Translational Diffusion Coefficients of Bovine Serum Albumin in Aqueous Solution at High Ionic Strength

Journal of Colloid and Interface Science 218, 167–175 (1999) Article ID jcis.1999.6401, available online at http://www.idealibrary.com on

Translational Diffusion Coefficients of Bovine Serum Albumin in Aqueous Solution at High Ionic Strength Nispa Meechai, Alex M. Jamieson, 1 and John Blackwell Department of Macromolecular Science, Case Western Reserve University, Cleveland, Ohio 44106-7202 Received January 11, 1999; accepted July 6, 1999

We report static and dynamic light scattering measurements on bovine serum albumin (BSA) solutions at high ionic strength (I) where potential and hydrodynamic interactions between BSA molecules are of comparable strengths. Measurements of the concentration dependence of the osmotic compressibility, (dp/dc), and the translational diffusion coefficient, D m, are presented for several solvent systems: (a) at the isoelectric pH 5 4.7 and I 5 0.1, where long-range electrostatic repulsions are absent; (b) at pH 5 7.4 and I 5 0.15, 1.5, and 3.3, where a well-screened electrostatic repulsion is present. The results are compared with theoretical predictions which involve a microscopic hard-sphere treatment of the potential and hydrodynamic interactions. At pH 5 7.4 and I 5 1.5, our experimental results for dp/dc are in good agreement with the hard-sphere prediction, and our values for D m are, likewise, consistent with a hard-sphere hydrodynamic analysis in which contributions from the divergence terms in the velocity field are neglected. At the isoelectric pH, similar agreement with theory is obtained, provided the contribution of an attractive potential is included; at pH 7.4 and I 5 0.15, the contribution from a longrange repulsion must be included; at pH 7.4 and I 5 3.3, onset of protein aggregation is observed. © 1999 Academic Press Key Words: bovine serum albumin; diffusion; hydrodynamic interactions.

D m 5 ~d p /dc!/f m,

where d p /dc is the concentration derivative of the osmotic pressure, which embodies the thermodynamic driving force for diffusion, and f m is the concentration-dependent friction factor which describes the resistance to collective translation of particles arising from the potential and hydrodynamic interactions between them. As pointed out by Phillies (14), experiment indicates that f m 5 f s, the friction factor for tracer diffusion of a single particle through the solution, but the theoretical basis for this observation remains unclear. Carrying out analysis of the concentration corrections to d p /dc and D m to first order, one obtains, for spheres of radius R, a result of the form (13)

1

To whom correspondence should be addressed.

D m 5 D 0 ~1 1 k mf ! 5 ~kT/f 0 !~1 1 ~k d 1 k h! f !,

[2]

d p /dc 5 kT~1 1 k df !

[3]

1/f m 5 ~1/f 0 !~1 1 k hf !

[4]

where

and

INTRODUCTION

Various studies of translational diffusion of globular proteins, such as bovine serum albumin (BSA) or hemoglobin (Hb), have been reported in the recent literature (1–12). The mutual or translational diffusion relates the mass flux to the local chemical potential gradient. Depending on pH, proteins can be highly charged so that their mobility is coupled strongly to each other through electrostatic or electrodynamic interactions. However, by adding small ions to the aqueous buffer, the strength of these interactions is substantially reduced. Consequently, the diffusion coefficient depends on protein concentration, the protein charge, and the ionic strength of the solution. The concentration dependence of the mutual diffusion coefficient, D m, is expressed in the form (13)

[1]

with f 0 5 6phR (where h is the medium viscosity). Here, concentration is expressed as particle hydrodynamic volume fraction, f 5 (4 p /3) R 3 . The coefficient k d is determined by integration of the radial distribution function of the particles, which depends on the form of the interaction potential between particles; k h is generally evaluated from the Smoluchowski equation describing particle motions and depends on both potential and hydrodynamic interactions (13–18). Using static and dynamic light scattering, it is possible to separately evaluate the concentration dependence of d p /dc and D m. In this paper we apply this approach to investigate the dependence of d p /dc and D m as a function of pH and ionic strength. While extensive studies of translational diffusion of BSA have been reported in the literature, our investigation is prompted by an apparent gap in the experimental record for BSA solutions. Specifically, we are aware of no study where

167

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dual measurements are made of D m and p or d p /dc for BSA solutions under conditions of ionic strength where long-range electrostatic interactions are highly screened. These solvent conditions are of interest because the influence of protein structure and the role of the hydrodynamic interactions at short interparticle distances become important. Since the scattering power of globular proteins is weak, static and dynamic light scattering experiments can be carried out to relatively high concentrations without problems from multiple scattering artifacts, as confirmed directly by experiment (5). As a framework for the discussion of the effects of ionic strength and pH on the diffusion coefficient of BSA, we utilize the theoretical calculations of Batchelor (13), Felderhoff (15), and Phillies (18), which assume an impermeable hard-sphere model. We compare data on D m and d p /dc at the isoelectric pH 5 4.7 and I 5 0.1, where BSA has a net charge of zero (19), with results at pH 5 7.4 and I 5 0.15, pH 5 7.4 and I 5 1.5, and pH 5 7.4 and I 5 3.3. Here, BSA is negatively charged (20), but the Debye screening length is short, , D 5 0.8 nm at I 5 0.15, , D 5 0.25 nm at I 5 1.5, and , D 5 0.17 nm at I 5 3.3. MATERIALS AND METHODS

Experimental Procedure Monomeric BSA (characterized as 98.6% monomer) was used in this study, as obtained from the Bayer Corporation (Kankakee). Four solvent systems were selected for measurements of scattered intensity and mutual diffusion coefficient: 1. pH 4.7, I 5 0.1 (0.1 M NaCH 3COO, isoelectric point) 2. pH 7.4, I 5 0.15 (0.01 M Na 2HPO 4, 0.13 M NaCl) 3. pH 7.4, I 5 1.5 (0.01 M NaH 2PO 4, 0.14 M Na 2HPO 4, 1.08 M NaCl) 4. pH 7.4, I 5 3.3 (0.004 M NaH 2PO 4, 0.074 M Na 2HPO 4, 3.061 M NaCl) BSA was dissolved first in distilled, deionized water and then concentrated salt solution was added to make up the desired ionic strength, pH, and protein concentration. To remove contamination by dust, the concentrated BSA stock solutions were filtered through Millipore filter (0.45-mm pore size). Concentration series were made by successive dilution of the concentrated stock solution with the filtered solvent. The scattering cells were precleaned with detergent and water in an ultrasonic bath, soaked in concentrated H 2SO 4, then thoroughly rinsed with water. A steam test was performed on the cells to check their clarity. Protein concentrations were determined via UV absorbance. The absorbance of the prefiltered- and filtered-concentrated stock solution were compared. The difference in absorbance was smaller than 1.0%. All scattering experiments were performed on a Brookhaven Instrument (BI) Corporation spectrometer with a Spectra Physics 15-mW He/Ne laser (l 5 632.8 nm). Sample cells were

mounted at the center of a temperature-controlled, refractive index matched bath. All measurements were made at 22°C. Absolute calibration of the spectrometer was made with double-distilled toluene for which the absolute Rayleigh ratio at l 5 632.8 nm for vertically polarized light was assumed to be R v 5 14 3 10 26 cm 21 as reported by Kaye et al. (21). The intensity measurements were carried out at a scattering angle of 90°C. For dynamic light scattering measurements the intensity autocorrelation function was measured at a 30° scattering angle with a 264-channel BI 2030 AT 4-bit correlator. Eight delay channels were also used to determine the measured baseline for a given autocorrelation function. The intensity–intensity time correlation functions, C( t ), were chosen for data analysis only when the difference between the calculated and measured baselines was less than 0.1%. The z average of the mutual diffusion coefficient (D mz ) was determined via cumulant analysis of C( t ) (22), ~1/ 2!ln @C~ t ! 2 B# 5 2G# t 1

S D

m2 ~G# ! 2 t 2 1 · · ·, ~G# ! 2

[5]

where the first cumulant G# 5 D mz q 2 , with q being the scattering vector and m 2 /(G# ) 2 the normalized variance. Typically, m 2 /(G# ) 2 , 0.05 except for systems 1 and 4 where m 2 /(G# ) 2 sometimes was ;0.1, indicative of a small degree of aggregation. The viscosity, h, of the aqueous solvents was measured at 22°C with a Cannon Ubbelohde viscometer by measuring the flow time and then calibrating relative to the flow time of pure water at the same temperature. The refractive index, n# 0, of each solvent was determined using an Abbe refractometer at 22°C. The refractive index increment at constant solvent chemical potential, (dn# /dc) ms, for each solution was determined using a differential refractometer at wavelength 632.8 nm. The instrument was calibrated using solutions of sodium chloride and bovine serum albumin solution in water for which the refractive index increment is known (23). All of the protein solutions used in this study were dissolved in the buffer solution and then dialyzed against this solution for 2 days (24). The refractive index increment was measured at constant solvent chemical potential by comparing the refractive index of the dialyzed solution versus the solvent. The concentration of the concentrated stock BSA solution after dialysis was checked by comparing the UV absorbance of the solution before and after dialysis. Table 1 summarizes the measured values of h , n# 0 , and (dn# /dc) ms. RESULTS

First, we present our results for static and dynamic light scattering from BSA in 0.1 M NaOAc at pH 4.7 and T 5 228C. Here, the net charge on the protein is zero (19), and long-range

DIFFUSION COEFFICIENTS OF BOVINE SERUM ALBUMIN

TABLE 1 Properties of the Solution Used in this Experiment

System pH pH pH pH

4.7, 7.4, 7.4, 7.4,

I I I I

5 5 5 5

0.1 0.15 1.5 3.3

Viscosity (cP)

Refractive index (n# 0 )

dn# /dc (ml/g)

0.9808 0.9724 1.1016 1.2384

1.3336 1.3340 1.3460 1.3620

0.1820 0.1830 0.1668 0.1452

electrostatic repulsions are absent. Figure 1 shows a plot of the static light scattering data in the form Kc/DR u 5 ~1/RT!d p /dc,

[6]

where we have used optical constant K 5 4 p 2 n# 02 (dn# /dc) 2 / N Al 4 5 2.41 3 10 27 , corresponding to wavelength l 5 632.8 nm, dn# /dc 5 0.1820 ml/g, and n# 0 5 1.3336. The data plotted in Fig. 1 represent the superposition of two completely independent sets of results. Evidently, from Fig. 1, within experimental error, a linear least squares fit to an equation of the form Kc/DR u 5 ~1/M w!~1 1 k df !

[7]

accurately describes the data and leads to the results M w 5 69,000 6 700 and k d 5 5.41 6 0.40. In constructing Fig. 1, and subsequent figures, we have elected to compute the hydrodynamic volume fraction, f, from the hydrodynamic radius, taking into account that the known structure of BSA in solution is not a sphere, but is a prolate ellipsoid of axial ratio (a/b) 5 3.5 (25). Thus, we compute the corresponding values of a and b using R5

a~1 2 b 2 /a 2 ! 1/ 2 . ln$~a/b!@1 1 ~1 2 b 2 /a 2 ! 1/ 2 #%

169

other systems are even less turbid at the highest concentrations studied. The corresponding values of D m for BSA at pH 4.7 and I 5 0.1 are plotted vs the hydrodynamic volume fraction in Fig. 2. The line with negative slope in Fig. 2 represents a least squares fit to Eq. [2], yielding a value for the hydrodynamic radius R 5 3.67 nm, in good agreement with the literature (27), and for the first-order correction, k m 5 21.42. Next, we report studies of BSA in phosphate-buffered saline at pH 5 7.4 and I 5 0.15. In Fig. 3, we display the static light scattering data which again can be well described by a linear least squares fit to Eq. [7], as shown by the solid line, from which we determine M w 5 68,000 6 1600 and k d 5 20.98 6 0.93. In Fig. 4, we exhibit the corresponding data for D m, together with a linear least squares fit to Eq. [2] which gives R 5 3.58 nm and k m 5 2.58 6 0.13. Finally, in Fig. 5, we show the static light scattering data from BSA solutions at pH 7.4 but with higher ionic strength, I 5 1.5. Again, within experimental error, a linear fit to Eq. [7] can be obtained, as shown, and gives values M w 5 67,000 6 200 and k d 5 12.09 6 0.44. The corresponding data for D m are shown in Fig. 6, and the linear least squares fit shown leads to R 5 3.65 nm and k m 5 0.46 6 0.04. As assurance that our experiments are measuring translational diffusion of the protein at this high ionic strength, we have confirmed that the mean decay rate G# is proportional to the square of the scattering vector, as shown in Fig. 7. We note, in passing, that analysis of the static light scattering data from BSA at pH 7.4 and even higher ionic strength, I 5 3.3, yields a slightly high molecular weight, M w 5 75,000, comparative to the calculated value 66,210 (28), and a k d 5 9.17. From the

[8]

With these values, and the measured molecular weight M w, we compute f as f 5 (cN A/M w)(4 p /3)ab 2 (26). The f values calculated in this way are 33.85% smaller than those calculated from the equivalent hydrodynamic sphere ( f 5 (cN A/ M w)(4 p /3) R 3 ), as pointed out by Dorshaw and Nicoli (26), and hence the k d value is higher (5.41 vs 3.60). In Table 2, we summarize the different k d, k m, and k h 5 k m 2 k d values computed for each system using the equivalent prolate ellipsoid and equivalent sphere assumptions for f, respectively. It should further be noted that, at f 5 0.060, the value of the Rayleigh ratio, R u 5 6.32 3 10 24 cm 21, corresponds to an exceedingly small turbidity, t 5 5.1 3 10 23 cm 21, which indicates, as confirmed by previous experiment (5, 12), that we can safely ignore contributions from multiple scattering. The

FIG. 1. The intensity of light scattered by BSA solutions in 0.1 M NaOAc at the isoelectric pH 4.7 at 22°C plotted against hydrodynamic volume fraction f for a prolate ellipsoid according to Eq. [7].

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TABLE 2 Experimental Results for the BSA Solutions Assuming spherical particle System pH pH pH pH

4.7, 7.4, 7.4, 7.4,

I I I I

5 5 5 5

0.1 0.15 1.5 3.3

Assuming prolate ellipsoid

D m0 3 10 7 (cm 2/s)

Rh (nm)

Mw

kd

kh

km

kd

kh

km

5.99 6.21 6 0.10 5.38 6 0.02 4.30

3.67 3.58 6 0.05 3.65 6 0.01 4.05

69,000 6 700 68,000 6 1600 67,000 6 200 75,000

3.60 6 0.26 13.88 6 0.62 8.00 6 0.29 6.06

24.54 212.17 27.69 25.59

20.94 1.71 6 0.08 0.31 6 0.03 0.47

5.41 6 0.40 20.98 6 0.93 12.09 6 0.44 9.17

26.83 218.40 212.14 28.47

21.42 2.58 6 0.13 0.46 6 0.04 0.70

diffusion data, the corresponding analysis leads to a high value for the hydrodynamic radius, R 5 4.05 nm, and a value of the first-order concentration correction k m 5 0.70. From the elevated values of M w and R it appears likely that some association is occurring at this salt concentration.

Pusey and Tough (16) have pointed out that, within the framework of the Felderhoff description, the effect of long-range Coulombic repulsions can be described by an effective hardsphere interaction of range R HS 5 gR, with g . 1, which leads to k d 5 8 g 3,

[12]

DISCUSSION

Theories that predict the concentration dependence of the diffusion coefficient of spherical particles have been given by Batchelor (13), by Felderhoff (15), and by Phillies (18). Batchelor (13) was the first to obtain a theoretical expression for the mutual diffusion coefficient which incorporates exact numerical knowledge of the two-body hydrodynamic interaction between particles. Under stick boundary conditions and using a hard-sphere interaction potential, the following result was obtained (13), D m 5 D 0 ~1 1 1.45 f !,

k h 5 26 g 2 1 1 2

15 9 75 1 . 3 1 8 g 64 g 256 g 4

[13]

Likewise, these authors note (16) that the effects of a shortrange attractive contribution to the interparticle potential can be characterized by adding an extra term to the hard-sphere radial distribution function, which leads to kd 5 8 2 a,

[14]

k h 5 26.44 1 0.49 a ,

[15]

[9]

corresponding to k d 5 8 and k h 5 26.55. Subsequently, Felderhoff (15) formulated a description based on the Smoluchowski equation which extends to general potential interactions and allows mixed stick–slip boundary conditions at the surfaces of the spheres. For mixed stick–slip boundary conditions, the single-particle diffusion coefficient becomes D 0 5 kT/6 ph R~1 2 z!,

[10]

where the parameter z characterizes the boundary condition, such that 0 # z # 1/3, where z 5 0 corresponds to stick and z 5 31 to pure slip. For hard spheres, the first-order concentration correction takes the form of Eq. [2], but where k h is now a function of z. Under stick conditions, Felderhoff (15) obtains a result for the first-order correction to D m which is essentially identical to that of Batchelor (13), D m 5 D 0 ~1 1 1.56 f !,

[11]

viz. k m 5 8 2 6.44 5 1.56. The slight discrepancy stems from a difference in the way the hydrodynamic interactions are evaluated (15).

FIG. 2. The mutual diffusion coefficient D m of BSA in 0.1 M NaOAc at the isoelectric pH 4.7 at 22°C plotted against hydrodynamic volume fraction f for a prolate ellipsoid.

DIFFUSION COEFFICIENTS OF BOVINE SERUM ALBUMIN

FIG. 3. The intensity of light scattered by BSA solutions in phosphatebuffered 0.15 M NaCl at pH 7.4 and 22°C plotted against hydrodynamic volume fraction f for a prolate ellipsoid.

171

FIG. 4. The mutual diffusion coefficient D m of BSA in phosphate-buffered 0.15 M NaCl at pH 7.4 and 22°C plotted against hydrodynamic volume fraction f for a prolate ellipsoid.

if the divergences in the hydrodynamic interactions are ignored, their calculation leads to

which gives k m 5 1.56 2 0.51 a .

[16]

Phillies (17) carried out the first calculation of the secondorder correction term to D m and found that three-body hydrodynamic interactions contribute significantly to this term. Subsequently, Carter and Phillies (18) extended this analysis to the third-order term in the concentration dependence of D m, incorporating three- and four-body hydrodynamic interactions and including certain divergence terms in the Smoluchowski equation which are neglected in the Batchelor and Felderhoff descriptions. Carter and Phillies (18) obtain the following results

D m 5 D 0 ~1 1 0.56 f 2 8.39 f 2 2 35 f 3 !.

[19]

The authors point out (18), in Eq. [19], that the difference between their value of the first-order correction coefficient,

D m 5 D 0 ~1 2 8.898 f 1 22.17 f 2 2 52 f 3 ! 3 ~1 1 8 f 1 30 f 2 1 72 f 3 !.

[17]

The first bracketed term represents the concentration dependence of the frictional coefficient calculated by Carter and Phillies (18); the second term is the concentration dependence of the inverse structure factor obtained from the literature (29). From Eq. [17], we obtain D m 5 D 0 ~1 2 0.898 f 2 19.01 f 2 2 70 f 3 !.

[18]

Thus, this analysis leads to the conclusion that the first-order correction term to D m is k m 5 20.898, corresponding to k d 5 8 and k h 5 28.898. Carter and Phillies (18) further note that

FIG. 5. The intensity of light scattered by BSA solutions in phosphatebuffered 1.50 M NaCl at pH 7.4 and 22°C plotted against hydrodynamic volume fraction f for a prolate ellipsoid.

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MEECHAI, JAMIESON, AND BLACKWELL

At the isoelectric pH (pH 5 4.7), from Table 2, the small value of k d 5 5.41, relative to the hard-sphere result, k d 5 8, is a strong indication that significant attractive interactions are present. The origin of the attractive interaction is expected to be predominantly van der Waals forces. Charge fluctuation interactions are also important at the isoelectric pH, but their strength diminishes rapidly as ionic strength increases. This conclusion invalidates the interpretation that the negative value of k m, k m 5 21.42, is consistent with the hard-sphere calculation of Phillies. Indeed, our results are in qualitative agreement with the predictions of the Felderhoff model incorporating an attractive potential, as presented in Eqs. [14]–[16], except that we include an additional term (21) in k h as a reference frame correction:

FIG. 6. The mutual diffusion coefficient of BSA in phosphate-buffered 1.50 M NaCl at pH 7.4 and 22°C plotted against hydrodynamic volume fraction f for a prolate ellipsoid.

k 5 0.56, and that obtained by Felderhoff (15), k 5 1.56, Eq. [11], arises because Eq. [19] implicitly incorporates a reference frame correction (30 –32), which requires that the volume flux of particles into a closed region must be exactly cancelled by the volume flux of fluid out of this region, and appears to be required for proper comparison of theory vs experimental data (18, 30 –33). The predictions embodied in Eqs. [9], [11], [18], and [19] are distinctly different, which suggests that the effect of the reference frame and divergence terms should be large enough to be observed by appropriate experiments. Here, a relevant point is that the divergence terms are most sensitive to the hydrodynamic interactions at short interparticle distances and hence are more important when long-range repulsions are absent (33). Experimental results on model hard-sphere dispersions have been reported which appear to support both the Batchelor and Felderhoff (34 –38) and Phillies (39) calculations. The comparison of protein diffusion data vs a hard-sphere model is not expected to be quantitatively accurate due to the fact that proteins are not exactly spherical, and in addition, may exhibit some porosity or even hydrodynamic slip, if they are very small. For our purposes, however, the hard-sphere calculations are a convenient framework to discuss our experimental results on the variation of the coefficients k d, k m, and k h for BSA solutions as a function of solvent conditions. We also note when comparing experiment vs theory, it is important to realize that the above theoretical expressions refer only to small f. Specifically, if terms to higher than linear in f are to be neglected, particle concentrations must be restricted to f , 0.05.

kd 5 8 2 a,

[20]

k h 5 27.44 1 0.49 a .

[21]

k m 5 0.56 2 0.51 a .

[22]

Hence,

Thus, from the experimental value k d 5 5.41, we deduce from Eq. [20] a value a 5 2.59 which corresponds, via Eq. [22], to a prediction k m 5 20.76, which is larger than the experimental result but represents satisfactory agreement, bearing in mind the nonspherical structure of BSA. Our results are consistent with earlier measurements on proteins near the isoelectric pH (5, 8, 11, 12). Previous studies of mutual diffusion of globular

FIG. 7. The mean decay rate plotted against the square of the scattering vector for 80 mg/ml BSA solutions in phosphate-buffered 1.5 M NaCl at pH 7.4 and 22°C.

DIFFUSION COEFFICIENTS OF BOVINE SERUM ALBUMIN

173

proteins have reported negative values of k m under isoelectric pH (5, 8, 11, 12). However, in most cases, evidence for the nature of the intermolecular potential in the form of osmotic compressibility data was not obtained. Veldkamp and Votano (11) reported k m ' 20.5 for oxyhemoglobin solutions near the isoelectric pH and I 5 0.15, and also observed k d ' 0, which indicates an important contribution from attractive interactions. Further support for this interpretation is the fact that analyses of osmotic pressures (20) from BSA solutions near the isoelectric pH, using a hard-sphere model, produce values of the contact radius which are substantially smaller than the hydrodynamic radius. At ionic strength 0.15 M and pH 7.4, Table 2 indicates k d 5 20.98 and k m 5 2.58. Preliminary inspection of these data indicates these data are qualitatively, but not quantitatively, in agreement with the Felderhoff model with incorporation of a long-range repulsion (16), viz. Eqs. [12] and [13], except that we again add a contribution (21) to k h as a reference frame correction, k d 5 8 g 3, 15 9 75 1 1 , k h 5 26 g 2 8 g 64 g 3 256 g 4 2

[23] [24]

and, hence, km 5 8g 3 2 6g 2 2

15 9 75 1 1 . 8 g 64 g 3 256 g 4

FIG. 8. Reduced light-scattering intensities for BSA solutions in the form KcM/DR u , where M 5 66,210, the known molecular weight for BSA, plotted against hydrodynamic volume fraction f calculated from the known hydrodynamic radius for BSA, R 5 3.6 nm, assuming a prolate ellipsoid. The solid curve is the normalized hard-sphere virial expansion, (M/RT)d p /dc 5 1 1 8 f 1 30 f 2 1 72 f 3 ; }, data at the isoelectric pH 5 4.7 and I 5 0.1; Œ, data at pH 5 7.4 and I 5 0.15; F, data at pH 5 7.4 and I 5 1.5.

[25]

Thus, application of Eq. [23] indicates a value g 5 1.4, which leads via Eq. [25] to a prediction k m 5 8.4, significantly larger than the experimental value. However, the contribution of the third virial term is certainly significant for this system, as discussed below, which will cause an overestimate of g. At pH 7.4, and higher ionic strength, 1.5 M, we determine k d 5 12.09 6 0.44, a little larger than the hard-sphere value, k d 5 8, and k m 5 0.46 6 0.04, a value intermediate between the first-order hard-sphere predictions of Felderhoff, k m 5 1.56 (Eq. [11]), and the calculation of Phillies, including both reference frame correction and divergence terms, k m 5 20.898 (Eq. [18]). In fact, our experimental result is numerically very close to the Felderhoff result with inclusion of a reference frame correction (k m 5 0.56). It remains for us to consider the fact that, in the concentration range covered in the present experiments, higher-order concentration terms are likely to be significant (40). Neal et al. (41) studied static and dynamic light scattering from solutions of bovine serum albumen (BSA). Their results demonstrate that when third-order virial terms become important in d p /dc, then the f 3 terms must also be included in the analysis of D m. In Fig. 8, we re-plot our results for d p /dc, normalized by the intercept I 5 RT/M, calculated from the known molecular

weight, M 5 66,210, and superpose the theoretical virial expansion (29): ~M/RT!d p /dc 5 ~1 1 8 f 1 30 f 2 1 72 f 3 !.

[26]

Here, for all systems, we compute the hydrodynamic volume fraction using the literature values R 5 3.6 nm (27), a/b 5 3.5 (25), and Eq. [8]. Clearly, Fig. 8 shows an evolution in the experimental data from the results at pH 7.4 and I 5 0.15, which fall well above the theoretical curve (Eq. [26]), indicative of a significant contribution from long-range electrostatic repulsions (cf. Eq. [23]), to the results at the isoelectric pH, which fall well below the hard-sphere prediction, indicative of the presence of a long-range attractive potential (cf. Eq. [20]). The data at pH 7.4 and I 5 1.5, where the electrostatic repulsions are more strongly screened, appear quite consistent with the hard-sphere prediction. In Fig. 9, we likewise re-plot our results for D m, normalized by an intercept, D m0, calculated using the literature value for the hydrodynamic radius, R 5 3.6 nm, and superpose the hard-sphere predictions according to Eqs. [18] and [19]. We also plot in Fig. 9 the expression found by Al-Naafi and Selim to accurately describe their experimental data on mutual diffusion of hydrophobic silica microspheres (38):

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MEECHAI, JAMIESON, AND BLACKWELL

FIG. 9. Normalized mutual diffusion coefficients, D m/D m0 for BSA solutions, plotted against hydrodynamic volume fraction f for a prolate ellipsoid (R 5 3.6 nm). The solid curve represents Eq. [18]; the broken line represents Eq. [19]; the dotted line represents Eq. [27]; }, data at the isoelectric pH 5 4.7 and I 5 0.1; Œ, data at pH 5 7.4 and I 5 0.15; F, data at pH 5 7.4 and I 5 1.5.

nonspherical, and also that the hydrodynamic interaction at short distances is not accurately described by the hard-sphere model embodied in Eq. [18]. The origin of the latter failure is not clear, but one possibility is that factors such as flexibility and porosity of the protein (42) have a significant influence on the hydrodynamics during close encounters, but relatively little effect on the interparticle potential. In summary, we have presented experimental data on both D m and d p /dc for BSA in three-solvent systems where the effects of potential and hydrodynamic interactions between BSA molecules have comparable impact on D m. Our light scattering intensity data at the isoelectric pH clearly indicate that attractive interactions make a major contribution to d p /dc and hence to D m. At pH 7.4, where the protein is negatively charged, our results likewise indicate that repulsive interactions are significant. As the ionic strength increases, hydrodynamic interactions contribute more importantly to the concentration dependence of D m. Under all solvent conditions, the concentration dependence of D m is adequately described by the hard-sphere hydrodynamic model of Batchelor (13) and Felderhoff (15), appropriately modified by inclusion of a longrange attractive potential at the isoelectric pH, and a screened electrostatic repulsion at pH 7.4. ACKNOWLEDGMENT We thank the Royal Thai Government for the award of a graduate fellowship to NM.

Dm @~1 1 2 f ! 2 1 ~ f 2 4! f 3 # 5 ~1 2 f ! 6.55 . D m0 ~1 2 f ! 4

[27]

In the limit of small f, Eq. [27] reduces to the result of Batchelor. It is quite evident in Fig. 9 that the experimental data for BSA in the three-solvent systems show differences which correlate with the trends apparent in the osmotic compressibility in Fig. 8. Accordingly, the D m/D m0 values at pH 7.4 and I 5 0.15 rise well above the hard-sphere predictions of Eqs. [18], [19], and [27], consistent with the observation that the osmotic compressibility is augmented by contributions from long-range electrostatic repulsions; the D m values at pH 5 4.7 and I 5 0.1 fall quite close to Eq. [18], but this is invalidated by the clear evidence for a substantial long-range attraction, as indicated in Fig. 8 by the deviation of the osmotic compressibility from the hard-sphere prediction. At pH 7.4 with I 5 1.5, where the osmotic compressibility data follows closely the hard-sphere prediction, the concentration dependence of D m deviates widely from the prediction of Eq. [18] of Phillies. Our D m values fall close to and between the predictions of Eqs. [27] and [19], which differ essentially only in the use of a reference frame correction in Eq. [19]. Thus, a hard-sphere calculation semiquantitatively describes our combined D m and d p /dc data at I 5 1.5, where the BSA retains predominantly its monomeric structure. The residual discrepancies presumably reflect, in part, the fact that BSA is

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