J. Phys. Chem. 1992,96, 905-912
905
Thermodynamic Properties of Aqueous a-Chymotrypsin Solutions from Membrane Osmometry Measurements C. A. Haynes, K. Tamura, H. R. Korfer, H. W. Blanch, and J. M. Prausnitz* Department of Chemical Engineering, University of California, Berkeley, California 94720 (Received: July 29, 1991)
Osmotic pressure data at 25 OC are reported for ternary aqueous solutions containing a-chymotrypsin and potassium sulfate or sodium phosphate buffer. Protein number-average molecular weights and apparent osmotic second virial coefficients are reported over a range of solution pH and ionic strength. The data show that significant aggregation of a-chymotrypsin occurs at pH 8.3. Protein osmotic second virial coefficients depend strongly on pH and ionic strength. The molecular theory of fluids is used to develop a potential of mean force expression useful for correlating osmotic pressures of dilute to concentrated aqueous a-chymotrypsin solutions.
Introduction Dilute to moderately concentrated aqueous protein solutions are found in nature and industry.'s2 In nature, for example, the periplasmic space of recombinant Gram-negative bacteria may contain a moderately concentrated mixture of proteins. In industry, fermentation broths are often complex mixtures of proteins, amino acids, and nutrients. Further, enzymes are finding increasing commercial applications in detergents, cosmetics, organic waste management, pharmaceuticals production, and analytical device^.^ The development of these and other applications is controlled, in part, by our understanding of the physicochemical properties of proteins in aqueous solutions. The literature is rich in experimental studies of catalytic activities in aqueous s ~ l u t i o n . However, ~ few fundamental studies have reported thermodynamic properties of enzymes in aqueous buffer solutions. This work reports osmotic pressure data at 25 OC for ternary aqueous solutions containing a-chymotrypsinand potassium sulfate or sodium phosphate buffer. For each ternary system, we have used membrane osmometry to determine protein number-average molecular weights and osmotic second virial coefficients over a range of solution pH and ionic strength. Theoretical studies of the thermodynamic properties of aqueous protein solutions have yielded a number of models useful for dilute solution^.^^^ However, in practice, these theories cannot be used for the description of concentrated protein solutions. A promising theoretical description for concentrated protein solutions is provided by the molecular theory of liquids, where the potential of mean force, W,plays a central role.' If W can be found from experimental data for dilute solutions, it may be possible to predict the properties of concentrated solutions. These properties, in tum, are useful in the development of molecular thermodynamic theories for protein precipitation. Vilker et a1.8 employed this concept to correlate osmotic pressures for aqueous solutions containing bovine serum albumin at concentrations to 400 g/L. In this work, we have used the molecular theory of fluids to develop a potential of mean force expression, similar to that of Vilker, useful for correlating osmotic pressures of aqueous achymotrypsin solutions. (1) Cantor, C. R.; Schimmel, P. R. Biophysical Chemistry Part 111 The Behavior of Biological Macromolecules; W. H. Freeman and Co.: New Yak, 1980. (2) Bailey, J. E.; Ollis, D.F.Biochemical Engineering Fundamentals, 3rd ed.;McGraw-Hill: New York, 1986. (3) Polastro, E. T.; Walker, A.; Teeuwen, H. W. A. Biotechnology 1989, 7, 1238. (4) Fersht, A. Enzyme Structure and Mechanism, 2nd 4.; W. H. Freeman and Co.: New York, 1985. (5) Tanford, C. Physical Chemistry of Macromolecules; Wiley: New York, 1961. (6) Morawetz, H. Macromolecules in Solution, 2nd ed.; Wiley: New York, 1975. (7) Hirschfelder, J. 0.; Curtis, C. F.; Bird, R. B. Molecular Theory of Gases and Liquids; Wilcy: New York, 1954. (8) Vilker, V. L.; Colton, C. K.; Smith, K.A. J. Colloid Interface Sci. 1981, 79, 548.
0022-3654/92/2096-905$03.00/0
Experimental Section Materisla Potassium sulfate and monobasic and dibasic sodium phosphate were purchased from Fisher Scientific (Pittsburgh, PA). Phenylmethylsulfonyl fluoride and a-chymotrypsin (Sigma C4129,lot 100H8275)were purchased from Sigma Chemical Co. (St. Louis, MO). Water used to prepare the protein solutions was filter4 through a Bamstead Nanopure water purification system. Analytical Methods. The concentration of a-chymotrypsin in each 10-mL sample was measured both gravimetrically and spectrophotometrically using either a Milton-Roy Model 1201 or Shimadzu Model W-160UV-visible spectrophotometer. The extinction coefficient for a-chymotrypin at 280 MI is 2.0cm*/mg. Solution pH was measured using a Sargent-WelchModel 8400 ion/pH meter with a Fisher Scientific Model SN9030116 electrode. Adjustments in pH for solutions containing potassium sulfate were made by adding small amounts of 0.01-1 M KOH or 0.01 M H2S04using a calibrated micropipet. Errors in pH measurements were estimated to be less than f0.02 units. In solutions above pH 6.0, phenylmethylsulfonylfluoride, which binds to serine 192 in the active center of a-chymotrypsin, was used to eliminate protein autodigestion. The protocol used to attach phenylmethylsulfonyl fluoride to a-chymotrypsin was that of Gold et al.9 Membrane 0s"etry.Osmotic pressure measurements at 25 OC were made with a Knauer Model 7310100000 membrane osmometer employing a 1 0-kDa water-soluble celluloseacetate membrane. Figure 1 shows a schematic of the osmotic pressure measurement for a ternary solution containing solvent (l), protein (2),and strong electrolyte (3). The osmometer consists of a sample solution cell (phase 8) and a reference solution cell (phase a) separated by a membrane that is impermeable only to the protein. Osmometry is used to measure the pressure difference between the sample solution and the reference solution at equilibrium. This pressure difference, II, is the osmotic pressure of the system; it depends on the natures of the solvent and solute components, the temperature, and the concentrations of the solutes. Figure 2 gives a schematic of the essential components of the Knauer membrane osmometer. The reference solution is contained in the lower half of the cell, which has a working volume of 0.2 mL. The upper half of the cell contains the sample solution and has a working volume of 20 pL. The cellulose-acetate disk membrane has a diameter of 12 mm. Membranes were conditioned by placing them in a pure water bath for 0.5 h. During this time, the measuring cell was cleaned using the procedure given in the Knauer Model 7310100000 membrane osmometer operations manual.'O The instrument was assembled and calibrated against a known hydrostatic pressure; for our systems, calibration was carried out using pure reference (9) Gold, A. M.; Fahrney, D. Biochemistry 1964, 3, 783.
(10) Knauer Model 7310I00000 Membrane Osmometer operations manual; Knaucr Instruments Inc.: 1990.
0 1992 American Chemical Society
Haynes et al.
906 The Journal of Physical Chemistry, Vol. 96, No. 2, 1992 TABLE I: Osmotic Pressure h t a at 25 Sulfate System c2,
a
3
PI(T,Pd = P,CT. Po+
n)
Figure 1. Schematic of the osmotic pressure measurement for a ternary solution containing solvent (l), protein macromolecule (2),and electrolyte (3).
1. 2. 3. 4.
5.
Head hemostat Channel for syringe Cahbration device with suction tube Calibration glass Clipillq position MEASURE-
hlENT 6.
7.
Capillary position CALLBRATION Tension screws
8. Upperhalfofcell 9. Smple inlet system 10. Teflon ual 1 1. S e m i - p e m r t k membrane 12. Lower half of :ell 13. Tukrosidei-Jerfvld pressuremeGsenng system 14. Cell thermosur 15. Suction orcLhation botde
Figure 2. Schematic of the essential components of the Knauer membrane osmometer.
solution in both cells at the working temperature. The instrument was then used to measure osmotic pressures of aqueous protein solutions. The duration of each measurement ranged from 5 to 15 min, depending on the composition and the viscosity of the sample solution. The operations manual provides detailed descriptions of membrane installation, instrument calibration, and sample measurement. Osmotic pressures were measured at 25 OC for ternary aqueous solutions containing a-chymotrypsin and a strong electrolyte (either potassium sulfate or sodium phosphate buffer). Tables
*,
mmH20 g/L 0.902 7.65 1.798 14.55 3.574 28.25 5.331 40.70 7.027 52.50 8.835 63.95 5.0 0.903 7.15 13.75 1.192 3.535 26.80 5.297 39.85 6.961 5 1.70 63.35 8.695 6.0 0.902 7.00 1.819 14.10 3.756 28.65 5.448 41.55 7.135 53.75 8.989 66.60 8.0 0.901 6.00 1.841 12.40 4.010 27.75 5.558 38.55 7.211 48.15 9.298 62.90 8.25 0.902 6.50 1.811 12.85 3.659 28.55 5.391 38.60 7.185 5 1.25 63.95 8.932 10.0 0.937 7.50 1.808 14.15 3.590 27.95 4.685 35.60 6.173 50.55 65.55 8.950 11.0 0.857 1.05 14.10 1.790 28.15 3.673 5.395 40.30 7.832 54.00 9.034 59.40 12.0 0.944 7.35 1.836 13.60 3.710 23.90 3 1.65 5.416 7.221 34.40 37.60 9.042 pH 4.0
r, Pa 74.8 142.3 276.2 398.0
513.3 625.3 69.9 134.4 262.0 389.6 505.5 619.4 68.4 137.9 280.1 406.3 525.6 651.2 58.7 121.2 271.3 376.9 476.7 615.0 63.6 125.6 259.6 377.4 501.1 625.3 73.3 138.4 237.3 348.1 494.3 640.9 68.9 137.9 275.2 394.1 528.0 580.8 71.9 133.0 233.1 309.5 336.4 367.6
O C
for the 0.1 M Potassium
u/c2,
Pa.L/g 82.92 79.14 77.29 74.65 73.05 70.77 77.43 75.01 74.13 73.55 72.56 11.24 15.88 15.18 74.58 14.58 13.66 72.45 65.10 65.85 66.62 67.82 66.10 66.15 70.46 69.37 70.95 10.02 69.14 70.01 78.24 16.54 16.12 74.31 72.97 71.61 80.45 77.04 74.93 73.04 67.42 64.29 76.16 12.44 62.98 56.51 46.58 40.66
mean-squared error 2.71 0.27 0.68 0.09 0.42 0.33 0.54 0.27 1.11 0.65
0.05 0.17 0.54 1.08 0.13 0.09 0.41 0.1 1 0.00 1.86 0.84 0.70 0.14 0.26 1.08 0.54 0.94 0.27 0.34 0.11 0.52 0.81 0.41 0.31 0.29 0.27 0.57 0.55 0.13 0.18 1.12 0.00 1.55 0.53 2.11 0.27 0.27 0.22
I and I1 l i t the experimental osmotic pressures, in the form n/cZ versus q (where c2 is the concentration of protein in g/mL), for the 0.1 M potassium sulfate system and for the 0.1 M potassium phosphate system, respectively. For a given ionic strength and solution pH, II was measured as a function of a-chymotrypsin concentration over the range 1-10 g/L. Osmotic pressure versus a-chymotrypsin concentration curves are reported at nine pH values for the sulfate salt containing solutions and at six pH values for the phosphate salt containing solutions. Each datum in Tables I and I1 is an average of two or more measurements. Meansquared errors in the experimental data are also reported in Tables I and 11. The data sets in Tables I and I1 all show a similar dependence on protein concentration: n/cZfalls with increasing c2. As discussed later, these data imply that chymotrypsin monomers are slightly attracted to each other in the dilute protein concentration regime. For the sulfate salt containing system at pH 3, II versus achymotrypsin concentration measurements were obtained at three different solution ionic strengths to determine the effect of electrolyte concentration on protein interactions in solution. As shown in Table 111, these data extend over a protein concentration range 1-40 g/L. As with the other systems, II/c2 falls with increasing c2 in the dilute protein regime. For protein concentrationsabove
Osmotic Pressure Data for a-Chymotrypsin
The Journal of Physical Chemistry, Vol. 96, No. 2, 1992 901
TABLE Ik Osmotic Pressure Dnta at 25 OC for the 0.1 M Sodium Phosphate Buffer Svstem c2,
PH 4.5
5.0
6.0
7.0
8.25
9.0
gJL 1.808 3.615 5.423 7.231 9.039 11.07 1.835 3.615 5.423 7.231 9.039 1.913 3.889 5.598 7.290 9.289 0.921 1.840 3.637 5.472 7.290 8.976 0.957 2.209 3.637 5.677 7.209 9.366 0.912 3.637 5.481 7.424 8.949
*,
~~
mm H ~ O 16.00 31.85 46.20 59.35 73.40 87.60 15.20 29.60 43.50 57.10 69.50 13.25 26.65 38.23 48.79 60.80 6.400 12.50 24.75 37.20 49.95 60.85 7.750 17.10 27.60 42.55 52.50 66.50 7.250 27.20 39.65 52.15 60.90
Pa
PaeLjg
*/c2,
mean-squared error
156.4 311.4 451.7 580.3 717.7 856.5 148.6 289.4 425.3 558.3 679.6 129.6 260.6 373.8 476.2 594.5 62.60 122.2 242.0 363.7 488.4 595.0 75.80 167.2 269.9 416.1 513.3 650.2 70.90 266.0 387.7 509.9 595.5
86.54 86.14 83.30 80.26 79.41 77.36 81.00 80.05 78.43 77.21 75.19 67.74 67.00 66.78 65.32 64.00 67.95 66.42 66.54 66.47 67.00 66.29 79.17 75.69 74.20 73.28 71.21 69.43 77.73 73.12 70.74 68.68 66.54
1.35 0.14 0.90 0.47 0.97 1.15 0.48 0.54 0.72 0.41 0.16 1.28 0.38 1.40 0.54 0.53 0.53 2.92 0.94 0.18 0.60 0.60 1.53 0.00 1.08 0.60 0.34 0.21 0.54 0.27 0.62 0.59 0.66
w,
TABLE IIk Osmotic Pressure Data for the Potassium Sulfate Svstem at 25 OC aad DH3 as a Function of Solution Ionic Streneth
TABLE I V Calculated Apparent Osmotic Second Vinal Coefficients and Number-Average Molecular Weights for a-Chymotrypsin in the 0.1 M Potassium Sulfate System at 25 OC B~~x 107, pH L-mol/g2 3.0 -6.75 f 0.31 4.0 -5.67 0.40 5.0 -2.72 f 0.29 6.0 -1.64 i 0.12 8.0 +0.45 f 0.35
B~~x 107, pH L-mol/g2 8.25 -0.12 & 0.24 10.0 -3.22 f 0.20 11.0 -7.37 i 0.38 12.0 -18.1 f 0.38
M~, kg/mol 27.4 f 0.1 30.0 0.2 32.2 f 0.2 32.5 0.7 37.7 f 0.3
M 2 9
kg/mol 35.3 f 0.2 31.6 0.1 30.4 f 0.2 30.9 h 0.2
*
pressure of the solution, n, which may be expressed as a power series in the concentration of solute, c2 (g/mL)
where VI is the molar volume of the solvent in mL/mol, M2 is the number-average molecular weight of the solute in g/mol, BZ2 is the osmotic second virial coefficient in mL.mol/g2, B222is the osmotic third virial coefficient in mL2.mol/g3, Tis temperature in Kelvin, R is the universal gas constant, and superscript o designates the standard state for the solvent (pure liquid water at system temperature). Equation 1 is the osmotic virial expansion, derived by McMillan and Mayer" by forming an analogy between the binary system of solute in solvent and that of gas in vacuum. Equation 1 is valid for solutions in which intermolecular forces are sufficiently short-range to ensure convergence of the cluster integrals. Thus, eq 1 is applicable to dilute solutions of proteins which contain sufficient electrolyte to provide Debye screening of the long-range coulombic interactions. At normal pressures, virial coefficientsare functions only of temperatureand the natures of the solvent and solute. Statistical-mechanical considerations indicate that B22is a measure of two-body interactions, while i3222 accounts for threebody interactions.I2 For description of dilute solutions, we can neglect three-body and higher order interactions in the expansion; eq 1 then becomes c
~~
*,
*,
*IC29
mean-squareci
I, M
g/L
mmH20
Pa
Pa-L/g
error
0.03
0.904 1.808 3.615 5.423 7.231 9.039 19.61 29.41 39.22 0.904 1.808 3.615 5.423 7.231 9.039 19.6 1 29.41 39.22 0.900 1.790 3.584 5.384 7.187 8.963 19.61 29.41 39.22
8.150 16.45 32.55 48.30 64.85 83.17 184.0 289.9 419.1 8.550 16.63 31.65 45.60 59.95 73.45 143.8 210.7 285.0 8.100 16.05 31.35 44.60 57.25 69.15 144.4 214.7 282.3
19.70 160.8 318.3 472.3 634.1 813.2 1799. 2835. 4098. 79.70 160.8 318.3 472.3 634.1 813.2 1406. 2060. 2786. 79.2 156.9 306.5 436.1 559.8 676.1 1411. 2099. 2760.
88.16 88.98 88.03 87.09 87.69 89.97 91.76 96.38 104.5 88.16 88.98 88.03 87.09 87.69 89.97 71.68 70.05 71.06 87.97 87.69 85.54 81.00 77.89 75.43 71.98 71.38 70.39
3.79 0.8 1 1.76 2.25 0.47 0.65 0.50 0.57 1.05 1.62 3.24 0.68 0.72 0.34 0.81 0.37 0.10 1.55 1.30 1.91 0.41 0.00 0.34 0.27 0.57 0.57 0.32
C2,
0.15
0.3
10 g/L, n/c2begins to rise with increasing c2, indicating that three-body (and possibly higher order) protein interactions are repulsive.
Results and Discussion Data Reduction. For a dilute binary solution, the chemical potential of the solvent, p i , is directly related to the osmotic
Thus,a plot of n/c2 versus c2 is linear for sufficiently small values of c2, with slope equal to RTBz2and intercept equal to RT/M2. To obtain reliable values of B22,it is necessary that the osmotic pressure data be of high accuracy in the dilute region. Table IV lists osmotic second virial coefficients and numberaverage molecular weights regressed from the data shown in Table I for a-chymotrypsin in sulfate salt solutions. The results in Table IV were obtained with eq 2 by viewing the system as a-chymotrypsin immersed in a pseudosolvent of aqueous 0.1 M potassium sulfate. The calculations are based on osmotic pressure data corresponding to protein concentrations below 10 g/mL. As shown in Table IV, the increase in the regressed number-average molecular weight of a-chymotrypsin as the pH approaches 8 indicates an increase in the concentration of protein dimers in solution. Thus, the Yapparent"osmotic second virial coefficients reported in Table IV include dimer-monomer and dimer4imer two-body interactions along with monomer-monomer interactions, Figure 3 shows n/c2versus c2 data for a-chymotrypsin in 0.1 M potassium sulfate solution at three different solution pHs. Figure 3 also shows the number-average molecular weight and apparent osmotic second virial coefficient regressed from each of these data sets. Similarly, Table V lists apparent osmotic second virial coefficients and number-average molecular weights regressed from the data shown in Table I1 for a-chymotrypsin in 0.1 M phosphate salt solutions. Finally, Table VI lists apparent osmotic second virial coefficients and number-average molecular weights as a function of ionic strength for the potassium sulfate system at pH 3. ( 1 1 ) McMiflan, W. G.; Mayer, J. E. J . Chem. Phys. 1945, 13, 276. (12) McQuarrie, D. A. Statistical Mechanics; Harper and Row: New York, 1976.
Haynes et al.
908 The Journal of Physical Chemistry, Vol. 96, No. 2, 1992
A pHII 022
i
Mf
I 8 1 x IO 3UYMi
i
Lmiilr,' *,,,"I
2
X
6
4
12
IU
Solution pH
Figure 4. Solution pH dependence of the apparent osmotic second virial coefficient for a-chymotrypsin in the 0.1 M potassium sulfate system at 25 "C.
TABLE V Calculated Apparent Osmotic Second Virirl Coefficients and Number-Average Molecular Weights for a-Chymotrypsinin the 0.1 M sodium Phosphate System at 25 OC
pH
B , ~x 107,
L.mol/g2
M,, kg/mol
4.5 -4.33 f 0.27 27.9 f 0.1 5.0 -3.24 i 0.14 30.0 i 0.1 6.0 -2.04 i 0.20 35.9 i 0.2
pH
B~~x 107,
L-mol/g2
45 -
M~, kg/mol
40
A
7.0 -0.40 f 0.23 36.9 i 0.2 8.25 -4.23 i 0.35 31.4 f 0.2 9.0 -5.49 i 0.20 31.6 f 0.1
TABLE VI: Comparison of Apparent Osmotic Second V i h l Coefficients and Number-Average Molecular Weights Calculated from Osmotic Pressure Data to Those Calculated from LALLS Data for the Potassium Sulfate System at 25 OC
I, M 0.05 0.15 0.30
osmometry data B~~x 107, M21 L-mol/g2 kg/mol +0.23 f 0.42 -6.45 f 0.55 -6.75 i 0.31
28.2 i 0.2 26.8 f 0.2 27.4 i 0.1
-6.48 f 0.26 -6.70 i 0.24
2
4
6
8
10
12
14
Solution pH
Figure 5. Solution pH dpendence of the number-average molecular weight for a-chymotrypsin in the 0.1 M potassium sulfate system at 25
28.5 27.9
O C .
(3)
where the molecular weight obtained from the inverse of the intercept is the weight-average molecular weight. K is an optical constant determined from experimental refractive index measurements and known physical constants. Equation 3 holds strictly only for a zero scattering angle, but it can be used accurately for scattering angles up to 7 O . I 3 As indicated in Table VI, there is good agreement between second virial coefficients and molecular weights obtained from membrane osmometry data and those obtained from LALLS data. Figure 4 shows a nearly parabolic dependence of the apparent osmotic second virial coefficient on solution pH for the 0.1 M sulfate system at 25 O C . B22is a maximum a t pH 8, near the isoelectric point of a-chymotrypsin (PI = 8.3). At pH 8, the value of BZ2is near zero, indicating that a-chymotrypsin has almost no mutual attraction (or repulsion) at solution pH values near its PI. However, dilute a-chymotrypsin monomers become inHugh, M., Ed. Lighr Scotreringfrom Polymer Solurions; Academic
Press: New York, 1972.
I
LALLS data B~~ x 107, M~. L.mol/a2 kg/mol
A more accurate method for experimentally determining second virial coefficients is provided by low-angle laser light scattering (LALLS). From experimental values of the reduced Rayleigh ratio, & = - R e d v ) ,at several solute concentrations, the osmotic second virial coefficient for a binary solution can be determined from
(1 3)
.
creasingly attracted to each other as the solution pH diverges from the protein's PI. Figure 5 shows the dependence of the number-average molecular weight of a-chymotrypsin on solution pH for the 0.1 M sulfate system at 25 O C . M zalso shows a maximum at pH 8, where its measured value of 37 700 Daltons is 30% higher than the molecular weight measured at the same pH by LALLS. As shown in Table V, a similar trend is seen in the molecular weights regressed from the 0.1 M phosphate buffer data. This implies that the presence of dimers and higher oligomers of cr-chymotrypsin in aqueous buffer solutions increases as the pH of the solution tends toward the PI of the protein, even when the protein solution is fairly dilute. Pandit and co-workers have studied the self-associationof a-chymotrypsin in aqueous solutions Over a wide range of pHs and ionic strengths.l4-I6 Using ultracentrifugation techniques, they showed that extensive association occurs a t pH 8.3 and ionic strength 0.05. This result is supported by protein solubility data, which show that most globular proteins in aqueous solution exhibit solubility minima at or near their isoelectric p0ints.I' Osmotic Second Virial Coefficients and tbe Potential of Mean Force. A decrease in BZ2,which depends on the properties of a ~~
(14) Murthy, B. S.N.; Pandit, M. W. Biochim. Biophys. Acra, 1990,1041,
285.
(15) Pandit, M. W.; Rao, M. S.N. Biochemistry 1975, 14,4106. (16) Pandit, M. W.; Rao, M. S.N. Biochemistry 1974, 13, 1048. (17) Foster, P. R.; Dunnill, P.; Lilly, M. D. Biotechnol. Bioeng. 1976.18,
545.
The Journal of Physical Chemistry, Vol. 96, No. 2, 1992 909
Osmotic Pressure Data for a-Chymotrypsin
TABLE VU: C O a M b u k to tbe Potmtirl of M a n Force for a-Chymotrypsin in A q u ” Buffer sdutiod type
mean-force potential m, r I 2a u 0, r > 2a + u
screening parameter, {(r) none
+
Wb ws-9
(z2e)2Slr)
exp[-K(r
er
(1
_2-(zze)2rc2W)
WG-P
3(1
e2kT@
3
(1
+ Kr) exp[-r(r - 2a)l
+ KU)[2 + 2KU + (Ka)’ + (1 + KU)t,/€] unknown (near unity)
(zd’a f(r)
wsir
- 20))
+ Kay
f2@
w#-#
unknown (near unity)
e2P wd
--[ 2 + - (y)] A 6
2
2
s2
-4
where s = r/a and A = (dz- dz)’
In
‘Symbol definitions are as follows: a is the equivalent hydrodynamic radius of a-chymotrypsin monomer (A), u is the minimum allowable surface separation distance between protein monomers (A), 6, is the effective dielectric constant at the surface of the protein, and fir) is the screening factor for electrostatic force contribution to FVz2 The screening factors for induction force contributions to Wzz are not known.
pair of solute molecules in pure reference fluid, can be attributed to several possible factors. The distribution and magnitude of charge on the surface of a protein changes significantlywith pH;18 this, in turn, changes the magnitudes of charge-charge repulsion, chargedipole attraction, dipoledipole attraction, dipole-indud-dipole attraction, and induced-dipole-induced-dipole attraction forces between protein pairs in solution. Changes in solution pH can also alter the conformation of a protein in so1uti0n.l~ These changes in protein conformation can cause variations in the magnitude of van der Waals attraction forces between protein pairs. Finally, binding of ions of a strong electrolyte onto a protein can change the protein’s net and local charges.20 As discussed later, nonuniform binding of ions to protein monomers adds a large attractive contribution to the protein-protein potential of mean force and to the osmotic second virial coeficient. We can estimate the magnitude of each of these contributions to the value of B22by relating B22to the potential of mean force, WZ2, which describes the force of interaction between two protein molecules in solution at infinite dilution. McMillan-Mayer solution theory provides the link between the osmotic virial expansion and potentials of mean force. For potentials which are spherically symmetric, BZ2is related to WZ2 through a volume integral2’
(4)
where aIois the activity of the pure solvent and NAv is Avogadro’s number. Similarly, the osmotic third virial coefficient, Bijk, can be calculated from potentials of mean force using the relation
Vis the system volume, and k is Boltzmann’s constant. For our system, i = j = k = 2. Following Vilker et al., the potential of mean force, WZ2,for a protein diluted in aqueous salt solution can be expressed as a sum of mean-force contributions W22(r22,a1°,T) =
W,,+ W,,+ W-,, + v,+, + W,,
Here, W, accounts for hard-sphere interactions between protein , for chargecharge interactions, W,, for charge-dipole pairs, W interactions, W + for charge-induced-dipole interactions, W,-, for dipoledipole interactions, W,+for dipole-induced-dipole interactions, and Wd for dispersion/van der Waals interactions between protein molecules. Table VI1 summarizes the expresions derived from molecular physics for each of these contributions to W2>The intermolecular potential functions shown in Table VI1 differ from those used by Vilker et al. in the expressions for W,, and W,+. The W,, and W,+ terms shown in Table VI1 were taken from the work of Phillres?z who derived an excess chemical potential expression for a dilute solution of spherical polyelectrolytes. As shown in Table VII, calculation of aqueous protein solution properties with eq 6 requires a number of physicochemical properties of the protein as a function of solution pH, including the protein’s equivalent hydrodynamic radius a, net charge z, dipole moment p, hydrated density p, and polarizability a. The equivalent hydrodynamicradius and the hydrated density for a-chymotrypsin were taken from the literature and were assumed to be invariant with solution pH; a was calculated using the Onsager-Kirkwood relation23 (e
- 1)(2e + 1) 9€
(18) Haynes, C. A., University of California, Berkeley. Unpublished results obtained from simulationson a-chymotrypsin using a modified version of Biograf software on the Stardent Titon supercomputer. (19) Cantor, C. R.; Schimmel, P. R. Biophysical Chemistry Part I: The Conformation of Biological Macromolecules; W. H. Freeman and Co.: New York, 1980. (20) Hill, T. L. Introduction to Statistical Thermodynamics; AddisonWesley: Reading, MA, 1960; Chapter 19. (21) Stigtcr, D.; Hill, T. L. J . Chem. Phys. 1959, 63, 55.
+ W,-i,, + W ,(6)
4 = -*(NlaIo - bN2 3
+ aN2)
(7)
where a10is the pure solvent polarizability in cm3 at system temperature, N 1is the number of solvent molecules/cm3, e is the dielectric constant for the solution, and b is a constant. The net charge of the a-chymotrypsin monomer was determined as a function of pH using the titration data of Marini et aleu and Shiao et aLZs (22) Phillies, 0. D. J. J. Chem. Phys. 1974, 60, 2721. (23) Gabler, R. Electrical Interactions in Molecular Biophysics;Academic Pres: New York, 1978. (24) Marini, M.; Wunsch, C. Biochemistry 1963, 2, 1454.
910 The Journal of Physical Chemistry, Vol. 96, No. 2, 1992
TABLE I X Cdculrted Hamnker Constants, A , for a-Chymotrypsin in the 0.1 M Potassium Sulfate System at 25 O C A X lozo.J PH form 1 form 2 form 3 3.0 7.1 7.0 5.2
TABLE VIII: Physicochemical Properties of a-Chymotrypsin at 25 O C property ref hydrated density, p 2 , /cm3 1.36 (25 O C ) 31 equivalent radius, a, 21.678 28 20 0.1261 x 10-l8 polarizability, a PH net charge dipole moment, D 3.0 +14.2 38 1.5
w
4.0 5.0 6.0 7.0 8.0 8.25 9.0 10.0 11.0 12.0
700
+10.2 +5.20 +3.10 +2.00 +0.20 0.00 -1 .oo -6.00 -12.0 -15.6
Haynes et al.
6.3 4.8 4.2 3.1 2.5 5.1 6.9 8.9
4.0 5.0 6.0 8.0 8.25 10.0 11.0 12.0
416.9 590.0 574.8 565.1 589.1 589.1 594.5 500.3 345.9 412.9
1
.
IJ1
5.0 3.9 3.5 2.5 2.0 4.4 5.6 7.1
0
0
A x lozo
650 -
6.0 4.2 3.7 2.5 2.0 4.8 6.8 8.7
4/
0
0
0
0 . 0
0 0
2-
Form2 0
Form3
04 .20
300
1 2
0
10
Figure 7. Solution pH dependence of calculated Hamaker constants for a-chymotrypsin in the 0.1 M potassium sulfate system at 25 OC. 4
X
6
10
I2
14
Solution pH
Figure 6. Solution pH dependence of the calculated dipole moment for a-chymotrypsin at 25 O C .
Linear dichorism experiments have provided limited information on dipole moments for a-chymotrypsin as a function of solution pH.26927 Experimental dipole moments for a-chymotrypsin at its PI range from 480 to 620 D. For our work, the dipole moment p was estimated as a function of solution pH using the method of South and Grant28with the dipole fluctuation term derived by K i r k w d and S h ~ m a k e r . ~Here, ~ p is determined from the crystal lattice structure of an a-chymotrypsin monomer30 and the knowledge of the pK,'s of the individual amino acids3' using the relation p
.IO
= Cz&i i - R) +
d -
(8)
where e is the unit charge on an electron, zi is the charge at ionized site i on the protein, R is a vector from the origin of coordinates to the protein's center of mass given by
Cmjrj
R=- J
Cmj
(9)
i
(25) Shiao, D. D. F.; Lumry, R.; Rajendee, S.Eur. J . Biochemistry 1972, 29, 377. (26) Antosiewicz, J.; Porschke, D. Biochemistry 1989, 28, 1072. (27) Lamotte-Brasseur, J.; Dive, G.; Dehareng, D.; Ghuysen, J. M. J . Theor. Biol. 1990, 245 (2), 183. (28) South, G. P.; Grant, E. H. Proc. R. Soc. London, A 1972,328, 371. (29) Kirkwood, J. G.; Shumaker, J. B. Proc. Narl. Acud. Sci. 1M2, 38, 863. (30) Tsukada, H.; Blow, D. M. J . Mol. Biol. 1985, 184, 703. (31) Stryer, L. Biochemistry, 3rd ed.; W. H. Freemand and Co.: New York, 1989.
m . is the mass of thejth atom, and rj is a vector from the origin oflcoordinates to the j t h atom of the protein. The sum in eq 9 is over all atoms in the protein. Table VI11 lists all of the physicochemical parameters for a-chymotrypsin including p values calculated using eqs 8 and 9. Figure 6 shows the dependence of the calculated dipole moments on solution pH; p has a fairly constant, maximum value of 590 D over the pH range 5-9. The dipole moment is somewhat lower at pH 9. Application of eq 6 to the calculation of aqueous protein solution properties also requires a value for the Hamaker constant, A, which was introduced by Hamaker to account for dispersion forces between rigid colloidal particles. The Hamaker constant appears in the van der Waals force term, wd, and depends on the size and chemical nature of the macromolecular solute and the nature of the intervening solvent. In general, Hamaker constants do not depend on solution pH. However, the Hamaker constant for a protein can show a modest pH dependence as a result of changes in the average conformation of the protein. Hamaker constants can be determined from B22data by inversion and solution of eq 4 with an assumed form for the potential of mean force, WZ2.Three different forms for W2, were considered. form 1: WZ2(r2J= wh, + Wq-q wd
+
form 2:
WZ2(rz2)=
form 3:
W22(r22)= wh,
wh,
+ Wqq + Wp-@+ Wp+ + wd
+ wqq + Wq-g + Wq-ip +
wp-p
+
wp-ip
+ wd
Except for wh,, form 1 is the potential-of-mean-force expression derived by Derjaguin, Landau, Verwey, and Overbeek (DLV0)32 which has found extensive use in the description of colloidal (32) Verwey, E. J. W.; Overbeek, J. Th. G. Theory of the Stability of Lyophobic Colloids; Elsevier: Amsterdam, 1948.
The Journal of Physical Chemistry, Vol. 96, No. 2, 1992 911
Osmotic Pressure Data for a-Chymotrypsin solutions. Form 2 includes the effects of dipoledipole and dipole-induced-dipole forces, and form 3 is similar to that proposed by Vilker et al. Inversion and solution of eq 4 is carried out in two steps. First, the integral in eq 4 is evaluated for r < 2a + U, where W , = m. This gives the excluded volume contribution to W,,. Next, the integral in eq 4 is evaluated for r 1 2a + u, where W,, is finite, and the result is added to the excluded-volume contribution. Both integrations are carried out numerically using a Newton-Raphson iteration scheme. Table IX reports Hamaker constants for achymotrypsin in the 0.1 M sulfate system calculated as a function of solution pH using eq 4, the data in Table VIII, and each of the three proposed forms for WZ2. As suggested by the success of the DLVO theory, charge-charge electrostatic repulsion forces and dispersion/van der Waals forces make the largest contribution to the potential of mean force for a-chymotrypsin. As shown in Figure 7, the calculated Hamaker constants show a pH (net charge) dependence for all three WZ2expressions. Comparison of results from forms 1 and 2 suggests that dipoledipole and dipole-induced-dipole forces make a relatively small contribution to protein-protein interactions in solution. Charge-dipole and charge-induced-dipole forces influence protein interactions to a much larger extent. A further of the contributions made by the individual terms to Wz2shows that the protein’s dipole moment influences W,, primarily through charge-dipole forces; a much smaller contribution is made by dipoledipole forces, and even smaller contributionsare made by charge-induced-dipole and dipole-induced-dipole forces. Amide I band Raman spectroscopy studies on a-chymotrypsin suggest that the conformation of the protein changes with pH34 but not to an extent which alone would account for the variations in the Hamaker constant shown in Figure 7. Indeed, Figure 7 suggests that a large attractive contribution to W2, has been neglected in eq 6. Effect of Ion Biading. For a protein in aqueous buffer solution at a given pH, the binding of ions onto the protein is described by a dynamic equilibrium with a very small time constant. Thus, when we speak of the net charge of the protein, we are referring to the ensemble average. Indeed, experiment and statistical mechanics indicate that the total charge carried by each protein monomer in solution varies rapidly with time.35 The existence of charge fluctuations and their influence on W,, were first noted by Kirkwood and S h ~ m a k e rwho , ~ ~used statistical-mechanical arguments to relate the time correlation between fluctuations in charge to the distribution of charge associated with fluctuations in the number and configuration of ions bound to the protein. Further analysis by Phillies2, led to the following expression for the charge fluctuation contribution to W,,
B (Q?QiZ)
- (QiQi)’
2
2
WAq-Aq= --
~xP[-~K~I
?
(10)
where
T(r) = Here,
K
4
[K ] exP(4
is the inverse Debye length, whose square is given by
Qi is the formal charge on protein monomer i, Ci is the molar concentration of component i. In eq 10, the quantity (Q?Q?) - (QiQi)2can be identified as [(z,)~- ( ~ ) ~= ]Az4e4, e ~ where z is the net charge on each protein monomer and Az is the variance in the average charge of the monomer. The negative sign in front of the right-hand side of eq 10 indicates that WAq-aqprovides an (33) Haynes, C. A., University of California, Berkeley. Unpublished results, 1991. (34) Przybycicn, T. M.; Bailey, J. E. Biochim. Biophys. Acta 1991, 1076, 103. (35) Hill, T. L. Starisrical Mechanics; McGraw-Hill: New York, 1956.
‘1
I
0
Form4
I
04 .20
-10
0
10
20
%2
Figure 8. Solution pH dependence of Hamaker constants for a-chymo-
trypsin in the 0.1 M potassium sulfate system at 25 O C calculated using form 4 for W22. 0
I’ll3
pllx
1
-
1
0
-+.
760
... ... ..
50 0
2
4
6
X
IO
c2 I giL I Figure 9. Calculated and experimental reduced osmotic pressure data for the 0.1 M potassium sulfate system at 25 OC. Solid lines indicate that calculations were performed using form 4 for W22.Dashed lines indicate that calculations were performed using form 3 for
attractive force contribution to W,,. We therefore consider a fourth form for W,,: form 4: W22(r22) = W,, + W,, + WAq-Aq+ Wq-,, + Wq-i,, + W,,-,, W,,-i,,+ Wd
+
We assume that the dependence of Az on pH is similar to that of net charge on pH. That is, Az is zero at the PI of the protein and increases as the pH diverges from the isoelectric point. However, Az is always lower than the absolute magnitude of z2 at a given pH (away from the PI). Applying these restrictions to Az values for a-chymotrypsin, we assumed that Az attains a maximum value of 5 at pH 12 and varies linearly down to pH 8, where Az equals zero. Similarly, Az was set equal to 5 at pH 2 and was assumed to vary linearly to pH 8. Figure 8 shows that the addition of Was-&to W,, drastically reduces the dependence of the calculated Hamaker constants on solution pH. Indeed, the modest dependence of A on pH shown in Figure 8 could be due to average conformational changes in the protein. The significant reduction in the pH dependence and magnitudes of the calculated Hamaker constants illustrates the importance of attractive charge fluctuation forces in aqueous protein solutions. Model Application and Iimitatiom. Figure 9 compares observed n/c, with those calculated using eqs 2 and 4 for the 0.1 M potassium sulfate system at pH 3 and at pH 8. The dashed curves in Figure 9 were calculated using Form 3 for W,,; the solid lines were calculated using Form 4 for W,, with Az adjusted to best fit each n/c, data set. For each of the calculated n/c, curves,
912 The Journal of Physical Chemistry, Vol. 96, No. 2, 1992 TABLE X Calculated Osmotic Third Virial Coefficients for a-Cbymotrypsin in the Potassium Sulfate System at 25 OC 8222
x 10'0,
I, M
L.mol/gz
0.03 0.15 0.3
4.1 43.1 18.2
a"w
0
1.0.03M 1:0.3M
4000-
0
- Model
0
10
20
30
40
I dL 1 F i p e 10. Calculated and experimental osmotic pressure data for the potassium sulfate system at 25 OC. Solid lines indicate that calculations were performed using form 4 for WZ2. e2
the Hamaker constant was set equal to 2.21 X 1020J, its average calculated value at the PI of a-chymotrypsin. The importance of attractive charge fluctuation forces is evident once again in Figure 9. n/c2values calculated with WM-asincluded in the expression for W2,agree well with experiment, whereas those calculated with Form 3 of W2,diverge quickly from experiment. Table X shows osmotic third virial coefficients, B222r calculated for a-chymotrypsin in the 0.03 M and 0.3 M potassium sulfate systems using eq 5 and Form 4 for W2*. Each B222calculation is based on a Hamaker constant of 2.21 X lozoJ and a value for Az regressed from dilute solution data (c2 < 10 g/L). Calculated osmotic third virial coefficients can be used in the osmotic virial expansion to predict osmotic pressures for concentrated a-chymotrypsin solutions. A truncated form of the osmotic virial expansion useful for calculating concentrated solution properties is
II = R T [ c ~ / M+ ~B22~2'+ B222~2~ 4- 0.2869(B22hS)3~24 + 0.115(B22hS)4~25] (11) where the fourth and fifth virial coefficient terms include only the hard-spherecontributionsto four- and five-body interactions, respectively. Hard-sphere contributions to osmotic virial coefficients were taken from Hirschfelder et al.,' who derived the hard-sphere osmotic virial expansion to the eighth virial coefficient term. The hard-sphere contribution to the osmotic second virial coefficient, B22hS, is given by
Figure 10 compares osmotic pressures calculated using eq 11 and the B222values shown in Table X with experimentaldata for the 0.03 M and 0.3 M potassium sulfate systems. For both systems, eq 1 1 slightly underpredicts osmotic pressures in the concentrated protein region (c2 > 20 g/L). Our objective is to describe the osmotic behavior of concentrated protein solutions with McMillan-Mayer solution theory, where the osmotic virial coefficients are calculated from an appropriate expression for the potential of mean force. This goal was realized in large part. However, as shown in Figure 10, predicted osmotic pressures are consistently lower than experiment in the concen-
Haynes et al. trated protein region. Many of the factors which could be responsible for this lack of agreement are discussed in detail by Vilker et ai.; therefore, they will only be mentioned briefly here. The equations shown in Table VI1 provide a simplistic representation of intermolecularinteractions between a-chymotrypsin monomers in aqueous buffer solutions. The model views each protein monomer as a hard sphere pogsessing an ideal point charge and an ideal point dipole, both located at the center of the hard sphere. Each individual force contribution to the potential of mean force is assumed to be spherically symmetric. Finally, the solvent is viewed as a dielectric continuum; thus, specific solvent effects are ignored in the potential of mean force. Calculated excluded-volume forces, which provide the largest contribution to the estimated second and third virial coefficients, are sensitive to the assumed hard-sphere geometries of the molecules in the system. X-ray crystallography studies show a-chymotrypsin to be globular but not perfectly spherical.28 This suggests that significant errors may arise in modeling a-chymotrypsin as a sphere, especially in the calculation of excluded volume forces. The expression for W shown in Table VI1 is based on DebyeH0ckel theory whicgfs strictly valid only at dilute solution conditions where the distance between neighboring protein molecules is large compared to the Debye length. Therefore, the accuracy of the expression of W,,shown in Table VI1 decreases rapidly with increasing protein concentration. The protein concentration range over which calculated values for W, remain accurate can be increased greatly through the use of a e mean spherical model of Blum et al.36in place of the expression for W,, shown in Figure 7. Protein dipole moment calculations require knowledge of the pK,'s of the amino acid residues making up the protein. In our dipole moment calculations, the pK, of each residue was set to its value in free solution. Further, calculated dipole moments are based on a static, ensembleaveraged structure for achymotrypin. Therefore, dynamic contributions to the protein dielectric constant are not accounted for in the model. Errors associated with these approximations effect the accuracy of the dipole force terms in the potential of mean force. Molecular physicists have not developed a quantitative theory for hydrophobic/hydrophilicforces. Thus, hydrophobic/hydrophilic forces between protein monomers are not adequately represented in the model. Any contribution hydrophobic/hydrophilic forces make to W2is therefore included indirectly in the dispersion and charge-fluctuation terms, since the Hamaker constant and Az are the only parameters regressed directly from osmotic pressure data. Hamaker theory3' has proven reliable in describing the properties of colloidal systems, but its applicability to the description of aqueous protein solutions is largely untested. The simplicity of the Hamaker theory suggests that it might be insufficient for describing dispersion forces between proteins in solution. For example, the Hamaker theory does not take into account heterogeneities in macromolecular structure or specific interactions between solvent and protein molecules. Finally, the truncated virial expansion shown in eq 1 1 may be insufficient for the description of concentrated protein solutions, where four-body and higher order interactions become significant. However, calculation of osmotic fourth virial coefficients from form 4 for W2,indicates that four-body interactions are not significant at a-chymotrypsin concentrations less than 50 g/L. Acknowledgment. This work was supported in part by the National Science Foundation through Grant #cTs8914849. For financial support, K.T. is grateful to the Japanese Ministry of Education and C.A.H. is grateful to the National Institutes of Health. Registry No. a-Chymotrypsin, 9004-07-3. (36) Blum,L. Mol. Phys. 1975,30, 1529. (37) Hamaker, H. C.Physic0 1937,4, 1058.
