10294
J. Phys. Chem. B 1999, 103, 10294-10299
Self-Diffusion and Cooperative Diffusion of Globular Proteins in Solution Christel Le Bon and Taco Nicolai*,† Chimie et Physique des Mate´ riaux Polyme` res, UMR 6515, UniVersite´ du Maine, 72085 Le Mans Cedex 9, France
Maxim E. Kuil*,‡ and Johan G. Hollander Leiden Institute of Chemistry, Leiden UniVersity, PO Box 9502, 2300 RA Leiden, The Netherlands ReceiVed: April 26, 1999
The globular protein β-lactoglobulin (β-lg) was studied in aqueous solution at pH ) 7.0 using pulsed field gradient nuclear magnetic resonance (PFG-NMR) and static (SLS) and dynamic light scattering (DLS). The concentration dependence of the self-diffusion coefficient (Ds) and the cooperative (or mutual) diffusion coefficient (Dc) were determined as a function of the concentration up to volume fraction φ ) 0.15. The effect of electrostatic interactions was investigated by comparing systems with 0.003 and 0.1 M added salt. The concentration dependence of Ds with 0.1 M added salt is found to be the same as for other globular proteins reported in the literature. Directly measured values of Dc using DLS are consistent with values derived from a combination of PFG-NMR and SLS assuming that the protein-protein friction coefficient is small compared to the protein-solvent friction coefficient.
1. Introduction Pulsed field gradient nuclear magnetic resonance (PFG-NMR) can be used to determine the self-diffusion coefficient (Ds) of polymers and colloids in solution.1 On the other hand, dynamic light scattering (DLS) can be used to determine the cooperative diffusion coefficient (Dc).2 For binary solutions, Ds depends both on the friction between the solute particles and the solvent and the friction between the particles themselves:3
Ds )
RTM fps + fpp
(1)
Here fps is the particle-solvent friction coefficient and fpp is the particle-particle friction coefficient both per unit of mass of the solute particles with molar mass M. R is the gas constant and T is the absolute temperature. Dc depends not only on the friction between the solute and the solvent but also on the osmotic compressibility (dπ/dC):2,3
Dc )
(1 - φ)2 dπ fps dC
( )
(2)
Here C is the concentration in weight per volume and φ is the volume fraction of the solute. Ds depends on both fps and fpp because the Ds characterizes the flow of the solute particles relative to each other. Dc depends only on fps because it characterizes the flow of the solvent past the solute particles. The osmotic compressibility can be determined independently using static light scattering (SLS)4 so that the combination of DLS, PFG-NMR, and SLS on the same system makes it possible to investigate the relative importance of fpp and fps. In older literature (e.g. refs 5 and 6) often (1 - φ) is used erroneously † ‡
E-mail:
[email protected]. E-mail:
[email protected].
in place of (1 - φ)2. The difference is significant at higher volume fractions. The concentration dependence of Ds has been measured for a number of globular proteins: hemoglobin,7,8 serum albumin,7 ovalbumin,5 and lysozyme.9 The concentration dependence of Dc has been measured for three of these globular proteins: hemoglobin,10 serum albumin,6 and ovalbumin.5 The concentration dependence of Dc was compared to the calculated concentration dependence utilizing the generalized Stokes-Einstein equation (eq 2). In refs 8 and 10, fps was derived from Ds assuming that fpp is negligible compared to fps. It was concluded that the experimental values of Dc were consistent with eq 2 using literature values of the osmotic pressure or the activity coefficient on the same protein. Gibbs et al.5 measured Dc using both a macroscopic boundary relaxation technique and DLS. Although for globular protein solutions the results from these two techniques are expected to be the same, they observed a weaker concentration dependence of Dc with DLS than with boundary relaxation measurements. Using the results from boundary relaxation measurements and osmotic compressibility data taken from the literature they calculated fps which they compared to values of (fps + fpp) obtained from self-diffusion measurements using PFG-NMR. As the results were close, they concluded that fpp is negligible compared to fps. Here we report a PFG-NMR, SLS, and DLS study of the main whey protein β-lactoglobulin (β-lg) in aqueous solution at pH 7. β-lg is a globular protein, with molar mass 18 400 g/mol and radius about 2 nm.11 At room temperature β-lg is present in the form of monomers and dimers with an association constant that depends on the pH and the ionic strength.12 We have measured Dc and Ds as a function of concentration up to C ) 0.18 g/ml, i.e., volume fractions up to φ ) 0.14 using 0.75 g/mL for the partial specific volume of β-lg.13 The purpose of the present study is three-fold. (1) We extend the investigation of the diffusion of globular proteins to another, industrially important, protein. We will show that the concentra-
10.1021/jp991345a CCC: $18.00 © 1999 American Chemical Society Published on Web 10/30/1999
Diffusion of Globular Proteins in Solution
J. Phys. Chem. B, Vol. 103, No. 46, 1999 10295
tion dependence of Ds is universal for all five globular proteins studied so far. (2) We combine in one set of experiments measurements on the concentration dependence of Ds, Dc, and dπ/dC for the same system with the objective to check the importance of fpp. (3) We investigate the influence of electrostatic interactions by comparing the results at high and low ionic strength. In addition, we believe that recent developments of the DLS technique allow for a more accurate determination of Dc. Especially, the method we use to analyze the DLS results makes it possible to detect and correct for the presence of aggregates in the protein solutions.
Rθ )
S(G) ) exp(-Ds(∆ - δ/3)γp2G2δ2) ) exp(-DsU) (3) S(0) with ∆ the diffusion delay time, γp the proton gyromagnetic ratio, G and δ the amplitude and the duration of the field gradient, respectively, and Ds the self-diffusion coefficient. For the experiments described here the diffusion coefficient was determined on samples at 25 °C using a fixed diffusion delay time ∆ of 60 ms and gradient duration δ of 2 ms. The signal was recorded with a spectral width of 5 kHz. The gradient currents (1.5-7.5 A) were chosen in such a manner that an exponential spacing in U was obtained. The magnetic field gradient varied between 0.054 and 0.27 T/m using this current range. A phase cycled stimulated echo sequence was optimal for the observed T1 and T2. Moreover, a longitudinal eddy current delay sequence with a 20 ms delay was added to reduce inductive artifacts.15 2.3. Light Scattering. If interactions are weak and the particles are small compared to the inverse scattering wave vector (q), the scattered light intensity I, is related to the osmotic compressibility as follows:4
dπ 1 KC ) Rθ dC RT
( )
(4)
with R the gas constant and T the absolute temperature. Rθ is the so-called the Raleigh factor:
(5)
where Ist is the scattered light intensity of a standard with refractive index nst and Raleigh factor Rst. Is is the scattered light intensity of the solvent with refractive index ns. We have used toluene as the standard with Rtol ) 2.8 × 10-5 cm-1 at the wavelength λ ) 532 nm and 20 °C.16 K is a constant given by
4π2ns2 dn 2 Naλ4 dC
K)
2. Experimental Methods 2.1. Sample Preparation. The β-lg used in this study was a gift from Besnier (batch no. 754). High-pressure liquid chromatography shows that the sample consists of equal fractions of genetic variants A and B. Solutions were prepared by dialyzing against the distilled and deionized water at pH 7 to which 200 ppm NaN3 was added to avoid bacterial growth which yields a ionic strength 0.003 M. For a second series of samples 0.1 M CH3COONH4 was added in order to screen electrostatic interactions. The solutions were filtered through 0.2 or 0.45 µm pore size Anatope filters depending on the concentration. The protein concentrations were determined after filtration by UV absorption at 278 nm using extinction coefficient 0.96 L g-1 cm-1.14 2.2. PFG-NMR. Proton PFG-NMR measurements were done on a Bruker AM 200 (4.7 T) wide bore NMR spectrometer controlled by an ASPECT 3000 computer system. The actively shielded gradient coil was designed and manufactured at Massey University, Palmerston North, New Zealand, by Callaghan and co-workers. It was mounted in a 5 mm proton probe (Bruker). The carefully balanced gradient and shielding currents were generated by a Techron 7570 amplifier. The NMR signals were transferred to a PC system and analyzed with a data-analysis program in a Matlab environment. In the case of a single diffusional process the signal attenuation S is given by
()
I - I s ns 2 R Ist nst st
( )
(6)
with dn/dC the refractive index increment and Na Avogadro’s number. The intensity autocorrelation function measured with the DLS technique is related to the normalized electric field correlation function, g1(t), by the Siegert relation.2 In the case of a binary solution
g1(t) ) exp(-q2Dct)
(7)
If the solute is polydisperse, g1(t) can be described by a sum of exponentials:
g1(t) )
∫A(τ) exp(-t/τ) dτ
(8)
The DLS results were analyzed in terms of eq 8 using the Laplace inversion routine REPES.17 Figure 1a shows an example of the intensity autocorrelation function of β-lg. The corresponding relaxation time distribution is shown in Figure 1b. Similar results were obtained at all concentrations. We note that in the q range available in light scattering experiments, q-1 is much larger that the radius of β-lg and that the results are independent of q. The relaxation distributions show a large peak at short times due to cooperative diffusion of native protein and a relatively small peak due to cooperative diffusion of small aggregates. In the representation of Figure 1b the surface area of each peak is proportional to the scattering intensity. The aggregates were probably formed during the isolation procedure and have a hydrodynamic radius of about 15 nm. SEC showed that the concentration of these aggregates is negligible. However, their contribution to the scattering intensity is not negligible since for a given weight concentration the intensity is proportional to the molar mass which is much larger for the aggregates than for the native proteins. In the calculation of KC/Rθ we have used the scattering by the native protein only which was obtained by multiplying the total scattering intensity with the relative amplitude of the peak at short relaxation times. Dc was calculated using the average relaxation rate of the fast peak: 〈τ-1〉 ) q2Dc. SLS and DLS measurements were made using an ALV-5000 multibit multi-tau correlator in combination with a Malvern goniometer and a Spectra Physics argon ion laser operating with vertically polarized light with wavelength λ ) 532 nm. The range of scattering wave vectors covered was 3.0 × 10-3 < q < 3.5 × 10-2 nm-1 with q ) (4πns/λ) sin(θ/2), θ being the angle of observation. The temperature was controlled by a thermostat bath to within (0.1 °C. The light scattering results shown below were done at 20 °C unless specified otherwise. 3. Results and Discussion 3.1. SLS. We have determined the concentration dependence of KC/Rθ obtained from SLS at ionic strengths 0.1 and 0.003
10296 J. Phys. Chem. B, Vol. 103, No. 46, 1999
Le Bon et al. TABLE 1: KC/Rθ of β-Lactoglobulin at pH 7 and Two Concentrations of Added Salt 0.1 M salt
Figure 1. (a) Intensity autocorrelation function of β-lg at pH7, 0.1 M salt, and C ) 0.0435 g/mL. The solid represents the result of an inverse Laplace transform using the REPES routine. (b) Relaxation time distribution corresponding to the correlogram shown in (a).
M; see Table 1 and Figure 2. In the calculation of K we used dn/dC ) 0.189 cm3 g-1.18 In the presence of 0.1 M salt the initial concentration dependence of KC/Rθ and thus the second virial coefficient is small because electrostatic interactions are screened. The strong increase at higher concentrations is due to higher order interactions. At low ionic strength KC/Rθ increases strongly at low concentrations up to 0.02 g/mL after which the dependence flattens off to increase again at higher concentrations. This peculiar concentration dependence can be understood by considering the contribution of counterions of β-lg to the total ionic strength. The net charge of β-lg at pH7 is about -611 which means that the contribution of dialyzed β-lg to the total ionic strength is 0.33C molar, with C expressed in g/mL and assuming that none of the counterions is condensed. Thus with increasing protein concentration the ionic strength increases leading to weaker electrostatic repulsion which explains the flattening off at 0.02 g/mL. Already at C ) 0.01 g/mL the total ionic strength is twice that of the solvent. The subsequent increase of KC/Rθ at higher concentrations is again due to higher order interactions. The extrapolation of KC/Rθ to C f 0 gives the inverse of the weight average molar mass (Mw). However, β-lg is present in the form of an equilibrium between dimers and monomers so that Mw varies with the protein concentration depending on the association constant. At pH 7 and 0.1 M salt added, the association constant is large and the value of KC/Rθ is close to
0.003 M salt
102C (g/mL)
KC/Rθ × 105 (g/mol)
0.24 0.25 0.39 0.46 0.49 0.69 0.91 1.09 1.11 1.38 1.50 1.85 3.54 3.66 5.01 5.37 5.61 6.73 7.62 7.81 8.45 9.39 10.15 11.09 12.98 13.49 15.15 17.90
2.80 2.95 2.87 2.84 2.91 2.99 2.92 2.81 2.94 3.00 2.95 3.02 3.04 3.01 3.30 3.19 3.27 3.58 3.97 3.79 4.03 4.36 4.55 5.29 6.53 5.84 7.76 9.84
102C (g/mL)
KC/Rθ × 105 (g/mol)
0.14 0.17 0.25 0.27 0.33 0.41 0.47 0.59 0.75 0.93 1.16 1.38 1.67 1.95 2.31 2.60 3.13 3.81 4.47 5.10 5.12 5.62 7.11 8.68 10.10 11.13 12.30
3.42 3.67 3.68 3.63 3.87 3.73 4.16 4.50 4.72 4.90 5.05 5.18 5.24 5.26 5.16 5.38 5.34 5.36 5.57 5.37 5.35 5.62 5.81 6.34 6.57 6.83 7.04
that expected for dimers even at the lowest concentrations at which accurate measurements could be done (see Figure 2b). At low ionic strength the association constant is smaller and monomers are still present at the lowest concentrations used in the experiments. 3.2. PFG-NMR. Figure 3a shows an NMR spectrum of β-lg at 0.073 g/mL with 0.1 M added salt. The spectra are identical for all samples investigated. The very strong signal from H2O at 5 ppm is reduced because it has relaxed already at the lowest gradient used in the experiment. The broad complex spectrum is due to the various protons on β-lg. Figure 4 shows the signal attenuation of the peak at 9 ppm. In all cases the signal attenuation could be well described by a single-exponential decay (see e.g. solid line in Figure 4) and Ds was obtained from a nonlinear least-squares fit to eq 3. Choosing any signal from protons on β-lg gives the same result. This is illustrated in Figure 3b where we have plotted Ds obtained from the echo attenuation at different chemical shifts. Figure 5 shows the concentration dependence of Ds with 0.1 and 0.003 M added salt (see Table 2). At high salt conditions the value extrapolated to C f 0 is D0 ) 8.65 × 10-15 m2/s. Using the Stokes-Einstein relation (D0 ) kT/6πηRh, with k Boltzmann’s constant and η the solvent viscosity) we obtain a hydrodynamic radius Rh ) 2.7 nm, in good agreement with early results obtained by Cecil and Ogston19 for β-lg dimers based on sedimentation experiments. At low ionic strength D0 ) 9.7 × 10-15 m2/s which gives Rh ) 2.4 nm. These results confirm the conclusion based on the SLS measurements that at high ionic strength β-lg is in the form of dimers over the whole accessible concentration range, while at low ionic strength a fraction of the proteins is in the form of monomers at low concentrations. The decrease of Ds with increasing concentration is due to an increase of the friction coefficient. At low ionic strength the initial decrease is faster due to an increase of the fraction dimers. At higher concentrations when all proteins are in the form of
Diffusion of Globular Proteins in Solution
Figure 2. (a) Concentration dependence of KC/Rθ for β-lg at pH 7 and with 0.1 M (circles) or 0.003 M (squares) salt added. The solid lines are guides to the eye. (b) Semilogarithmic representation of the same data as in (a).
dimers Ds is somewhat smaller at low ionic strength. However, it is clear that the effect of electrostatic interactions on Ds is weak. The same conclusion was drawn by Gibbs et al.6 based on PFG-NMR measurements on ovalbumin at two pH values, i.e., two different charge densities of the protein, and two different ionic strengths. For the comparison of our results with literature results on other globular proteins we use the results at high ionic strength to avoid the complications of a concentration-dependent composition. Figure 6 shows that the dependence of Ds on the volume fraction is very close to that of four other globular proteins at high ionic strength. The concentration dependence of Ds for ovalbumin at low ionic strength is slightly weaker. The partial specific volume for all five globular proteins is close to 0.75 mL/g. Apparently, the concentration dependence of Ds is not sensitive to variation in shape and charge density of the proteins. Up to a volume fraction of 10% Ds/D0 has a linear dependence on C: Ds/D0 ) 1 - aφ, with a ≈ 5.5. a is 2-3 times larger than the value predicted and found for uncharged hard spheres or charged spheres in the presence of salt.20 The difference is partly due to the hydration of the proteins which leads to a larger effective volume fraction. An additional reason for the discrepancy could be the anisotropic structure and the charge of the proteins. However, it is remarkable that Ds has the same concentration dependence for five globular proteins with different degrees of anisotropy and charge density.
J. Phys. Chem. B, Vol. 103, No. 46, 1999 10297
Figure 3. (a) NMR spectrum of β-lg at 25 °C, pH 7, 0.1 M added salt, and C ) 0.073 g/mL. (b) Self-diffusion coefficients obtained from the signal attenuation at different chemical shifts indicated by the dots in (a).
Figure 4. Signal attenuation of the peak at 9 ppm as function of the field strength for β-lg at pH 7, 0.1 M salt, and C ) 0.073 g/mL. The solid line represents a nonlinear least-squares fit to a single-exponential decay.
3.3. DLS. Figure 7 shows the concentration dependence of Dc with 0.1 and 0.003 M added salt (see Table 3). In order to facilitate the comparison with Ds we have corrected the data for the difference in solvent viscosity and thermal energy between 20 and 25 °C: Dc(25 °C) ) Dc(20 °C) × (278/273) ×
10298 J. Phys. Chem. B, Vol. 103, No. 46, 1999
Le Bon et al.
Figure 5. Concentration dependence of the self-diffusion coefficient of β-lg at pH 7 and 0.1 M (circles) or 0.003 M (squares) added salt. The solid lines are guides to the eye.
Figure 7. Concentration dependence of the cooperative diffusion coefficient of β-lg at pH 7 and with 0.1 M (circles) or 0.003 M (squares) salt added. Dc was measured at 20 °C, but corrected for the difference of the solvent viscosity and thermal energy between 20 and 25 °C to facilitate comparison with Ds data. The solid lines represent the cooperative diffusion coefficient deduced from SLS and PFG-NMR; see text.
TABLE 3: Cooperative Diffusion Coefficients of β-Lactoglobulin at 25 °C, pH 7, and Two Concentrations of Added Salt 0.1 M salt 102
C (g/mL)
Figure 6. Comparison of the volume fraction dependence of Ds for five globular proteins at high ionic strength.5,7,9 The partial specific volume was assumed 0.75 mL/g for all proteins.
TABLE 2: Self-Diffusion Coefficients of β-Lactoglobulin at 25 °C pH 7 and Two Concentrations of Added Salt 0.1 M salt 102C
(g/mL)
0.24 0.47 0.90 1.74 3.45 4.92 7.28 9.09 9.52 10.80 13.60 14.90 17.90
0.003 M salt
1011D
c
(m2/s)
8.33 8.55 8.12 8.07 7.23 6.86 5.91 5.05 4.75 4.48 3.80 3.53 2.58
102C
(g/mL)
0.33 0.76 1.40 2.97 5.00 7.34 10.22 12.30
1011Dc (m2/s) 9.56 9.28 8.19 7.80 6.47 5.18 4.01 3.52
(η25/η20). The difference between the concentration dependence of Dc at high and low ionic strength reflects the influence of thermodynamic interactions on Dc as expressed by eq 2. In the presence of 0.1 M salt the value of the diffusion coefficient extrapolated to C ) 0 is the same as found in PFG-NMR. This observation confirms that the fraction monomeric β-lg is small
0.14 0.24 0.24 0.37 0.46 0.46 0.69 0.91 1.09 1.11 1.38 1.50 1.76 2.58 3.39 3.54 3.66 4.50 5.40 5.61 6.76 7.40 7.81 8.50 9.39 9.95 11.09 12.98 13.57 15.15 17.90
0.003 M salt
11
2
2
10 Dc (m /s)
10 C (g/mL)
1011Dc (m2/s)
8.56 8.39 8.56 8.28 8.30 8.28 8.53 8.37 8.19 8.23 8.21 8.10 7.98 7.74 7.28 7.50 7.11 7.00 7.11 6.94 6.78 6.68 6.50 6.52 6.31 6.11 5.97 5.86 5.84 5.53 5.29
0.08 0.14 0.25 0.27 0.33 0.41 0.47 0.59 0.75 0.93 1.16 1.38 1.67 1.95 2.31 2.60 3.13 3.81 4.47 5.10 5.12 5.62 7.11 8.68 10.10 11.13 12.30
9.61 10.40 10.88 10.35 11.58 10.70 11.80 12.03 12.36 12.48 12.59 12.48 12.25 12.14 11.58 10.68 10.41 9.97 9.27 8.87 8.85 8.58 7.83 7.10 6.44 6.17 5.70
under these conditions. An important fraction of monomeric β-lg would lead to a smaller value of D0 from PFG-NMR than from DLS because the former gives a weight-averaged value while the latter gives a z-averaged value. The steep initial rise at low ionic strength precludes an accurate extrapolation to C ) 0. If we assume that fpp , fps then from eqs 1, 2, and 4 it follows that the concentration dependence of Dc can be deduced from
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J. Phys. Chem. B, Vol. 103, No. 46, 1999 10299
the concentration dependence of Ds and KC/Rθ:
KC Dc ) (1 - φ)2 DsM Rθ
(9)
The solid lines in Figure 7 show the concentration dependence of Dc deduced from Ds and KC/Rθ. In the comparison, we have used Mw obtained from SLS and we have calculated φ using the partial specific volume of β-lg. The directly measured values using DLS are within 10% of the values deduced from Ds and KC/Rθ. A more precise quantitative comparison is not possible for two reasons. First, for polydisperse systems there is a difficulty with the comparison of experimental data of the different techniques since the averaging of the different terms is not the same. The light scattering signal is proportional to the concentration times the molar mass, while the NMR signal is proportional to the concentration. This effect introduces an ambiguity for the low ionic strength system for which a small fraction of monomers is present at low concentrations. In addtion, with increasing temperature the association constant decreases12 so the fraction of monomers is higher at 25 °C than at 20 °C. To check whether this effect is significant we did static and dynamic light scattering measurements at 15 and 30 °C on a number of concentrations. We found that the results at 0.1 M are the same within the experimental error after correction for differences in viscosity and thermal energy. At 0.003 M the values of KC/Rθ at the lowest concentrations were slightly higher at 30 °C, but the effect remains small. Second, in using the partial specific volume of β-lg to calculate φ we ignore hydration of the proteins which increases their effective volume fraction and leads to somewhat smaller values of Dc at higher concentrations. Nevertheless, in spite of these uncertainties it is clear that their is no need to invoke an important contribution of fpp to Ds. Note that the neglect of fpp in eq 9 leads to an underestimation of Dc. 4. Conclusion In the presence of 0.1 M salt the second virial coefficient of β-lactoglobulin at pH 7 is negligible. Higher order interactions become important for C > 0.05 g/mL. With only 0.003 M salt, added, electrostatic repulsive interactions are strong at low concentrations, but weaken with increasing concentration due to the contribution of counterions to the total ionic strength. At high ionic strength most β-lg is present in the form of dimers even at the lowest concentration investigated (0.002 g/mL). Electrostatic interactions reduce the association constant so that
at low concentrations a significant fraction of β-lg is in the form of monomers if only 0.003 M salt is added. The concentration dependence of the self-diffusion coefficient of β-lg at pH 7 and with 0.1 M salt added was found to be very close to that of four other globular proteins reported in the literature. The effect of electrostatic interactions on Ds is small for globular proteins. The concentration dependence of Ds is stronger than predicted for hard spheres. The cooperative diffusion coefficient is sensitive to electrostatic interactions. Experimental values of Dc are consistent with values derived from a combination of osmotic compressibility and the friction coefficient utilizing the generalized StokesEinstein relation (eq 1). The protein-protein friction coefficient is small compared to the protein-solvent friction coefficient at least up to φ ) 0.1. References and Notes (1) Price W. S. Concepts Magn. Reson. 1997, 9, 299. Price W. S. Concepts Magn. Reson. 1998, 10, 197. (2) Berne B.; Pecora R. Dynamic Light Scattering; Wiley: New York, 1976. (3) Vink, H. J. Chem. Soc., Faraday Trans. 1 1984, 81, 1725. (4) Huglin M. B., Ed. Light scattering from polymer solutions; Academic Press: London, 1972. (5) Gibbs, S. J.; Chu, A. S.; Lightfoot, E. N.; Root, T. W. J. Phys. Chem. 1991, 95, 467. (6) Phillies, G. D. J.; Benedek, G. B.; Mazer, N. A. J. Chem. Phys. 1976, 65, 1883. (7) Keller, K. H.; Canales, E. R.; Yum, S. I., J. Phys. Chem. 1971, 75, 379. (8) Everhart, C. H.; Johnson, C. S., Jr. J. Magn. Reson. 1982, 48, 395. (9) Coffman, J. L.; Lightfoot, E. N.; Root, T. W. J. Phys. Chem. 1997, 101, 2218. (10) Hall, S. H.; Oh, Y., S.; Johnson, C. S.; J. Phys. Chem. 1980, 84, 756. (11) McKenzie, H. A. In Milk Proteins in Chemistry and Molecular Biology; Academic Press: New York, 1971; Vol. 2. Pessen, H.; Kumosinski, F.; Farrel, Jr., H. M. J. Ind. Microbiol. 1988, 3, 89. (12) See e.g.: Aymard, P.; Durand, D.; Nicolai, T. Int. J. Biol. Macromol. 1996, 19, 213. Verheul, M.; Pedersen, J. S.; Roefs, P. F. M.; de Kruif, K. G. Biopolymers 1999, 49, 11 and older literature cited in these references. (13) Liquori, A. M. In Structural Order in Polymers; Ciardelli, F., Giusti, P., Eds.; Pergamon Press: Oxford, UK, 1981. (14) Townend, R.; Winterbottom, R. J.; Timasheff, S. N. J. Am. Chem. Soc. 1960, 82, 3161. (15) Gibbs, S. J.; Johnson Jr., C. S. J. Magn. Reson. 1991, 93, 395402. (16) Finnigan, J. A.; Jacobs, D. J. Chem. Phys. Lett. 1970, 6, 141. Moreels, E.; De Ceunick, W. J. Chem. Phys. 1987, 86, 618. (17) Jakes, J. Collect. Czech. Chem. Commun. 1995, 60, 1781. Stepanek, P. In Dynamic Light Scattering; Brown, W. Ed.; Oxford University Press: London, 1993; Chapter 4. (18) Perlmann, G. E.; Longsworth, L. G. J. Am. Chem. Soc. 1948, 70, 2719. (19) Cecil, R.; Ogston, A. G. Biochemistry 1949, 44, 33-35. (20) Van Blaaderen, A.; Peetermans, J.; Maret, G.; Dhont, J. K. G. J. Chem. Phys. 1992, 96, 4591. Pusey, P. N.; Tough, R. J. A. Dynamic Light Scattering; Pecora, R., Ed.; Plenum Press: New York, 1985; Chapter 4.
