J . Phys. Chem. 1989, 93, 474-419
414
The Role of Supporting Electrolytes in Protein Diffusion Derek G . Leaist Department of Chemistry, University of Western Ontario, London, Ontario N6A 5B7,Canada (Received: April 25, 1988)
A conductometric technique is used to measure mutual ternary diffusion coefficients for aqueous solutions of bovine serum albumin (BSA) + NaCl at 25 OC and pH 7-8. NaCl is found to sharply reduce the rate of diffusion of the BSA component. The electric field induced by diffusion of charged BSA with Na' counterions produces a large cocurrent flow of NaCl. Neglect of the coupled flow of the NaCl supporting electrolyte causes the measured apparent diffusivity of BSA to differ from the protein's true diffusivity. Equations are presented to estimate the ternary diffusion coefficients for charged proteins in supporting electrolyte solutions. The equations can be used to estimate the diffusion coefficients as functions of the pH, ionic strength, protein concentration, and counterion binding.
component such as a charged protein with attendant counterions, the mutual diffusion coefficient is a complicated average of the This paper presents a study of the effects of supporting elecdiffusivities of the protein and counterions. Since the driving force When trolytes on the diffusion behavior of proteins in for mutual diffusion is the gradient in solute chemical potential, charged proteins undergo mutual diffusion with small mobile the mutual diffusion coefficient is proportional to the activity counterions, such as Na+, charge separation is prevented because coefficient factor 1 + C d In y,/dC, which reflects variations in the diffusion-induced electric field slows down the counterions the driving force for nonideal solution^.'^ S e l f - d i f f u s i ~ n , ' ~by- ~ ~ and speeds up the protein ions. If a supporting electrolyte such contrast, can be measured by tagging some of the protein ions as NaCl is added, the electric field produces a coupled flows of with radioactive or magnetic labels and then following the motion relatively mobile Cl- ions at the expense of speeding up the slower of the tagged ions in a chemically uniform solution. Self-diffusion protein anions. Supporting electrolytes should therefore reduce involves interchange of tagged and untagged species which form the rate of protein diffusion. The coupled flow of supporting thermodynamically ideal mixtures. Therefore, the activity electrolyte will produce a gradient in that component's concencoefficient factor vanishes from the expression for the self-diffusion tration. Therefore, even if the initial distribution of the supporting coefficient. In addition, the diffusivity of the counterions makes electrolyte is uniform, the protein will soon be diffusing in an no direct contribution to the self-diffusion coefficient of the protein electrolyte gradient. The electrolyte gradient might in turn ions. For these and other reasons, it is not unreasonable to expect produce coupled transport of the protein. A flow of protein driven significant differences between the mutual and self-diffusion by the supporting electrolyte would lead to a systematic errorg coefficients of a protein. The mutual diffusion coefficient may in the measured protein diffusivity. be larger because in mutual diffusion the protein ions are "pulled Gosting l o and Tanford" have suggested that the interaction along" by the relatively mobile counterions to maintain electrobetween diffusing proteins and simple electrolytes might be imneutrality. portant. To examine this possibility, equations are reported here Keller et al.7 have used the diaphragm cell method to measure to estimate multicomponent diffusion coefficients for protein both the mutual and self-diffusion coefficientsof BSA in an acetate solutions with added salt. Ternary mutual diffusion coefficients buffer. Remarkably, no significant difference between the mutual for bovine serum albumin (BSA) + NaCl water have been and self-diffusion coefficients was detected over a wide range of measured conductometrically to supplement the analysis. Some composition. It is known, however, that the self-diffusion coefmeasurements were made at pH 7 to examine the effects of ficient of an ion and the mutual diffusion coefficient of the parent supporting electrolytes on the protein's diffusivity under normal electrolyte component are identical if a large excess of supporting physiological conditions. electrolyte is p r e ~ e n t . ' ~ J ~(For , ' ~ >example, ~~ the mutual diffusion In previous studies, Gouy interferometry12and a conductimetric coefficient for small amounts of KI in supporting KC1 solutions t e c h n i q ~ ewere ' ~ used to measure accurate binary mutual diffusion equals the self-diffusion coefficient of the I- ions.Ig) The present coefficients for BSA H20. Apparent binary diffusivities for work explores the possibility that supporting electrolytes are reBSA in multicomponent solutions containing added salts or buffer electrolytes have been determined by using the Gouy m e t h ~ d , ~ ? ~ sponsible for the reported equivalence of the mutual and selfdiffusion coefficients of BSA. Since diffusion of the protein is light scattering,'-" and the diaphragm cell technique.' used as a model to help interpret the diffusion of other charged Mutual diffusion involves interchange of solvent and solute by macromolecules, it is important to understand the factors that random thermal motions which cause concentration gradientsI4 govern its diffusion behavior. or concentration fluctuation^'^^ to decay in time. For an electrolyte Introduction
+
+
Experimental Section (1) Doherty, P.; Benedek, G. B. J. Chem. Phys. 1974, 61, 5426. (2) Raj, T.;Flygare, W. H. Biochem. J. 1974, 13, 3336. (3) Phillies, G. D. J.; Benedek, G. B.; Mazer, N. A. J. Chem. Phys. 1976, 65, 1883. (4) Neal,D. G.; Punch, D.; Cannell, D. S. J. Chem. Phys. 1984,80, 3469. ( 5 ) Creeth, J. M. Biochem. J. 1952, 51, 10. (6) Wagner, M. L.; Scheraga, H. A. J. Phys. Chem. 1956, 60, 1066. (7) Keller, K. H.; Canales, E. R.; Yum, S. I. J. Phys. Chem. 1971,75,379. (8) Leaist, D. G. J. Phys. Chem. 1986, 90, 6600. (9) Leaist, D. G. J. Colloid Interface Sci. 1986, 1 1 1 , 240. (10) Gosting, L. J. Adu. Profein Chem. 1956, ZI, 429. (1 1) Tanford, C. Physical Chemistry of Macromolecules; Wiley: New York, 1961; p 353. (12) Tinoco, I., Jr.; Lyons, P. A. J. Phys. Chem. 1956, 60, 1342. (13) Leaist, D. G. J. Solufion Chem. 1987, 16, 805. (14) Robinson, R. A.; Stokes, R. H. Electrolyte Solufions, 2nd ed. (revised); Butterworths: London, 1965; Chapter 11, Appendix 6.1.
0022-365418912093-0474$01 SO10
BSA (Sigma Biochemicals no. A-3912) solutions were prepared with distilled deionized water in volumetric flasks. Molar concentrations were calculated by using 66 000 g mol-' for the protein's molecular weight4 The protein solutions were deionized and made isoionic (pH -5.4) on 50-cm ion exchange columns.1u1*22The correct amount of carbonate-free NaOH solution (15) Stokes, R. H. J. Phys. Chem. 1965, 69, 4012. (16) Dunlop, P. J. J. Phys. Chem. 1965, 69, 1693. (17) Harris, K. R.; Mills, R.; Back, P. J.; Webster, D. S. J. Magn. Reson. 1978, 29, 473. (18) Braun, B. M.; Weingartner, H. J. Phys. Chem. 1988, 92, 1342. (19) Stokes, R. H.; Woolf, L. A.; Mills, R. J. Phys. Chem. 1957,151, 1634. (20) Leaist, D. G.; Lyons, P. A. J. Phys. Chem. 1982.86, 564.
0 1989 American Chemical Society
Supporting Electrolytes in Protein Diffusion
The Journal of Physical Chemistry, Vol. 93, No. 1, 1989 475
+
TABLE I:" Measured and Predicted (in Parentheses) Ternary Mutual Mffusion Coefficients for Aqueous Na,BSA (C,) NaCl (C,) Solutions at 25 O C 103Cl/mM 103C2/mM 1O901, 10~0~~ 109~21 1090~~ 0.38 0.0 0.288 (10.004) 0
0.38
5.0
0.38
10.0
0.76
0.0
0.76
5.0
0.76
10.0
(0.430)' 0.21 (f0.02) (0.16) 0.15 (fO.02) (0.12) 0.237 (fO.005) (0.430) 0.168 (*0.004) (0.206) 0.108 (f0.004) (0.156)
-0.009 (fO.005) (-0.006) -0.013 (f0.003) (-0.004)
1.4 (f0.2) (4.1) 2.3 (f0.2) (4.6)
1.42 (f0.03) (1.70) 1.48 (f0.03) (1.66)
0.0
1.0 (f0.1) (3.3) 2.1 (fO.l), (4.1)
-0.020 (f0.004) (-0.010) -0.005 (*0.005) (-0.006)
1.38 (f0.02) (1.76) 1.35 (f0.03) (1.70)
Du in m2 s-l.
+
TABLE 11:" Measured and Predicted (in Parentheses) Ternary Mutual Diffusion Coefficients for Aqueous Na18BSA(C,) NaCl (C,)Solutions at 25 OC ClhM C2/mM 1 0 ~ 0 ~ ~ 1090,~ 1 0 ~ 0 ~ ~ 1 0 ~ 0 ~ ~ 0.38 0.0 0.378 (fO.005) 0
0.38
5.0
0.38
10.0
0.76
0.0
0.76
5.0
0.76
10.0
(0.707) 0.21 (f0.02) (0.40) 0.015 (f0.02) (0.29) 0.341 (f0.004) (0.707) 0.18 (f0.02) (0.51) 0.12 (fO.O1) (0.40)
-0.008 (f0.003) (-0,010) -0.005 (*0.005) (-0.007)
6.5 (f0.2) (6.53) 9.1 (f0.2) (8.8)
1.33 (f0.02) (1.83) 1.35 (*0.03) (1.76)
0
5.8 (f0.3) (4.3) 7.8 (f0.2) (6.5)
-0.016 (f0.003) (-0.01 3) -0.006 (f0.002) (-0.0 10)
1.28 (f0.02) (1.90) 1.48 (f0.02) (1.83)
" Dn in m2 s-l. was then added from a microburet to prepare protein samples with 7 or 18 mol of Na+ counterions per mole of protein, designated by Na7BSA and Na18BSA. The NaOH was delivered slowly with rapid stirring to minimize local excesses. Ternary solutions were prepared by dissolving weighed amounts of NaCl (BDH reagent grade) in the Na,BSA solutions. The NaCl had been dried in a vacuum oven at 120 OC. A simplified versionZo of the Harned restricted diffusion techniquez3 was used to determine ternary diffusion coefficients for the Na,BSA NaCl solutions at 25.00 f 0.01 OC. At each mean cell composition, separate experiments were performed in triplicate with the initial concentration gradient in the Na,BSA or NaCl components. The rate of diffusion was followed by monitoring changes in the electrical conductance with pairs of of the height of the small electrodes positioned at '/a and diffusion column. Moments analysisz4of the conductance versus time data gave the ternary diffusion coefficients. The cells used in this work had cylindrical diffusion channels (3.048 cm height, 1.270 cm diameter) that were milled from high-density polyethylene and fitted with 0.1 cm diameter Pt electrodes. Fresh solutions were prepared for each run.
+
Results Nu$SA(C1) + HzO.A few binary diffusion measurements were made on NaC1-free solutions of Na7BSA or Na18BSA. At the solution pH values (7-8) and protein concentrations (0.38 or 0.76 mM, 25 or 50 g L-I) that were used, the concentrations of free H+and OH- ions were negligible. The major solute species were Na+ counterions and the protein ions ..., BSA("+')-, BSA", BSA("'b, ..., which carried an average of -n proton charges. The protein ions equilibrate rapidly by fast proton exchanges.21 Under these conditions electroneutrality and the local equilibria of protein (21) Tanford, C.; Swanson, S.A.; Shore, W. S.J . Am. Chem. SOC.1955, 77, 6414. (22) Timasheff, S.N.; Dintzis, H. M.;Kirkwood, J. G.; Coleman, B. D. Proc. Natl. Acad. Sci. U S A . 1952, 38, 863. (23) Harned, H. S.;Nuttall, R. L. J . Am. Chem. SOC.1949, 71, 1460. (24) Leaist, D. G. Can. J . G e m . 1985, 63, 2933.
ions dictate that only one solute flow is independent." As a result, mutual diffusion of the NaC1-free protein can be described by using Fick's equation -JI = DlIVCl
(1)
where J1 denotes the molar flux density of the Na,BSA component and Dll is its binary diffusion coefficient. The description of diffusion in terms of the flow of the neutral Na,BSA component rather than charged protein ions has the advantage that electric potential gradient terms vanish from the transport equation. The measured values of Dll for the Na7BSA sample were 0.288 (f0.004) X 10" and 0.237 (f0.003) X l p m2 s-' at the respective protein concentrations 0.38 and 0.76 mM. The corresponding values of Dll for Na18BSAwere 0.378 (f0.005) X 10" and 0.341 (f0.003) X 10" m2 s-l. The latter sample diffused more rapidly due to the larger number of mobile counterions. For comparison, the limiting diffusion coefficient of the neutral BSA molecule obtained by extrapolation of optical data5*12 for the isoionic protein is 0.075 X m2 s-I. Nu$SA(C,) NuCl(C2) HzO.The addition of NaCl to a solution of Na,BSA gives a ternary mixture of two electrolytes with a common ion. To allow for interactions between the flows of the two electrolytes, mutual diffusion in the solutions is described by the coupled Fick e q u a t i o n ~ ~ J ~ . ~ ~
+
+
-J1 = DllVC1
+ D12VC2
(2) (3)
Mutual ternary diffusion coefficient DNC gives the flux density of component i produced by the gradient in the concentration of component k. Equations 1 and 2 have been used in previous studies to describe diffusion of colloidal electrolytes with added sa1t.9,26.27 (25) Stockmayer, W. H. J. Chem. Phys. 1960, 33, 1291. (26) Vitagliano, V.; Laurentino, R.; Costantino, L. J . Phys. Chem. 1969, 73, 2456. (27) Leaist, D. G.; Lyons, P. A. J . Phys. Chem. 1982, 86, 1542.
476 The Journal of Physical Chemistry, Vol. 93, No. 1, 1989 TABLE III: Comparison of Measured Ternary Mutual Diffusion Coefficients (Dll) and Apparent Diffusivities ( D ) for Na,BSA and Nal,BSA in Aqueous NaCl (C,) Solutions at 25 OC sample C,/mM CdmM 109D1,/m2s-' 109D/m2s-Ia 0.168 0.126 Na,BSA 0.76 5.00 10.00 0.108 0.106 Na,BSA 0.76 Na,8BSA 0.76 10.00 0.12 0.154 a
Interpolated values from ref 1.
Ternary diffusion coefficients were measured a t 0.38 or 0.76 mM Na,BSA and 5.00 or 10.00 mM NaCl. The results are listed in Tables I and I1 together with the binary values of Dll that were determined for the NaC1-free solutions. The relatively large molar conductance of the Na,BSA components allowed the values of D l l and D I 2to be determined more precisely than the values of Dzl and Dz2for the NaCl component. The concentration of NaCl was kept relatively low to minimize binding of C1- ions to the protein.28 This point will be discussed later. The values of the ternary diffusivities of the Na,BSA components with added NaCl are 3 0 4 5 % smaller than the binary values of D l l for the NaC1-free solutions. The substantial reductions in the rate of protein diffusion are caused by relatively small mass fractions of salt, 6-23 mg of NaCl per gram of protein. Unless proteins are purified by ion exchange or extensive dialysis, they usually contain a few tenths of a percent by mass of ionic impurities (largely NaCl). This level of contamination is sufficient to provide several moles of electrolyte impurity per mole of protein, and thus sharply reduce the diffusivity of the protein relative to the value for the pure material. The binary diffusivity of NaClZ9in the concentration range 5-10 m M NaCl takes values from 1.54 X to 1.56 X mz s-l. In the Na,BSA + NaCl solutions that were studied, the ternary diffusivity of NaCl is 5 1 5 % smaller. The reported values of the cross-coefficient DZ1show that the gradients in the Na,BSA components produce significant coupled flows of NaCl. As anticipated in the introduction, the coupled flows of NaCl travel cccurrent to the flows of Na,BSA. The ratio D21/D11, which gives the number of moles of NaCl cotransported per mole of Na,BSA,8 takes values in the range 6-60. Since NaleBSA has more counterions than Na,BSA, the gradient in the former generates a stronger electric field and hence it produces larger coupled flows of NaCl. As the ratio NaCl-Na,BSA decreases, so do the values of Dzl because there can be no coupled flow of NaCl in a NaCl-free solution. The reported values of D12show that the NaCl gradient produces countercurrent flows of Na,BSA. The mobility of the Na+ is lower than the mobility of C1- ions.14. Therefore, the electric field along the NaCl gradient tends to slow down the C1- co-ions and to generate counterflow of the BSA" ions. Benedek and Doherty' have used laser light scattering to measure apparent mutual diffusion coefficients, denoted here by D, for aqueous Na,BSA NaCl solutions a t compositions overlapping those used in the present study. In Table I11 values of b obtained by interpolation of their data are compared with the ternary mutual diffusion coefficients that were determined conductometrically for Na7BSA and Na18BSA. The fact that good agreement is not obtained is not surprising since the two sets of measurements were made on different protein samples. In addition, it has been established that light scattering measurements on electrolyte solutions yield operationally defined diffusion Coefficients that differ from the Dik phenomenological coefficients that are defined by Fick's equation and measured in classical diffusion experiments.3w36 The diffusion coefficient obtained by
+
Leaist light scattering for a dilute polyelectrolyte such as Na,BSA is believed to be approximately (1 n)DBsA' rather than the phenomenological values of DI1defined by eq 1. The meaning of diffusion coefficients determined by light scattering for polyelectrolytes in solutions with added salt is less clear.3e36 Although fluctuations in the concentration of simple salts scatter light relatively weakly, they can drive coupled fluctuations in the macroion's concentration, thereby contributing indirectly to the spectrum of the scattered light. The strength of the coupling between the macroion and salt increases with the macroions charge. This may explain why the agreement between D l l and the apparent D values shown in Table I11 is reasonably good for Na7BSA, but poorer for Na18BSA. T ~ r r e 1 has 1 ~ ~discussed the difficulties in the interpretation of electrolyte diffusivities that are determined by light scattering.
+
Discussion To interpret the effects of added NaCl on diffusion of the Na,BSA components, it is convenient to assume that each protein carries the average charge of -n units. The diffusing components can then be treated as a ternary mixture of 1:n and 1:l electrolytes. Even with this simplification, however, a detailed treatment of diffusion should include the electrophoretic interactions between the migrating ions,37the effects of changes in the solution viscosity, and departures from ideal solution through the activity coefficient factor mentioned earlier. Unfortunately, quantitative estimates of these contributions to the Dikcoefficients are not available. Even for simple unsymmetrical electrolytes such as aqueous CaC12,37 the electrophoretic effect is poorly ~ n d e r s t o o d . ~Moreover, ~ the device of multiplying limiting diffusion coefficients by 7"/7,where ' 7 and 7 are the respective viscosities of the pure solvent and the solution, tends to overcorrect for viscosity changes.39 Predicted Dik Values. Despite these unresolved difficulties, qualitative estimates of the effects of added salt on the diffusion properties can be obtained by using the limiting expressions10,20 Dll
= DBSA + tBSA(DNa - DBSA)
(4)
O12 = t B S A ( D N a - D C l ) / n
(5)
DZl = ntCl(DNa - DBSA)
(6)
D22
DCl
+ tCI(DNa -
(7)
+
for the ternary diffusion coefficients of the Na,BSA( 1) NaCl(2) components. DBSA, Da,and DNa denote the limiting diffusion coefficients of the BSA", C1-, and Na+ ions. The ionic transference numbers ti = c , z i 2 D i / ~ c m z m z D m m
(8)
give the fraction of the total electric current due to ion i in an externally applied electric field. zi and ci denote the valence and molar concentration of the ions. Due to the omission of terms for electrophoresis, activity coefficients, and viscosity changes, eq 4-8 become exact only as the solute concentrations drop to zero. Nevertheless, electrostatic coupling between the ions, the major source of interaction, is included as shown by the nonzero values for cross-coefficients DI2 and Dzl. Previous studies have shown that limiting expressions such as eq 4-8 provide a qualitative guide to the diffusion behavior of dilute ternary e l e c t r ~ l y t e s .Equation ~ ~ ~ ~ ~4,~ for ~ example, can be used to estimate changes in the diffusivity of the protein component as the solute composition changes from pure protein to pure salt. At the level of approximation used here, the selfdiffusion coefficients of the Na+, BSA", and C1- ions are, respectively, DNa,DBsA,and Dcl.
(28)Scatchard, G.;Coleman, J. S.; Shen, A. L. J . Am. Chem. Soc. 1957, 79, 12.
(29) Harned, H. S.;Hildreth, C. L. J. Am. Chem. SOC.1951, 73, 650. (30) Clarke, J. H. R.; Hills, G. J.; Oliver, C. J.; Vaughan, J. M.J. Chem. Phys. 1974, 61, 2810. (31)Czworniak, R.J.; Andersen, H. C.; Pecora, R. Chem. Phys. 1975,11,
.__.
A41
(32)Phillies, G.D.J. J . Chem. Phys. 1974, 60, 983 (33)Stephen, M. J. J. Chem. Phys. 1971, 55, 3878.
(34)Friedhof, L.; Berne, R. Biopolymers 1976, 15, 21. (35) Weissman, H. B.; Ware, B. R. J . Chem. Phys. 1978,68, 5069. (36)Tyrrell, H. J. V.; Harris, K. R. Dgfusion in Liquids; Butterworths: London, 1984;p 208. (37)Stokes, R.H. J . Am. Chem. SOC.1953, 75, 4563. (38)Leaist, D.G.Can. J . Chem. 1984, 62, 1692. (39)Steel, B. J.; Stokes, J. M.;Stokes, R. H. J . Phys. Chem. 1958, 62, 1514.
The Journal of Physical Chemistry, Vol. 93, No. 1 . 1989 477
Supporting Electrolytes in Protein Diffusion
1
0-
D22
1
2
3
7
loOD,,
00
I
,
,
05
c,/ (C,+C,)
,
-
,
10
+
against solute fraction of NaC1. The dashed curves give calculated ternary mutual diffusion coefficients for the aqueous Na,BSA (C,) + NaCl (C,)Components by assuming the 2C1- ions bound to each BSA7ion. In Figure 1 values of Dlk calculated from eq 4-8 for Na,BSA (C,) + NaCl (C2)and NalsBSA (C,) + NaCl (C2) mixtures are plotted against the solute fraction of NaCl. The plotted values of D l l and D12have been multiplied by 10 and 100, respectively, to help visualize the diffusion behavior. In the calculations it was assumed that the diffusion coefficient of the BSA" ions was identical with the limiting diffusivity of the uncharged BSA m o l e c ~ l e , 0.075 ~ ~ ' ~ X lov9m2 s-l at 25 OC. This approximation was justified on the grounds the protein does not undergo conformational changes in the pH range that was u s e d . ' ~ ~Also, .~ removal of a few H+ ions from the protein molecule is not expected to alter its diffusivity significantly. The limiting diffusivities of mz s-I, reand 2.03 X the Na+ and C1- ions, 1.33 X spectively, were evaluated from accurately knowing ionic conductances at 25 "C.14 As the solute fraction of NaCl increases from 0 to 1, the electric field accompanying the protein gradient produces a coupled flow of C1- ions instead of speeding up the BSA" ions. This causes the diffusivity of the protein component to drop from the binary Nernst value
for NaC1-free solutions to the self-diffusion coefficient of the BSA" ion (10)
for solutions containing a large excess of NaCl (tmA = 0). There is a corresponding increase in DZl,the cross-coefficient that measures the coupled flow of NaCl produced by the protein gradient
Dzl(Cz/C1 = 0 ) = 0
(11)
The diffusivity of the NaCl component, Dz2, is predicted to decrease from Da, 2.03 X mz s-', to the binary value 2 h a D C I / ( h a DCl),1.61 X lo4 m2 s-I, as the fraction of NaCl increases from 0 to 1. D12measures the flux of Na,BSA produced by the NaCl gradient. If the solution contains a large excess of NaCl over Na,BSA, the electric field along the NaCl gradient can produce only a negligible flow of BSA" ions, and hence D l z vanishes as
+
I
7
I
9
PH Figure 2. Calculated values of the ternary diffusion coefficient Dll for Na,BSA (pH >5.4) or BSA(Cl), (pH 5.4) counterions then substantially increases the mutual diffusivity of the protein. The values of the protein's mutual diffusion drop sharply as the ratio of salt to protein drops because the electric field produced by the protein gradient becomes effective in speeding up the protein ions. The behavior depicted in Figure 3 agrees qualitatively with the diffusion data for BSA obtained by light-scattering studies.'+ The increase in the apparent protein diffusivities at low ionic strength has been attributed to hydrodynamics4 or to protein-protein* interactions. The present work suggests it is due in large measure to strong electrostatic coupling between the protein ions and the counterions.' Mutual and Self-Diffusion of BSA. In many studies of protein diffusion the tacit assumption is made that the measured mutual diffusion coefficient of the protein component (e.g. Na,BSA) equals the diffusion coefficient of the protein ion (BSA"). This assumption is correct provided the solution under investigation
478 The Journal of Physical Chemistry, Vol. 93, No. 1, 1989
-1
-008Iw,
00
,
,
,
,
,
,
05
,
4 10
c,/ (C,+C,)
Figure 3. Calculated values of the relative error in the diffusivities of the Na7BSAor BSA(Cl)7components caused by neglecting the coupled flow of the BSA components driven by the NaCl gradient.
contains a large molar excess of supporting electrolyte. In this case the electrophoretic effect vanishes.14 In addition, the excess supporting electrolyte ensures that the activity coefficient of the diffusing electrolyte is effectively constant along the diffusion path, in which case the activity coefficient factor 1 C , d In y,,/dCI is unity. The mutual protein diffusivities D l l calculated from eq 4 and plotted in Figure 1 indicate that at least a 100-fold excess of NaCl (Le. 100 mol of NaCl per mole Na,BSA) is required to ensure that D,, is identical with DwA+ within a few percent. If less than owing this amount of NaCl is present, D, I will be larger than hAto electrostatic speedup of the protein ions by the Na+ counterions. The amount of supporting electrolyte required to ensure D l l N DmA increases with protein’s charge and concentration. Keller, Canales, and Yum’s diaphragm cell measurements7 indicate that the mutual and self-diffusion coefficients of BSA are identical within experimental error. The measurements were made at protein concentrations from 0.1 to 2.3 mM (5-150 g L-I) at pH 4.7 in an acetate buffer containing 100 mM sodium acetate 100 mM acetic acid. In this buffer the average protein charge is about 7 proton units. Since the protein was diffusing in solutions containing a 40- to 1000-fold molar excess of the acetate salt, the reported equivalence of the mutual and self-diffusion coefficient agrees with the theory developed in the present paper. Apparent BSA Diffusivities. In several previous studies,’-7 diffusion of BSA was treated as a binary process even though the measurements were made on multicomponent solutions that contained added salts or buffer electrolytes. Suppose that a gradient in the concentration of the BSA component, solute 1, is prepared in a solution with a uniform concentration of solute 2. Diffusion of the BSA component will then produce a coupled flow of solute 2. Eventually, the flow of solute 2 driven by the BSA gradient will be balanced by the flow of solute 2 back down its own gradient:@ J2 N 0 -D21VC1- D22VC2.The induced gradient in solute 2, V C , N -(D21/D22)VC1, will in turn drive a coupled flux of the BSA component given by -DI2VC2 N (D12D21/D22)VC1. This means that a portion of the total flow of the BSA component is in fact driven by a gradient in solute 2, even though there was no initial gradient in that solute. Accordingly, the apparent diffusivity of the BSA component will differ from its true value, D l l , by about -D,2D21/D22. The dimensionless ratio D12D21/D11D22 gives the relative error in the measured diffusivity of the BSA component that results from the neglect of multicomponent transport. For purposes of
+
+
(40) Noulty, R. A.; Leaist, D. G. J . Solution Chem. 1987, 16, 813.
Leaist illustration, predicted values of D12D21/D11D22 for solutions of Na7BSA + NaCl or BSA(Cl), + NaCl are plotted in Figure 3. For solutions containing a 10- to 20-fold excess of NaCl, the predicted magnitude of the error peaks at about 10%. This calculation suggests that it will be difficult to make accurate measurements of protein diffusion in multicomponent solutions unless allowance is made for coupled transport. (The analogous problem of measuring the diffusivity of ionic micelles in supporting NaCl solutions has been discu~sed.~) If diffusion is monitored by following changes in a nonspecific solution property such as the refractive index, the induced gradient in the supporting electrolyte will cause changes in the refractive index and thus make further contributions to the protein’s apparent diffusivity. Counterion Binding. Up to this point in the analysis, only electrostatic interactions between the diffusing Na,BSA and NaCl components have been considered. It is known, however, that C1and other anions bind to BSA.28 In the solutions that were used here, each protein ion may have bound up to one to two C1- ions.28 This would lead to further interactions between the diffusing components. Suppose that each protein ion binds m C1- ions, in which case the concentration of free C1- ions drops from C2 to C, - mC1. Diffusion can then be described in terms of the molar fluxes J,’ and J i of the incompletely dissociated Na,+,BSA(CI),( 1’) NaCl(2’) components as follows:
+
-Jl’
-52’
+ DIiVC2‘ = D21‘VCi + DZiVC2‘ = D11’VCI’
(13) (14)
where JI’ = J , , J2’ = J2 - mJI, CI’= C1and C{ = C2 - mC1. The D,; coefficients, by analogy with eq 4-7, are given by
Dll’ = DBSA(Cl)m + t$SA(Cl)m(DNa- DBSA(Cl)m)
(15)
O12’ = t’BSA(Cl)m(DNa- D C l ) / ( n + m )
(16)
- DBSA(Cl)m)/(n+ m )
(17)
021’ =
tb(DNa
O22’ = DCl + t’CI(DNa - DCI)
(18)
Comparison of eq 2-3 and 13-14 reveals that the ternary diffusion coefficients for the Na,BSA + NaCl components, with m C1- ions attached to each BSA ion, are given by41
D I , = Dll’ - m D l i 4 2 021
(19)
= 012’
(20)
+ D2,‘- mD22’ 0 2 2 = m o l ; + D2i
= mDl,’ - m2D12/
(21) (22)
The dashed CUNM in Figure 1 give values of the Dikcoefficients that were calculated from eq 15-22 for Na7BSA NaCl components by assuming two bound C1- ions per protein ion. The diffusivity of the BSA(Cl),’ ion was assumed to equal the limiting value of DESA, 0.075 X mz s-l. Study of Figure 1 shows that chloride binding is predicted to increase the mutual diffusivity of the Na7BSA component. The bound chloride ions increase the magnitude of the charge on the protein ion, and hence the flow of the protein ions couples more strongly to the flow of mobile Na’ counterions. On the other hand, chloride binding is predicted to reduce the cocurrent flow of C1co-ions driven by the protein gradient, as reflected by the drop in the values of D2,. This behavior can be understood by considering diffusion in response to a protein gradient prepared in an otherwise uniform solution of NaCl. In this case some C1- ions will diffuse up the protein gradient, countercurrent to the protein, to replace the free C1- ions consumed by binding to the protein. Although the present calculations are necessarily qualitative, they suggest that a few bound anions per protein ion can cause significant changes in the multicomponent diffusion properties of the mixture. The effects of anion binding will be most important at pH values below the isoelectric value where electrostatic forces
+
(41) Noulty, R. A,; Leaist, D. G. J. Phys. Chem. 1987, 91, 1655.
J . Phys. Chem. 1989, 93,479-484 tend to increase the binding strength.
Conclusions Electrostatics and ion binding cause strong interactions between diffusional flows of charged proteins and supporting electrolytes. Owing to the large difference in the mobilities of protein ions and typical counterions, a gradient in the concentration of a charged protein generates a relatively strong electric field to prevent charge separation. The diffusion-induced electric field along the protein gradient produces a coupled flow of the ions of the supporting electrolyte. Neglect of the simultaneous flow of the supporting
419
electrolyte can lead to a systematic error in the measured diffusivity of the protein. Multicomponent diffusion coefficients for the mixtures can be estimated as functions of the pH, ionic strength, and protein concentration provided the diffusivities of the species and the number of bound ions are known.
Acknowiedgment. The author thanks G. Samardzic for technical assistance. Acknowledgment is made to the Natural Sciences and Engineering Research Council for financial support of this research. Registry No. NaCl, 7647-14-5.
Salt-Induced Conformational Changes in DNA: Analysis Using the Polymer RISM Theory Fumio Hirata and Ronald M. Levy* Department of Chemistry, Rutgers University, New Brunswick, New Jersey 08903 (Received: May 9, 1988)
A RISM theory developed for solvated polymers is applied to polyelectrolyte solutions modeled as an infinite helical array of charged spheres immersed in a primitive-type electrolyte solution. The relation to earlier theories by Manning and Iwasa based on the same model is studied. As an application of the theory, salt effects on conformations of DNA in solution are examined over a wide range of added salt concentrations. It is found that the counterion distribution around the phosphates is not solely determined by the axial charge density of the polyion but depends on a more detailed characterization of the polyion charge distribution. A calculation is performed for the free energy dependence of B- and Z-DNA helical forms on salt concentration over a wide range of added salt concentration. A transition in the stable DNA form from B- to Z-DNA is predicted at -3.6 M added salt in qualitative accord with experiment.
Introduction The sustained intense interest in the physical chemistry of polyelectrolyte solutions is motivated in part by its importance in biological systems.’ A major conceptual breakthrough in our understanding of polyelectrolytes was made by Manning with his development of an ionic strength limiting law; the associated “counterion condensation” theory is based on a simple two-state thermodynamic While the theory has enjoyed a status similar to the Debye-Huckel theory for simple electrolyte solutions, it has left many questions unanswered, such as the detailed structure of counter- and co-ion atmospheres around polyions. Attempts to answer these questions and to place the theory upon a more solid statistical mechanics foundation have been made during the past decade. These include studies using the mean spherical model (MSM),S hypernetted chain (HNC),6 PoissonBoltzmann (PB),’ and Monte Carlo (MC)*-’O methods. With few exceptions, most theoretical studies have been carried out for a simplified model of a polyion for which charges are distributed ‘*12 While the cylindrical uniformly along the cylindrical axis.zS96*8,1 rod description with a continuous charge distribution has been successfully used to describe many polyion proper tie^,'^ models (1) Anderson, C. F.; Record, M. T., Jr., Annu. Rev. Phys. Chem. 1982, 33, 191. (2) Manning, G. S.J . Chem. Phys. 1969, 51, 924. Ibid. 1969, 51, 3249. (3) Manning, G. S. Biophys. Chem. 1977, 7, 95. (4) Manning, G. S.Q. Reu. Biophys. 1978, ZZ, 179. (5) Fixman, M. J . Chem. Phys. 1979, 70,4995. (6) Baquet, R.; Rossky, P. J. J . Phys. Chem. 1984, 88, 2660. (7) Le Bret, M.; Zimrn, B. H. Biopolymers 1984, 23, 287. (8) Le Bret, M.; Zimm, B. H. Biopolymers 1984, 23, 271. (9) Murthy, C. S.; Baquet, R.; Rossky, P. J. J. Phys. Chem. 1985,89,701. (10) Vlachy, V.; Hayrnet, D. J. J . Chem. Phys. 1986, 84, 5874. (1 1) Fuoss, R. M.; Katchalsky, A.; Lifson, S. Proc. Nutl. Acad. Scz’. U.S.A. 1951, 37, 579. (12) Kotin, L.; Nagasawa, M. J . Chem. Phys. 1962, 36, 873.
0022-365418912093-0479$01SO10
that can incorporate more realistic features of the polyion structure are of great interest. One such model, which may be described as “beads on a string”, was first investigated by Rice using a Poisson-Boltzmann equation,14 later by Manning and Zimm15 based on the Mayer cluster expansion,16 and by Iwasa” by a mode expansion.’* The latter two methods lead to the same result for the excess chemical potential of a polyion due to interaction with small ions as a leading term of the expansions. It is of interest to develop an integral equation method for this model, because the evaluation of higher order terms after the first few terms in the expansions is generally very difficult. A primary goal of a statistical mechanics polyelectrolyte theory is to establish the dependence of interionic distribution functions on the polyion structure. This can be achieved within the framework of the so-called RISM or sitesite Ornstein-Zernike (SSOZ) e q ~ a t i o n , which ~ ~ a is one of the most successful statistical mechanics approaches for the study of molecular liquids in which molecules are modeled as collections of several interaction (1 3) Manning, G. S. Annu. Reu. Phys. Chem. 1972, 23, 1 17. (14) Lapanje, S.;Haebig, J.; Davis, T.; Rice, S. A. J . Am. Chem. SOC. 1961, 83, 1590. (15) Manning, G. S.; Zimm, B. H. J . Chem. Phys. 1965, 43, 4250. (16) Mayer, J. J . Chem. Phys. 1950, 18, 1426. (17) Iwasa, K. J . Chem. Phys. 1975,62, 2967. (18) Andersen, H. C.; Chandler, D. J . Chem. Phys. 1970, 53, 547. Chandler, D.; Andersen, H. C. Ibid. 1970,5426. Andersen, H. C.; Chandler, D. Ibid. 1971, 55, 1497. (19) Chandler, D.; Andersen, H. C. J . Chem. Phys. 1972, 57, 1930. Chandler, D. Mol. Phys. 1976, 31, 1213; J . Chem. Phys. 1977, 67, 1113. (20) Cummings, P. T.; Stell, G. Mol. Phys. 1982, 46, 383. (21) Lowden, L. J.; Chandler, D. J . Chem. Phys. 1973, 59, 6587; 1974, 61, 5228. (22) Hirata, F.; Rossky, P. J. Chern. Phys. Lett. 1981, 84, 329. Hirata, F.; Pettitt, B. M.; Rossky, P. J. J . Chem. Phys. 1982, 77, 509. Hirata, F.; Rossky, P. J.; Pettitt, B. M. J . Chem. Phys. 1983, 78, 4133. (23) Pettitt, B. M.; Rossky, P. J. J . Chem. Phys. 1982, 77, 1451; J . Chem. Phys. 1983, 78, 7296.
0 1989 American Chemical Society
