Measurement and Modeling of Protein Crystal Nucleation Kinetics

a preexponential factor, A, and an exponential factor, B, related to the surface energy between crystal ... In the homogeneous range, the preexponenti...
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CRYSTAL GROWTH & DESIGN 2002 VOL. 2, NO. 5 395-400

Articles Measurement and Modeling of Protein Crystal Nucleation Kinetics Venkateswarlu Bhamidi, Sasidhar Varanasi, and Constance A. Schall* Department of Chemical & Environmental Engineering, University of Toledo, 2801 West Bancroft Street, Toledo, Ohio 43606-3390 Received January 24, 2002

ABSTRACT: Nucleation kinetics of tetragonal hen egg-white lysozyme obtained by the method of initial rates are reported. Sodium chloride (NaCl), ranging in concentration from 2% (w/v) to 7% (w/v), was used as the precipitant. Data are modeled using an empirical kinetic expression based on classical nucleation theory. The expression contains a preexponential factor, A, and an exponential factor, B, related to the surface energy between crystal and solution. The parameters A and B were evaluated from the experimental data using standard nonlinear regression techniques. The trends in the experimental data suggest that two nucleation mechanisms may exist: heterogeneous nucleation at lower protein concentrations and homogeneous nucleation at higher protein concentrations. Accordingly, the data were split into two regions and the model parameters were estimated separately in individual regions. Within both regions, parameter B varied little with ionic strength. In the homogeneous range, the preexponential factor, A, increased monotonically with salt concentration and correlated with trends observed in osmotic second virial coefficients and solubility. The implications of such a correlation to globular protein nucleation kinetics are discussed. 1. Introduction Development of drugs and other pharmaceutical products to treat various diseases can be accelerated with structural information of enzymes and other proteins that are involved in the disease processes. Currently, X-ray or neutron diffraction methods are widely employed to obtain the structural information for these macromolecules. These techniques require the availability of high-quality crystals that are of suitable size. The growth of protein crystals is also important in the separation and purification of protein products. Obtaining such crystals can be a challenging task, owing to the complexity of these molecules and existent understanding of the protein crystallization process. Crystallization is comprised of two processessnucleation and crystal growth. The dependence of the kinetics of these processes on solution conditions and supersaturation differs greatly. For biological molecules, crystals are most often grown with no seeding, where primary nucleation must precede crystal growth. Thus, an understanding of the nucleation process and the effect of various parameters on nucleation is essential for successful production of protein crystals. Among factors that are thought to influence protein crystal nucleation are protein concentration, solution * To whom correspondence should be addressed. Tel: (419) 5308097. Fax: (419) 530-8086. E-mail: [email protected].

pH, temperature, and precipitant type and concentration. In some nucleation studies the number of crystals formed in a crystallizing batch after a prolonged incubation period was assumed to be a rough measure of nucleation.1,2 In other studies, population balances or material balances were used to estimate nucleation rates in a batch system.3,4 Protein concentration in solution continuously decreases in these batch systems due to the increase in solid mass with subsequent variation in nucleation rates during the batch run. Hence, estimates of nucleation rates in refs 3 and 4 depended upon crystal growth rate models. Galkin and Vekilov5,6 used a temperature-jump technique to obtain nucleation kinetics of hen egg-white lysozyme (HEWL) at 12.6 °C in very small droplets. Dixit et al.7 compared these nucleation rates to the predictions of classical nucleation theory and a kinetic theory based on an approach developed by Ruckenstein et al.8,9 They observed that the models overpredicted the number of protein crystals formed by several orders of magnitude. Additionally, they concluded that the assumptions used in interpreting the data were subject to many uncertainties and may have resulted in a large underestimation of nucleation rates.7 Given the sparse amount of nucleation kinetic data available, we have measured the nucleation kinetics of tetragonal hen egg-white lysozyme obtained by the

10.1021/cg025504i CCC: $22.00 © 2002 American Chemical Society Published on Web 08/07/2002

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method of initial rates at various protein and precipitant concentrations, as explained in our previous work.10 Since both nucleation and growth are simultaneously occurring in a crystallizing solution, it is difficult to decouple these processes and obtain information about nucleation kinetics experimentally. Detecting crystals during the early stages of growth (when they are a few microns in size and protein concentration is essentially unchanged) allows one to obtain nucleation rates under the initial conditions. Assuming every critical nucleus grows into a crystal, one can obtain the nucleation rate by counting crystals as they are forming. Sodium chloride (NaCl) was used as the precipitant. Both the protein and salt were buffered in 0.1 M sodium acetate buffer at pH 4.5. Nucleation kinetics were measured at 4 °C, and the salt concentration was varied between 2% (w/v) and 7% (w/v) (∼0.34-1.2 M). Data were modeled using an empirical relation based on classical nucleation theory. The apparent correlation of a model parameter with osmotic second virial coefficient is discussed.

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Figure 1. Nucleation rate data versus protein concentration are plotted at variable NaCl concentrations of 2% (b), 2.5% ([), 3% (2), 4% (9), 5% (O), 6% (g), and 7% (0) (w/v). Note the differences in scales on x and y axes. Smooth variation of data at NaCl concentrations from 2% to 4% (w/v) can be seen (right-hand side). Data obtained at salt concentrations above 4% show significant scatter. These batches formed “needle”like crystals along with the tetragonal form10 and, hence, were not considered for further analysis.

2. Experimental Section 2.1. Solution Preparation and Protein Solubility. Hen egg-white lysozyme (HEWL) from chicken (Gallus gallus) was purchased from Seikagaku and was dissolved in 0.1 M sodium acetate buffer at pH 4.5. The protein was desalted and concentrated using a 250 mL pressure concentrator (Vivascience) with three to four buffer exchanges using 5000 molecular weight cutoff membranes. The concentrated protein stock was filtered through 0.2 µm syringe filters (Corning). The concentration of protein in solution was determined by measuring its absorbance at 280 nm using an extinction coefficient of 2.64 mL/(mg cm).11 Sodium chloride solutions were prepared by dissolving the required amount of sodium chloride in 100 mL of acetate buffer. Protein solutions of required concentrations were prepared by diluting the concentrated stock with buffer so that the diluted solution contains twice the required amount of protein in the final crystallizing batch. All the solutions were equilibrated at 4 °C for at least 1 h before they were mixed. An 8 mL portion of crystallizing batch was prepared by further diluting 4 mL of this protein solution with 4 mL of NaCl solution so that the final crystallizing batch contains the required amount of protein and salt. The mixture was then filtered through a 0.1 µm syringe filter (Millipore) into a clean scintillation vial. At that point the vial was placed on the particle counter stage. The solubility of tetragonal HEWL was obtained by measuring the concentration of protein in the supernatant of the batches that were incubated at 4 °C in 0.1 M sodium acetate buffer at pH 4.5 for about 6 months. These solubility values were found to be 3.02, 1.64, 0.934, and 0.575 mg/mL for 2%, 2.5%, 3%, and 4% NaCl, respectively.10 These values were slightly less than those reported by Cacioppo and Pusey.12 The different sodium chloride concentrations studied were 2%, 2.5%, 3%, 4%, 5%, 6%, and 7% (w/v) (1% w/v ) 0.171 M). With salt concentrations of 5% and higher, “needle”-like crystals formed in addition to tetragonal crystals.10 In general, a supersaturation range of 10-20 was explored, where supersaturation is defined as the ratio of bulk protein concentration to tetragonal HEWL solubility (C/C*). This corresponds to protein concentrations of approximately 27-60, 19-32, 1023, and 7-13 mg/mL for 2%, 2.5%, 3%, and 4% (w/v) NaCl concentrations, respectively. Several of the experimental conditions were replicated three to five times to verify reproducibility of results. Other conditions were run once or twice. All data points were used in the analysis. 2.2. Particle Counting. A PC2000 particle counting instrument (Spectrex Corp.), which uses the principle of nearangle light scatter to detect and count particles, was used. The

particle counter was placed in an incubator for temperature control ((0.1 °C), and data were collected at 4 °C. The details of the particle counting instrument and procedure were discussed in a previous paper.10 The counter was calibrated using particle standards containing known numbers of particles having radii of 2, 5, and 10 µm (Duke Scientific). Solutions containing mixtures of different sizes of these particles in various proportions were also used in calibration. This exhaustive calibration procedure resulted in a slightly altered calibration as compared to the single particle size calibration performed in our earlier work and narrowed the particle size range used in analysis of nucleation rate. Particle counts acquired after any particle reached a size of 10 µm were not used in nucleation rate calculations. As a result of the minor change in calibration and data treatment, some nucleation rates previously reported increased or decreased randomly.10

3. Results 3.1. Kinetic Data. Figure 1 shows the nucleation rate data for tetragonal hen egg-white lysozyme crystals obtained at various salt concentrations. This includes the data we reported in an earlier paper.10 Nucleation rates obtained for 2-4% (w/v) NaCl varied monotonically with protein concentration. At high salt concentrations, 5% and above, this regularity was absent in the data and needles were formed along with the expected tetragonal crystals. Crystallizing batches at 7% NaCl formed needles exclusively. Galkin and Vekilov reported similar deviations in nucleation rates at high salt concentration.6 They also observed this morphological change to needlelike crystals in some batches. The probable role of liquid-liquid phase separation in causing these changes was discussed in our previous paper and by Galkin and Vekilov.6,10 Only nucleation kinetic data obtained at NaCl concentration less than 5% (w/v) were modeled, as outlined below, where no evidence of liquid-liquid phase separation was observed. 3.2. Modeling. Data were modeled using an empirical relation based on classical nucleation theory. The classical nucleation theory13 expresses the homogeneous

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nucleation rate J for a spherical nucleus as

J)

( ) (

)

∆Ga 2vxkTσ -16πσ3v2 exp 3 3 N1 exp h kT k T (ln S)2

(1)

where v is the molecular volume, k is the Boltzmann constant, T is absolute temperature, σ is the surface energy per unit area of the nuclei, h is Planck’s constant, N1 is the number density of monomeric species in the solution, ∆Ga is the energy barrier to diffusion from bulk solution to the cluster, and S is the supersaturation, often expressed as C/C*, the ratio of bulk concentration of solute to equilibrium solubility. It should be noted that the activity coefficients at bulk and equilibrium concentrations are assumed to be approximately equal by considering C/C* as the driving force for phase change instead of the ratio of activities. One can lump the parameters in eq 1 into preexponential and exponential terms. The number density of monomeric species is directly proportional to the bulk concentration of the solute, C, and one can transform eq 1 into the twoparameter empirical expression

J ) AC exp

{

-B [ln(C/C*)]2

}

Figure 2. Estimated parameters of eq 2 for nucleation rate data as a function of NaCl concentration. Solid circles represent values of parameter A, and solid squares show parameter B. Error bars indicate 1 standard deviation of uncertainty in the parameters. Parameter B appears to be unchanging with increasing ionic strength, suggesting that the surface energy is relatively independent of salt concentration. No trend can be perceived in parameter A, the preexponential factor.

(2)

3.2.1. Parameter Estimation. Exponential models similar to eq 2 are vulnerable to covariance of the estimated parameters A and B.14 In models exhibiting parameter interaction, estimates of the parameters are highly interdependent. An appropriate transformation reduces the parameter interaction and uncertainty in parameter estimates. Equation 2 was transformed into

Y′ ) A′ exp(-BX′)

(3)

by defining Y′ and X′ as

Y′ ) X′ )

{

J C

(4)

} {

}

1 1 2 [ln(C/C*)] [ln(C/C*)]2

(5) avg

where the subscript “avg” in (5) refers to the arithmetic average value of [ln(C/C*)]-2 for all experimental data points at a given NaCl concentration. The transformed parameter A′ is related to parameter A of eq 2 through

{ (

A′ ) A exp -B

1 [ln(C/C*)]2

)}

(6)

avg

The parameters A′ and B of (3) were estimated using a Marquardt-Levenberg nonlinear least-squares estimation algorithm.14 3.2.2. Kinetic Parameters. Figure 2 gives values of the parameters and their estimated errors as a function of salt concentration. Figure 3 shows the model fit to the data. From Figure 2, it can be seen that the estimated parameter B appears to be fairly constant over the range of salt concentrations studied. The temperature was constant for all experiments, and the monomer molecular volume is not expected to be a strong function of salt concentration. A constant value for parameter B indicates that the surface energy may

Figure 3. Symbols representing individual nucleation rate measurements for 2-4% NaCl concentration, as indicated on the graph. The lines represent the nucleation rate model (eq 2) with the estimated parameters from Figure 2. Systematic deviation of model predictions toward the lower protein concentration region can be seen and are particularly evident when the model is linearized and plotted as in the inset for 4% NaCl nucleation rate data.

be either weakly dependent or independent of salt concentration. In contrast, the preexponential factor, A, changes with increasing ionic strength with no apparent trend. It can be observed from Figure 3 that the model agrees well with experimental data at high protein concentrations but deviates significantly from the experimental values at lower protein concentrations in all cases. Linearizing eq 2 and plotting ln(J/C) vs [ln(C/ C*)]-2 for the experimental data did not yield a straight line as expected but indicated the possible presence of two intersecting straight lines (Figure 3). These observations and the lack of an apparent trend in the preexponential factor suggest that the nucleation kinetic data may span a range of concentrations where heterogeneous nucleation predominates at low solute concentration and homogeneous nucleation is the dominant mechanism at higher concentrations. In heterogeneous nucleation, nucleation is catalyzed by surfaces or par-

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Figure 4. Model fit to data split into two regimes for 2-4% NaCl concentration. Solid symbols and lines indicate data points in the homogeneous (high protein concentration) range, and open symbols with dashed lines belong to the heterogeneous (low protein concentration) range. The division between regimes was based on minimization of the sum of squared errors of the model compared to experimental data with variable placement of the division point.

ticles that lower the surface energy of solute clusters that form nuclei. In homogeneous nucleation, the surface energy between solute clusters and the bulk phase is unaffected by surfaces other than the growing clusters of solute.15 Hence, the data were split into two regions: a heterogeneous regime at lower protein concentrations and a homogeneous nucleation regime at higher protein concentrations. The point of division was adjusted to minimize the estimated parameter errors and the total sum of squared errors between the model and data points over the two regions. The split between the heterogeneous and homogeneous nucleation regions ranged from a supersaturation, C/C*, of 16-18. The model parameters were then estimated for the individual regions as detailed in section 3.2.1. A graph of the revised model and data is shown in Figure 4. Better agreement between data and the model can be seen. Parts a and b of Figure 5 show the parameter values and their estimated errors in these two regions. In both regions, the parameter B is fairly constant with changes in ionic strength. However, the value of B in the heterogeneous range is about half of that in the homogeneous range. Since this parameter is directly related to the surface energy, this suggests a lower surface energy at low protein concentrations. This is consistent with the assumption that heterogeneous nucleation may be the dominant nucleation mechanism at low solute concentration. The preexponential factor in the heterogeneous range appears to vary little from 2% to 3% NaCl and was slightly higher at 4% NaCl. In the homogeneous range the preexponential factor increased monotonically with salt concentration. 4. Discussion The development of the classical nucleation model (eqs 1 and 2) involves several assumptions that may or may not be valid for protein systems. These assumptions include isotropic surface energy of solute clusters and nuclei and surface energies independent of cluster size. However, this theory gives a good starting point for empirical modeling. On the basis of this theory, the

Figure 5. Regressed parameters of the model fit to nucleation rate data divided into two regions. The preexponential factor in the heterogeneous region is fairly constant, whereas that in the homogeneous region shows an asymptotic trend. The surface energy parameter B is relatively constant in both regions with variation in ionic strength.

surface energy of the crystal nucleus can be calculated from the value of the estimated parameter B in eq 2 at a temperature of 4 °C using a molecular volume of lysozyme of 2.97 × 10-26 m3.16 The calculated surface energies are 0.673, 0.677, 0.700, and 0.688 erg/cm2 for 2%, 2.5%, 3%, and 4% (w/v) NaCl, respectively, when nucleation data are not split into two regimes. When the rates are separated into two ranges, these are calculated to be 0.547, 0.539, 0.536, and 0.549 erg/cm2 for the heterogeneous range and 0.687, 0.696, 0.702, and 0.696 erg/cm2 for the homogeneous range for 2%, 2.5%, 3%, and 4% (w/v) NaCl, respectively. Land et al. reported similar values obtained through atomic force microscopic studies of Canavalin crystal growth.17 From crystal growth studies, Malkin et al.18 obtained a surface energy value of 0.26 ( 0.09 erg/cm2 for 2D nucleation of satellite tobacco mosaic virus, which is of the same order of magnitude as our calculated surface energies. When the nucleation rate data of the present study were split into two regimes, the surface energy values obtained for the low protein concentration range were consistently less than those at higher protein concentrations, supporting the hypothesis that heterogeneous nucleation may dominate at lower supersaturation.

Protein Crystal Nucleation Kinetics

Figure 6. Generalized correlation between the solubility and the second virial coefficient for tetragonal hen egg-white lysozyme, redrawn from Guo et al.19 Solid circles represent the data points used in interpolating or extrapolating the value of B22 under our experimental conditions. The open diamond indicates the B22 value measured in this work at 4% (w/v) NaCl and 20 °C, where the solubility is 3.1 mg/mL.20

More importantly, separating measured nucleation rate data into high and low concentration regions resulted in a trend in the parameter A, the preexponential factor, with salt concentration. In the low protein concentration region, this parameter is almost constant, with some deviation at 4% (w/v) NaCl (Figure 5). In the high protein concentration region, the value of this parameter is orders of magnitude greater and appears to increase asymptotically with increasing ionic strength. This parameter can be viewed as a collisional term in the original expression. Intermolecular interactions are expected to affect the sticking probability of molecules when they collide. One way to quantify these interactions is by considering the osmotic second virial coefficient (B22). Hence, we examined the trend in the values of the second virial coefficients with change in salt concentration. Published values for second virial coefficients were not available for all of our experimental conditions. Wilson and co-workers19 presented a generalized correlation of second virial coefficients with solubility for tetragonal hen egg-white lysozyme. However, few data points were available at the low values of solubility under our experimental conditions. It should be noted that it is extremely difficult to measure the second virial coefficient under these low solubility conditions using light scattering techniques, owing to the possibility of crystallization at the minimum concentration of protein required in solution. For this reason, the second virial coefficient of tetragonal HEWL was measured at a solubility of 3.1 mg/mL (corresponding to 4% (w/v) NaCl in 0.1 M sodium acetate buffer at pH 4.5 and 20 °C20) using static light scattering. This value of B22 was found to be -6.57 × 10-4 mol mL g-2. Figure 6 shows this measured value of the second virial coefficient with respect to the generalized correlation presented by Wilson et al.19 It is also worth noting that this solubility value of 3.1 mg/mL is close to the solubility of HEWL at 2% (w/v) and 4 °C (3.0 mg/mL), which was one of our experimental conditions. Using these data points and

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Figure 7. Comparison of variation in the preexponential factor, A (2), and the osmotic second virial coefficient of hen egg-white lysozyme (O) normalized to their values at 4% NaCl. The preexponential factors were estimated from the data in the homogeneous nucleation range. Error bars on A indicate the estimated error in the value of the preexponential factor.

the B22 data at the low protein solubility portion of the generalized correlation, the values of B22 under our experimental conditions were determined by interpolation or extrapolation. Figure 7 presents the comparison of normalized values of B22 and the normalized preexponential factor for the homogeneous nucleation range. It appears that the parameter A of the model in the homogeneous range correlates with the osmotic second virial coefficient. Parameter B in eq 2 is relatively constant at different salt concentrations (and, hence, different solubilities). The preexponential factor, A, appears to correlate with the osmotic second virial coefficient and subsequently with protein solubility. If B remains invariant over a wide range of solution conditions and the preexponential factor is correlated with solubility, then equal solubility conditions are expected to produce equal nucleation rates at a given protein concentration. This gives us a basis for comparing nucleation rates obtained at different salt concentrations, temperatures, or buffer conditions. Galkin and Vekilov5,6 reported the nucleation kinetics of hen egg-white lysozyme at 12.6 °C and a range of NaCl concentrations in 0.05 M sodium acetate buffer with a pH of 4.5. Our nucleation rate measurements were performed at the same pH at 4 °C in 0.1 M sodium acetate buffer. We attempted to compare nucleation rate data under similar solubility conditions. Attempts to compare our nucleation rates with these data revealed several important points. Solubility data for Galkin and Vekilov’s work under their precise conditions5,6 were either unavailable in the references cited therein or incorrectly applied in their analysis of their nucleation rate data. Galkin and Vekilov reference solubility data published by Rosenberger et al.21 and Cacioppo et al.12 Rosenberger et al. reported tetragonal hen egg-white lysozyme solubility of 3.3 mg/mL at 2.5% (w/v) NaCl concentration in 0.05 M sodium actetate buffer (pH 4.5).21 Interpreting Galkin and Vekilov’s nucleation rate data (from plots as a function of both protein concentration and supersaturation) at this NaCl concentration yields a solubility of 5.0 mg/mL. For the other conditions, Galkin and Vekilov apparently used the lysozyme solubilities reported by Cacioppo et al.,12 3.0 and 1.5 mg/mL, as interpreted from

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their graphs at 3% and 4% (w/v) NaCl, respectively, which match closely with the solubilities of 3.29 and 1.49 mg/mL from Cacioppo et al.12 However, it should be noted that the solubility data presented by Cacioppo et al. were for a sodium acetate buffer concentration of 0.1 M, not the 0.05 M sodium acetate concentration used in Galkin and Vekilov’s experiments. Forsythe et al.22 indicate that the sodium acetate buffer concentration has a significant effect on the solubility of lysozyme at pH 4.0, particularly between 0.05 and 0.1 M. Similar differences in solubility at pH 4.5 may exist between these two buffer concentrations. Using solubility as the basis for comparison of nucleation rates between our work and that of Galkin and Vekilov yielded only one condition where solubility data were readily available in the literature: 0.05 M acetate buffer and 12.6 °C with 2.5% (w/v) NaCl, with a reported solubility of 3.3 mg/ mL.21 This is close to the solubility of 3 mg/mL under our experimental conditions of 2% (w/v) NaCl. Hence, we compared the data under these conditions. The overlap in the protein concentration range studied in our work and Galkin and Vekilov’s6 was small (50-60 mg/mL). At 50, 56, and 60.7 mg/mL protein concentrations, Galkin and Vekilov report nucleation rates of 0.60, 1.96, and 3.35 nuclei/(mL min), respectively, whereas our estimates from the model under these protein concentrations give 83, 204, and 369 nuclei/(mL min). Galkin and Vekilov’s experimental method may have resulted in a large underestimation of nucleation rates due to reasons detailed by Dixit et al.7 5. Conclusions The parameter B in the exponential term of the classical nucleation expression (eq 2) is a function of molecular volume, nucleus shape, surface energy, and temperature. The molecular volume and shape are not expected to vary significantly with pH, ionic strength, and the narrow temperature ranges used in crystallization of a selected protein molecule. The surface energy parameter was found to be relatively invariant with ionic strength in heterogeneous and homogeneous nucleation kinetic regimes for HEWL and may vary little within the pH range employed in protein crystallization. If it can be assumed that the ratio of surface energy to temperature is fairly constant, then the parameter B in the exponential term in the classical nucleation expression (2) would be constant for tetragonal HEWL crystallization over a wide range of solution conditions and solubility values. In the homogeneous nucleation range, the preexponential parameter, A, of the model correlates with the osmotic second virial coefficient, B22. Wilson et al. has correlated B22 with the protein solubility of HEWL and other globular proteins.19 Thus, the trend in the preexponential factor, A, of the classical nucleation kinetic expression is expected to correlate directly with HEWL solubility. It is hypothesized that

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equal solubility conditions will produce the same nucleation rates, independent of solution pH and possibly temperature. If the correlation between solubility and second virial coefficients is valid for a wide range of globular proteins, a measure of solubility of a globular protein may provide a method for estimation of the preexponential factor, A, and subsequent prediction of homogeneous nucleation rates. Acknowledgment. This research was partially funded by a National Aeronautics and Space Administration grant, No. NAG8-1578. We thank Prof. John Wiencek and Dr. Vladimir Aseyev for providing the equipment and expertise necessary to carry out the static light scattering experiments at the University of Iowa. We also thank Dr. Donald White of the University of Toledo for his advice and assistance in the nonlinear regression analysis. References (1) Burke, M. W.; Leardi, R.; Judge, R. A.; Pusey, M. L. Cryst. Growth Des. 2001, 1, 333-337. (2) Judge, R. A.; Jacobs, R. S.; Frazier, T.; Snell, E. H.; Pusey, M. L. Biophys. J. 1999, 77, 1585-1593. (3) Saikumar, M. V.; Glatz, C. E.; Larson, M. A. J. Cryst. Growth 1998, 187, 277-288. (4) Ataka, M. Prog. Cryst. Growth Charact. Mater. 1995, 30, 109-128. (5) Galkin, O.; Vekilov, P. J. Phys. Chem. B 1999, 103, 1096510971. (6) Galkin, O.; Vekilov, P. G. J. Am. Chem. Soc. 2000, 122, 156163. (7) Dixit, N. M.; Kulkarni, A. M.; Zukoski, C. F. Colloids Surf., A: Phys. Eng. Apsects 2001, 190, 47-60. (8) Dixit, N. M.; Zukoski, C. F. J. Colloid Interface Sci. 2000, 228, 359-371. (9) Nowakowski, B.; Ruckenstein, E. J. Colloid Interface Sci. 1990, 139, 500-507. (10) Bhamidi, V.; Skrzypczak-Jankun, E.; Schall, C. A. J. Cryst. Growth 2001, 232, 77-85. (11) Sophianopoulos, A. J.; Rhodes, C. K.; Holcomb, D. W.; VanHolde, K. E. J. Biol. Chem. 1962, 237, 1107. (12) Cacioppo, E.; Pusey, M. L. J. Cryst. Growth 1991, 114, 286292. (13) Walton, A. G. In Nucleation; Zettlemoyer, A. C., Ed.; Marcel Dekker: New York, 1969; p 238ff. (14) Himmelblau, D. M. Process Analysis by Statistical Methods; Wiley: New York, 1970. (15) Nielsen, A. E. Kinetics of Precipitation; Pergamon Press: New York, 1964. (16) Nadarajah, A.; Pusey, M. L. Acta Crystallogr., Sect. D: Biol. Crystallogr. 1996, 52, 983-996. (17) Land, T. A.; Malkin, A. J.; Kuznetsov, Y. G.; McPherson, A.; De Yoreo, J. J. Phys. Rev. Lett. 1995, 75, 2774-2777. (18) Malkin, A. J.; Land, T. A.; Kuznetsov, Y. G.; McPherson, A.; DeYoreo, J. J. Phys. Rev. Lett. 1995, 75, 2778-2781. (19) Guo, B.; Kao, S.; McDonald, H.; Asanov, A.; Combs, L. L.; Wilson, W. W. J. Cryst. Growth 1999, 196, 424-433. (20) Forsythe, E. L.; Judge, R. A.; Pusey, M. L. J. Chem. Eng. Data 1999, 44, 637-640. (21) Rosenberger, F.; Howard, S. B.; Sowers, J. W.; Nyce, T. A. J. Cryst. Growth 1993, 129, 1-12. (22) Forsythe, E. L.; Pusey, M. L. J. Cryst. Growth 1996, 168, 112-117.

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