Quantitative Characterization of Nonspecific Self- and Hetero

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Quantitative Characterization of Nonspecific Self- and HeteroInteractions of Proteins in Nonideal Solutions via Static Light Scattering Di Wu† and Allen P. Minton* Section on Physical Biochemistry, Laboratory of Biochemistry and Genetics, National Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health, U.S. Department of Health and Human Services, Bethesda, Maryland 20892, United States S Supporting Information *

ABSTRACT: The dependence of static light scattering upon the compositions of solutions including hen egg white ovalbumin, hen egg white ovomucoid, ribonuclease A, and binary mixtures of these proteins at total concentrations of up to about 40 g/L were measured at different values of the pH and ionic strength. At the pH values of measurement, ovalbumin and ovomucoid have a net negative charge and ribonuclease A has a net positive charge. The observed dependence of scattering intensity upon solution composition may be accounted for by an extension of previously formulated equivalent hard particle models that allows for the presence of both repulsive interactions between like species and attractive interactions between unlike species in mixtures of positively and negatively charged proteins.

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scattering,6 sedimentation equilibrium,7 and osmotic pressure.8 This model has been found to quantitatively describe, with a very small number of parameters (the mass and equivalent volume of each solute species) the composition dependence of all three colligative properties of a variety of protein solutions over the entire experimentally accessible range of composition, which in some cases can reach total protein concentrations of several hundred g/L.9 Comparison of the covolumes of two best-fit equivalent particles representing two molecular species with the covolumes calculated on the basis of steric repulsion between the two species yields a quantitative measure of the magnitude of attractive or repulsive long-range interaction between the two solute species.10 The measurement and analysis of static light scattering of protein solutions has long been widely used as a standard method to characterize the molecular weights and states of association of proteins.11 Apparatus and techniques have been developed in our lab allowing us to measure the light scattering of protein solutions over a broad range of concentration.12 In a previous study, the composition-dependent static light scattering of bovine serum albumin, hen egg white ovalbumin (oval), hen egg white ovomucoid (ovo), and their binary mixtures at different weight fractions has been measured using a sequential diluting device.13 Multicomponent static light scattering theory together with an equivalent hard sphere model was employed to analyze the experimental data. It was

INTRODUCTION Interest in the characterization of nonspecific interactions between biological macromolecules has increased in recent years along with recognition of the important role that they play in modulating the kinetics and equilibria of biochemical reactions in complex physiological media.1−3 The two-body interaction coefficient, or second virial coefficient, has been widely used to quantify nonspecific interactions between like molecules and to describe first-order deviations from thermodynamic ideality in solutions of a single solute species.4 However, a single “second virial coefficient” does not suffice to describe compositiondependent thermodynamic properties in solutions containing multiple solute species at moderate or higher concentrations. Expansions of the logarithm of thermodynamic activity in powers of concentration can be carried out to higher powers of the concentration of a single solute species, but the number of self-interaction and hetero-interaction coefficients required to completely characterize thermodynamic behavior in multicomponent solutions can rapidly expand with total solute concentration to the point where individual coefficients cannot be even approximately evaluated on the basis of analysis of experimental data. In order to circumvent this problem, an equivalent hard particle model has been introduced, according to which each solute species (normally a protein or protein complex) is represented by a hard convex particle, typically a sphere or a spherocylinder.5 The scaled particle theory of hard convex particle fluids is then used to model measured concentration-dependent thermodynamic activities and their derivatives required to account for experimentally measurable composition-dependent colligative propertiesstatic light This article not subject to U.S. Copyright. Published 2015 by the American Chemical Society

Received: October 31, 2014 Revised: January 8, 2015 Published: January 9, 2015 1891

DOI: 10.1021/jp510918d J. Phys. Chem. B 2015, 119, 1891−1898

The Journal of Physical Chemistry B

Article

Rayleigh scattering theory for a single species in nonideal solution, the scattered intensity is given by19

found that the best-fit values of the equivalent specific volume of each protein were significantly larger than the experimentally measured partial specific volume of the protein (the volume excluded to solvent), indicating the presence of significant electrostatic repulsion between each pair of protein molecules. This finding is not unexpected, since the three proteins investigated in the study have similar isoelectric points around pH 4.5,14,15 and would all carry negative charge of comparable magnitude at the pH of measurement (7.0), leading to approximate additivity of pairwise interactions. The present study was undertaken to explore whether the equivalent hard sphere model can be generalized to treat binary solution mixtures of proteins bearing opposite charge, such that like molecules repel each other and unlike molecules attract each other, a situation that is likely to occur in heterogeneous physiological media. The composition dependence of light scattering of solutions of three binary mixtures of two negatively charged proteins and one positively charged protein has been measured over a range of concentrations greater than that which can be accounted for by two-body interaction coefficients. The results have been analyzed in the context of a thermodynamic model employing two-body and three-body interaction coefficients and in the context of an extension of the equivalent hard sphere model.

⎛ n ⎞2 M2c R =⎜ ⎟ K ⎝ n0 ⎠ 1 + c ∂ ln γ ∂c

(1)

where R denotes the excess Rayleigh ratio, n and n0 denote the refractive index of the solution and solvent, respectively, and M, c, and γ represent the molar mass, molar concentration, and activity coefficient of the scattering species, respectively. The optical constant K is given by K=

4π 2n0 2(dn/dw)2 λ 0 4NA

(2)

where dn/dw denotes the refractive increment for the scattering species, λ0 the vacuum wavelength of incident light, and NA Avogadro’s number. For most polypeptides in aqueous buffer, the refractive increment is approximately 0.185 cm3/g.20 According to multicomponent scattering theory,11,13,21 the excess Rayleigh scattering for two species can be written as shown below: R = K

■

∂ ln γ

(

)

∂ ln γ

(

)

2 2 2 1 ⎛ n ⎞2 M1 c1 1 + c 2 ∂c2 + M 2 c 2 1 + c1 ∂c1 − 2M1M 2c1c 2 ⎜ ⎟ 2 ∂ ln γ ∂ ln γ ∂ ln γ ⎝ n0 ⎠ 1 + c 2 ∂c 2 1 + c1 ∂c 1 − c1c 2 ∂c 1

EXPERIMENTAL METHODS Chemicals and Reagents. Hen egg white ovalbumin (LS003048), hen egg white ovomucoid (LS003086), and ribonuclease A (LS003433) were obtained from Worthington Biochemical Corporation. Proteins were used without additional purification. All proteins were dialyzed against working buffers: 0.05 M phosphate buffer at pH 6.0, 7.0, 8.0, and 8.8 and 0.05 M phosphate buffer at pH 7.0 plus 0.15 M sodium chloride. Pierce 10000MWCO Slide-A-Lyzer Dialysis cassettes (Thermo Science, MA) were used for the buffer exchange. Final protein concentrations were determined by absorbance of each protein at 280 nm. The standard values of absorbance of the three proteins for 1 g L−1 cm−1 are 0.75 for ovalbumin,16 0.41 for ovomucoid,17 and 0.71 for ribonuclease A.18 All of the protein solutions and buffers were prefiltered through 0.1 μm Whatman Anotop filters (Whatman, U.K.). All samples were centrifuged at 50000g (Optima TLX, TLS 55 rotor, Beckman, CA) for 15 min at 20 °C for degassing and removal of residual particulates. Measurement of Light Scattering of Protein Solution at Gradient Concentrations. The dependence of the intensity of 90° light scattering at 690 nm by protein solutions upon concentration was measured automatically at 20 °C using a MiniDAWN Tristar light scattering detector (Wyatt Technology, Santa Barbara, CA), modified as described in ref 12. Data were acquired from solutions of three individual proteins and their binary mixtures in various proportions (Table S1, Supporting Information). The initial total concentration of protein of each sample was about 30 g/L. The protein solution was then sequentially diluted by the dialysis buffer used for the protein buffer exchange. The scattered intensity at 90° of each dilution was recorded using ASTRA (v4.9, Wyatt Technology, Santa Barbara, CA) and subsequently exported for analysis using scripts written in MATLAB (Mathworks, Natick, MA).

(

2

)(

1

)

∂ ln γ1 ∂c 2

( ) 2

(3)

In order to calculate the concentration derivatives of the logarithms of the activity coefficients appearing in eqs 1 and 3, the composition dependence of these quantities must be specified. In the present work, we employed two models for doing so, described below. Model 1: Statistical-Thermodynamic Model. On the basis of statistical-mechanical solution theory,22 the natural logarithm of the thermodynamic activity coefficient of the ith species in a multiple species solution may be expanded in powers of the concentrations of solute species ln γi =

∑ Bijcj + ∑ Bijk cjck + ... j

(4)

j,k

where Bij, Bijk, ..., denote interaction coefficients. (These coefficients are not identical to the virial coefficients Aij, Aijk, ..., used to define the power series expansion of osmotic pressure but are related to them by Bij = 2Aij, Bijk = 3/2Aijk, ....) For a solution containing a single protein species (labeled 1), eq 4 reduces to ln γ1 = B11c1 + B111c12 + ...

(5a)

and the concentration derivative to ∂ ln γ1/∂c1 = B11 + 2B111c1 + ...

(5b)

For a binary solution mixture, eq 4 reduces to ln γ1 = B11c1 + B12 c 2 + B111c12 + 2B112 c1c 2 + B122 c 2 2 + ... (6a) 2

2

ln γ2 = B12 c1 + B22 c 2 + B112 c1 + 2B122 c1c 2 + B222 c 2 + ... (6b)

■

where the following equalities have been taken into account: B12 = B21, B112 = B121 = B211 and B122 = B212 = B221. The concentration derivatives appearing in eq 3 are straightforwardly obtained from eq 6:

THEORETICAL METHODS Analysis of Concentration-Dependent Light Scattering of Nonideal Protein Solutions. According to the excess 1892

DOI: 10.1021/jp510918d J. Phys. Chem. B 2015, 119, 1891−1898

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Figure 1. Dependence of normalized intensity of scattered light upon concentrations of ovalbumin (oval, panel A), ovomucoid (ovo, panel B), and ribonuclease A (RNase, panel C) under different conditions. Circles are experimental results. Curves are best-fits according to model 1 (black solid lines) and model 2 (red dotted lines).

Table 1. Best-fit Values And Uncertaintiesa of Parameters Obtained by Fitting Models 1 and 2 to Concentration-Dependent Light Scattering Data from Solutions of Individual Proteins oval

ovo

RNase

45.4 ± 1.5

26.1 ± 0.3

16.5 ± 0.4

438/213

343/48.2 396/44.6 510/36.4 458/55.0 355/27.3

78.9/4.61 2.58/4.87 31.9/7.65 72.7/16.0 40.6/7.11

46.1 ± 1.1 2.36

26.2 ± 0.2 1.96

16.7 ± 0.2 1.68

1.44 ± 0.06 1.56 ± 0.02 1.84 ± 0.06 1.79 ± 0.04 1.29 ± 0.05

0.02 ± 0.17 −1.20 ± 0.36 −0.40 ± 0.29 0.23 ± 0.12 −0.34 ± 0.22

Model 1 M (kg/mol) Bii/Biiib

pH 6 pH 7 pH 8 pH 8.8 pH 7 + NaCl

820/162 869/165

Model 2 M (kg/mol) calculated ri̅ (nm) δriic (nm)

pH 6 pH 7 pH 8 pH 8.8 pH 7 + NaCl

2.30 ± 0.15 2.37 ± 0.23 1.58 ± 0.17

Errors indicate uncertainties represent 95% confidence limits. bThe unit of Bii is M−1; the unit of Biii is ×103 M−2. cCalculated according to the bestfit values of molar masses of each protein using eq 8. a

⎛ 3M v ⎞1/3 ri ̅ = ⎜ i i̅ ⎟ ⎝ 4πNA ⎠

∂ ln γ1/∂c1 = B11 + 2B111c1 + 2B112 c 2 + ... ∂ ln γ2/∂c 2 = B22 + 2B122 c1 + 2B222 c 2 + ... ∂ ln γ1/∂c 2 = B12 + 2B112 c1 + 2B122 c 2 + ...

(8)

where vi̅ denotes the partial specific volume of the protein, i.e., the volume excluded to solvent molecules. The experimentally measured partial specific volume of most polypeptides, which is the volume excluded to solvent per unit mass, is approximately identical within the experimental uncertainty, and equal to approximately 0.73 cm3/g.23 Therefore, the steric radius of each equivalent spherical particle ri̅ can be estimated as a function of its molar mass. We define the effective contact distance between the centers of two equivalent spheres of species i and j as follows

(7)

It will be shown below that, over the range of concentration of each protein utilized in the present study, the logarithm of the activity coefficient of each pure protein may be precisely calculated using the expansions indicated in eq 6 truncated after the quadratic terms. Therefore, only two- and three-body interaction terms were used in the analysis to follow. Using eqs 3 and 7, the dependence of scattered intensity upon the concentrations of each protein in a binary protein mixture may therefore be calculated given the values of nine parameters: M1, M2, B11, B12, B22, B111, B112, B122, and B222. Model 2: Generalized Equivalent Hard Sphere Model. According to the previously formulated hard particle model, each globular protein species is represented by an equivalent sphere with a steric radius equal to

σij = ri ̅ + rj̅ + δrij

(9)

where ri̅ + rj̅ represents the steric or “hard” contribution to the total two-body interaction between species i and j and δrij represents the contribution of the long-range or “soft” interaction between the two molecules. A positive δrij indicates net repulsive soft interactions, while a negative δrij indicates net 1893

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Table 2. Best-Fit Values of Two-Body and Three-Body Hetero-Interaction Coefficients Obtained by Fitting Model 1 to Composition-Dependent Light Scattering Data Obtained from Binary Mixture Solutions, with Molar Masses and Self-Interaction Coefficients Constrained to Best-Fit Values Given in Table 1 oval−ovo B12 pH 6 pH 7 pH 8 pH 8.8 pH 7 + NaCl

a

B112

a

444 708

122 99.3

283

196

oval−RNase B122

a

217 60.3 87.8

B12

B112 −21.0 −16.0

168 78.2 182

18.3

ovo−RNase B122 −7.31 −0.443 −6.62

B12

B112

B122

−153 39.3 131 172 147

11.5 11.1 −25.8 −3.59 7.71

8.42 22.0 25.9 10.6 −3.70

The unit of B12 is M−1; the unit of B112 and B122 is ×103 M−2. The first protein in each interacting pair is species 1, and the second protein in each interacting pair is species 2. a

Figure 2. Dependence of normalized intensity of scattered light upon concentrations of binary mixtures of ovalbumin (oval) and ovomucoid (ovo) in the first column, oval and ribonuclease A (RNase) in the second column, ovo and RNase in the third column, in 0.05 M phosphate buffer at pH 6.0 (first row), 7.0 (second row), 8.0 (third row), pH 8.8 (fourth row), and 0.05 M phosphate buffer at pH 7 plus 0.15 M sodium chloride (bottom row). Circles are experimental results. Curves are best-fits according to model 1 (black solid lines) and model 2 (red dotted lines). wtot denotes total protein concentration.

attractive soft interactions between molecules. The two-body

Bij =

and three-body interaction coefficients for a binary mixture of

4π NAσij 3 3

(10)

equivalent hard particles may then be obtained from results of Kihara.24 1894

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Table 3. Best-Fit Values and Uncertaintiesa of Parameters Obtained by Fitting Model 2 to Composition-Dependent Light Scattering Data Obtained from Binary Mixture Solutions M (kg/mol) calculated ri̅ b (nm) δrii (nm)

pH 6 pH 7 pH 8 pH 8.8 pH7 + NaCl

oval−oval

ovo−ovo

46.2 ± 1.1 2.37

26.2 ± 0.7 1.96

16.7 ± 0.7 1.69

1.43 ± 0.18 1.57 ± 0.18 1.84 ± 0.21 1.82 ± 0.21 1.29 ± 0.21

−0.04 ± 0.42 −0.93 (+0.88, −0.73c) −0.35 (+0.55, −0.63) 0.22 ± 0.68 −0.41 (+0.65, −1.03)

2.28 ± 0.25 2.39 ± 0.21 1.63 ± 0.25

RNase−RNase

oval−ovo

oval−RNase

ovo−RNase

2.09 ± 0.36 2.10 ± 0.31

−0.46 (+0.76, −1.35) −1.11 (+1.01, −1.26c)

1.38 (+0.34, −0.41)

−0.13 (+0.59, −0.81)

−7.8 ± 0.75 −0.36 (+0.58, −0.93) −0.04 (+0.50, −0.58) 0.22 ± 0.60 0.10 (+0.65, −0.87)

a

Errors indicate uncertainties represent 95% confidence limits. bCalculated according to the best-fit values of molar masses of each protein using eq 8. cErrors indicate uncertainties represent 50% confidence limits.

⎧ ⎛ π ⎞2 ⎪ ⎜ ⎟ NA 2(σii 6 − 18σii 4σij 2 when σii ≤ 2σij ⎝6⎠ ⎪ ⎪ 3 3 Biij = ⎨ + 32σii σij ) ⎪ ⎪ 8 N 2σ 6 when σii > 2σij ⎪ A ij ⎩9

equivalent specific volume of each protein can be estimated given the best-fit value of the apparent contact distance between the centers of two equivalent spherical particles of the same species: 3 veff ii = πσii NA/(6Mi). We note that, in phosphate buffered saline (pH 7.0, 0.15 M NaCl), the best-fit values of the calculated equivalent volumes of both ovalbumin and ovomucoid are approximately 1.7 cm3/g, agreeing closely with results obtained in a previous study of the same proteins under very similar conditions.13 Binary Protein Mixtures. The composition-dependent light scattering intensities of all protein mixtures were simultaneously modeled using eqs 3 and 7. In order to globally fit all of the data collected in this study with model 1, a total of 62 undetermined parameters are required (3 molecular weights, 24 two-body interaction coefficients, and 35 three-body interaction coefficients). It is inconvenient and time-consuming to allow all 62 parameters to vary simultaneously in order to minimize the sum of squared residuals. Therefore, we adopt a two-stage minimization strategy. First, the data sets for all solutions of individual proteins are globally modeled to obtain best-fit values of the molecular weights and two-body and three-body selfinteraction coefficients (i.e., Bii and Biii), thus obtaining the bestfit values of 29 parameters shown in Table 1. The remaining data sets obtained from binary mixture solutions were then modeled by constraining the values of these parameters to the best-fit values so obtained, and allowing only the values of the 33 heterointeraction coefficients Bij, Biij, and Bijj to vary to achieve a best-fit to the data. The best-fit values of these parameters so obtained are presented in Table 2. The dependence of scattering intensity calculated using eqs 3 and 7 together with the best-fit values of M1, M2, B11, B111, B22, B222, B12, B112, and B122 obtained under each set of conditions are plotted with data in Figure 2. In order to explicitly take into account the qualitative difference between self- and hetero-interactions in mixtures of oppositely charged proteins, the data were modeled using model 2, the extended equivalent hard sphere model, in which the parameter δrij represents the contribution of soft interactions to the total interaction between the two spheres representing molecules of species i and j. Since the total number of adjustable parameters in model 2 (27) is significantly lower than that in model 1 (62), we did not need to constrain the values of any of these parameters to the values obtained by modeling data obtained from solutions of individual proteins. Therefore, the combined data obtained under all conditions was modeled by allowing all 27 parameter values in model 2 to float. The best-fit values of M1, M2, δr11, δr22, and δr12 so obtained are presented in Table 3 together with the values of ri̅ of each protein calculated as described above using the best-fit values of Mi and σii presented in

(11)

For the case of self-interaction, eqs 10 and 11 reduce to

Bii =

4π NAσii 3 3

(12)

Biii =

5π 2 2 6 NA σii 12

(13)

Using eqs 3 and 7−13, the dependence of scattered intensity upon the concentrations of each protein in a binary protein mixture may therefore be calculated given the values of five parameters: M1, M2, δr11, δr22, and δr12. The abilities of each of these models to account quantitatively for the data to be presented will be compared below.

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RESULTS Solutions of Individual Proteins. The normalized light scattering intensity of solutions of pure ovalbumin (oval), ovomucoid (ovo), and ribonuclease A (RNase) at different values of pH and ionic strength are plotted against protein concentration in Figure 1. The data obtained under all conditions were simultaneously modeled using eqs 1 and 5b of model 1, constraining the value of molar mass of each protein to be uniform for all data sets while permitting the interaction coefficients to vary with conditions. The best-fit values of all parameters so obtained are presented in Table 1. We then simultaneously modeled the data obtained under all condtitions using eqs 1 and 5b and eqs 8, 9, 12, and 13 of model 2, constraining the value of the molar mass of each protein to be uniform for all data sets while permitting δrii to vary with conditions. The best-fit values of all parameters and the calculated ri̅ of each protein based on the best-fit values of the molar mass of each protein are presented in Table 1. The concentration-dependent light scattering intensities calculated according to both models are plotted together with the data in Figure 1. As shown in the figures and table, both models fit the data well, and yielded similar and reasonable molar masses. Model 1 requires two parameters to describe the concentration dependence under a given set of conditions, whereas model 2 requires only one. According to the equivalent hard sphere model, the 1895

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Table 4. Calculated Bija for Each Pairwise Interaction According to eqs 9 and 10 Using Best-Fit Parameters in Table 3 for Model 2 oval−oval pH 6 pH 7 pH 8 pH 8.8 pH 7 + NaCl a

872 914 651

ovo−ovo

RNase−RNase

386 417 482 477 356

93.9 37.1 70.1 118 66.0

oval−ovo

oval−RNase

667 670

146 84.1

469

187

ovo−RNase −180 89.8 119 146 133

The unit of Bij is M−1.

Table 5. Bij/Bijsteric for Each Pairwise Interaction, Calculated Using Best-Fit Parameters Obtained from Model 2 oval−oval pH 6 pH 7 pH 8 pH 8.8 pH 7 + NaCl

3.24 3.40 2.42

ovo−ovo

RNase−RNase

2.53 2.74 3.16 3.13 2.34

0.96 0.38 0.72 1.20 0.68

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DISCUSSION The two-body interaction coefficient Bij (or second virial coefficient) is widely used to describe the interaction of solute species i and j. According to the equivalent hard sphere model,25 the value of Bij is equal to the volume excluded by species i to the center of species j, and vice versa, as follows: 4π 4π NAσij 3 = NA( ri ̅ + rj̅ + δrij)3 3 3

oval−RNase

3.25 3.27

0.69 0.38

2.29

0.90

ovo−RNase −1.46 0.74 1.03 1.19 1.08

interactions between species i and species j. If the ratio is significantly greater than 1, a net repulsive soft interaction between the two species is indicated, and a ratio significantly less than 1 indicates the presence of a net attractive soft interaction. Examining the results shown in Table 5 according to best-fit values from model 2, it is observed that all of the Bij/Bijsteric values for ovalbumin, ovomucoid, and their mixtures are significantly greater than 1 under all conditions of measurement, indicating that soft self-interactions and hetero-interactions are both net repulsive. This result is reasonable because, at around neutral pH, both of the proteins are negatively charged, causing significant electrostatic repulsion between protein molecules. As the concentration of salt increases, the magnitude of Bij/Bijsteric is reduced, in accordance with our expectation that electrostatic interactions between proteins are increasingly shielded with increasing ionic strength.26 On the other hand, the values of Bij/ Bijsteric for the mixtures of RNase with ovalbumin in both pH 7 and 8 are smaller than 1, indicating the presence of attractive hetero-interactions between RNase with ovalbumin. This is in accord with intuition, since RNase is a basic protein27 that is positively charged at pH values below ∼9. We also find that the interaction between ovomucoid and RNase is attractive at pH 6, close to steric at pH 7 and 8, and a little repulsive at pH 8.8. These results are confirmed by an independent experimental measurement of the net charge of RNase upon pH described in the Supporting Information, which indicates that, in phosphate buffer at pH 6, RNase is significantly positively charged, bears little net charge at pH 7 and 8, and is slightly negatively charged at pH 9 (Figure S1, Supporting Information). The results of both measurements clearly indicate that our measurements and analyses of the composition-dependent light scattering provide accurate and sensitive indications of both the sign and magnitude of soft interprotein interactions, which in the present study are dominated by electrostatic effects. The values of Bii/Biisteric obtained for the self-interaction of RNase indicate that the self-interaction of RNase is net attractive at pH 7 and 8. This result agrees qualitatively with the finding that RNase can form reversible oligomers under comparable conditions,28,29 since a specific attraction leading to the formation of oligomers may offset both steric and nonspecific electrostatic repulsion.30 The observation that the best-fit molar mass of RNase slightly exceeds the sequence-based value may be due to the presence of a small amount of irreversibly cross-linked dimer.31,32

the table. The composition-dependent light scattering of a binary protein mixture calculated using eqs 3 and 7−11 together with the best-fit values of M1, M2, δr11, δr22, and δr12 obtained under each set of conditions are plotted together with the data in Figure 2. As indicated in Figures 1 and 2, the best-fits of both models 1 and 2 to all data sets are almost indistinguishable.

Bij =

oval−ovo

(14)

By fitting model 1 to the data, we obtain best-fit values of Bij directly. By fitting model 2 to the data, we obtain best-fit values of δrij directly, and Bij indirectly via eq 14, as shown in Table 4. In this way, we can compare the best-fit values of Bij obtained using the different models. Tables 1 and 4 show that the two models yield similar values of Bij for each protein pair under each set of measurement conditions. Small differences between the results obtained from the two models may be due to (1) the approximation in model 2 that each protein species be represented by equivalent spherical particles, (2) adjustment and/or compensation of the values of independently variable parameters in order to better match small systematic errors of measurement, or (3) a minor contribution to light scattering of multibody (n > 3) interactions neglected in our models. Nevertheless, the equivalent hard sphere model still suffices to account for the nonspecific protein interactions within the uncertainty of measurement over the range of protein concentrations in our experiments (wi + wj < 40 g/L). The interaction coefficient Bij is a measure of the total interaction between the two interacting molecules, i.e., the sum of “soft” long-range interaction and “hard” steric repulsion. On the basis of the hard sphere model, the “hard” steric repulsion can be estimated by ri̅ and rj̅ according to the molar masses of species i and j using eq 8, assuming that the partial specific volumes (v)̅ are identical for all of the proteins. We denote the theoretical interaction coefficient associated with steric interactions only by Bsteric ≡ (4π/3)NA(ri̅ + rj̅ )3. Thus, the ratio of Bij to Bsteric is an ij ij alternative measure of the type and magnitude of the soft 1896

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Figure 3. Calculated composition-dependent logarithm of the activity coefficient of each species in each binary mixture solution at pH 8.0, calculated using best-fit parameters of model 1. Panel A: ovalbumin (black), ovomucoid (blue). Panel B: ovalbumin (black), RNase (red). Panel C: ovomucoid (black), RNase (red).

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The excess free energy of solvation, which is the equilibrium free energy of interaction per mole between a molecule of species i and all other protein molecules in solution, is equal to RT ln γi, where R denotes the molar gas constant and T the absolute temperature. Given the best-fit values of parameters characterizing intermolecular interactions in each model, the composition dependence of the excess free energy of solvation may be estimated. For model 1, this quantity is calculated directly and analytically using eq 6. For model 2, eqs 8−11 are used to calculate the several two-body and three-body interaction coefficients, followed by use of eq 6 to calculate RT ln γi. As an example, the dependences of ln γ1 and ln γ2 upon composition at pH 8 according to model 2 are plotted in Figure 3 for all three binary protein mixtures. A similar plot calculated using best-fit parameter values obtained from model 1 is essentially indistinguishable from that shown.

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Corresponding Author

*Address: Building 8, Room 226, National Institutes of Health, Bethesda, MD 20892-0830. Phone: +1-301-496-3604. E-mail: [email protected]. Present Address †

(D.W.) Institute of Molecular Biophysics, Florida State University, Tallahassee, FL. Author Contributions

D.W. designed the experiments, performed the experiments, interpreted the data, and wrote the manuscript. A.P.M. designed the experiments, interpreted the data, and wrote the manuscript. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank Dr. Xiao-nan Zhao (NIDDK) for the electrophoresis experiments and thank Dr. Peter McPhie (NIH) for a critical reading of the draft manuscript. This research was supported by the Intramural Research Program of the National Institute of Diabetes and Digestive and Kidney Diseases.

CONCLUSIONS

In the present study, we have shown how the composition dependence of light scattering in binary solution mixtures of proteins can yield information regarding the nature and strength of nonspecific interactions between all three protein−protein pairs in the mixture, even when homo- and hetero-interactions are of opposite sign. We have also shown how the equivalent hard spherical model can be extended to provide a compact description of the thermodynamics. However, the extension presented here applies only when interactions of order higher than three-body are negligible.

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AUTHOR INFORMATION

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REFERENCES

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ASSOCIATED CONTENT

S Supporting Information *

Figure S1 shows electrophoresis gels indicating the pH dependence of net charge of ovomucoid and ribonuclease A. Native proteins were mixed with 0.5% bromothymol blue and 10% glycerin, respectively, and loaded in 0.5% agarose gel. Electrophoresis was performed at 100 V in phosphate buffer (pH 6, 7, 8, and 9) at 4 °C for 30 min. Gels were stained with Coomassie blue. Table S1 lists the weight fraction of each protein in protein mixtures. Table S2 lists the number of data points in each measured concentration gradient, corresponding to the symbols plotted in Figures 1 and 2. This material is available free of charge via the Internet at http://pubs.acs.org. 1897

DOI: 10.1021/jp510918d J. Phys. Chem. B 2015, 119, 1891−1898

The Journal of Physical Chemistry B

Article

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