2355
J. Phys. Chem. 1992, 96, 2355-2359 A VZ a
B K
Pmar
volume change associated with the exchange of the alcohol (cm' mol-I) ultrasonic absorption coefficient (Np cm-') degree of micelle ionization specific conductance (S cm-l) absorption per wavelength for the surfactant exchange pro-
uA2 us2 0AS2 7i
variance in the distribution of (5 for the mixed micelles variance in the distribution of ii for the mixed micelles covariance in ii and cf distributions for mixed micelles relaxation time associated with process i (s)
Registry No. Pr, 71-23-8; DTAB, 2082-84-0.
cess
Dlffuslon of Bovine Serum Albumin in Aqueous Solutions A. K. Gaigalas, J. B. Hubbard,* M. McCurley, and Sam Woo Center for Chemical Technology, National Institute of Standards and Technology, Gaithersburg, Maryland 20899 (Received: July 3, 1991; In Final Form: October 17, 1991)
The diffusion coefficient of bovine serum albumin (BSA) was measured in aqueous solutions of varying temperature, pH, BSA concentration, and ionic strength. The measurements were carried out using dynamic light scattering with the photon detector set at a 90' angle. The measured diffusion coefficients were compared to calculated values using phenomenological models which account for the screened Coulomb interaction between the charged proteins, as well as hydrodynamic corrections to the friction factor. The dimensions of BSA were obtained from structural data, and the charge on the protein was estimated using titration data. Although the measured and calculated values of the diffusion coefficient are in general agreement, significant discrepancies are observed. No single theoretical model seems capable of accurate predictions for all ranges of ionic strength and protein concentration.
Introduction This paper deals with the physical principles of globular protein diffusive motion and explores the accuracy of existing models for estimating the mutual diffusion coefficients of compact macromolecules under a variety of solution conditions. The mutual diffusion coefficient relates the mass flux to the local chemical potential gradient. In most cases, the proteins are in the presence of other species and the protein motion is coupled to the motion of these particles. Furthermore, the proteins are charged so that their motions are strongly coupled to each other as well as to other mobile ions. A complete description of the diffusive motion of a protein needs to account for the coupled system involving the motion of other species and electric charge currents. Consequently, the observed mutual diffusion coefficient is a function of the concentration and ionic character of other components. The mutual diffusion coefficient should be distinguished from the self-diffusion and tracer diffusion coefficients. The mutual diffusion coefficient describes the relaxation of concentration gradients. Tracer diffusion characterizes the random motion of a tagged molecule in a solution of identical untagged molecules, while the self-diffusion coefficient is the dilute concentration limit of both mutual and tracer diffusion coefficients. Bovine serum albumin (BSA) was chosen for thii study because it is a readily available protein that has served as a model in many physical and chemical studies. Its ability to bind many ligands has made it important in separation and drug delivery methods. Its stability and reproducibility have made it ideal for use in calibration of many biological assays. In the present work, we use BSA to characterize a variety of phenomena which can influence diffusive transport of proteins in solution. The mutual diffusion coefficient scales linearly with temperature and inversely with solvent viscoSity. At sufficiently high temperature, the protein structure can be modified, leading to large change in diffusion coefficient. The pH can have a strong effect on the diffusion coefficient by controlling the amount of surface charge on the protein; the surface charge determines the Coulomb interaction with other charged species. The pH may also influence the conformational state of the protein. For example, BSA goes through a conformational transition as the pH is lowered from 4.5 to 4.0. The solution in which the protein is dissolved usually contains many small ions which screen the protein charge and mediate the Coulomb interactions between proteins. As a result, there is a significant dependence of the mutual diffusion coefficient
on the ionic strength. Other interactions that are important include short-range repulsion and modification of the friction factor through hydrodynamic coupling. All of the above phenomena lead to a concentration dependence of the observed mutual diffusion coefficient. A good starting point for estimating the mutual diffusion coefficient, D, is the generalized Einstein-Stokes expression's2
D=- aa/ a c P , T
f
(1)
where the concentration derivative, ac, of the osmotic pressure, T,a t constant pressure and temperature gives the effect of all interactions between the proteins and other molecules. For the case of charged proteins, screened Coulomb interactions are dominant. The friction factor,f, represents the drag on the protein exerted by the liquid. For dense suspensions the drag is influenced by the flow induced by nearby proteins. Therefore, a hydrodynamic correction to the basic Stokes friction drag is required. Estimates for the effect of protein interactions and the hydrodynamic corrections lead to an expression for the mutual diffusion coeficient which is good to first order in the protein concentration. The expression is of the form D = DO(1 + klc + k 2 ~ ) (2) where c is the concentration of proteins in g/L and kl and k2 are approximate contributions from the thermodynamic particle interactions and the hydrodynamic corrections, respectively. Dois the mutual diffusion coefficient at infinite dilution. The above expression provides a framework for the discussion of the relative importance of the changes in the diffusion coefficient of BSA brought about by temperature, ionic strength, and pH. The results should be most useful for globular proteins, but the general theory applies to any class of solvated macromolecules.
Experimental Technique Dynamic Ught Scattering. Dynamic light scattering has become an important technique for the measurement of diffusion coefficients of proteins in aqueous solutions. The foundations of this technique have been discussed in many work^.^,^ We will sum(1) Tanford, C . Physical Chemistry of Macromolecules; John Wiley and Sons: New York, 1967. (2) Phillies, G. D. J. J . Chem. Phys. 1974, 60, 976-982.
This article not subject to U S . Copyright. Published 1992 by the American Chemical Society
2356 The Journal of Physical Chemistry, Vol. 96, No. 5, 1992 Argon Ion Laser L
I
Sample
I u
Autocorrelator
Computer
Light trap
Figure 1. A schematic diagram of the experimental apparatus. The argon ion laser operated at 514 nm with 30-60 mW of power. The low-noise photomultiplier was set at a 90’ angle to the incident beam. The optics were arranged to provide the largest possible photon correlation (small scattering volume and small detector area).
marize the applicable results for the sake of completeness. The emphasis will be on low-level scattering where photon counting techniques are utilized. The primary measured quantity is the conditional probability that if a photon is detected at time zero, then another photon will be detected at time t seconds. Let C(t) denote the measured conditional probability. Then under the assumption that the scattered photons can be described by a Gaussian random variable, it is possible to relate C(t) to the fluctuations of the electric field at the surface of the detector. The explicit relation is given by
[ + ;:;]
C(t) = (N)’ 1
-g
(3)
where ( N ) is the average photon counting rate and Z,(t) is the autocorrelation function of the electric field at the detector. The brackets ( ) denote equilibrium ensemble averaging. Here, g describes the effects due to finite detection efficiency and the loss of photon correlation due to the finite scattering volume and the finite extent of the detector sensitive surface. A general analysis of the autocorrelation was developed by K ~ p p e l . In ~ this treatment, which is applicable to small times, the terms on the left side of eq 1 are written as an exponential with an exponent which is a polynomial in time. The linear term in this expansion is proportionate to the diffusion coefficient while the quadratic term gives the variance of the diffusion coefficients. This approach is used in the present work for the analyses of data where the BSA concentration is 0.5 g/L. At such a small protein concentration, interactions can be assumed to be small. At higher protein concentrations, it is necessary to include interactions between particles. In that case, the interpretation of the autocorrelation data has to follow the procedure developed by Ackerson.6 His analysis shows that the linear term in the cumulant expansion is still proportional to the average diffusion coefficient, but the meaning of higher order terms is less clear. For interacting particles the convergence of the expansion at large times may be poor. In this case, it is necessary to develop a set of coupled transport equations whose solution may contain several collective modes. The autocorrelation function can then be represented as a sum of fast and slow decaying exponentials corresponding to the different modes. The above discussion implies that the connection between the measured autocorrelation function and protein transport parameters is not a t all simple. Various assumptions have to be made before the ensemble averages can be carried out, and these assumptions must be justified before an analysis can be deemed reliable. (3) Berne, B. J.; Pecora, R. Dynamic Light Scattering John Wiley and Sons: New York, 1976. (4) Ware, B. R. In Optical Techniques in Biological Research; Rousseau, D. F., Ed.; Academic Press: New York, 1984; pp 1-64. (5) Koppel, D. E.J . Chem. Phys. 1972.57, 4814-4820. (6) Ackerson, B. J. J . Chem. Phys. 1979, 70, 242-246.
Gaigalas et al. Light Scattering Apparatus. The diffusion coefficient of BSA was measured using photon correlation spectroscopy. The apparatus, shown in Figure 1, consisted of a argon ion laser operating at 514 nm, a cuvette containing the protein solution, and a photomultiplier detector at a 90° angle. BSA does not absorb strongly at 5 14 nm so that the laser beam does not perturb the protein. The measured photon autocorrelation response of a BSA solution may contain two components: a fast decay associated with the BSA protein and a slow decay associated with the “dust” particles. In practice, it is very difficult to eliminate dust particles from the solution. Yet, if the tilt in the background caused by the dust particles is ignored, it can lead to large errors in the value of the fast decay constant extracted from the rapidly decaying part of the autocorrelation fun~tion.~~* The effects due to dust particles come at discrete times in the form of a large increase in photon counting rate. After such a burst, we find the presence of a slow decay component in the autocorrelation function. Data acquisition procedures were modified to separate the data into a part with a dust component and a part without dust. This was done by counting the photons for a 30-s period. After the 30-s period, the last 16 bins of the data were compared with 16 delayed bins. The delay was set at lo00 times the sampling time. If the two numbers were within two standard deviations of each other, the data were accumulated as dust-free data. Otherwise, the data were tagged as having a dust component and stored in another array. All of the reported measurements used the data which were free of dust. Sample Preparation. Sample preparation is a critical component for measurements of light scattering from protein solutions. The bovine serum albumin (BSA), obtained from Dr. Jess Edwards of the National Institute of Standards and Technology, is a standard reference material (SRM) of known concentration (7% solution) and purity. It is of known ionic strength (25 mM) and pH 6.66. The buffers used were phosphate (range 5.7-8.0) and citratephosphate (range 2.6-7.0) at 100 mM. In order to maintain 25 mM ionic strength of the SRM BSA, each buffer was diluted 4-fold. These 25 mM buffer solutions were made and stored without addition of BSA. Each cuvette sample was generated by adding 20.1 pL of SRM BSA, from sealed singleuse ampules, to 3.0 mL of buffer solution which had been filtered through a 0.2-pm nitrocellulose membrane filter. The final BSA concentration in each cuvette was 0.5 g/L for the temperature, pH, and ionic strength tests. For the protein concentration tests, the investigated concentrations were 0.5, 10, 20, and 35.0 g/L; the pH was measured for the final solution. The conductivity of all buffer solutions was -5 mS. All cuvettes were centrifuged at 2500 rpm for 30 min prior to the scattering experiment.
Results The diffusion coefficient of BSA was measured at selected values of temperature, pH, salt concentration, and BSA concentration. The data a t low protein concentrations were analyzed using the cumulant expansion technique which parametrizes the measured autocorrelation in terms of an average diffusion coefficient and a variance in diffusion coefficients. The data from samples containing larger protein concentrations were analyzed using a two-exponential model. Figure 2 shows the observed autocorrelation function in the case of a solution with a 35 g/L protein concentration. The autocorrelation function is described by a rapidly decaying component which can be described by a single exponential and a background with a small linear slope. The background slope seems to be independent of measurement parameters and is most likely instrumental. The background slope could be characteristic of collective modes which may exist in concentrated samples. A similar slope has been observed9 and interpreted as clustering of BSA molecules. We have measured (7) Licinio, P.; Delaye, M. J . Phys. Chem. 1987, 91, 231-235. (8) Stelzer, K. J.; Hastings, D. F.; Gordon, M. A. Anal. Biochem. 1984, 136. 251-257. (9) Giordano, R.; Maisano, G.; Mallamace, F.; Micali, N.; Wanderlingh, F. J . Chem. Phys. 1981, 75, 4770-4775.
The Journal of Physical Chemistry, Vol. 96, No. 5, 1992 2351
Diffusion of BSA in Aqueous Solutions
I c
490000
a3
n
L, 390000 Y
5
290000
90000
E
TABLE I D x 107, ( ? / T ) D x 109, p 2 x 105, T, K cm2/s erg/(K cm) l/s2 283 4.36 f 0.04 0.0202 1.82 296 6.32 f 0.07 0.0200 4.36 310 8.83 f 0.09 0.0198 11.59 316 10.20 f 0.13 0.0201 14.61 323 8.93 f 0.10 0.0152 18.05
BSA in pH 7 25mM buffer
3.5% protein by weight D = (9.38 f 0.14)x 10'7cmz/s Sampling time = 2 microseconds
€\
dP2/P1 0.187 f 0.002 0.199 f 0.002 0.205 f 0.002 0.225 f 0.003 0.298 f 0.002
gOO
-looooO
, ,
20
40
60
80
100
Temperature
i
23 degrma Cdclus
120
TIME IN SAMPLING UNITS
Figure 2. Measured autocorrelation function for a sample containing 35 g/L protein. Beyond 20 sample units, the correlation function is almost flat; the small slope is most likely instrumental. Dust particles give rise to larger slopes and introduce error in the measured short time decay constant.
BSA in 25mM Phosphate Buffer pH 6.66
-_ I-
t
F 3 x
. VI
-t N
pH
-
7.0
IS mM bu1l.r
D = 8.83 E-07 r " / s .-'
0
Figure 3. Measured autocorrelation function expressed as a log of the ratio of the function at time t to that at time zero. The full lines are the fits to the data using a second-order polynomial in time. The three curves correspond to different temperatures.
this slope only to eliminate small biases in the measurement of the fast decay component of the autocorrelation function. Temperature Variation. The diffusion coefficient of BSA was measured in the range 10-50 "C, a t a concentration of 0.5 g/L of BSA. The measured autocorrelation functions for three temperatures are shown in Figure 3. The data are characterized by a slope and a curvature, both of which increase with temperature up to 50 OC at which time there is a marked decrease in the measured slope. The diffusion coefficient of BSA was obtained from the data in Figure 3 by fitting to a quadratic function in time (cumulant expansion): (4)
The average diffusion coefficient is given by ( D ) = Pl/(2q2), where q is the magnitude of the scattering vector. The resulting values of the average diffusion coefficient are tabulated in column 2 of Table I. As expected, the measured diffusion coefficients scale with temperature T and viscosity, r). The product q D / T , shown in the third column of Table I, is constant up to 50 OC. The sharp decrease in the diffusion coefficient observed at 50 OC suggests a large change in the average dimension of the BSA protein such as would occur due to a partial unfolding of the three sphere and link structure of BSA.IO The values of P,, shown in column 4 of Table I, scale with the square of the viscosity and (10) Peters, T., Jr. In Advances in Protein Chemistry; Anfinsen, C. B., Edsall, J. T., Richards, F. M., Eds.; Academic Press: New York, 1985; pp 161-245.
10
20
30
40
Concentration BSA (glL)
SAMPLE TIME
Figure 4. A comparison of the measured diffusion coefficient as a function of protein concentration in various solutions to the values calculated by two formalisms. The solid lines correspond to Felderhofs formalism, and the dotted lines are those due to Phillies. The experimental points are represented by full circles, rectangles, and triangles, each corresponding to different solution conditions.
temperature. The values of P, lead to an estimate of the variation of the BSA diffusion coefficient according to The variation of the diffusion coefficient relative to the average value can be obtained by taking the ratio P 2 / P l ,which is tabulated in the last column of Table I. In the context of this analysis, BSA is characterized by a 19% dispersion in the measured diffusion coefficient at 10 OC; the dispersion increases to 30% at 50 O C . The origin of this dispersion is most likely in the natural variability of the BSA dimension, as a result of conformational fluctuations. Dependence on Ionic Strength, pH, and Protein Concentration. The ionic strength, pH, and protein concentration affect the protein diffusion coefficient by modulating the electrostatic interaction between proteins. The magnitude of the Coulomb repulsion depends on the total charge on the protein which in turn depends on the pH. Small ions in the solution screen the Coulomb interaction between the charged proteins so that at salt concentrations of 0.1 M the Coulomb field vanishes beyond a small fraction of the protein linear dimension. Concentration determines the probability of encounters during which the proteins come close enough to feel the Coulomb repulsion as well as other short- and long-range mutual interactions. Thus, pH, ionic strength, and protein concentration are factors which together influence the magnitude of the diffusion coefficient. The results of our measurements are summarized in Figure 4. The diffusion coefficient was measured as a function of protein concentration in deionized water at pH 6.2 (solid spheres) and
2358 The Journal of Physical Chemistry, Vol. 96, No. 5, 1992 TABLE 11"
6 5 4 3
6.45 f 0.1 6.79 0.2 6.47 f 0.2 5.36 0.03
* *
0.20 f 0.01 0.22 0.01 0.18 f 0.01 0.25 f 0.01
*
6.68 f 0.1 6.37 f 0.1 6.74 f 0.1
'T = 23 OC; 25 mM phosphate buffer. 100 mM buffer at pH 5.2 (solid rectangles). The two sets represent conditions where the Coulomb repulsion is strong and weak, respectively. The solid triangle is a measurement at intermediate conditions. The measurements performed at relatively low protein concentration of 0.5 g/L show that the diffusion coefficient does not change when the ion concentration goes from 0 to 0.1 M. This can be explained by noting that although the protein charge is not screened at low ionic strength, the protein concentration is sufficiently low so that the Coulomb interactions are not significant, and thus the addition of more salt does not change the relative importance of the Coulomb interactions. At higher protein concentrations the effects of added salt are pronounced as seen in Figure 4. This behavior has been observed by others."-'5 Measurement by Barish,I4in 1.1 mM salt solution, gives a diffusion coefficient of 15 X cm2/s a t concentration of 40 g/L. The measurements of PhilliesIs give an increase of 20%;their measurements were at 0.15 M NaCl, which is a considerably higher ionic strength than that of Barish. Our value a t 35 g/L and 25 mM buffer is 9.38 X lo-' cm2/s and falls between the two above values as does the ionic strength. In an experiment by Weissman,I2 the BSA concentration was set at 4% (40 g/L) and the ionic concentration was varied. The reported diffusion coefficient varied from the high salt value of 5.4 X cm2/s (at 1 M NaCl) to approximately 21 x cm2/s at low ionic concentrations (no excess salt). Previous measurements done at pH values around 5 show a decrease in the diffusion coefficient with increasing concentration of BSA.I5,l6 The isoelectric point of BSA is pH 5.2, where the net protein charge is expected to be small. Our measurements at pH 5.2 show a small dependence of the diffusion coefficient on protein concentration up to 30 g/L. At sufficiently low protein concentrations, if protein interactions are negligible, it may be possible to observe the ionic friction effect caused by the fluctuations of the ionic atmosphere surrounding individual proteins. Ionic friction tends to reduce the diffusion coefficient as the ionic strength decreases.I7 The observed constancy of the diffusion coefficient at 0.5 g/L suggests that at this protein concentration the ionic friction effect either is offset by interactions between proteins or is very small. The protein charge can be varied by changing the pH. Thus, it is possible to study the effects of the protein charge by varying the pH while keeping the ionic strength and protein concentration constant. The concentration was set to 0.5 and 10 g/L in a 25 mM citric acid-phosphate buffer with the value of pH ranging from 3 to 7. Table I1 summarizes the measured results. Between pH 4 and 7 there is little change in measured diffusion coefficient for the 0.5 g/L sample as would be expected if Coulomb interactions were not important. Raj and Flygarel* reported measurements of the dependence of the diffusion coefficient on pH for BSA concentration of 1% and ionic strength of 0.03 and 0.1 M. They observed that the diffusion coefficient remains constant 1) Doherty, P.; Benedek, G. B. J . Chem. Phys. 1974, 61, 5426-5434. 2) Weissman, M. B.; Pan, R. C.; Ware, B. R. J . Chem. Phys. 1979, 70, -2903. 3) Neal, D. G.; Purich, D.; Carnell, D. S. J . Chem. Phys. 1984, 80, -3477. 4) Barish, A. 0.;Gabriel, D. A.; Johnson, Jr., C. S. J . Chem. Phys. 1987, 594-3602. 5) Phillies, G. D. J.; Benedek, G. B.; Mazer, N. A. J . Chem. Phys. 1976, 883-1892. 6) Oh, Y. S.; Johnson, Jr., C. S. J . Chem. Phys. 1981,74, 2717-2720. 7) Schurr, J. M. Chem. Phys. 1980, 45, 119-132. 8) Raj, T.; Flygare, W . H. Biochemistry 1974, 13, 3336-3340.
Gaigalas et al. TABLE III: Diffusion Coefficients of Globular Proteins at 20 "C and Infinite Dilution dimension, Do nm D, x 107, (calcd), MW cm2/s maior minor cm2/s protein BPTI 6500 14.4 f 0.2" 2.9 1.68 20.5 egg white 14600 12.3c 4.4 3.0" 12.4 1ysozy m e sperm whale 17000 10.2/ 4.4 0' 9.7 myoglobin 64500 6.75 f 0.089 6.5 5.8" 7.1 human hemoglobin 70000 6.09 f 0.04* 14.1 4.2b 6.0 BSA rabbit muscle 150000 4.21 6.5 g' 3.8 aldolase
'Protein structure data base. bReference 10. 'Syqusch, J.; Boulet, H.; Beaudry, D. J . Biol. Chem. 1985,260, 15286-15290. dReference 22. 'Reference 19. 'Reference 23. 8Reference 20. hPresent work.
with decreasing pH for the 0.1 M solution and increases slightly in the 0.03 M solution. Their observations at 1% protein and 0.1 M ionic strength can be compared with ours at 0.05% protein and 0.025 M ionic strength since in both cases the mutual interactions are not important. At pH 3, both Raj and Flygare and our results show a decrease in the measured diffusion coefficient. This decrease is most likely due to a change in protein conformation.I0 For the 10 g/L sample, the diffusion coefficient is at a minimum at pH 5 and increases at pH 4 and 7, which is consistent with the presence of Coulomb repulsion. Our measurements are consistent with others in the literature and with results for other globular proteins. They demonstrate that the magnitude of the observed diffusion coefficient depends strongly on ionic strength, pH, and protein concentration. These three variables appear to be related through their effect on the magnitude of the protein electrostatic interaction. Discussion At a fixed temperature, the measured diffusion coefficient of BSA depends strongly on pH, ionic strength, and protein concentration. The pH regulates the protein charge and thus the magnitude of the Coulomb interaction, which is further screened by the ions in solution. The protein concentration determines the likelihood of mutual encounters. Thus, the three solution parameters are interrelated in their effect on the magnitude of the diffusion coefficient. The behavior observed in the case of BSA, as discussed above, is most likely a general property of globular proteins. Similar behavior has been observed in lysozymeI9 where the mutual diffusion coefficient increases with concentration, but the increase becomes less as the salt concentration increases further. Measurements on hemoglobin at the isoelectric point20 indicate that the diffusion coefficient decreases with increasing concentration. A similar decrease is observed for BPTI at pH 7 where the expected charge is 6.2',22 Estimates indicate that the Coulomb repulsion is offset by hydrodynamic interactions, which become important when the repulsive Coulomb forces are screened. In addition, there may be other attractive forces; e.g., BPTI has a tendency to dimerize. By properly estimating the hydrodynamic and Coulomb interactions, it seems possible to predict the main trends of the mutual diffusion coefficient of globular proteins. Table I11 lists some well-studied globular proteins, and the diffusion coefficients at infinite dilution are presented in column 3. The values at infinite dilution can be obtained from tracer meas u r e m e n t ~ . The ~ ~ dimensions of the proteins, listed in columns (19) Nystrom, B.; Johnsen, R. M. Chem. Scr. 1983, 22, 82-84. (20) Hall, R. S.; Oh, Y. S.; Johnson, Jr., C. S. J . Phys. Chem. 1980,84, 156-161. (21) Wills, P. R.; Georgalis, Y . J . Phys. Chem. 1981. 85, 3978-3984. (22) Gallagher, W. H.; Woodward, C. K. Biopolymers 1989, 28, 200 1-2024. (23) Muramatsu, N.; Minton, A. P. Proc. Nutl. Acud. Sci. U.S.A. 1988, 85, 2984-2988.
Diffusion of BSA in Aqueous Solutions
The Journal of Physical Chemistry, Vol. 96, No. 5, 1992 2359
TABLE IV Slope of the Linear Dependence of the Mutual Diffusion Coefficient on the Concentration of ProteinsC
$. -10
1
0
3
BPTI lysozyme hemoglobin
BSA
MW 6500 14600 64 500 70000
measd -3.4” -1.7b -0.56V -0.96d
calcd Felderhof +0.56 +1.33 +1.67 +1.75
I Experimental
0 Phillles A Felderhof 25mY buffer l0gA proteln
slope, L/kg Drotein
I
I
calcd Phillies -0.77 -0.32 -0.96 -0.84
io
Reference 22. Reference 19. e Reference 20. dPresent work. ‘Conditions are set to minimize electrostatic interactions.
4 and 5, were obtained from the protein structure data bank. The
dimensions, when used in Pemn’s expressionZ4for orientationally averaged Brownian ellipsoids of revolution, give the predictions shown in column 6 . They are close to the measured tracer values as well as the values obtained from empirical correlations of protein diffusion coefficients with molecular ~ e i g h t . 2 Perrin’s ~ formulas also explain the decrease in the observed diffusion coefficient of BSA at pH 3 (see Table 111). This decrease can be ascribed to a conformation change in BSA which occurs at a pH of 3.5. The predicted value of Dois 6.62 X lo-’ cm2/s at pH 7 and 5.87 X lo-’ cm2/s at pH 3. The predicted values were calculated using geometric changes as reported by Peters.’O The values of the diffusion coefficient at infinite dilution can be combined with an estimate of interaction effects which depend only on the average charge of the protein and the ionic strength to arrive at a correction depending on protein concentration. Thus, from the available structure data and estimate of protein charge it should be possible to calculate a reasonably accurate value of the diffusion coefficient in various environments. The effects of mutual interaction on the diffusion coefficient of BSA can be qualitatively understood by considering the generalized Stokes-Einstein expression for the diffusion coefficient; see eq 2. The form of eq 2 suggests that the interaction effects come from two distinct sources. The osmotic derivative suggests that it is necessary to include both protein hard-core repulsion and screened Coulomb interactions. The friction factor has to be modified because in addition to Stokes drag the proteins will experience each other’s hydrodynamic disturbance with the resultant modification of the friction factor. The implied independence of the two corrections is approximate. The osmotic derivative contribution and the hydrodynamic interaction have been examined in a self-consistent way by PhilliesZ6and Feld e r h ~ f . ~ ’Both authors convert the Smoluchowski equation for interacting particles into an equation for noninteracting particles with a renormalized diffusion coefficient. The differences in procedure between Phillies and Felderhof is that the divergence of the particle velocity field is nonzero in the Phillies treatment. Other procedures such as that of Anderson and ReedZ8are not self-consistent; however, they provide a clear picture of the underlying physics. Felderhof s original formalism has been simplified by Dorshow and N i ~ l i and , ~ it~ is this form that we use. The resulting phenomenological expression for the protein diffusion coefficient is expected to be good to first order in c. The explicit form is given by eq 2. The detailed forms of kl and k2 are presented in Phillies (ref 26, eq 11) and Dorshow (ref 29, eqs 5 and 6 ) . Figure 4 shows the correspondence between measured values and those calculated from Phillies (dashed lines) and Felderhof-Dorshow (solid line). In both cases, the conversion between concentration and volume fraction was obtained by assuming that the protein has a prolate volume with dimensions given in Table 111. The agreement is reasonably good for both formalisms. The quality of the agreement (24) Perrin, F. J . Phys. Radium 1936, 7, 1-11. (25) Young, M. E.; Carroad, P. A,; Bell, R. L. Eiotechnol. Bioeng. 1980, 22,947-955. (26) Phillies, G. D.J. J . Colloid Interface Sci. 1987, 119, 518-523. (27) Felderhof, B. U.J . Phys. A: Math. Gen. 1978, 11, 929-937. (28) Anderson, J. L.; Reed,C. C. J . Chem. Phys. 1976, 64, 3240-3250. (29) Dorshow, R.;Nicoli, D. F. J . Chem. Phys. 1981, 75, 5853-5856.
4
5
6
7
PH Figure 5. Measured diffusion coefficient (solid circles connected by a solid line) as a function of pH in a 25 mM phosphate buffer and a protein concentration of 10 g/L. The charge on the protein was set equal to the net charge as given in titration measurements on BSA (dashed curve). Calculated (open circles from Phillies, open triangles from Felderhof, and a connecting solid line to emphasize trends) values are in reasonable agreement except at pH 4, where the very large titration charge gives predictions which are much too high. Charges based on linear extrapolation of the titration curve give a more reasonable prediction as indicated by the dot-dashed curve which was obtained using Felderhofs formalism.
can be checked for the case where the Coulomb interaction is minimized (large ionic strength and isoelectric pH). Table IV compares the measured and calculated dependence of the diffusion coefficient of several proteins on the concentration. The correspondence with values calculated using Phillies formalism (column 5 ) is good for the high molecular weight proteins. In the case of BPTI, there seems to be more attractive interaction than included in the model. This has been discussed by Gallagher.22 Felderhof s analysis (column 4) always leads to a positive slope. Although the Phillies theory is slightly superior at high ionic strength, the Felderhof theory yields better predictions at low salt concentrations. Cichocki and Felderhof30 suggest that the transport coefficients can be used as a sensitive test of the interaction potential. In our opinion, the discrepancy between the theories of Felderhof and Phillies must be resolved in order for this program to have a chance of success. The condition that the Coulomb interaction is minimal is very important. Figure 5 shows data taken for four values of pH. The dashed line shows the protein charge as estimated from titration data.” For pH values close to the isoelectric point, the data are in reasonable agreement with the predictions of Phillies and Felderhof. At pH 4 both predictions are off. The discrepancy is most likely due to an overestimate of the protein charge. The titration charge measures the number of sites accessible to binding, and its value is most likely larger than the charge which characterizes the coupling of the protein to other charges.32 If one simply extends the relationship between pH and charge as determined near the isoelectric point to pH 4, then the predictions, shown in Figure 5, are in better correspondence with the data. For practical estimates, this ad hoc prescription for estimating protein charge may suffice. However, the uncertainties in protein charge present an obstacle for predicting the diffusion coefficient.
Acknowledgment. We are grateful to Dr. Jess Edwards for providing samples of BSA. We are also grateful to B. Robertson for useful suggestions and discussions. We thank Dr. Gilliand for access to the protein structure data bank. (30) Cichocki, B.;Felderhof, B.U.J . Chem. Phys. 1990,93,4427-4432. (31) Tanford, C.; Swanson, S.A.; Shore, W. S.J. Am. Chem. SOC.1955, 77,6416-6421. (32) Schmitz, K. S.;Lu, M. Proc. Natl. Acad. Sci. U.S.A. 1983, 80, 425-429. (33) Schor, R.; Serrallach, J. Chem. Phys. 1979, 70, 3012-3015.
