by LCM-DIM. C is the bulk concentration of lysozyme, and Ce is the solubility. In the figure legend, Ca corresponds to an agarose concentration. Growth conditions: 99.99% pure lysozyme, NaCl 25 mg/mL, in 50 mM sodium acetate buffer (pH 4.5).
repeated nucleation nor growth of multilayer islands was observed in purified gelled solutions. 2D nucleation rates measured for different supersaturations and gel concentrations are shown in Figure 2. Two clearly different regimes are observed for the 2D nucleation rate from ungelled solutions (filled disks in Figure 2) as indicated by the two dotted regression lines. The crossover between these regimes is around σ ) 0.8. In the case of tetragonal lysozyme this behavior has been explained as due to the transition from heterogeneous to homogeneous nucleation as supersaturation increased.22 This crossover was also observed in the gelled solution experiments (open symbols in Figure 2) as indicated by the two dashed-dotted regression lines fitted to the data of agarose concentration Ca ) 0.175%. The presence of this crossover for both gelled and ungelled solution indicates that this transition is not due to the presence of gel fibers, but would probably be related to impurities adsorbed on the crystal surface as was discussed previously for heterogeneous 2D nucleation on crystals growing from free solution.22 The 2D nucleation rate from purified solutions in gelled and ungelled experiments were fit to linear relations accounting for the expected supersaturation dependency of the homogeneous nucleation rate given by eq 3. Analysis of variance32 of the data factorized by agarose concentration was used to test for statistically significant differences in slopes. Only the homogeneous nucleation part of the data under a higher supersaturation range was used for the regression. Three distinct groups of data showing statistically significant differences of slopes within the 5% confidence level were identified by this analysis. They correspond to (a) the free solution data (dotted line in Figure 2), (b) the highest agarose concentration (Ca ) 0.175%) (dash-dotted line), and (c) the low and middle agarose concentration experiments together (dashed line). Consequently, there is a clear effect of gel in the exponential term of the nucleation rate equation; this effect has the same magnitude for agarose concentrations up to 0.125% and is larger for Ca ) 0.175%. Our experimental data is not enough to ascertain the discrete or continuous character of this change. Figure 3 shows measured average step velocities V for different supersaturations and agarose concentrations in crystals growing from purified lysozyme solutions. Step velocity is intrinsically anisotropic, typically 6 times larger in the (fast) direction than in the (slow) direction. Only data
in the direction are plotted for clarity, but the same conclusions are derived from the data in the direction. In the Chernov model25 step velocity is expected to be a linear function of the supersaturation (eq 1); the series for ungelled solutions (filled disks in Figure 3) shows this linear dependency for C - Ce > 10 mg/mL, and below this limit, the slope continuously decreases for decreasing supersaturation probably as an effect of impurities,33 even in the highly purified samples. Step velocity values for all the experiments in gelled solutions (open symbols) are also linearly dependent on the supersaturation with the same trend but with slightly smaller velocity, at least at low supersaturation. These data were fit and tested for significant differences as before. No significant slope differences were found within the 5% or even 10% confidence limit meaning that the effect of gel, if any, only accounts for a small decrease in the step velocity that is not statistically significant. 3.2. Effect of Impurities. A second series of experiments were performed using the same setup and procedures, but this time protein solutions were prepared from commercial grade lysozyme (98.5% pure) to test for the effect of protein impurities. All the results reported in this subsection refer to these “commercial grade lysozyme” experiments. Figure 4a shows the typical elliptical shape of 2D islands growing from an ungelled commercial grade lysozyme solution. This shape is clearly different from the pointed lens shape shown in Figure 1a-d due to the anisotropic effect of impurity incorporation that results in a larger retardation of the steps in the fast (pointed) direction.33 Surprisingly, the morphology of 2D islands growing from gelled commercial grade lysozyme solution (Figure 4b-d) is closer to that of purified solutions (Figure 1a-d). This lens-like morphology is systematically observed in gelled solutions at C - Ce > 20 mg/mL despite the presence or absence of impurities. As in the experiments using purified lysozyme, 2D nucleation was observed to occur randomly over the entire crystal surface without repeated nucleation or formation of multilayer islands. As shown in Figure 5, the 2D nucleation rate is also increased in these experiments by the presence of agarose gel, but this effect is almost independent of the gel concentration and, in overall, smaller than for purified lysozyme. 2D nucleation rates slightly increase from agarose concentration Ca ) 0.025 to 0.075% and then stays constant, but this difference is not
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Figure 4. Photomicrographs of 2D islands morphology on {110} faces of tetragonal lysozyme crystals. (a) no gel, (b) 0.075% (w/v) gel, (c) 0.125% (w/v), and (d) 0.175% (w/v). Growth conditions: commercial grade (98.5% pure) lysozyme 40 mg/mL (a) and (d), 38 mg/mL (b), 41 mg/mL (c), NaCl 25 mg/mL, in 50 mM sodium acetate buffer (pH 4.5), at 25.0 °C (a), 20.0 °C (b) and (c), 19.0 °C (d).
Figure 5. Changes in ln J as a function of 1/[T2 ln(C/Ce)], measured on {110} faces of tetragonal lysozyme crystals. J is the 2D nucleation rate, T absolute temperature, C the bulk concentration of lysozyme, and Ce the solubility. In the figure legend, Ca corresponds to an agarose concentration. Straight lines are the results of regression by eq 3. Growth conditions: commercial grade (98.5% pure) lysozyme, NaCl 25 mg/ mL, in 50 mM sodium acetate buffer (pH 4.5).
Figure 6. Step velocities V in directions measured on {110} faces of tetragonal lysozyme crystals under various supersaturations and agarose gel concentrations. C is the bulk concentration of lysozyme, and Ce is the solubility. In figure legend, Ca corresponds to the agarose concentration. Growth conditions: commercial grade (98.5% pure) lysozyme, NaCl 25 mg/mL, in 50 mM sodium acetate buffer (pH 4.5).
statistically significant so all data from gelled experiments are fitted together by the dashed regression line in Figure 5. We observed that crystals grown from a free solution of commercial grade lysozyme (solid disks in Figure 6) experience a significant reduction of step velocities with respect to purified solutions (solid disks in Figure 3 also represented in Figure 6 as a reference line labeled “purified ungelled”) especially in the lower supersaturation range indicating a strong impurity effect. Above a critical supersaturation, C - Ce ) 25 mg/mL, a rapid increase in step velocity occurs and around C - Ce ) 35 mg/
mL step velocities become similar to these of purified solution experiments. Figure 6 also shows the step velocities found for commercial grade lysozyme in gelled solution (open symbols and dotted regression line). There is a clear influence of the gelled solution on the step velocity for commercial grade lysozyme solutions. The velocity of the step is still reduced but lies between the values for purified free solutions and commercial grade free solutions. A rapid increase in step velocity occurs above a critical supersaturation of C - Ce ) 17 mg/ mL. Step velocities become similar to the purified solution
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Crystal Growth & Design, Vol. 8, No. 10, 2008 3627
m(θ) ) cos θ ) (γsf - γsc) ⁄ γcf
Figure 7. Variation of cos(θ) (solid curves) as a function of supersaturation for the four agarose concentrations used in the experiments. The cos(θ) ) 0 line (dashed) and the f ) 0.35, 0.40,... 0.65 curves (dotted curves labeled at the right margin) are shown for reference.
experiments around C - Ce ) 28 mg/mL. No significant effect of the gel concentration in the step velocity was observed.
4. Discussion The birth and spread mechanism for crystal growth, of widespread application to most growth situations at intermediate supersaturation, explains the growth of the crystal interface as a consequence of successive events of 2D nucleation and lateral spread of such nuclei to cover the crystal surface. Both the nucleation and the growth of 2D islands are modified by the presence of impurities. In our experiments the presence of agarose increases 2D nucleation rates (Figure 2). Because supersaturation and interactions in solution are not affected by the presence of agarose gel,19,34 this effect must be due to heterogeneous nucleation, that is, a reduction of the 2D nucleation barrier induced by the agarose fibers. Heterogeneous 2D nucleation has been addressed seldomly in the literature, despite the wide field of application in real crystal growth experiments. The basic idea is that an impurity present at the crystal surface can promote or reduce 2D nucleation depending on the energies of heterogeneous nucleation. Impurities that reduce the surface energy contribution for the 2D nuclei will increase the 2D nucleation frequency while impurities increasing this surface energy will reduce 2D nucleation, poisoning the surface growth. The analysis of this problem by Liu and co-workers35 predicts an effect of impurities leading to a nucleation frequency
(
J ) Aδ exp -
)
πsκ2 f 2k2T2 ln(C ⁄ Ce)
(4)
where the terms f and δ, ranging from 0 to 1, account for all heterogeneous effects in 2D nucleation. The limiting case f ) δ ) 1 reduces eq 4 to eq 3. Since f e 1 and ln δ e 0, f and δ accounts for an exponential increase and a linear decrease of the nucleation rate, respectively. In principle, both f and δ can be obtained from the regression analysis shown in Figure 2, but the absence of data for very high supersaturation values makes δ values unreliable. Equation 11 from Liu et al.35 shows that f is a function of the cosinus of the wetting angle θ and the ratio between the radius of the impurity provoking heterogeneous nucleation (Rs) and the radius of the critical 2D nucleus (rc). Because f can be obtained from the experimental data, rc can be computed and Rs is available in literature, it follows that a estimation of cos(θ) can be obtained. This value is central for the understanding of the interactions between the growing crystal and the gel fibers. Young equation
(5)
states that cos(θ) is a function of the specific energies of the fiber/solution interface (γsf), the fiber/crystal interface (γsc) and the crystal/solution interface (γcf). When cos(θ) ≈ 1, complete wetting exists between the nuclei and the gel fiber and incorporation of the fiber into the crystal lattice is thermodynamically favored because γsc is negligible. The opposite case, cos(θ) ≈ -1, corresponds to the situation were γsf is negligible and no wetting is possible, and thus, the fiber will be excluded from the crystal lattice. Finally, the case cos(θ) ≈ 0 corresponds to situations where γsf ≈ γsc and, therefore, trapping of fibers neither increase nor reduce significantly the energy of the interface. f values were computed from the slope of the regression lines in Figure 2, normalized by the corresponding slope of the ungelled experiment. The resulting f factors for the different agarose concentrations are f0.025 ) 0.64, f0.075 ) 0.58, f0.125 ) 0.62 and f0.175 ) 0.39. The radius of the agarose gel Rs was estimated to be around 6 nm from values provided in literature36 for low agarose concentration. The radius of the critical nuclei rc was computed from the equation22
rc )
sκ 2kT ln(C ⁄ Ce)
(6)
and, within our experimental conditions (0.8 e ln(C/Ce) e 1.4), corresponds to 1.7 nm e rc e 3.2 nm. These f, Rs and rc values lead to -0.132 e cos(θ) e -0.056 for Ca ) 0.025, -0.033 e cos(θ) e 0.046 for Ca ) 0.075, -0.099 e cos(θ) e -0.021 for Ca ) 0.125 and 0.266 e cos(θ) e 0.348 for Ca ) 0.175. The variation of cos(θ) within the supersaturation range used in the experiments is shown in Figure 7 as computed from the variation of rc with supersaturation and assuming that both f and Rs are independent of supersaturation. These cos(θ) values are close to zero, except for the highest agarose concentration, and decreases with increasing supersaturation, which indicates that the energy involved in the gel/crystal interaction on the crystal surface is weak and the incorporation of gel fibers is energetically favorable at least for high agarose concentrations and low supersaturation because the specific surface energy of the fiber/solution interface is higher than the specific surface energy of the fiber/crystal interface. This is consistent with the previously reported incorporation of gel fibers into the lysozyme crystal lattice16 and with the almost negligible effect of agarose fibers on the step velocity (Figure 3). Gel fibers will interact weakly with the advancing steps and no strong step pinning will occur. For lower agarose concentrations and higher supersaturation, cos(θ) becomes slightly negative. If this energy is larger than the elastic energy involved in stretching or shrinking agarose fibers, fibers will be rejected from the crystal lattice. The higher values for Ca ) 0.175 are consistent with the distinctively different slope of this data set in figure 2 and can be due to a continuous or sudden change in the agarose fiber diameter at a concentration 0.125 < Ca < 0.175 or to a change in the connectivity of the agarose network that modifies the mechanical properties of the fibers. The observed reduction of the nucleation rate in gelled impure solutions (Figure 5) with respect to gelled pure solutions (Figure 2) is more difficult to explain with the data available from these experiments. One plausible hypothesis is that impurities have more affinity for the gel fibers than the lysozyme molecules and, therefore, attach preferentially to the gel fibers. This could decrease cos(θ) either via γsc or γsf leading to an increase of f and consequently a reduction of J, but no data are currently available to check this hypothesis.
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Table 1. Ledge Free Energies and Surface Free Energies of Steps on the {110} Faces for Crystals Growing from Free Solutions and Apparent Ledge Free Energies and Surface Free Energies of Steps on the {110} Faces for Crystals Growing from Gelled Solutions (1.09 ( 0.09) × 10 (6.07 ( 0.31) × 105 (4.17 ( 0.44) × 105 (9.3 ( 1.30) × 105 (7.75 ( 0.89) × 105 (6.62 ( 0.49) × 105 6
purified free solution gelled solution 0.025 - 0.125% (w/v) 0.175% (w/v) Seikagaku free solution gelled solution 0.025% (w/v) 0.125-0.175% (w/v)
R and Reff (mJ/m2)
κ and κeff (J/m)
slope
-12
(3.5 ( 0.1) × 10 (2.6 ( 0.1) × 10-12 (1/f ) 1.3) (2.2 ( 0.1) × 10-12 (1/f ) 1.6) (3.3 ( 0.2) × 10-12 (3.0 ( 0.1) × 10-12 (2.8 ( 0.1) × 10-12
0.63 ( 0.03 0.47 ( 0.01 0.39 ( 0.02 0.58 ( 0.03 0.53 ( 0.03 0.50 ( 0.01
a Ledge free energies κ [J/m] and surface free energies R [mJ/m2] were obtained by linear curve fittings of the experimental data shown in Figure 2 using eqs 3. R was calculated as R ) κ/h, where h is a step height (5.6 nm for {110} faces and 3.4 nm for {101} faces); apparent ledge free energies κeff [J/m] and surface free energies Reff [mJ/m2] are obtained by the linear curve fittings of the experimental data shown in Figure 5 using eq 3 taking into account that κ ) 1/fκeff and Reff ) κeff/h.
To check for this effect, we must see a reduction of Γi in gelled systems. Rearranging eq 7, we find
Vst(Γi ) 0) 1 2 σ ) Vst(Γi ) 0) - Vst(Γi) 4d2Γ
(8)
0 i
Figure 8. Plot of the inverse step deceleration as a function of supersaturation (eq 8). The slope of the plot is inversely proportional to the concentration of surface adsorbed impurities.
Nucleation rate data are commonly used to estimate the ledge free energy (κ) using eq 3. When heterogeneous nucleation is significant, this procedure can produce misleading κ values or, more precisely, “apparent” values (κeff) that can be used to predict the macroscopic growth rate but not to investigate or discuss the molecular level processes. Comparing eqs 3 and 4, it is clear that κ ) 1/f κeff. This 1/f factor must be taken into account when analyzing molecular level processes in a system showing heterogeneous 2D nucleation. Table 1 shows the values of ledge free energies (κ) and surface free energies (R) obtained from our experiments. There are basically two (possibly coexistent) mechanisms for impurities to affect the step growth rate: the local modification of the chemical potential close to charged impurities and the pinning of steps.37 The strength of both processes is proportional to the impurity concentration, but the first process only operates at high impurity concentration while the second (pinning) can be very important at low concentration of impurities. Despite some quantitative differences, all models for the effect of impurities on step velocity predict a reduction of the step velocity by a factor proportional to the density of impurities on the surface and the step stiffness37
(
Vst(Γi) ) Vst(Γi ) 0) 1 -
4Γi (K/)2
)
(7)
where Γi is the surface concentration of impurities (number of impurity molecules per unit area), and K* ) σ/d0 is the critical curvature for which steps stop advancing due to stress accumulation. Here σ ) (C/Ce - 1) is the relative supersaturation and d0 is the edge capillary length. As shown in Figure 6, the agarose gel plausibly has an influence on the growth process by reducing the pinning effect.
and plotting the data in this way we can find Γi by linear fit. As shown in Figure 8, data are reasonably fitted by straight lines and the slopes of these lines show a clear difference in Γi. The slope of the plots is 0.041 ( 0.005 for the ungelled experiments and 0.284 ( 0.076 for the gelled experiments. Because d0 depends mainly on the step stiffness37 which is independent of the presence of gel, we can conclude that the presence of agarose gels has a “filtering effect” reducing the impurity concentration at the crystal surface by a factor of approximately 7. Previous studies of crystal growth in gels38 and microgravity39 already indicated the possible existence of an impurity depletion zone, but this is the first time the impurity filtering has been quantified, indicating that low concentration agarose gels are an effective method for reducing the impurity concentration at the crystal-solution interface.
5. Conclusions The duality of agarose gels as either impurity or impurity filter in protein crystal growth experiments has been discussed frequently. The results on crystal growth kinetics in terms of 2D nucleation rate and step velocity reported in this paper show that agarose plays both roles simultaneously, promoting 2D nucleation up to a 25% as an impurity but reducing the concentration of other impurities on the crystal surface by a factor around 7 at the same time. The overall effect on growth kinetics is a net increase of the growth rate due to the increased 2D nucleation and a reduced pinning effect. In the range of supersaturations and agarose concentration tested, the benefits of the impurity filtering effect are not counterbalanced by the agarose/crystal interactions due to a close value of the specific surface energy of the solution/agarose and agarose/crystal interfaces, which is also relevant for the incorporation of gel fibers within the crystal lattice. Acknowledgment. The authors were grateful for the support by Grant No. ESP 2006-11327 of the Ministry of Education and Science (MEC), Spain (F.O.), the Consolider-Ingenio 2010 project “Factorı´a Espan˜ola de cristalizacio´n” and the partial support by Grants-in-Aid (Nos. 17034007 and 18360003) of Scientific Research of the Ministry of Education, Science and Culture Japan (G.S.) and Grant Intramural-200730I013 of the Consejo Superior de Investigaciones Cientı´ficas (J.A.G.).
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