3623
J . Phys. Chem. 1993,97, 3623-3630
The Titration Curve of Weak Polyacids Charles P . Woodbury, Jr. Department of Medicinal Chemistry and Pharmacognosy (MfC 781), University of Illinois at Chicago, Chicago, Illinois 60680 Received: October 22. 1992
We present a simplified model of the titration curve of weak polyacids, based on a combination of limiting law expressions for the extent of counterion binding and a nearest-neighbor model for interactions among titrated sites on the polyion. We derive expressions for the intrinsic acid dissociation constant and the nearest-neighbor anticooperativity that can be related to experimental data in the middle of the titration curve, a region readily accessible by experiment, so avoiding extrapolation of the titration curve into regions where data are hard to collect or where errors are magnified. The shape and salt dependence of the titration curve are quantitatively predicted very simply in our model. In analyzingdata from the literature for different weak polyacids (hyaluronic acid, poly(acry1ic acid), carboxymethylcellulose) at a variety of ionic strengths, we obtain satisfactory agreement of our model with experiment. Extensions of the model to the more general case of a polyion binding a ligand of arbitrary size and charge are also discussed.
Introduction An important problem in polyelectrolytetheory is the prediction of the titration curve of weak polyacids, in particular predicting the salt dependence of pK,,,, the apparent acid dissociation constant of the polyacid, as a function of the degree of titration and the concentration of salt in the solution. A related problem is that of obtaining the intrinsic acid dissociation constant, pK0, from experimental data. It is well-known that the apparent acidity of a weak polyacid decreases with progressive ionization of the polymer.Il2 In the absence of any conformational change in the polymer, a plot of pK,,, versus a,the degree of ionization of the polyacid, typically shows a monotonic increase in pK,,, with a. Furthermore, the slope and curvature in such plots depend on the concentration of added salt, with higher salt concentrations suppressing the rise in pK,,,. It is common practice to obtain pK0 by backextrapolation of the titration curve to a = 0, but because of anticooperativity and salt dependence, the extrapolated values for pKo may be in doubt. A reliable theoretical guide or method would be useful here. There are at present two major analytic theories for the prediction of titration curves for weak polyacids: (1) Manning's counterion condensation (CC) m0de1~9~ and its later developmentss-IOand (2) models using the Poisson-Boltzmann (PB) equationt1-I3(see Anderson and Recordt3for a recent review and comparison of these models). Both have their advantages and their disadvantages. In the CC model, 'condensation" of counterions refers to a certain type of binding of counterions by a polyion of suitably high charge density. Counterion condensation occurs if the charge density parameter 4 exceeds unity (for monovalent counterions); 4 is given by
4 = e2/(4reoekT6)
(1)
using SI units and standard symbols for physical constants. The parameter b is the (average) axial separation between structural charges on the polyion. In the original CC m ~ d e l ,for ~ . ~4 above unity a fraction 1 of a counterion was hypothesized to be bound (condensed) per structural charge on the polyion. From this hypothesis various limiting-law results were deduced for the thermodynamics of the solution. Subsequently, Manning developed a second treatment?,' based on a two-phase model of the 0022-3654193f 2091-3623SO4.00/0
polyelectrolyte solution, that extended the basic CC model results to solutions containing moderate salt concentrations. The theory has subsequently been improved and elaborated in various ways. In particular, Manning and co-workers9~t0have combined the CC model with the McGhee-von Hippelt4 model for ligand binding on linear polymers to treat binding of large, multivalent ligands. They have also discussed how the release of bound counterions will decline smoothly throughout a titration as a result of charge neutralization on the polyion. This causes a continuous reduction in the binding affinity of the ligand, which can be thought of as a sort of polyelectrolyte-induced anticooperativity that is distinct from the nearest-neighbor anticoop erativity used in the McGhee-von Hippel model. A drawback to the CC model for weak polyacid titrations is that at acertaincriticaldegreeofionization,wherethenet polyacid charge density Enetpasses unity, the model produces a discontinuity in the titration curve of about 0.24.4 pK units. Furthermore, the model tends to overstate the salt dependence of pKapp. According to the CC model, at u = 1 a plot of pK,,, - pK0 versus log C3 should be linear with a slope of -4 for 4 < 1 and -1 for 4 > 1 (C3 is the salt concentration). Experiments2JS-'*are consistent with the predictions of linearity and negative slope but show magnitudes of the slope that are appreciably smaller than those predicted by the CC model. The PB equation approach requires the solution of a nonlinear second-order differential equation for the electrostatic potential around the polyion. It is common to assume that the polyion solution can be modeled using a cylindrical cell of radius R with the polyion at its center (the PB cell model). The polyion is taken to be a rod that is infinitely long, with radius (I and a uniform charge density. Solution of the PB equation for rodlike polyions must be done by machine, since (with added salt present) there is no exact analytic expression for the potential. The potential V(a) at the polyion's surface is directly proportional to the shift in p~,,,.I,2,tt.l2 ApK,,,
PK, - PK,
4.434eV(~)/kT
(2)
using standard symbols for physical constants. The potential V(a) must be re-evaluated continuously over the course of the titration as the charge density on the polyion changes. The main difficulties with the PB approach to polyacid titration modeling are that it requires knowledge of the polyion's radius a, it is moderately demaadingin termsofcomputation (i.e.,solving 0 1993 American Chemical Society
Woodbury
3624 The Journal of Physical Chemistry, Vol. 97,No. 14, 1993 the PB equation numerically), and predicted curves typically do not follow the experimental data at low extent of titration. The PB model does, however, produce smooth continuous titration curves that follow experiment at moderate to high extent of titration, for a wide range of salt concentrations and polyion charge densities. Record and co-w0rktrs~~-22 have shown that under limitinglaw conditions the PB cell model yields useful analytic expressions for thermodynamic quantities that are very similar to thosederived from the CC model. The derivation of these expressions from the PB cell model, however, does not require the assumption of counterion condensation. An important quantity here is the Donnan salt exclusion parameter,23J4 which is connected to the thermodynamic degree of dissociation of counterions from the polyion. Under limiting-law conditions, there are simple expressions in the PB cell model for these parameters in terms of the charge density parameter [. These formulas agree with those from an earlier treatment25*26 that combined ionic screening and counterion condensation concepts to treat the total extent of counterion binding and release by the polyion in terms of a parameter For our purposes, the result is that a fraction J, of a counterion may be thought of as thermodynamically bound for each structural charge of the polyion. The parameter J, is given by one of two formulas, depending on the polyion's value of (:
+.
instead of simple concentrations; it also takes into account very specifically the release or uptake of counterions by the polyion. KT varies only with pressure and temperature, not with the concentration of salt; on the other hand, in polyelectrolyte systems KobS typically varies strongly with the salt concentration. K o b is what is usually reported in studies of drug-DNA or protein-DNA binding; when binding isotherms for such systems are analyzed by using e.g., the McGhee-von Hippel modelI4 and Scatchardtype plots, Kobs is equated to the binding constant K in the McGhee-von Hippel model, and KobSis obtained by extrapolation of the binding curve to zero binding density. The origin of the salt dependence of Kobs in terms of both the CC model and the PB cell model has been reviewed several times.21~22~2**29 For a z-valent oligocation binding to a polyanion such as DNA, the thermodynamic equilibrium can be written 8828.29
Lz++ P
LP + z( 1 + 2r)M'
where Lz+is a cationic ligand of valence +z, P is the set of z contiguous sites on the polyion that will bind the ligand, LP is that same set of sites after ligand binding (and with their charges neutralized by the ligand), I' is the preferential interaction coefficient of the polyion (per charged residue), and z( 1 2r) is the number of counterions released thermodynamically by the polyion in the course of binding the ligand. The neutralized residues in the L P complex are assumed to have r = -0.5, the ideal value. The observed equilibrium constant is defined here as
+
Here the parameter le (the Bjerrum length) is defined by (4) so that the polyion charge density parameter [ is just Ie/b. Again,
for the PB cell model these formulas are only limiting-law relations and are not rigorously correct for finite polyion concentrations or for any but dilute salt conditions. The above relations have been widely used in analyzing binding and conformational change data in nucleic acid systems,25-28and they work surprisingly well under conditions that are far from thelimiting-law regime. It is worth noting here that, in Manning's later development of the CC model,6q7 the limiting-law level of counterion binding is predicted to persist to salt concentrations well above the dilute regime. In this paper we use Manning's9.10 concept of polyelectrolyteinduced anticooperativity and his idea of combining polyelectrolyte theory with the McGhee-von HippelI4treatment of ligand binding on a linear lattice, to model the potentiometric titration of weak polyacidsof both sub- and supercritical chargedensity. We depart from the Manning theory in the treatment of the effects of counterion release by the polyion. Our approach yields simpler formulas for the binding isotherm than those proposed by Manning and co-workers, and we avoid the discontinuity in the titration curve predicted by the CC model. We also show how data from the middle range of the titration can be used to obtain the parameters pK0 and nearest-neighbor cooperativity w , without a priori knowledge of such quantities as the polyion radius or the spacing of charged residues, as are required in other treatments. Theory
The Polyelectrolyte Effect. Record and co-workers2'.22.25-29 have discussed very carefully the connection between an obserued equilibrium binding constant Kok and the equilibrium thermodynamic binding constant KT for the binding of a charged ligand by a polyion. Kobs is a ratio of Observed concentrations of the ligand and the free and liganded macromolecular species, and as it is usually defined, it does not explicitly include a term for the counterion species. KT is a ratio of thermodynamic activities
(square brackets denote concentration), and the thermodynamic equilibrium constant is
(7) where a3 is the activity of the counterions, and the various subscripted y are activity coefficients for the indicated species. have shown that, for an Record and oligocation with charge +z binding to a polyanion, the observed binding constant Kobs will change with added salt according to
With the insertion of the limiting-law expression for r and the assumption that the counterion activity a3 can be replaced by counterion concentration C3, this becomes
a relation also derived in the context of the CC model.2s.26 For the PB cell model, eq 8b is based on limiting-law relations for r, under conditions of high dilutions of all ionic components and with salt in excess over polyele~trolyte.~~.~~.~~ However, eq 8b appears to work well even for salt concentrations approaching the molar level. A formula similar to eq 8 has been obtained for a polyion undergoing a conformational ~ h a n g e . ~ I .For ~~N J ~contiguous residues changing from conformation 1 to conformation 2, the result is
where 2Ar is the difference in thermodynamic degree of dissociation between final and initial conformational states of a
The Titration Curve of Weak Polyacids residue. In terms of the $ parameter, the relation is
Here A$ is $2 - $I. Again, in the PB cell model eq 9b is rigorous only for limiting-law conditions, but it appears to work well even for relatively high salt concentrations. Equation 9b has also been obtained through the C C where its applicability at low to moderate salt concentrations is expected. Wilson and Lopp30 used CC model resultsZSto treat the combined effects of charge neutralization and conformational change induced by a ligand. They expressed the salt dependence of the observed binding constant Kobsin a differential form that is basically a sum of eqs 8b and 9b (see their eq 15). The result, written in our notation, is
(
%)T,p
H
-zJI
+ NA$
This relation was used to analyze data on the binding of intercalating drugs to DNA, in the limit of zero binding density.30 We wish to obtain a correction to the apparent acidity constant, pK,,,, in polyacid systems. A form for this correction is suggested by the differential relation, eq 10. Denoting the ionized polyacid sites by P and the corresponding neutralized sites by LP, and continuing to use a generalized ligand Lz+,we write the thermodynamic equilibrium as
L ~+ + P + LP +YM+
(1 1) where the parameter y represents the amount of counterions released thermodynamically during the binding of the ligand and the concomitant conformational change. The thermodynamic equilibrium constant for eq 11 can be written as
The Journal of Physical Chemistry, Vol. 97, No. 14, 1993 3625 conformations. We suppose that the polyion has M units, each with unit charge. Let the bound ligand neutralize z charges (out of the M charges total on the polyion). Further, let the bound ligand change the spacing of Nunits around the binding site from the original spacing of hl to a different value h2. For a polyion with X ligands bound to it, the net average charge spacing b,,, will then be
b,,, = [ ( M - N X ) h , + N X h J / [ M - X z ]
(14)
or, in terms of the binding density u = X / M ,
bn,,= [ N v ( h , - h , ) + h , I / [ l - v z ]
(15)
This expression for b,,, can now be used in eq 1 to find the net polyion charge density parameter tnel at any point in the titration. Then with b,,, replacingb ineqs 3aand 3b, at any point throughout the titration curve we can calculate the net degree of counterion binding and so follow how this modifies the binding curve. Local Anticooperativity in Binding. The term KPEwill account for global electrostatic sources of anticooperativity in ligand binding. There may also be Coulombic repulsions between bound ligands if they are electrically charged. This local interaction would introduce further anticooperativity into the binding isotherm. Since this is a local effect, a reasonable first approximation would be to attribute it to interactions between nearest neighbors and so to use the McGhee-von Hippel cooperativity parameter w to account for it. To do this, in eq 15 of McGhee and von Hippel14 we replace their ligand binding constant K with Kbbs: V = K6,,(1 - nu) C
+
I-’
o - 1)( 1 - nv) v - R x 2(0 - 1 ) ( 1 - nv) 1 - ( n l ) ~R 2(1 - nu)
+
+
where The form of eq 12 is chosen in order to separate terms commonly observed in potentiometric titrations from those connected particularly with the polyelectrolyte. The first term in parentheses (which we shall denote as Kbb) includes Kobsas well as the activity coefficient for the ligand species, and the logarithm of this term corresponds to pK0 for a polyacid when the ligand is a proton. Rearranging eq 12 to isolate the observed variables in the titration, we get
Here the term KPE,which equals 439, specifically represents the effects of counterion release during the reaction. Record and c o - ~ o r k e r s 2 ~have ~ ~ * proposed that the free energy change corresponding to K ~ represents E the “polyelectrolyte effect” in binding, by analogy to the well-known hydrophobic effect. The other term in eq 13, KNPE,we will assume to represent all the other factors involved in binding, apart from the polyelectrolyte effect. Turning now to the parameter y, we note that, in terms of the PB cell model, y = r(1 + 2r) - 2NAI’, and AT = rZ- rl for initial conformation 1 and final conformation 2. We shall assume that we may use the limiting-law forms for r and AT even for salt concentrations approaching the molar level and so write y in terms of the $ parameter (see eq 3). This is of course equivalent to using the CC model results for counterion binding. We shall also assume that we can write the $ parameter at any point in the titration in terms of a net charge spacing b,,,, based on an average (and uniform) charge density on the polyion. We now need a general expression for the net charge spacing b,,, a t any degree of titration, in order to find for both
R = ( [ l - (n
+ 1)vI2+ 4wv(l - n ~ ) ] ” ~
(17)
Here u is the binding density of the ligand on the polyion, n is the site size, o is the nearest-neighbor cooperativity as defined by McGhee and von Hippel,I4 and C i s the concentration of free or unbound ligand. Polyacid Titration Theory. Polyacid systems offer a test of the theory for a simple ligand where there are no conformational changes in the polyion, electrostaticeffectsdominate,and binding site exclusion effects are minimal (the ligand, H+, can reasonably be presumed to have a site size of 1). Furthermore, accurate data are available over a wide range of binding densities on several different polyacid systems, over a considerable range of polyelectrolyte charge densities and bulk salt concentrations. We shall restrict our discussion tosystems where conformational changes in the polyion are negligible; that is, we will take hl = hz in eq 15. The relation’defining the net charge spacing b,,, can then be simplified to
bn,, = h l / ( 1 - UZ)
(18)
Potentiometric titration data are characteristically represented in graphs of apparent acid pK versus degree of acid ionization a. The parameter a is equal here to 1 - u, since the site size n = 1. The apparent acid pK is given by PK,,, = PH
+ log((1 - 4/4
(19)
We can easily recast the McGhee-von Hippel isotherm (eqs 16 and 17) into a form suitable for polyacid titrations while allowing for the polyelectrolyte correction to the binding constant. Assuming a site size n = 1, substituting for K6b, taking logarithms
Woodbury
3626 The Journal of Physical Chemistry, Vol. 97, No. 14, I99 3
of eq 16, and rearranging, we find (20) PKapp = log K N p , + log Kp, + 2 lOg/l wherefi represents the effects of nearest-neighbor interactions. fi is given by f l=
(2a - 1
+ R)/2a
(21)
R = [(2a - 1)2 + 4wr~(l-
(22) In eq 20 we have implicitly assumed that log K N ~represents E the intrinsic proton binding constant (the part free of polyelectrolyte effects) for a residue in the polyacid. Thus we will identify log K N ~(the E logarithm of a binding constant) as pK0 (the negative of the logarithm of a dissociation constant) for the polyacid. In eq 20 we have essentially a representation of the (excess) free energy changes of the polyacid system, because of the logarithmic relation between equilibrium constants and AG. The values of pKappat the two extremes of the titration, at a = 0 and a = 1, can be readily interpreted in terms of interaction free energies. Consider first the intercept of the titration curve a t a = 1, where the polyacid is fully ionized. No protons have yet bound, so there is no contribution to the free energy change from nearest-neighbor interactions among bound protons on the polyacid; the logfi term is zero. This leaves pK0 + log KPE= log K’,,b as the intercept a t a = 1. Because of the polyelectrolyte term log K P E this , intercept will shift with changes in the bulk salt concentration. Consider now the other intercept, a t a = 0. Here the polyacid has been almost completely neutralized and has a negligiblecharge density. The polyelectrolyte term log KPEvanishes, leaving pK0 2 log w (we have used the limiting value of w forfi at a = 0). Interestingly, the predicted intercept is not simply pK0 but instead has a correction for nearest-neighbor interactions corresponding to the binding of the last proton to a site where it interacts with its neighbors on both sides. Only when the binding is completely noncooperative, Le., when w = 1, does this nearest-neighbor term vanish. This is apparently a different interpretation of the a = 0 intercept than is commonly used in the literature, where the intercept is usually taken to be simply pK0. Behind this latter interpretation is the tacit assumption that there is negligible nearest-neighbor cooperativity in the binding of protons a t a = 0, an assumption that may or may not be justified. Useful information can also be obtained from the value of pKapp.atthe midpoint of the titration and from the slope of the titration curve there. At the midpoint, from eq 21 the quantity fi is simply w1I2, and for a = 0.5 we can write eq 20 as
+
PKl/2=P~o-zrlI/2log~3+log~ (23) (the subscript 1/2 indicates that the quantity is to be evaluated at a = 0.5). Thus a plot of the midpoint pKap, versus log C, should be linear, with a slope of and an intercept (for C3 1 M) of pKo log W . With respect to the slope of the titration curve, from eqs 20-22 it is straightforward to find a general formula for the slope in the case where the ligand’s charge z = 1 and the ligand’s site is also 1:
+
where Wncl/aa = l B / ( 2 b n c l ) for
> 1.
0
02
04
06
08
10
V
where now
I1 or bnet/(2l~a*)for
tncl
For strictly noncooperative binding of protons by a polyacid (that is, when w = l), in eq 24 the nearest-neighbor term vanishes
Figure 1. Titration of hyaluronic acid, in Scatchard-type plot. Data from Cleland et aI.1’ Key: 0, 0.01 M NaNO,; 6 , 0.04 M NaNO,; 0, 0.10 M NaNOI; . , 0.18 M NaNOI; A, 0.45 M NaNO,. loo0
200 0
0
0.2
0.4
0.6
0.8
I 1.0
U
Figure 2. Correction of data in Figure 1 for polyelectrolyte effects. Symbols as in Figure 1.
and there is only the simple polyelectrolyte term. However, for w # 1 the nearest-neighbor interaction contribution to the slope does not vanish. Furthermore, as CY approaches zero, this slope contribution diverges (though the corresponding free energy term of course remains bounded by 2 logfi). At the other end of the titration curve, as a approaches 1, the free energy contribution from nearest-neighbor interactions vanishes, but the slope contribution goes to 4(1 - 0). The slope a t the midpoint of the titration curve, where a = 0.5, also has a simple and useful form. From eq 24 where ( a \ l . / d ~ ) , = / ~lB/(2b) for 5 1 or 2 b / ( l ~ )for t 1 / 2 > 1. Since data are usually plentiful in this middle region as opposed to the extremes at either end of the titration curve, the midpoint slope should be readily obtained by experiment. Thus eq 25 may be of practical use in supplementing the limiting forms of eq 20 and in deducing the extent of nearest-neighbor or polyelectrolyteinduced (anti)cooperativity in the titration behavior of a given polyacid.
Results The systems we have modeled are (1) hyaluronic acid, using the data of Cleland et al.;I7 (2) poly(acry1ic acid), using data from several sources;15.31-34and (3) carboxymethylcellulose,again using data from several different S O U ~ C ~ S . ~ . ~ These * J ~ J ~systems have been studied over a considerable range of salt and polymer concentrations, and the data for each system are plentiful and precise. HyaluronicAcid. Figure 1 presents titration data on hyaluronic acid (HA) from Figure 2 of Cleland et al,I7 recast in the form of a Scatchard plot. Note how the titration curves for different salt concentrations are well separated from one another. The upward curvature in all of them is typical of anticooperative binding. In Figure 2 we have divided the ordinate values by the appropriate value of &E, to correct data in the Scatchard plot for the polyelectrolyte effect. There is no indication of any conformational change in H A upon titration, so we have used a simplified form of KpE that drops the nz’A+n,lterm in theexponent of C,. The charge density of HA is subcritical at all degrees of titration, with a maximal value f o r t of 0.7; thus eq 3a was used to calculate $. As can be seen from Figure 2, the corrected Scatchard plot at all salt concentrations is now linear, indicating
The Titration Curve of Weak Polyacids
The Journal of Physical Chemistry, Vol. 97, No. 14, 1993 3627
TABLE I: Binding Parameters for Hyaluronic Acid by Statistical Curve Fitting' salt concn, M
pK0
w
salt concn, M
pKo
w
0.01 0.04
2.89 2.86 2.82
0.92 0.91 0.90
0.18 0.45
2.82 2.87
0.93 0.84
0.10
Data from Cleland et aI.l7
I 11 0
TABLE 11: Midpoint Titration Curve Parameters for Hvaluronic Acid' ~~~~
~
salt concn, M 0.01 0.04 0.10
PKI 3.20 3.07 2.97
~
salt ~( d p K / d a ) , - o ~ concn, M 0.72 0.18 0.54 0.45 0.45
11
1
11
In 6
1
11
8
1
I)
a
Figure 3. Titration of poly(acrylic acid). Data of Nagasawa et al.I5 Solid lines are theoretical curves with values for pK0 and w derived by pKlj2 ( d p K / d ~ ~ ) ~ , ~ statistical curve fitting with e q s 3, 18, and 20-22 (see Table 111). Key: 0 , 0.01 M NaCI; +, 0.02 M NaCI; 0,0.05 M NaCI; m, 0.10 M NaCI. 2.93 0.36 2.86 0.26
Data from Cleland et aI.l7
no residual anticooperativity. Furthermore, the data taken at different salt concentrations are virtually superimposable. The linearity of the plot and thevalue of the intercept on the horizontal axis are consistent with a site size of 1 for the protonic ligand. The intercepts on the vertical axis indicate a binding constant of (6.3-7.3) X 102 M-1; this corresponds to pK0 = 2.80-2.86 for the acid, which is consistent with that reported by Cleland et al.I7 These results are consistent with all or nearly all of the anticooperativity being due to the global effect of charge neutralization of the polyacid. We have also performed nonlinear least-squares fitting of eq 20 to the H A titration data at each salt concentration, with pKo and w as the parameters to be fitted. We used FORTRAN routines for the Levenberg-Marquardt method37 in the curve fitting, assuming a spread of 0.02 in values of pK and a. Uncertainties in the fitted values of pK0 and in w were generally 0.05 units or less. Table I gives the parameters found for each titration. The values of pK0 range from 2.83 to 2.89, again consistent with those reported by Cleland et a1.I' The values for the cooperativity parameter w range from 0.84 to 0.93, corresponding to free energy changes of about 0.2-0.5 kJ/mol. This indicates a slight residual anticooperativity beyond that accounted for by the polyelectrolyte effect, but it is clear that local anticooperativity is not a major effect here. We have also evaluated the midpoint pK values and slopes of the titration curves at the midpoint, using a linear least-squares routine on the data in the range 0.4 < a < 0.6 for each curve. Table I1 summarizes the values found this way. A linear leastsquares fit to the midpoint slopes as a function of the logarithm of the salt concentration gives a line with a slope of 0.29 and an intercept a t 1 M salt of 0.14. This corresponds to a value of [ I : 0.57 for the fully ionized polyacid (somewhat lower than the expected value of 0.7), and an average cooperativity w = 0.93. A similar fit of the midpoint pK values leads to a value for pKIl2 a t 1 M salt of 2.76. Combining this with the value w = 0.93 and using eq 23 yield a value for pK0 = 2.79. The value for w found by analyzing the data in the middle of the titration curve is in good agreement with those found by the nonlinear fitting method and reaffirms the general absence of any nearest-neighbor anticooperative interactions in this system. Further, we see that there is also good agreement among the values of pK0 found by the different methods and with the values found by Cleland et a1.1' The disagreement of predicted and expected values for [ we regard as a consequence of forming two successive numerical derivatives (the slope of a slope of the experimental titration curve), a process that can easily lead to numerical errors when there is scatter in the data. We conclude that use of eq 25 and plots of midpoint slope as a function of log C,probably are not an accurate or reliable method for obtaining [. Poly(acrylic acid). Figure 3 shows the potentiometric titration data of Nagasawa et al.15 on poly(acry1ic acid) (PAA) at salt
TABLE III: Bindin Parameters by Statistical Curve Fitting for Poly(acrylic acidfa salt concn, M
PKO
w
saltconcn. M
pKo
w
0.01 0.02
5.32 5.31
0.48
0.05 0.10
5.30 5.31
0.37 0.35
0.44
Data from Nagasawa et aL15
concentrations of 0.01, 0.02, 0.05, and 0.10 M. Nagasawa et al.I5 also reported a titration at 0.005 M NaCl, with a polymer concentration of 0.008 29 N; since our equations are based on a model that assumes salt is in excess over p o l y i ~ n , ' ~we - ' ~have not modeled this last titration. Assuming a spacing of 2.5 A between residues for syndiotactic PAA, the fully ionized polyacid has a value of [ = 2.84; the charge density goes below the critical value at a = 0.35. We have fitted eqs 20-22 to these titration data using our nonlinear least-squares routine. The resulting values for pKo and w are given in Table 111, and the corresponding theoretical titration curves are shown in Figure 3. The values for pKo agree quiteclosely, ranging from 5.30 to 5.32. Thevalues for w indicate an increasing degree of local anticooperativity as the salt concentration rises, w dropping from 0.48 to 0.35 as the salt concentration is increased 10-fold. These values for w correspond to free energy changes of 1.8-2.5 kJ/mol and indicate moderate amounts of local anticooperativity. The value of pKappat a = 0 is readily found using eq 20. The quantityfi becomes simply w here. Using the values for pK0 and w from Table 111, we can predict pKappa t a = 0 to be 4.68,4.60, 4.44, and 4.40 respectively for salt concentrations of 0.01,0.02, 0.05.and0.10M. Figure4ofNagasawaet al.I5seems toindicate a salt-independent pKappofaround 4.3 a t a = 0, rather lower than what we find. We note that the PB theory curves shown by Nagasawa et a1.I5 do not fit the data well at low degrees of ionization, so the disagreement here is not unexpected. As with the H A titration curves, to find the midpoint slopes and pKvalues we have fitted the PAA titration data of Nagasawa et al.I5 around the midpoint using a linear least-squares routine. Table IV presents theseresults. We havealsoestimated midpoint slopes and values for pKappusing titration curves reported by Kono and Ikegami,31Pederson et al.,32Kawaguchi and Nagasawa,33and Mande1;34the results are summarized in Table IV and plotted in Figure 4. Despite some scatter in the estimated slopes and values for pKappat the midpoint, the linearity in the plots is clear. The extrapolated value for the midpoint pK,,, ranges from 4.61 to 4.73 at 1 M salt, for reasonably good agreement among the different experiments. From eq 25 and the extrapolated value of the midpoint slope at 1 M salt, we calculate that the effective nearest-neighbor anticooperativity lies in the range from 0.53 to 0.59. It is clear that there is appreciably more local anticooperativity in the PAA system than in the H A system. Possibly the closer proximity of thecarboxylic acid groups on PAA lead to greater local Coulombic repulsions here.
3628 The Journal of Physical Chemistry, Vol. 97, No. 14, 1F'93 TABLE I V Midpoint Titration Curve Parameters for Poly(acry1ic acid) salt concn, M
0.01 0.02 0.05 0.06 0.75 0.10 0.20
pKip
2.07," 2.89,"2.33,d2.10" 2.35," 1.83h 1.96," 2.01 q h 2.02' 1.82' 1.86 1.79," 1 .72,h 1 .80d 1 .5 1 ,h 1.67'' 1.591 I.60 1 .30h 1.31"
5.10'
4.96' 4.90h 4.91'
0.50 0.60
TABLE V Midpoint Titration Curve Parameters for Carboxymethylcellulose at Subcritical Cbarge Density'
(apK/aa),,=o3
6.19,a6.08,".90,"5.88" 5.95," 5.82h 5.62.O 5.60,h5.50 5.52'' 5.32' 5.41," 5.38,h 5.2Id 5.25,h 5.12''
0.30 0.33
Woodbury
0.006 0.016 0.032
3.98 3.87 3.76
1.31 1.12 0.94
0.064 0.30
3.66
3.50
Data from Pals and Hermans.j5
Data of Nagasawa et Data of Kono and Ikegami." Data of Pederson et aL3? Data of Kawaguchi and Nagasawa for syndiotactic polyacrylic acid." Data of MandeLJJ I'
b 0
44
i -25
-20
-1s
-10
-05
00
Log [ Salt ]
A m
i -2
II
-1 5
bo
-1 n
06:
0
0
0
J
10 -2 5
JpKY
-05
00
Log [ Salt 1 Fipre4. Salt dependenceof titration midpoint parameters for poly(acry1ic acid): (a, top) titration midpoint p K I p ; (b, bottom) titration midpoint slope (apK.,,,/aa)ip Key: 0 , data from Nagasawa et a].;" +, Kono and Ikegami;" 0, Pederson et ai.;)? W, Kawaguchi and Nagasawa;)) A, MandeL'j
From Figure 4, using eq 23, we calculate that pK0 for poly(acrylic acid) is respectively 4.96 (data of Kono and Ikegami3I), 4.97 (data of M a t ~ d e l ~and ~ ) , 4.88 (data of Nagasawa et al.I5). This is in reasonable agreement with values for pK0 of 4.72-4.85 reported by these investigators. However, our method of determining pKo allows for local anticooperativity, while the literature values were obtained from intercepts of the titration curves at a = 0. For comparison we can use eq 20 and the estimated values for w to predict pK,,, at a = 0. We calculate values for this intercept of 4.50 (data of Kono and Ikegamijl), 4.44 (data of Mande134), and 4.34 (data of Nagasawa et al.I5). These values are substantially lower than the literature pK0 values cited above, but they do agree fairly well with the intercepts predicted using our nonlinear least-squares fit parameters. We note here that in the figures of Nagasawa et al.I5and Kawaguchi and Nagasawa3j we can see apparent values for the intercepts at a = 0 of 4.3-4.5, while in Figure 1 of M a t ~ d e the l ~ ~extrapolated intercepts range from 4.9 to 5.2. From this survey of PAA titration curves, our impression is that there is general agreement on the midpoint slopes and the overall shape of the titration curves among the investigations. However, at any given salt concentration the individual titration curves are often offset from one another by 0.2 or more pKunits. Experimental details in the type of electrode used, its calibration,
0.68 0.16
The Titration Curve of Weak Polyacids
The Journal of Phy, al Chemistry, Vol. 97, No. 14, 1993 3629 polyelectrolyte theory accounts quite well for the preponderance of anticooperativity in carboxymethylcellulose titrations.
SO1
Discussion and Conclusion
3
0 00
L
-
n?
I '
'
11.1
in
118
06
cc
.,
Figure 6. Titration of carboxymethylcellulose. Data of Muroga et al.ln Solid lines are theoretical curves with values for pKo and w derived by statistical curve fitting with eqs 3, 18, and 20-22 (see Table VI). 0 , O . O l M NaCI; +, 0.02 M NaCI; 0,0.05 M NaCI; 0.10 M NaCI; A, 0.20 M NaCI.
TABLE VI: Binding Parameters by Statistical Curve Fitting for Carboxvmethvlcellulose of Supercritical Charge Densitva ~
~~
~
~~~~
salt concn, M
PKo
w
salt concn, M
pK0
w
0.01 0.02 0.05
3.53 3.54 3.53
1.09 0.97 0.91
0.10 0.20
3.58 3.58
0.77 0.70
(i
Data from Muroga et a1.l"
TABLE VII: Midpoint Titration Curve Parameters for Carboxymethylcellulose at Supercritical Charge Density" salt concn, M 0.01 0.02 0.05
salt p K ~ p (apK/aa),,=o.r concn, M 4.61 4.40 4.17
I .58 1.42 1.15
0.10 0.20
pK1/2 ( d p K / d a ) , , = o5 3.97 3.77
0.97 0.80
[' Data from Muroga et a1.I"
cellulose with a degree of substitution of 1.54. This corresponds to a value of [ = 2.1 for the fully ionized polyacid; the critical charge density is obtained at a = 0.48. Unlike the highly linear titration curves reported by Pals and Hermans,js the titration curves here show quite definite concave-down curvature. We have applied our nonlinear least-squares program to the fitting of the data reported for the lowest polymer concentrations used by Muroga et al.I6 Figure 6 shows the data of Muroga et a1.I8 that we used, along with our best-fit theoretical curves. Table VI summarizes the results of this analysis. We find negligible salt-dependent variation in the pK0 values, which range from 3.53 to 3.58. The best-fit pK0 values for this case are somewhat higher (by about 0.2pKunits) than thosereportedin theliterature and those found in our analysis of the (subcritical charge density) data of Pals and her man^,^^ discussed above. The best-fit cooperativity parameter here also shows a saltdependent variation, from 1.09 at 0.01 M salt to 0.70 at 0.20 M salt; the average value is 0.89, and we conclude that it too indicates that very little positive or negative cooperativity remains to be accounted for, once the polyelectrolyte-induced anticooperativity is included. We have also analyzed the middle of the titration curves of Muroga et a1.,I8as we did for the systems discussed above. Our estimates for pKIi2 and the slope at the midpoint are collected in Table VII, while Figure 5 shows the dependence of these quantities on the salt concentration. Analyzing these data as before, we estimate a value for pKl,2 at 1 M salt of 3.33, an average anticooperativity of 0.84, a value of pK0 = 3.41, and a value of pK,,, = 3.26 at a = 0. The value of w = 0.84 indicates that there is relatively little anticooperativity in this system beyond that due to polyelectrolyte effects. Furthermore, it is not very far from the average value found by statistical curve-fitting, nor is it far from that found for the (subcritical) CMC-73 sampleof Pals and her man^.^^ Taken as a whole, these values for w consistently show that our
The simplified theory presented here has in it a number of assumptions and approximations. First, we have factored the observed binding constant KLbs into a term specifically representing the salt-dependent polyelectrolyte effect and a term independent of the polyion's characteristics and of the salt concentration. Such a factorization has been used and justified by Record and co-workers for various DNA-ligand systems.2s-*8 We justify this assumption here by the good agreement of our theoretical isotherms with experimental titration curves for the systems analyzed here. Furthermore, our relations for the midpoints and midpoint slopes of thecurves, relations that depend strongly on the assumed form of the polyelectrolyte term, seem to give quite reasonable agreement with experiment. Finally, in plots of pK,,, versus a for each polyacid system, the apparent pK values extrapolate reasonably well to a common intercept on the ordinate for a = 0. This agreement implies that this intercept, which is proportional to the polyion-independent free energy change of binding in our model, truly is independent of ionic interactions, a conclusion that is consistent with our assumed factorization. Second, we have assumed a particular form for the polyiondependent term KPEthat is based on the limiting-law results for counterion binding by a rodlike polyion of infinite length. Since the three polyacids analyzed here are themselves reasonably stiff polymers, and since the experiments used polymers of high molecular weight, we believe it is reasonable to ignore end effects and flexibility of the polyion in polyion-counterion interactions, that is, to assume the (conventional) infinitely-long rod model for the polyion. Whether it is reasonable to apply limiting-law formulas to experiments done with finite concentrations of polyelectrolyte and moderate to substantial concentrations of simple electrolyte is another question. The best justification we can give here is (1) our success in fitting the entire titration curve at quite different salt concentrations with the three polyacids, using just the two parameters pK0 and w and (2) the consistency of parameter values obtained by statistical curve fitting with one another and with parameter values derived by analyzing midcurve data with our formulas. It is worth noting that, in DNA-ligand systems, the limiting-law forms for KobSalso work quite well from the dilute range up to at least several tenths molar in added saIt.26.28.29 There is also the question of the applicability of our formula for K P Eover the entire titration curve and not just in the limit of zero binding density of protons. To justify this, we offer the view that the binding of protons to the polyacid can be considered as a sort of charging process in reverse that is accompanied by a change in the activity of the proton species. That is, at any given point in the titration curve we view all the previously-bound protons as mere neutralizing electrical charge that is uniformly distributed over the polyion, so that we have a partially-charged but hypothetically unliganded polyion ready to bind a proton. This corresponds to proton binding to our hypothetically unliganded polyion in the limit of zero binding density, an event to which we can apply the appropriately-corrected form of KPE. Implicit in our formulas for b,,, is our assumption of random binding of the protons; since the values of w found for the various polyacids are all relatively close to 1, there appears to be little bias away from random binding. Third, we have ignored corrections for activity coefficients, lumping the coefficients for the polyion and the complex into KNPkon the one hand and using C3 instead of the counterion activity in KpEon the other hand. In their original treatment of the binding problem, Record and c o - w o r k e r ~assumed ~~ that activity coefficient corrections for the charged ligand and
3630 The Journal of Physical Chemistry, Vol. 97, No. 14, 1993 counterion species would cancel each other out, or at least leave a correction that was negligible by comparison to experimental error in determining the binding. This approximation has worked well for DNA systems,25-29and we make the same approximation here. Fourth, we have used a simple nearest-neighbor model14 for the interaction of bound ligands (protons). Other workers3944 have proposed models that take into account second and even third nearest neighbors, but with quite different treatments of the ionic effects. Our combined treatment of local and global effectson the titration curve is simpler and provides a satisfactory fit to the data. Manning and co-workers,9JO in the context of the CC model, have recently presented corrections to the McGhee-von Hippel14 isotherm for the anticooperativity and salt dependence expected for binding large, charged ligands to a polyion. Data from several polyelectrolyte systems were used to test the theory’s predictions. Although qualitative and even quantitative agreement with experiment were found, the theory predicted a nonphysical discontinuity in the dependence of the apparent binding constant on the degree of titration (see Figure 1 of Friedman and Manning’). This discontinuity seems to be caused by a switch in formulas for counterion binding as the effective charge density 4 passes the critical value of 1. Our combination of eqs 3, 13, 15, and 20 does not lead to such a discontinuity in the titration curve. Furthermore, our formulas are considerably simpler than the formulas presented by Manning and co-w~rkers.~~~O We anticipate that the relations presented here may be extended to the binding of small ligands to DNA, and in a later report we will examine the accuracy of our treatment in modeling such systems. To summarize, we are able to account for both local and global electrostatic sources of anticooperativity in the potentiometric titrations of a variety of polyacids, over a range of salt concentrations and polyion charge densities. Our formulas are a simple extension of widely-used models for DNA-ligand binding. Our treatment agrees with chemical intuition on the respective roles of global and local interactions in anticooperative binding. Furthermore, the computations in our theory are much easier than those required for solving the nonlinear Poisson-Boltzmann equation11J2.20-45 or for solving an extended Ising model with second- and third-nearest-neighbor interactions.3944 Our formulas for obtaining pK0 and w use data from the middle region of the titration curve, the region most readily accessible by experiment. They do not require extrapolations of the titration curve into regions where experimental errors are magnified and where data are typically difficult to collect. Finally, we find quantitativeagreement of our theory with experiment for polyacid systems of both s u b and supercritical charge density, in predicting values of the intrinsic acidity, in predicting shifts of pK,,, with salt concentration, and in matching the titration curve at low, intermediate, and high degrees of titration.
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Woodbury Manning, G. S. J. Chem. Phys. 1969,51, 924. Manning, G . S. J . Chem. Phys. 1969,51, 3249. Manning, G . S.;Holtzer, A. J . Phys. Chem. 1973,77, 2206. Manning, G. S.Biophys. Chem. 1977,7,95. Manning, G. S. Q.Reo. Biophys. 1978,1 1 , 179. (8) Manning, G. S.J . Phys. Chem. 1981,85, 870. (9) Friedman, R. A. G.; Manning, G. S . Biopolymers 1984,23, 2671. (IO) Friedman, R. A. G.; Manning, G. S.;Shahin, M. A. In Chemisrry und Physics of DNA-Ligand Interactions; Kallenbach, N. R., Ed.; Adenine Press: Guilderland, NY, 1988. ( I I ) Marcus, R. A. J . Chem. Phys. 1955,23, 1057. (12) Kotin, L.; Nagasawa, M. J . Chem. Phys. 1962,36, 873. ( 13) Anderson, C. F.; Record, M. T., Jr. Annu. Reu. Phys. Chem. 1990, 19, 423. (14) McGhee, J. D.; von Hippel, P. H. J. Mol. Biol. 1974,86, 469. (15) Nagasawa, M.; Murase, T.; Kondo, K. J. Phys. Chem. 1965,69, (3) (4) (5) (6) (7)
4005. (16) Olander, D. S.; Holtzer, A. J . A m . Chem. SOC.1968, 90,4549. (17) Cleland, R. L.; Wang, J. L.; Detweiler, D. M. Macromolecules 1982, I S , 386. (18) Muroga, Y .;Suzuki, K.; Kawaguchi, Y .;Nagasawa, M. Biopolymers 1972,1 1 . 137. (19) Anderson, C. F.; Record, M. T., Jr. Biophys. Chem. 1980,II,353. (20) Klein, B. K.; Anderson, C. F.; Record, M. T.. Jr. Biopolymers 1981, 20, 2263. (21) Anderson, C. F.; Record, M. T., Jr. In Srrucrure and Dynamics:
Nucleic Acids und Proteins; Clementi, E., Sarma, R. H., Eds.; Adenine Press: New York, 1983. (22) Anderson, C. F.; Record, M. T., Jr. Annu. Reu. Phys. Chem. 1982, 33, 191. (23) Gross, L. M.; Strauss, U. P. In Chemic41 Physics of ionic Solutions; Conway, B. E., Barradas, R. Q.,Eds.; Wiley: New York. 1966. (24) Katchalsky, A. Pure Appl. Chem. 1971,26, 327. (25) Record, M. T., Jr.; Lohman, T. M.; de Haseth, P. J . Mol. Biol. 1976, 107,145. (26) Record, M. T., Jr.; Anderson, C. F.; Lohman,T. M. Q. Reo. Biophys. 1978,1 1 , 103. (27) Record, M. T., Jr.; Anderson, C. F.; Mills, P.; Mossing, M.; Roe, J.-H. Adv. Biophys. 1985,20, 109. (28) Record, M. T., Jr.; Mossing, M. In R N A Polymerase und rhe Regulurion of Trumcriprion; Reznikoff, W., Ed.; Elsevier: New York, 1987. (29) Record, M. T., Jr.; Olmsted, M.; Anderson, C. F. In Theorericul Biochemistry and Molecular Biophysics; Beveridge, D. L., Lavery, R., Eds.; Adenine Press: Guilderland, NY, 1990. (30) Wilson, W. D.; Lopp, I. G. Biopolymers 1979,18, 3025. (31) Kono, N.; Ikegami, A. Biopolymers 1966,4, 823. (32) Pederson, D.; Gabriel, D.; Hermans, J., Jr. Biopolymers 1971,10, 2133. (33) (34) (35) 513. (36) (37)
Kawaguchi, Y.; Nagasawa, M. J . Phys. Chem. 1969,73, 4382. Mandel, M. Eur. Polym. J . 1970,6 , 807. Pals, D. T. F.; Hermans, J. J. R e d . Trau. Chim. Puys-Bus 1952,71,
Mandel, M. J . Polym. Sci. 1970,6 , 1841. Press, W. H.; Flannery, B. P.; Teukolsky, S.A.; Vetterling, W. T. Numerical Recipes, The Arr of Scientific Computing, Cambridge University: Cambridge, U.K., 1986. (38) Schneider, N. S.;Doty, P. J. Phys. Chem. 1954,58, 762. (39) Harris, F. E.; Rice, S.A. J . Phys. Chem. 1954,58, 725. (40) Marcus, R. A. J . Phys. Chem. 1954,58,621. (41) Lifson, S. J . Chem. Phys. 1957,26, 727. (42) Sasaki, S.;Minakata, A. Biophys. Chem. 1980,1 1 , 199. (43) Cleland, R. L. Mucromolecules 1984,17, 634. (44) Nishio, T. N. Biophys. Chem. 1991,40, 19. (45) Stigter, D. J. Colloid interface Sci. 1975,53, 296.
