ARTICLE pubs.acs.org/IECR
Diffusion of Lysozyme in Buffered Salt Solutions Moriamou K. Antwi,*,† Allan S. Myerson,‡ and Walter Zurawsky§ †
St. Joseph’s College, 245 Clinton Avenue, Brooklyn, New York 11205, United States Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, United States § Polytechnic Institute of NYU, Six Metro Tech Center, Brooklyn, New York 11201, United States ‡
ABSTRACT: The diaphragm cell technique was used to measure the diffusion coefficient of lysozyme in buffered solutions with and without NaCl. In the absence of salt, the pseudobinary diffusion coefficient decreases with increasing lysozyme concentration. In the presence of NaCl, the main term D11 decreased slowly with increasing NaCl concentrations and exceeded the pseudobinary values by 410%. The main term D22 did not vary significantly with NaCl concentration and was slightly less than the binary diffusion coefficient for NaCl. The cross-term D21 increased sharply with salt concentration. The cross-term D12 is small and decreased slowly with salt concentration. These results are consistent with the expected electrostatic interactions between lysozyme and salt in the diffusion process.
’ INTRODUCTION The study of diffusion is fundamental to understanding the mechanism of protein crystal growth. Crystallization from solution is a two step process. The first step is the formation of aggregates or nuclei in supersaturated solutions and the second step is the growth of the existing nuclei. Previous work on lysozyme includes the work done by Myerson and Kim1 using the Gouy-interferometer method. They found that the diffusion coefficient of lysozyme in a salt-free buffer solution first increased and then slowly decreased with increasing lysozyme concentration. In a salt-buffer solution, however, lysozyme diffusivity did not change much with lysozyme concentration in the under-saturated region but decreased slowly in the supersaturated region. Two studies done by Albright et al.2,3 reported the diffusion coefficients of lysozyme using the Rayleigh interferometry method. In their first paper, the system lysozyme/NaCl/water was studied over a wide range of concentrations of NaCl and at a concentration high enough to be relevant to crystallization. The ternary diffusion coefficients were reported at several mean concentrations of NaCl and at a single mean concentration of lysozyme. The second paper reported the ternary diffusion coefficients at different pH. They found that the diffusion coefficients of lysozyme decrease slightly with increasing pH of the solution. In this study, a diaphragm cell technique based on a simplified and improved model introduced by Asfour4 was used. To our knowledge, the proposed method was used for the first time to determine the ternary and binary diffusion coefficient of lysozyme with and without buffered salt solution. The knowledge of the diffusion coefficient of proteins is limited and this method can be extended to other proteins.
Anson5 and elaborated by Stokes6 and Asfour4 and many others. The technique has been shown to give results of good reproducibility and therefore gives confidence in the use of the technique. The diaphragm cell method uses the difference of the concentrations of the two solutions to determine the diffusion coefficients of the solutes. For binary systems, the diaphragm cell equation is
’ DIAPHRAGM CELL METHOD The diaphragm cell method is one of the simplest methods for studying isothermal diffusion. It was introduced by Northrop and
Received: February 10, 2011 Accepted: August 5, 2011 Revised: August 2, 2011 Published: August 05, 2011
r 2011 American Chemical Society
D̅ ¼
1 c01B c01A ln βt cf1B cf1A
ð1Þ
where t is the time at the end of the experiment; c01B c01A is the initial concentration difference between the lower (B) and top compartment (A) of the diffusion cell; cf1B cf1A is the final concentration difference between the top and lower compartment (the value at time t); β is the cell constant; and D is the integral diffusion coefficient. The diaphragm cell yields an average or integral diffusion coefficient. Gordon7 has shown that under quasi-steady state conditions, the differential diffusion coefficient is related to the integral diffusion coefficient D by Z ̅ cB Dð D dc ð2Þ ̅ c̅ B c̅ A Þ ¼ ̅ cA
where c̅ B ¼
c0B þ cfB 2
ð3Þ
c̅ A ¼
c0A þ cfA 2
ð4Þ
and D is the differential diffusion coefficient.
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For runs of very short duration and with the top cell concentration equal to zero, we can define an integral diffusion coefficient D0 as Z 1 cB D̅ 0 ðcB Þ ¼ D dc ð5Þ cB 0 cA ð6Þ D̅ 0 ðc̅ B Þ þ D̅ ̅ ðD̅ D̅ 0 ðc̅ A ÞÞ c̅ B with D measured, cB and cA known and using an iterative technique, the experimental data for D can be used along with eq 6 to obtain a curve of D (cA) versus concentration. Equation 2 is then differentiated to yield D ¼ D0 ðcB Þ þ cB
d 0 D ðcB Þ dcB
ð7Þ
Equation 7 can be used to calculate the differential diffusion coefficient, D. For ternary systems, a material balance combined with the more general form of Fick’s law811 gives ∂ci ¼ ∂t
∂2 cj Dij 2 ∂x j¼1 2
∑
Figure 1. Schematic diagram of a diffusion cell.
i ¼ 1, 2
ð8Þ
The expressions found for the solute concentration differences at time t are c1B c1A ¼ P1 eβσ1 t þ Q1 eβσ2 t
ð9Þ
c2B c2A ¼ P2 eβσ1 t þ Q2 eβσ2 t
ð10Þ
where P1 ¼
ðD11 σ 2 Þðc01B c01A Þ þ D12 ðc02B c02A Þ σ1 σ2
ð11Þ
Q1 ¼
ðD11 σ 1 Þðc01B c01A Þ þ D12 ðc02B c02A Þ σ2 σ1
ð12Þ
P2 ¼
D21 ðc01B c01A Þ þ ðD22 σ 2 Þðc02B c02A Þ σ1 σ2
ð13Þ
Q2 ¼
D21 ðc01B c01A Þ þ ðD22 σ 1 Þðc02B c02A Þ σ2 σ1
ð14Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 D11 þ D22 þ ðD11 D22 Þ2 þ 4D12 D21 σ1 ¼ 2 ð15Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 D11 þ D22 ðD11 D22 Þ2 þ 4D12 D21 σ2 ¼ 2
divides it into two compartments (the top compartment, A, and the lower compartment, B) of approximately 50 mL each. The diaphragm is a fritted glass disk. The pores are approximately 2.5 μm in diameter. The disk is 23 mm thick and has diameter of 40 mm. It was established by Gordon7 that too large of a pore size introduces a bulk flow through the diaphragm. The pores should be big enough for the solute molecule to pass through freely but not allow bulk flow through the diaphragm. The cell was oriented so that the diaphragm was horizontal. A schematic diagram of the diaphragm cell is shown in Figure 1. The cells are secured in a cell support, which, in turn, fits snuggly into a water bath kept at 25 °C. The cell support allows the cell to be located reproducibly.
’ EXPERIMENTAL PROCEDURES The four cells used in this study were calibrated with 0.1 M KCl at 25 °C. Solid KCl, purity 99%, purchased from Sigma was dissolved in deionized, distilled water and outgassed to remove air bubbles introduced during the dissolution. The KCl solution was used to fill the lower compartment and the pores of the diaphragm, and the lower stopcock was fixed in place. Similarly, the upper compartment was filled with deionized, distilled water and the upper stopcock was placed. The filled cells were placed into a water bath, and the stirring apparatus placed outside the cells was turned on. The KCl solution in the diffusion cell was allowed to prediffuse for 13 h to establish a pseudosteady state inside the diaphragm. A rule to estimate a safe time to reach steady state was established by Gordon7 and given by T g 1:4
ð16Þ Concentrations are measured at various times, and the Dij are obtained by nonlinear least-squares fits to the data.
’ EXPERIMENTAL APPARATUS The Pyrex diaphragm cell used in this study was constructed by South-Eastern Laboratory Apparatus Inc., (North Augusta, SC). The cell is cylindrical with a diaphragm in the center that
L2 D
ð17Þ
In eq 17, L is the effective length of the diffusion path, approximately equal to 0.4 cm, D is the diffusion coefficient of the solute used for calibration, and T is the diffusion time. After 23 h of prediffusion, the stirring apparatus was stopped. The upper compartment was emptied carefully while the cell remains in the water bath, rinsed quickly 23 times with fresh solution, and then refilled with fresh solution. The upper 10314
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Table 1. Measured and Estimated Values for the Diffusion Cell Constants, β (cm2)a cell
a
trials
measured
estimated
avg deviation
deviation from
(%)
estimated (%)
Table 2. Differential, Integral and DebyeHuckel (DH ) Diffusion Coefficients (106 cm2/s) for the Binary System Lysozyme/Water at Different Lysozyme Concentrationsa concn
1.8 0.2
(g/100 mL)
integral
differential
DH
1 2
4 5
0.2279 0.2379
0.2323 0.2384
0.003 0.035
0.5503
4.45
4.34
4.19
3
5
0.2982
0.2357
0.034
20
0.5501
4.77
4.66
4.19
4
5
0.2396
0.2294
0.010
4
0.5508
4.16
4.05
4.19
1.1341 1.1226
4.06 4.43
3.84 4.21
3.94 3.94
1.1404
4.17
3.94
3.94
2.2817
4.57
4.15
3.60
2.2976
4.14
3.72
3.59
2.3011
4.04
3.62
3.59
3.0895
4.01
3.46
3.41
3.0736
3.57
3.02
3.41
4.0286 4.0246
3.67 4.16
2.96 3.46
3.22 3.22
4.0552
3.64
2.93
3.21
5.4208
3.89
2.97
2.97
5.3383
3.40
2.48
2.99
6.9617
3.50
2.31
2.73
6.8674
3.16
1.98
2.75
Average deviation is the based on the mean value.
stopcock was placed, and the stirring apparatus was started again. The test run was considered to start at this time. During KCl calibration, each run lasted approximately 24 h. After completion of the test, the cells were removed from the bath. The lower and upper solutions were quickly removed and carefully transferred into clean and dry beakers. The concentration of each compartment after completion of the calibration experiment was determined by titration of chloride by silver nitrate.12 Chicken egg white lysozyme chloride, catalog number 10837059001 (95100% purity) purchased from Roche Diagnostic, Inc., was used in this work. Deionized, distilled water and 0.1 M sodium acetate buffer solutions were prepared as solvents. The pH of the buffer solution was adjusted to 4.5 with concentrated hydrochloric acid as needed. Four sets of experiments were constructed so that four data points were obtained at each solute composition for both the ternary and binary experiments. The diffusion experiments for lysozyme were conducted using the same procedure described above except that the diffusion experiments were conducted for 13 days. To avoid bacterial attack on the solutions during that period of time, freshly boiled distilled water is used to dissolve the buffer (0.1 M sodium acetate solution). The cells and the stopcocks were regularly sterilized using an autoclave in case bacteria are trapped inside the cell and the diaphragm. Before and after each experimental run, the cells are cleaned with dilute acid or LIQUI-NOX solution and distilled water and dried thoroughly with acetone or alcohol. After completion of the experimental run, the solution of each compartment was analyzed. The sodium chloride solution was determined by titration with silver nitrate.12 The protein concentration was determined spectrophotometrically at 280 nm.
’ EXPERIMENTAL RESULTS Cell Constant. The cell factors, β, for each diffusion cell were determined by calibration with 0.1 M KCl at 25 °C using the data of Stokes13 to establish the integral diffusion coefficient of 0.1 M potassium chloride solution against distilled water. The experimental results are shown in Table 1. For cell number 1, the results of four runs yielded a mean value of 0.2279 cm2. For cell numbers 2, 3, and 4, the results of five runs each yielded a value of 0.2379, 0.2982, and 0.2396 cm2, respectively. The theoretical values of the cell constants were calculated using eq 18: A 1 1 β¼ þ ð18Þ l VA VB
where l is the length of the diaphragm pores, A is the crosssectional area of the pores, and VA and VB are the volumes of the upper and lower compartments of the diffusion cell. According to
a
Integral and differential values are estimated to be (4%.
the manufacturer, the pores occupy 48% of the total surface area of the diaphragm. With l = 0.4 cm and A = 4.8396, 4.9668, 4.9318, and 4.7792 cm2 for the cells used in these experiments, it is possible to estimate the theoretical values (Table 1) for the cell constants, β, for each cell. For cells 1, 2, and 4, the measured and estimated values agree. The cell constant for cell 3, however, exceeds the calculated value by 20%. This discrepancy can arise from variations in either the porosity, or the thickness, or the area of the glass frit. However, it does not affect our results since for all cells, the measured cell constants were used in all subsequent diffusion coefficient calculations. The measured cell constants are reproducible, as seen from the small average deviation from the mean in Table 1. Lysozyme Diffusion without NaCl. The experimental values of the binary diffusion coefficient for the lysozyme/water system at different concentrations of lysozyme were obtained at 25 °C and pH 4.5. Table 2 shows the results of those measurements and includes the initial concentrations of lysozyme and the integral and the differential diffusion coefficients. Mass balances agreed within 3% for lysozyme diffusion measurements, and the estimated uncertainty in the diffusion values in Table 2 are based on propagation of this 3% uncertainty in the concentration differences. The values of the integral diffusion coefficients are plotted in Figure 2 and are compared to the differential diffusion coefficient. Binary diffusion coefficients from the DebyeHuckel equation are also included in Table 2 and are plotted in Figure 1. Figure 3 shows a comparison to previously reported integral diffusion data. Lysozyme Diffusion with NaCl. Because of the presence of buffer species, the diffusion of lysozyme in the presence of NaCl is complicated and 16 diffusion coefficients would be required to fully describe the system. To elucidate the effects of NaCl on the diffusion of lysozyme, we treated these systems as ternary. To obtain the four ternary diffusion coefficients, a minimum of four 10315
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Table 3. Ternary Diffusion Coefficients (106 cm2/s) for the System LysozymeSalt-Water at pH 4.5 and 25 °C at Different Average Salt Concentrations (C2, g/100 mL) C2
D11
D12
D21
D22
1
4.74 ( 0.77
2.99 ( 0.78
0.43 ( 0.24
15.10 ( 0.17
2
4.82 ( 0.05
0.21 ( 0.06
1.42 ( 0.08
12.61 ( 0.14
3
3.92 ( 0.50
0.07 ( 0.40
2.35 ( 0.04
12.41 ( 0.19
4
3.79 ( 0.11
0.04 ( 0.13
5.11 ( 0.43
12.80 ( 0.44
5 6
4.28 ( 0.27 3.75 ( 0.22
0.12 ( 0.10 0.14 ( 0.23
9.85 ( 0.61 6.03 ( 0.76
12.85 ( 0.66 13.30 ( 1.25
Figure 2. Integral (O), differential (]), and theoretical (2) diffusion coefficient for the binary system lysozyme/water.
Figure 4. Main term diffusion coefficient of lysozyme (D11) in the ternary system lysozyme/NaCl/water: this work (]) and Albright et al. (0). The lines are the results of the NP calculations: with buffer (solid) and without buffer (dotted).
N (where N is the number of experiments at each salt concentration) sets of diffusion coefficients from the results of the N 1 remaining measurements. The reported error (Table 3) is the standard error, based on the best fit value, from these N sets of diffusion coefficients. The main terms (D11 and D22) and the cross terms (D12 and D21) are plotted in Figures 48 and compared to previously reported data. Figure 3. Comparison of experimental integral diffusion coefficients obtained from this study to literature values: this work (integral value, ]), Albright et al. (9), and Myerson and Kim (2).
’ DISCUSSION
experiments were performed at each average salt concentration. The experiments were performed at pH 4.5 and 25 °C at a mean concentration of lysozyme of approximately 1% but with different mean concentrations of NaCl (1, 2, 3, 4, 5, and 6%). Table 3 shows the best fit results of all of the measurements at each salt concentration. The mass balances agreed to within 3% for lysozyme and 1.5% for NaCl, on average. This suggests that any losses of lysozyme to the walls of the cell are not significant. To estimate the uncertainty in the diffusion coefficients at each salt concentration, one set of results was dropped, in turn, to find
Lysozyme Diffusion without NaCl. Figure 2 shows that the diffusion coefficient of lysozyme decreases with increasing lysozyme concentration, and Figure 3 shows a comparison of our lysozyme diffusion coefficients with values from the literature. The lysozyme diffusion coefficient is in the same range as previously reported values and shows a decrease with increasing lysozyme concentration, as is also shown by Albright2,3 and Myerson.1 It is somewhat surprising that the values and trends are similar to those reported by Albright since our solutions contain acetate buffer and are more appropriately described as pseudoternary. 10316
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Figure 5. Comparison of the binary diffusion coefficient of NaCl, D2, in the system NaCl/water to the ternary diffusion, D22, in the system lysozyme/NaCl/water: ternary D22 this work (]), ternary D22 Albright (0) and binary D2 Stokes (2). The lines are the results of the NP calculations: with buffer (solid) and without buffer (dotted).
Figure 6. Cross term diffusion coefficient of salt (D21) in the ternary system lysozyme/NaCl/water: this work (]), Albright (0). On the scale of this figure, the error bars are smaller than the symbol size. The lines are the results of the NP calculations: with buffer (solid) and without buffer (dotted).
For a binary electrolyte, the DebyeHuckel equation gives the concentration dependence of the diffusion coefficient: d ln γ 0 D¼D 1 þ ð19Þ d ln c The activity coefficient, γ, is given by ln γþ ¼ Ajz1 z2 jI 1=2
ð20Þ
where A = 1.174 (L/mol)1/2 and I is the ionic strength defined as I ¼
1 2
∑i mi z2i
ð21Þ
For the present study involving aqueous lysozyme solution at pH 4.5 and 25 °C, lysozyme has a positive net charge n and eq 21 becomes I ¼
1 cr ðn2 þ nÞ 2
ð22Þ
After substitution of eq 22 into eq 20 and taking the first derivative of the expression, the thermodynamic correction term becomes 1=2 d ln γ 1 1 2 ð23Þ ¼ An ðn þ nÞc d ln c 2 2 Equation 19 therefore becomes " 1=2 # 1 1 2 0 D ¼ D 1 An ðn þ nÞc 2 2
ð24Þ
With the use of eq 24, the charge, n, and infinite dilution diffusion coefficient, D0, are treated as adjustable parameters. The value for n obtained by this procedure, 5.14, is much smaller than the
Figure 7. Cross term diffusion coefficient of lysozyme (D12) in the ternary system lysozyme/NaCl/water: this work ([), Albright (0). The line is the result of the NP calculations: the values calculated with and without buffer overlap on the scale of this figure.
actual value, which should be 11 at pH 4.5. As suggested by Rozenberger14 and by Albright et al.,2,3 this may be due to the presence of counterions that screen some of the protein charge. The best fit obtained for D0 was 4.750 106 cm2/s. The DH values for the diffusion coefficient are shown in Figure 2. The DH equation is only expected to be valid for very dilute solutions, and we see from Figure 2 that the DH values 10317
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Figure 8. Diffusion coefficients of the ternary system lysozyme/NaCl/ water at pH 4.5 and 25 °C. The lines in the figures are visual guides, not calculated values.
and the measured values are diverging with increasing lysozyme concentration. A comparison to previously reported integral and differential diffusion data by Albright et al.2,3 and Myerson and Kim1 is shown in Figure 3. Albright used Rayleigh interferometry, and their values are approximately 1520% larger than those reported in this study. Albright’s values are 5060% larger than the experimental values reported by Myerson and Kim using Gouy interferometry for lysozyme at pH 4.75. Nevertheless, all of the results in Figure 3 show a decrease in the diffusion coefficient with an increasing lysozyme concentration. There are several differences that might account for this variation. First, the diaphragm cell used in this work requires 13 days of diffusion time. Albright’s studies lasted less than 16 h. Diffusion coefficients in supersaturated solutions tend to decrease with solution age as shown by Myerson and Kim.1 Second, Albright relied upon the self-buffering capabilities of lysozyme and did not use any additional buffer. In this work, acetate buffer was used to maintain the pH at 4.5 and this difference could reflect the influence of the buffer. Lysozyme Diffusion with NaCl. For systems with added NaCl, we treated the system as ternary. Table 3 and Figure 4 show that the ternary main term diffusion coefficient, D11, for the flux of lysozyme (solute 1) due to its own gradient decreases with increasing salt concentration, as expected. However, comparing Tables 2 and 3, we can see that the ternary main terms D11 over our range of study exceed our experimental binary diffusion coefficients by approximately 4 to 15%. On the other hand, the ternary main term diffusion coefficient D22, for the flux of sodium chloride (solute 2) due to its own gradient, lies within approximately 1% to 16% of the Stokes13 binary diffusion data as shown in Figure 5. We can also see from Table 3 and Figure 6 that D21, the ternary cross term diffusion coefficient for the flux of solute 2 (NaCl) driven by a gradient in solute 1 (lysozyme), increases sharply as the sodium chloride concentration increases. For example at 3%
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salt concentration, the ratio D21/D22 is 146, which tells us that during the diffusion process, with a gradient of lysozyme alone, each lysozyme molecule would transport 146 molecules of sodium chloride. In crystallization, this can be an important factor. As lysozyme diffuses to a growing crystal, the concentration of salt near the growing crystal interface will exceed its value in the bulk solution. Under those conditions, the flux of NaCl will be driven from higher to lower concentration regions, which results in an increase of the cross term diffusion coefficient D21. The cross term diffusion coefficient, D12, for the flux of lysozyme due to the NaCl gradient is very small and decreases with increasing NaCl concentration as shown in Table 3 and Figure 7. At the same composition (3% salt concentration), the ratio D22/D12 is 41 272, which indicates that the flux of NaCl during the diffusion process does not interfere much with the diffusion of lysozyme. In Figures 48, our ternary values are shown as a function of NaCl concentration. In Figures 47, the ternary values are compared with the values obtained by Albright.2,3 The figures show similar trends and a fair agreement between Albright’s data and the experimental values of the ternary diffusion coefficients. Diffusion coefficients for electrolyte mixtures can be calculated from the NernstPlanck equations as described by Leaist,15 and values calculated from the NP equations are included, both with and without buffer, in Figures 47. The NP equations include a concentration driven flux and a flux driven by the electric field that arises due to the different mobilities of the ions. The NP equations are expected to be valid for dilute solutions so it is not surprising that there are discrepancies here. Nevertheless, the agreement between the measured and NP values is reasonably good (correct magnitude) for the main terms and the NP equations correctly predict a small (near zero) value for D12 (lysozyme diffusion due to salt gradients). In the case of D21 (NaCl diffusion due to a lysozyme gradient), the NP calculations do not agree with the measured values. This discrepancy suggests that the lysozymebuffer system may not be well-described by the NP equations at these concentrations.
’ CONCLUSIONS A diaphragm cell has been used for this work to experimentally determine ternary and binary diffusion coefficients of lysozyme. Although it has been proven to be one of the simplest methods for measuring diffusion coefficients with good accuracy, it requires frequent calibrations. The results of our experimental work shows that the binary diffusion coefficient decreases with increasing lysozyme concentrations, the main term D11 decreases with salt concentration and exceeds the binary diffusion coefficient D1 by approximately 415%, and the main term D22 lies within 116% of the binary diffusion coefficient, D2. The large cross-term diffusion coefficients D21 and the smaller cross-term D12 give an idea of the magnitude of the interaction between electrolyte solutes in a multicomponent diffusion process. A gradient in lysozyme has a large influence on the diffusion of salt, but a gradient in salt has only a small effect on the diffusion of lysozyme. ’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected]. 10318
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’ REFERENCES (1) Myerson, A. S.; Kim, Y. C. Diffusivity of Lysozyme in Undersaturated, Saturated and Supersaturated Solutions. J. Cryst. Growth 1994, 143, 79. (2) Albright, J. G.; Annunzita, O; Miller, D. G.; Paduano, L.; Pearlstein, A. J. Precision Measurements of Binary and Multicomponent Diffusion Coefficients in Protein Solutions Relevant to Crystal Growth: Lysozyme Chloride in Water and Aqueous NaCl at PH 4.5 and 25 °C. J. Am. Chem. Soc. 1999, 121, 3256. (3) Albright, J. G.; Annunzita, O; Miller, D. G.; Paduano, L.; Pearlstein, A. J. Extraction of Thermodynamic Data from Ternary Diffusion Coefficients. Use of Precision Diffusion Measurements for Aqueous Lysozyme Chloride-NaCl at 25 °C to Determine the Change of Lysozyme-Chloride Chemical Potential with Increasing NaCl Concentration Well into the Supersaturated Region. J. Am. Chem. Soc. 2000, 122, 5916. (4) Asfour, A. A. Improved and Simplified Diaphragm Cell Design and Analysis Technique for Calibration. Rev. Sci. Instrum. 1983, 54, 1392. (5) Northrop, J. H.; Anson, M. L. A Method for the Determination of Diffusion Constants and the Calculation of the Radius and Weight of the Hemoglobin Molecule. J. Gen. Physiol. 1929, 12, 543. (6) Stokes, R. H. An improved Diaphragm-Cell for Diffusion Studies, and Some Tests of the Method. J. Am. Chem. Soc. 1950, 72, 763. (7) Gordon, A. R. The Diffusion of Electrolytes and Macromolecules in Solution. Ann. N.Y. Acad. Sci. 1945, 46, 285. (8) Onsager, L. The Diffusion of Electrolytes and Macromolecules in Solution. Ann. N.Y. Acad. Sci. 1945, 46, 241. (9) Kirkwood, J. G.; Baldwin, R. L.; Dunlop, P. J.; Gosting, L. J.; Kegeles, G. Flow Equations and Frames of References for Isothermal Diffusion in Liquids. J. Chem. Phys. 1960, 33, 1505. (10) Cussler, E. L. Multi-component Diffusion; Elsevier: New York, 1976. (11) Cussler, E. L. Mass Transfer in Fluid Systems, 2nd ed.; Cambridge University Press: New York, 1997. (12) Slowinski, E. J.; Wosley, W. C.; Masterton, W. L. Chemical Principles in the Laboratory, 7th ed.; Hartcourt: Fort Worth, TX, 2000. (13) Stokes, R. H. The Diffusion Coefficients of Eight Uni-univalent Electrolytes in Aqueous Solution at 25 °C. J. Am. Chem. Soc. 1950, 72, 2243. (14) Rosenberger, F. Protein Crystallization. J. Cryst. Growth 1966, 166, 40. (15) Leaist, D. G.; Hao, L. Diffusion in Buffered Protein Solutions: Combined Nernst-Planck and Multicomponent Fick Equations. J. Chem. Soc. Faraday Trans. 1993, 89, 2775.
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