Interactions of polyelectrolytes with simple electrolytes. II. Donnan

U. P. STRAUSS,C. HELFGOTT, AND H. PINK

2550

Interactions of Polyelectrolytes with Simple Electrolytes. 11. Donnan Equilibria Obtained with DNA in Solutions of 1-1 Electrolytes1

by U. P. Strauss, C. Helfgott,2and H. Pink3

Downloaded by UNIV OF NEBRASKA-LINCOLN on September 7, 2015 | http://pubs.acs.org Publication Date: July 1, 1967 | doi: 10.1021/j100867a024

School of Chemistry, Rutgers, The State University of NEWJersey, New Brunswick, NEWJersey (Received December 89,1966)

Donnan equilibrium studies have been carried out with DNA in aqueous solutions of lithium, sodium, potassium, and tetramethylammonium bromide. For each of the four simple electrolytes, the dependence of the membrane equilibrium parameter, I?, on the salt concentration follows the general trend predicted by theoretical calculations based on the Poisson-Boltzmann equation for a uniformly charged, completely dissociated stiff rod model of the macroion; however, the experimental values are significantly smaller than the theoretical ones. Satisfactory agreement between experiment and theory is obtained if one allows for specific counterion binding governed by the appropriate form of the mass action law. The binding constants are of the order of magnitude expected on the basis of results obtained with other phosphates. The theoretical and experimental results for I’ examined in the body of this paper are compared critically with predictions of a recent alternate theoretical treatment in the Appendix.

Introduction Most solution properties of polyelectrolytes are critically affected by the distribution of small ions in the neighborhood of the macroion. While the distribution cannot be observed directly, it is predictable, in principle, by statistical mechanics. In practice, the theoretical approach meets with formidable obstacles and only more or less approximate treatments have been developed so far. Of these, those involving the solution of the intact Poisson-Boltzmann (PB) equation for a uniformly charged cylindrical rod model of the macroion, either by computer or by analytical approximation methods, have shown considerable promise.”’ However, in view of both the oversimplification of the physical model employed, e.g., the neglect of solvation effects, and the somewhat uncertain statistical mechanical foundation of the PB equation,&’O it is difficult t o appraise the reliability of the theoretical results within the scope of the theory alone. At present the most promising means of assessing the adequacy of the PB method is to test its power t o predict appropriate experimental results.” Recent investigations with flexible anionic polyelectrolytes have shown that ion distribution data obtained from The Journal of Physical Chemistry

Donnan membrane equilibrium experiments are particularly suitable for this purpose.6J2 However, with (1) Presented before the Division of Colloid and Surface Chemistry a t the 151st National Meeting of the American Chemical Society, Pittsburgh, Pa., March 1966. The research was supported by a grant from the National Institutes of Health of the U. S. Public Health Service. (2) The results presented here will be contained in a thesis presented by C. Helfgott to Rutgers, The State University of New Jersey, in partial fulfillment of the requirements for the Ph.D. degree. (3) National Science Foundation Undergraduate Research Participant, (4) L. Kotin and M. Nagasawa, J . Chem. Phys., 36, 873 (1962). (5) 2. Alexandrowicz and A. Katchalsky, J . Polymer Sei., A l , 3231 (1963). (6) L. M. Gross and U. P. Strauss in “Chemical Physics of Ionic Solutions,” B. E. Conway and R. G. Barradas, Ed., John Wiley and Sons, Inc., New York, N. Y., 1966,p 361. (7) For earlier treatments, see, for example, S. A. Rice and M. Nagasawa, “Polyelectrolyte Solutions,” Academic Press Inc., New York, N. Y., 1961. (8) J. G.Kirkwood, J . Chem. Phys., 2 , 767 (1934). (9) J. G.Kirkwood and J. C. Pokier, J. Phys. Chem., 58, 591 (1954). (IO) R. H. Fowler and E. A. Guggenheim, “Statistical Thermodynamics,” Cambridge University Press, London, 1960, Chapter 9. (11) An alternate approach based on a comparison with a recent theory having more rigorous statistical mechanical foundations is explored in the Appendix. (12) U. P. Strauss and P. Ander, J . A m . C h m . SOC.,80, 6494 (1958).


IXTERACTIONS OF POLYELECTROLYTES WITH SIMPLE ELECTROLYTES

the stretched chain serving as the rod model and with all the neutralized acid groups assumed to be completely dissociated, the measured differences in the co-ion concentrations on the two sides of the membrane were generally smaller than those predicted by the theory. Agreement between theory and experiment could be reached by two alternate adjustments of the theoretical model. One, used by Alexandrowicz and Katchalsky, was to increase the charge density of the rod modeL5 This procedure was justified on the basis that the real macroion is coiled. The other, used by Gross and Strauss, was to assume incomplete dissociation (or “site-binding”).6 A small ion was defined as bound whenever it was so close to the macroion that desolvation took place. Direct experimental evidence for such desolvation had been obtained independently from dilatometry experiment^.'^ The choice between these assumptions may be narrowed down, or hopefully eliminated, if a rigid polyelectrolyte is used whose structure is similar to that of the model. Deoxyribonucleic acid (DNA) comes close to this specification. It is well characterized and both its charge density and its cylinder radius are k n ~ w n , ’ ~thus , ‘ ~ precluding most of the arbitrary adjustment of molecular parameters which is possible with flexible polyelectrolytes. For this reason we have investigated membrane equilibria of the lithium, sodium, potassium, and tetramethylammonium salts of DNA in solutions of simple electrolytes having the cation in common. The results are presented in this paper and compared with the predictions of the theoretical treatments.

Experimental Section Materials. The DNA used for most of the work was a salmon sperm DXA (SDNA) sample obtained as the sodium salt from the Worthington Biochemical Gorp. (Lot S o . 6118). According to the manufacturer the N/P weight ratio was 1.42 (theoretical 1.6716). We determined the absorptivity per g-atom of phosphorus per liter, to be 6500. The protein content was determined by the method of Lowry, et al.,17 to be 0.8yG,.No denaturation could be detected by the method of Hotchliiss.‘* The molecular weight was reduced by sonification a t 20,000 cps carried out in dilute ice-cold 0.1 N alkali bromide solution under nitrogen. As has been previously observed by others,lg the sonification did not affect the extent of denaturation. For most of the membrane equilibrium runs involving sodium ion, a DKA sample isolated in this laboratory from calf thymus (CD?JA) was used. A conventional isolation procedure was employed involving the

2551

sodium salt of ethylenediamine tetraacetic acid (EDTA) to deactivate undesirable enzyrnes,*O sodium dodecyl sulfateJ21and chloroform-amyl alcohol for deproteinationZ2and sonification to lower the molecular weight.lg The protein content of the final product was less than l%17 and no denaturation could be measured by Hotchkiss’ method.’s All the inorganic chemicals used were reagent grade obtained from J. T. Baker or RiIatheson Coleman and Bell. The phenol reagent used for the protein analysis was from A. H . Thomas Go., the lysozyme used as a protein standard was a crystallized product from Armour, and the tetramethylammonium bromide (TMABr) was from Eastman Organics. The two DNA preparations gave identical membrane equilibrium results. Membrane Equilibrium. A previously described method12 was modified as follows. Instead of equilibrating each polyelectrolyte solution against an external salt solution in a separate container, a number of solutions of different DNA concentrations were equilibrated against the desired external solution in one container. In each run two dialysis bags containing no DNA were included t o serve as blanks. The simple electrolytes were LiBr, NaBr, KBr, and TRIABr. Their concentrations ranged generally from 0.002 t o 0.2 N , in one case to 1 N . The DYA concentrations were chosen so that the normality ratios of DNA to simple electrolyte were approximately 0.4, 0.6, and 0.8, except for the 0.002 N salt solutions where normality ratios up t o 2 were necessary in order to determine the differences between the external and internal bromide concentrations with the desired precision. In each case the DNA was converted to the desired cation form by exhaustive dialysis at the highest concentra(13) U. P. Strauss and Y. P. Leung, J . Am. Chem. Soc., 87, 1476 (1965). (14) F. H. Crick and J. D. Watson, Proc. R o y . SOC.(London), A223, 80 (1954). (15) R. Langridge, D. A. Marvin, W. E. Seeds, H. R. Wilson, C. W. Hooper, 14. M. F. Wilkins, and L. D. Hamilton, J . Mol. Biol., 2, 38 (1960). (16) J. E. Hearst, Biopolymers, 3, 57 (1965). (17) 0. H. Lowry, N . J. Rosebrough, L. A . Farr, and R. J. Randall, J . B i d . Chem., 193, 265 (1951). (18) R. D. Hotchkiss in “Methods of Enzymology,” Vol. 111, S. P. Colowick and N. 0. Kaplan, Ed., Academic Press Inc., New York, N. Y., 1957, p 710. (19) P. Doty, B. D. hicGill, and S. A. Rice, Proc. Natl. Acad. Sci. U . S.,44, 432 (1958). (20) S. Zamenhof, G. Griboff, and N. Narullo, B w c h i m . B w p h y s . Acta, 13, 459 (1954). Kay, ‘I. N. S . Simmons, and A. L. Dounce, J . Am. Chem. (21) E. R. & SOC.,74, 1724 (1952). (22) E. Chargaff in “The Nucleic Acids,” Vol. 1, E. Chargaff and J. N. Davidson, Ed., Academic Press, Inc., New York, N. Y., 1955, 324.

Volume 7 1 , Xumber 8

July 1967

U. P. STRAUSS, C. HELFGOTT, AND H. PINK

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tion employed for the desired electrolyte and the completeness of the counterion exchange was confirmed with a Perkin-Elmer Model 146 flame photometer. In all runs the external solution was changed at least twice before final equilibration was attained 0.1". To eliminate by tumbling for 72 hr a t 25.0 impurities and biological growth, the cellulose-casing dialysis bags were boiled in three successive portions of distilled water and all solutions were passed through a 0.22-p pore size Millipore filter prior to equilibration. Immediately after final equilibration the interior solutions were removed from the bags and bromide, DNA, and density determinations were performed on internal and external solutions. Bromide ion was determined by potentiometric titration with silver nitrate using silver-silver bromide and glass electrodes. By adding about 85% of the titrant by weight and delivering the remainder with a microburet, a precision of better than two parts per thousand was usually obtained. Under the conditions employed, DKA did not interfere with the analysis. DNA was determined from the optical density at 259 mp of solutions prepared by diluting the sample solutions with a known excess of a 4% NaCl solution. The extinction coefficient, ~ ( P ) 2 5 9 , needed for this purpose was obtained from orthophosphate analysis on DNA degraded with alkaline p e r s ~ l f a t e . ~Densi~ ties of the sample solutions accurate to better than 0.02% were obtained at 24.78 f 0.01" with precision pycnometers.

I

I

I

'1


*

Results The quantity desired is the membrane equilibrium parameter, r, defined by the relation

r

=

lim(n,' - n,)/n,

nD=O

(1)

where np and n, are the concentrations, expressed in equivalents per liter, of polyelectrolyte and simple electrolyte, respectively. Primed quantities refer to external solutions, unprimed to internal solutions. For convenience, we prefer to use the equivalent definition

Figure 1. Graphical determination of r, for KDNA in 0.095 N KBr from plot of w sus. nP: 0, internal solutions; 8, external solution.

A feature apparent in this figure and common to all such runs was that the bromide concentration was slightly lower in the external solution than in the blanks. This happened regardless of from which side equilibrium was approached. It was not caused by the DXA, as ascertained in trial runs where only simple electrolyte was present in external and internal solutions. The phenomenon thus appeared to be caused by the membrane and since the same casing stock was used in all experiments we did not include the external bromide concentrations in our least-squares calculations. With this procedure no curvature could be discerned in any of the plots. Since all our runs were carried out at constant chemical potential of simple electrolyte, it was convenient to define a parameter rWas the negative slope of the straight lines, i.e.

(3) The desired quantity, relation

where p, is the chemical potential of the simple electrolyte. Experimentally we determined ws,the bromide concentrations expressed in equivalents per kilogram of solution. For a given run, the values of w swere then plotted against n, and the slope was calculated by the method of least squares. A typical plot representing KDNA in 0.095 N KBr is shown in Figure 1. The Journal of Physical Chemistry

r,

was then obtained by the

where the density l and its derivative refer to the simple electrolyte solution in the limit of infinite dilution of polyelectrolyte.24 (23) J. Kolmerten and J. Epstein, Anal. Chem., 30, 1536 (1958).

ISTERACTIONS OF POLYELECTROLYTES WITH SIMPLE ELECTROLYTES

2553

The results are given in Table I. The values of ( d ~ / h , ) , , in the fourth column were obtained from the linear plots of the densities of the internal solutions against the D S A concentration. In a few instances where it was obvious that the second term on the righthand side in eq 4 would be negligible, no density determinations were performed.


Table I : Membrane Equilibrium Parameters for D N A in Electrolyte Solutions

Salt

LiBr

NaBr

KBr

TMABr

n8’ X 102

rw

0.199 0.212 1.672 1,840 8.91 22.14

0.064 0.099 0.127 0.128 0.185 0.266

0.12 0.12 0.12 0.12

0.064f0.011 0.099 f 0.012 0.125f0.008 0.126f0.028 0.172 f 0.020 0.242 f 0.027

0.95 8.89 23.16 98.2

0.124 0.216 0.302 0,620

0.17 0.15 0.15 0.14

0.122 f 0.004 0.204 f 0.006 0.272 f 0.008 0.538 f 0.037

0.208 1.826 9.50 23.40

0.094

...

0.119 0.209 0.292

0.20 0.16 0.15

0.094 f 0.004 0 . 1 1 5 f 0.012 0.194 f 0.014 0.262 f 0.012

0.221 1.830 9.07 22.10

0.111 0.142 0.229 0.327

0.12 0.12 0.11

... ...

...

0 . 1 1 1 f 0.007 0 . 1 4 0 f 0.005 0.218f0.007 0.304 & 0.018

The precision limits of r which are included in the last column of Table I were obtained by a standard methodz5 from the deviations of the experimental points from t)he least-squares lines of w B against np used to determine rU. It is seen in Table I that the first two values of r which were obtained at essentially the same LiBr concentration (0.002 N ) differ from each other somewhat more than might be anticipated from the precision limits. However, this discrepancy should not be considered to be representative of the size of error expected in general. Much better reproducibility was observed at the higher salt concentrations where the demands on the analytical method were less exacting.

Discussion The experimental values of 2r are compared with the theoretical curves of Gross and Strauss in Figure 2. The theoretical curves are essentially the ones given in Figure 10 of ref 6. However, the abscissas are given here as logarithmic scales of the salt concentrations

:n

( LI+, NO+, TMA+)

Figure 2. Comparison of experimental values of 2 r with theoretical curves of Gross and Strauss6corresponding t o Q = 8.5 and t o the constant values of CY shown. The experimental points for LiDNA (o),NaDNA (0),and TMADNA (A) correspond t o t h e bottom scale which is based on a = 13.2 A and the points for KDNA (0)t o the t o p scale which is based on a = 11.6 A.

rather than in terms of xu, where x is the DebyeHuckel parameter of the external salt solution and u is the sum of the radii of the cylinder representing the macroion and of the hydrated counterion. The radius of DNA has been established as 9.7 A.14r16 Because of the essentially qualitative nature of the concept of a hydrated ion, some arbitrariness is unavoidable in the choice of the values for the radii of the alkali metal ions. Depending on the basis used for these estimates, a range of possible values is obtained for each ion.26-2s For the sake of convenience in the graphical presentation, we chose a common value of 3.5 A for the radii of Li+, Na+, and Th4A+.z9 The corresponding scale for these cations is given on the bottom of the graph, while the scale for K+, corresponding to an ion radius (24) Equation 4 follows directly from the definitions and the relation n. = wB{, (25) A. G. Worthing and J. Geffner, “Treatment of Experimental Data,” John Wiley and Sons, Inc., New York, N. Y . , 1946, p 250. (26) J. Kielland, J . Am. Chem. Soc., 5 9 , 1675 (1937). (27) H. 9. Harned and B. B. Owen. “The Phvsical Chemistrv of Electrolytic Solutions,” 2nd ed, Reinhold Pubiishing Corp., New York, N. Y., 1950, p 381. (28) R. A. Robinson and R. H. Stokes, “Electrolyte Solutions,” 2nd ed, Academic Press Inc., New York, N. Y., 1959, Chapter 6.

Volume 71 Number 8 ~

J u l y 1967

U. P. STRAUSS,C. HELFGOTT, ASD H. PINK

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of 1.9 A, is given on the top. The axial projection of the average distance between phosphate groups, b,30 has been reported to be 1.7 A for DNA, which leads to a value of 8.5 for the charge parameter Q. Q has been defined by the expression6


Q

=

2e2/&Tb

(5)

where e is the electronic charge, E the dielectric constant of the medium, and k and T the Boltzmann constant and the absolute temperature, respectively. The value of Q was used to calculate the theoretical curves, each of which represents a different degree of dissociation, cy, and is accordingly labeled. As in the case of flexible polyelectrolytes, the great majority of the points fall considerably below the cy = 1 curve.31 Since with DNA a linear charge density higher than the one chosen-ie., b < 1.7 A-cannot be justified on the basis of any realistic model, it appears that another explanation is needed. The most simple such explanation is the assumption of partial dissociation. Such a procedure must be viewed with some caution, as it may involve the lumping into a binding parameter, 1 - cy, of short-range forces not explicitly included in the theory along with compensations for any basic inadequacies of the PB method. Yet the following considerations lend support to an interpretation in terms of counterion site binding. First, the values of cy are seen to be close to values of the degree of dissociation obtained from electrical transport measurements by means of quite different assumption^.^^ Second, the points for lithium are seen to be on the bottom, those for TAIA on the top in Figure 2 , consistent with the binding order Lif > Na+ > K + > Ti\IA+ observed with many other simple and complex phosphate compound^^^-^^ and also expected on theoretical Third, the observation that cy decreases with increasing counterion concentration is qualitatively in accord with the mass action law.as These new findings encourage us to carry the interpretation of the results in terms of counterion binding one step further by testing for a quantitative fit with the law of mass action. For polyelectrolytes this law takes the form

K =

1--a! m,’ exp4E

where K is the intrinsic association constant of a counterion with a phosphate binding site, nI8 exp& is the effective counterion concentration, and 4~ the absolute value of e+/kT at the macroion surface, J. being the potential. Following a method described previo u ~ l y we , ~ construct a series of theoretical curves of The Journal of Physical Chemistry

2I’ against log xu, each corresponding to a different value of K‘, defined by the relation

(7) Such a series of theoretical curves is given in Figure 3 in which, however, the abscissas have been converted to logarithmic scales of the salt concentration, using for the parameter a the same values as were used in Figure 2 . We have also included in Figure 3 the same data points as in Figure 2, so that the two figures differ only in that the theoretical curves in Figure 2 correspond to constant values of cy while those in Figure 3 correspond to constant values of K’. It is seen that the data points for each cation appear to fit a curve of constant K’ quite closely. The values of K’ so obtained are given in Table I1 together with the corresponding values of K calculated by eq 7. In view of the difficulty of locating exact values of the radii of the hydrated alkali metal ions, I’h, we have studied the effects of varying ?“h and hence a on our results. We have found that for all reasonable values of a, good fits with curves of constant K’ could be obtained. In general, K‘ decreased with decreasing I’h. Since the original values of ?“h chosen by us corresponded closely to the high ends of the generally accepted ranges, we have given in Table I1 values corresponding to the low ends of these ranges also (in parentheses). The values of K in Table I1 may thus be viewed as upper and lower bounds. For the alkali metal ions, these values (29) Had we chosen any other reasonable value for a , no substantial difference would be observable because of the small scale of the graph. The effects of variations in a will be given in Table I1 and treated in the subsequent discussion. (30) This quantity is denoted by 1 in ref 6. (31) For comparison with previous data for Na+. see J. Shack, R. J. Jenkins, and J. hl. Thompsett, J . Biol. Chem., 198, 85 (1952). (32) P. D. Ross and R. L. Scruggs, Biopolymers, 2, 233 (1964). (33) R. M. Smith and R. A. Alberty, J . Phys. Chem., 60, 180 (1956). (34) U. P. Strauss and P. D. Ross, J . Am. Chem. Soc., 81, 5295, 5299 (1959). (35) P. H. Teunissen and H. G. Bungenberg de Jong, Kdloid Beih., 48, 33 (1938). (36) E. L. King, J . Chem. Educ., 30, 71 (1953). (37) G. Eisenman, “Symposium on Membrane Transport and Metabolism,” Academic Press Inc., New York, N. Y., 1961, p 163. (38) Reports by several authors that a increases with electrolyte concentration (see, for example, D. 0. Jordan, “The Chemistry of Nucleic Acids,” Butterworth Inc., Washington, D. C., 1960, p 220) are based on incorrect interpretation of experimental data. For example, the effect of DNA on activity coefficients was neglected in calculating a from membrane potentials. (In our case the same incorrect procedure would lead to a = 2I’ and hence t o a value of a which would also increase with ionic strength.) Other results cited by Jordan, obtained from electrophoresis data, are based on values of molecular dimensions now known to be incorrect (P. D. Ross, Biopolymers, 2, 9 (1964)). A more recent interpretation of electrophoresis measurements leads t o values of a in substantial agreement with ours.32

IKTERACTIONS OF POLYELECTROLYTES WITH SIMPLEELECTROLYTES

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Table 11: Mass Action Law Parameters K, Cation

Li + Na+

K+


TMA +

lo 0'

'

& .' I;:

'

n;

" ".05 " " .I

I

(ti+, NO+, TMA+)

' ' '.5' ' ' 1I '

i'

Figure 3. Comparison of experimental values of 2 r with theoretical curves of Gross and Strausse corresponding t o Q = 8.5 and to the constant values of K' shown. Experimental points and scales are the same as in Figure 2.

are seen to be of the magnitude expected on the basis of results obtained with other phosphate compounds. 3 3 , 8 4 The value of K for TAIA+, on the other hand, is considerably larger than the value obtained with inorganic polyphosphates. 34 This may indicate some interaction between the T N A + and the hydrophobic portion of the DNA molecule.

Appendix Comparison with Manning-Zimm Theory. An alternate approach to the theoretical problem discussed in this paper has recently been formulated by Manning and Zimm.3g,40 Since the cluster-integral method of JIayer which they used has a more rigorous statistical mechanical foundation than the PB method, a comparison of the results of the two methods might serve as a theoretical test of the latter. Unfortunately, because different approximations are made in the two treatments, such a comparison is valid over a limited range of experimental conditions only. In both theories the model is a cylindrical rod. However, the Manning-Zimm theory treats the ionic groups of the macroion as discrete whereas the PB method treats the charge as uniformly smeared out over the surface of the rod. JIoreover, the short-range repul-

(moles/ 1.) -1

Th

a

K'

(2.5) 3.5

(12.2) 13.2

(0.13)

(2.2)

(11.9) 13.2

(0.048)

3.5 (1.5) 1.9

(11.2) 11.6

(0.056)

3.5

13.2

(2.1) 4.1

0.21

0.09

(0.7) 1.6

(0.77)

0.07

1.0

0.04

0.76

sions between the macroion and the small ions which are included in the PB treatment in terms of the distance of closest approach, a , are neglected in the Manning-Zimm treatment. These differences lead to different dependences of r on the ionic strength. In the PB method the Debye-Huckel parameter, x , appears solely in the dimensionless quantity xa. An unlimited rise of with increasing concentration of simple electrolyte is predicted as shown in the theoretical curves of Figure 2. The Manning-Zimm treatment leads to the closed-form expression41

2r

=

a[i -

1)]

(8)

Here, too, F rises with increasing H, but it approaches an upper limit and the ionic strength dependence is controlled by the distance between nearest fixed charges, b/a. The physical significance of this is that with increasing ionic strength the fixed charges are increasingly shielded from one another so that the "polyelectrolyte effect" diminishes and r approaches its "ideal" value of 1/2. In the limit of zero ionic strength these differences disappear and a comparison between the two theories becomes possible. At low linear charge densities (a& 5 2), both theories agree and predict that r is given by the expression 2r

=

-

$)

5

2

In aqueous solution at 25", the condition a&

(9)

5 2

(39) G. S. Maiming and B. H. Zimm, J . Chem. Phys., 43, 4250 (1965). (40) G. S. Manning, ibid., 43, 4260 (1965). (41) The expression of Manning has been modified to include the degree of dissociation, a. When a = 1, eq 8 reduces t o eq 4.10 of ref 40. The procedure for including a in theoretical expressions for r has been given in ref 6.

Volume 7 1 , Xumber 8 J u l y 1967

C . HEITNER-WIRGUIS ASD ROSELCOHEN


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means that the average distance between nearest charges is greater than 7 A. It might appear that the agreement between the theories covers the range where 4

Interactions of polyelectrolytes with simple electrolytes. II. Donnan

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