4292
J. Phys. Chem. 1996, 100, 4292-4304
Competitive Electrostatic Binding of Charged Ligands to Polyelectrolytes: Planar and Cylindrical Geometries Ioulia Rouzina and Victor A. Bloomfield* Department of Biochemistry, UniVersity of Minnesota, 1479 Gortner AVenue, St. Paul, Minnesota 55108 ReceiVed: September 1, 1995; In Final Form: NoVember 21, 1995X
The strong salt dependence of the binding of multivalent ligands to DNA indicates the predominantly electrostatic nature of that binding. To a good first approximation, the salt dependence can be expressed as a competition between two point counterion species in screening the highly charged macroion surface. This basic problem is solved in terms of the Poisson-Boltzmann (P-B) equation for a macroion of arbitrary shape and charge, in a solution of arbitrary salt composition and ionic strength. Various binding regimes are defined by the relative magnitudes of a, the radius of curvature of the macroion; rd, the Debye-Hu¨ckel screening length, which characterizes exponential decay in the linear screening regime; and λ, the decay length of the counterion distribution close to the macroion surface in the nonlinear screening regime. Only the nonlinear regime (λ , rd) produces relatively high free energy of ligand binding (>kBT per ion). This regime is very similar in planar and cylindrical geometries. For this case we suggest a simple new method of calculating the amount of each species bound, which avoids numerical solution of the P-B equation for each set of species concentrations and further integration of the charge. Instead, we follow changes in the surface concentration and the decay length of each counterion distribution in the course of titration. The product of these two quantities yields a reasonably accurate estimate for the amounts bound. It also yields a closed form of the binding isotherm in both geometries. The apparent ligand (species 2) binding constant obtained in this way has a conventional dependence on the salt (species 1) concentration to the -z2/z1 power. However, we obtain a new expression for the magnitude of the electrostatic binding constant in terms of the macroion surface charge density, Bjerrum length, and ion charges z1 and z2. Binding of counterions to the DNA double helix changes qualitatively from nonlinear cylindrical, through nonlinear planar, to linear planar behavior as the ionic strength of the solution is raised. Our results are comparable in many ways to those of counterion condensation theory and the McGhee-von Hippel site binding model, but show some different behavior and suggest different interpretations of similar behavior.
1. Introduction Much of the biological activity of DNA is modulated by its interaction with multivalent cations: histones, basic regulatory proteins, polyamines, and the like. Such interaction generally takes place in a background of lower valence (often monovalent) counterions. The multivalent ligands are more strongly attracted to the negative DNA phosphate backbone, but the monovalent cations, being present at much higher molar concentrations, provide effective competition. Thus the apparent equilibrium binding constant for the ligand to the DNA depends on the salt concentration. Numerous experimental studies have characterized this dependence, e.g., refs 1 and 2. The dominating influence on binding of the charge relative to all other characteristics of the ligand indicates an overall delocalized electrostatic, rather than a site-specific, mechanism. Nevertheless, such binding data are conventionally analyzed in terms of the McGhee and von Hippel site binding model,3 applied to the reaction of z2/z1 (ratio of ligand to salt cation charges) salt cations replaced by each ligand.1 The reasons for the relative success of this one-dimensional excluded volume model, and the variation of “site size” with salt concentration, are not clear; it does not connect the value of the binding constant Kobs with the charge characteristics of the polyion and the salt solution. Indeed, Ray and Manning4 have argued that excluded volume along the linear DNA lattice should be treated as a perturbation on electrostatic effects, rather than Vice Versa. * To whom correspondence should be addressed. FAX: (612) 625-6775. E-mail:
[email protected]. X Abstract published in AdVance ACS Abstracts, February 15, 1996.
0022-3654/96/20100-4292$12.00/0
Most analytical treatments of the competitive electrostatic binding of multivalent ligands to polyions have been based on Manning’s5 counterion condensation (CC) theory, e.g., ref 6. This theory is remarkably successful in predicting the total fraction of polyion charge neutralized by a single counterion species. However, it does not yield an adequate description of such important properties of the counterion distribution as its free energy or spatial distribution.7,8 Therefore, its validity for describing counterion competition is unclear although it has recently been subjected to successful experimental test.9 In fact, we shall see below that its results are different in some important respects from those following from the Poisson-Boltzmann (PB) treatment. Numerical studies over the past two decades have proved the P-B equation to provide a reliable basis for studying polyelectrolytes. For reviews, see refs 10-12. Recently several numerical P-B studies have addressed the problem of competitive electrostatic binding.13-18 It is indeed possible to achieve a very good agreement with experiment through this approach. However, it has the significant drawback that extensive calculations must be performed at each set of ionic concentrations. This makes the construction of the desired theoretical curvesgenerally, a plot of the log of the apparent binding constant vs the log of the salt concentrationsa tedious task. The numerical approach is sometimes necessary to account for the fine details of the charge distribution on large ligands, like globular proteins.17,18 However, the general trends of counterion competition should first be understood for the simplest model of two point counterion species screening a uniformly charged © 1996 American Chemical Society
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J. Phys. Chem., Vol. 100, No. 10, 1996 4293
cylinder. This problem, though basic, has not been comprehensively solved, since no explicit solution of the P-B equation for macroion screening in a mixed salt solution is available even in planar geometry. In this paper, we find an explicit though approximate solution of that problem. We compare two exact analytical solutions for a single salt component interacting with planar19 and cylindrical20,21 surfaces of similar charge densities, and find essential similarities in the counterion distributions. This forms the basis for developing a semianalytical description of counterion competition near a plane and applying it to the cylinder. Our approach is similar to that of Gue´ron and Weisbuch.22-26 These workers also emphasize the thin surface layer of counterions which exists near a highly charged surface independent of surface curvature or bulk ionic strength. They prove this result for finite salt.22 In this study we show that screening of the cylinder in the low salt limit (which is in fact appropriate for DNA up to about 0.4 M salt) is in many ways similar to the planar low salt case, but has some definite distinctions. The problem of a polyelectrolyte in mixed salt is also addressed in ref 24. However, it concentrates on the surface potential rather than the description of competitive binding. The potential at a curved surface in pure, and then in mixed, salt is obtained as a perturbation to the analytical, planar, pure salt case. The resultant explicit expression for the potential, an expansion with respect to small parameters of the system, is a very good numerical approximation over a wide range of conditions. Perhaps because these papers are mathematically complex and not in a form readily compared with experiment, they have not (to the best of our knowledge) been used by other groups in subsequent experimental or numerical studies of the problem. Therefore we feel that, although most of our conclusions can be derived from the free energy expression of Gue´ron, our treatment of the problem is useful. We focus more on the qualitative features of the counterion distribution of each species and changes in the distribution over the course of titration. This provides qualitative insight and helps to interpret the partial successes of the McGhee-von Hippel and Manning approaches. Our results are presented in the form of simple expressions for the amounts of counterions bound and the binding isotherms or Scatchard plots, with explicit expressions for the apparent binding constants and slopes of their logarithmic salt dependences. Another original feature of this paper is the treatment of competitive electrostatic binding to a highly charged cylinder in the more general context of an arbitrary curved and charged macroion in a solution of arbitrary composition. This is important since the character of DNA-ligand binding changes considerably with ionic strength. In general, six different counterion competition regimes can be distinguished, which can be classified with respect to the relative values of three fundamental length scales, described in the next section. 2. Length Scales in Macroion Screening by Electrolytes There are three important length scales in our treatment of polyelectrolyte theory. The first is the radius of curvature, a, of the macroion. The second is the Debye screening length rd, which characterizes the exponential decay of the counterion profile in the Debye-Hu¨ckel (D-H), or linear, regime:
rd2 )
kBT 8πqe2I
)
1 8πlbI
(2-1)
where I is the ionic strength of the solution (summing over only the small ion species i)
I ) 1/2 ∑ zi2 nbi
(2-2)
i
and the other parameters are dielectric constant , Boltzmann constant kB, temperature T, electron charge qe, ion charge z, bulk ion concentration nb (ions/cm3), and Bjerrum length
lb ) qe2/kBT
(2-3)
The third characteristic length scale is not so generally recognized. It is the decay length λ of the counterion concentration profile in the nonlinear screening regime which will pertain close to a highly charged surface of charge density σ:
λz )
kBT 1 ) 4πσqez 4πσ/qelbz
(2-4)
According to Gauss’s law in cgs units, a charged surface generates near itself an electrostatic field E ) 4πσ/. Thus, the physical significance of λz is the average distance from this surface of a z-valent ion with energy kBT. We will use below the notation λ ) λz)1. The relative values of the lengths a, rd, and λz define different regimes of the macroion screening, associated with the different competitive behaviors. These regimes allow us to make two major distinctions: linear vs nonlinear behavior of the PoissonBoltzmann equation and effectively planar vs cylindrical geometry. Nonlinear behavior occurs if the macroion surface is so highly charged, or the ionic strength is so low, that λz < rd. As we shall see in the next section, this ratio can be expressed as
( )
λz I ) 2 rd z ns
1/2
(2-5)
where
ns ) 2π(σ/qe)2lb
(2-6)
is the counterion concentration at the charged surface in the nonlinear screening regime. In other words, screening switches from the nonlinear to the linear, Debye-Hu¨ckel, type when the ionic strength becomes greater than I′:
I > I′ ) z2ns
(2-7)
Both ns and λz are independent of bulk salt concentration. Therefore, in the nonlinear regime the macroion charge is substantially screened within a small volume close to the surface, and this short-range screening does not disappear with dilution of the bulk solution. This effect has been termed “counterion condensation”,5 and has often been thought of as specific for the cylinder. However, Gue´ron and Weisbuch24 proved in a general way that the short range screening environment is rather insensitive to the surface curvature at finite ionic strength, if the surface charge density is high enough. Despite these general similarities between geometries, we want to understand the differences between them with respect to competitive binding. Thus our second major distinction is between planar and cylindrical surfaces. A surface is effectively planar if its radius of curvature a is greater than both λz and rd. The advantage of a planar geometry is that the full, nonlinear P-B equation can be solved analytically. Linear, pseudoplanar behavior is observed if rd < a < λz.
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TABLE 1: Regimes for Solution of the Poisson-Boltzmann Equation case
condition
description
1 2 3 4 5 6
rd < λz < a λz < rd < a λz < a < rd rd < a < λz a < rd < λz a < λz < rd
linear, planar nonlinear, planar nonlinear, cylindrical linear, cylindrical (pseudoplanar) linear, cylindrical (weakly) nonlinear, cylindrical
Nonlinear screening of a curved surface is possible if its radius of curvature is larger than λz. In the special case of a cylindrical surface with length per charge b along the cylinder axis,
σ ) qe/2πab
(2-8)
a/λz ) 2zlb/b ) 2zξ
(2-9)
and
f2 )
1
∑i nbi(ez ψ - 1) i
I
(3-1b)
to yield the potential profile:
x(ψ) ) I1/2 ∫ψ
dψ′
ψs
x∑
(3-1c)
nbi[exp(ziψ′) - 1]
i
Here ψ ) -φqe/kBT is the reduced potential, φ is the potential, x ) r/rd, and nbi and zi are the bulk concentrations and valences of the ith species, and f ) -∂ψ/∂x is the dimensionless electrostatic field, which is fixed at the surface at fs ) rd/λ ) (ns/I)1/2. This boundary condition substituted in (3-1b) yields
ns ) ∑ nsi
(3-2)
nsi ) nbi[exp(ziψs) - 1]
(3-3)
i
where
where
ξ ) lb/b
(2-10)
Viewed in this way, the condition a/2λz > 1 is the same as Manning’s5 well-known criterion for counterion condensation, zξ > 1. We summarize these possibilities, which correspond to different approximate solutions of the P-B equation and to different binding behavior, in Table 1. All of these inequalities should be strong ones in order for distinctions between the regimes to be pronounced. Weak inequalities correspond to intermediate binding regimes. Specifically, B-DNA, with radius a ) 10 Å and charge/unit length b ) 1.7 Å, has surface charge density in qe units σ/qe ) 9.4 × 10-3 Å-2. Then in aqueous solution at room temperature, ns ) 6.65 M and λz ) 1.12, 0.75, and 0.38 Å for mono-, di-, and trivalent cations. This is strongly in the λ < a regime. The fact that the counterion radius is considerably greater than λz does not invalidate our approach. For finite size counterions, λz should be measured from the distance of closest approach. The low value of λz simply reflects the fact that DNA is very highly charged, and therefore exhibits nonlinear screening properties over a wide range of salt concentrations. The condition of the cylindrical radius being equal to rd,
a/rd ) ξ(I/ns)1/2
(2-11)
I ) I′′ ) ns/ξ2
(2-12)
distinguishes between the nonlinear planar and cylindrical cases, if a is still much less than rd. Thus, increasing the salt concentration from physiological or below to about 0.4 M changes the solution condition from case 3 (nonlinear, cylindrical) to case 2 (nonlinear, planar). Only when I reaches several molar (see (2-5)) is there a transition to case 1 (linear, planar). We proceed now to a systematic consideration of these various cases. 3. Binding to the Plane The P-B equation in planar geometry
2
∂x
)
1 2I
∆ni(x) ) nbi[exp(ziψ(x)) - 1]
(3-4)
The relationship (3-2) seems not to be commonly recognized in the current polyelectrolyte literature, but has been known for a long time by electrochemists as Grahame’s equation.27 Together with (3-3), it allows definition of the surface excess concentrations of all species as functions of the bulk concentrations, without solving the P-B equation. The solution of (3-1c) takes very different forms depending on the value of ψs, which is in turn defined by fs, i.e., by the ratio (2-5). If the latter is small, then ψs , 1, and linear screening takes place. Case 1: Linear Planar Screening: rd < λz < a. This is the case when the ionic strength is high or the surface charge is low (I > I′). (3-1b) then can be expanded with respect to small ψ in this case to give
f2 )
1
nbi(ziψ + zi2ψ2/2) ) ψ2 ∑ I i
which results in the D-H exponential decay of the concentration profile
or
∂ 2ψ
is the surface value of the excess concentration profile
∑i zinbiez ψ i
can be integrated through the intermediate stage
(3-1a)
∆ni(x) ) nbi(eziψ - 1) ≈ zinbiψ0e-x ) zinbi(ns/I)1/2e-r/rd (3-5) and the fraction of the surface charge screened by the ith species is
ziΘi ) zi∫d3r ∆ni(r) ) zi2nbi(ns/I)1/2rd/σ ) zi2nbi/2I (3-6) where Θi is the fraction of an ion “bound” per unit surface charge. Thus each counterion species participates in screening the plane in an amount proportional to its contribution to the ionic strength. The definition of ionic strength, (2-2), shows that the surface is completely neutralized by all species taken together:
∑ziΘi ) 1
(3-7)
Complete neutralization is a general property of planar geometry, independent of surface charge, solution composition or ionic
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J. Phys. Chem., Vol. 100, No. 10, 1996 4295
strength.7 This is proved for the highly charged surface in the next paragraph. Physically it comes from the fact that the ion entropy increases more slowly than the electrostatic energy upon dissociating from the surface. A planar surface therefore keeps all of its neutralizing charge even upon infinite dilution, while the sphere loses all of it, and the cylinder keeps only the fraction (1 -1/zξ). On the other hand, at high enough salt (rd < a), screening of the highly charged surface is practically independent of its curvature.22 Our main goal in this paper is to calculate the electrostatic binding of a multivalent ligand L (counterion species 2) to polyion sites P in the presence of a lower valent salt (counterion species 1). The binding constant is defined as
K2 )
Θ2 [LP] ) [L][P] nb2
(3-8)
where the second equality assumes that the concentration of neither polyion sites nor ligands is significantly disturbed by the binding: [P] ≈ [P]b, [L] ≈ nb2. Then in the linear (high salt) regime according to (3-6),
K2 ) z2/2I
(3-9)
so K2 depends on the salt concentration through I. This dependence is conventionally measured by the slope
S)-
∂ log K2 z1(z1 + 1)nb1 ) ∂ log nb1 2I
(3-10)
where we have used the fact that, in most experiments, there is a common univalent coion, so I ) 1/2[z2(z2 + 1)nb2 + z1(z1 + 1)nb1]. If the salt is in considerable excess, then S ) 1. In the less common case that multivalent ligand is in excess, S tends to zero. In any case the absolute value of the slope is not more than unity, in contrast to the following nonlinear screening regime. Case 2: Nonlinear Planar Screening: λz < rd < a. This is the case of a highly charged surface, or low ionic strength (I < I′). Then ψs . 1 and the screening profile (3-1) has three regions of qualitatively different decays. The first is very close to the surface, where ψs - ψ e 1 and the decay of the potential is rapid. The second is at intermediate distances, where 1 < ψs - ψ e ψs and the decay of the potential is slower because the surface charge is largely screened by the counterions in the first region. The third is at larger distances, where ψ , 1 , ψs and normal D-H screening occurs. Close to the surface (r e λz), the potential can be expanded in a Taylor’s series
ψ ) ψs - r/λ
(3-11)
Figure 1. (a) Excess concentration of a single counterion species as a function of distance from a charged plane in the strongly nonlinear screening regime nb/ns ) 10-3. Coordinates are normalized by the nonlinear screening distance λz (2-4) and the counterion concentration at the surface ns (2-6). The inset shows the very long range dependence. The exact P-B profile (solid line) is compared with simple analytical approximations at short range, (3-12), long dashes; intermediate range, (3-13), short dashes; and long range, (3-14), dotted line. (b) Fractional charge neutralization as a function of distance from the plane, from numerical integration of the P-B profile in part a.
This decay also does not depend on the bulk ionic strength, being fully determined by the surface charge density and lb. The third region, within r > rd, is essentially a D-H type exponential tail of the distribution:
∆ni(r) ) nbi[exp(ziψ) - 1] ≈ nbiziψ ≈ 4nbi exp(-r/rd) (3-14) An example of such a profile obtained with (3-1) for a solution with a single counterion and a complementary coion of the same valence (pure salt) is given in Figure 1a, together with its approximate forms, (3-12), (3-13), and (3-14), in the various ranges. Figure 1b presents the charge neutralized by the single z-valent counterion species within a given distance from the surface
leading to the concentration profile of the ith species
∆ni(x) ≈ ni(x) ) nsi exp(-r/λzi)
(3-12)
In the second region, confined to the layer λz < r e rd, the decay of the potential is slower because of extensive screening which occurs in the first region. If there is just a single species of counterion, its concentration decays as the second power of r:
n(x) ) nb
ns
(2x) ) 2πl1z r ) (r/2λ ) 2
2 2
b
z
2
(3-13)
g(r) )
()
zqe 1 r/λz r′ ∆n(r′/λz) d3r′ ∆n(r′) ) ∫0 d (3-15) ∫ σ 2 λz ns
This notation emphasizes that, for r < rd, g depends on r/λz. Here we have combined (2-4) and (2-6) to show that
2zqensλz/σ ) 1
(3-16)
independent of counterion charge. Thus one λz layer contains about 32% of the neutralizing charge, since zθ(r e λz) ) g(λz) ) 1/2(1 - e-1) ) 0.316, while 10 λz layers always contain about 80% of the neutralizing charge.
4296 J. Phys. Chem., Vol. 100, No. 10, 1996
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Figure 3. (a) Effect of competition by monovalent counterions on excess concentration profile of trivalent counterions near a cylindrical surface with the charge density of B-DNA: ns ) 6.65 M and λ3 ) 0.385 Å. Results are from numerical integration of the cylindrical P-B equation. Profiles are normalized by their maximum surface values, which differ by more than four-fold. Monovalent ion concentration nb1 ) 10-2 M. Bulk trivalent ion concentration nb2 from top to bottom ) 10-2, 10-4, 10-6, 10-8 and 10-10 M. The curves for the two highest concentrations coincide. (b) Normalized excess concentration profiles of the monovalent counterions at the same conditions as on previous plots. The line styles denote the same trivalent ion concentrations as in part a. The additional dashed-dotted line is for nb2 ) 0. Figure 2. Excess counterion concentration as a function of distance from a charged surface in various screening regimes for (a) planar and (b) cylindrical macroions. Coordinates are normalized by (a) λz and ns and (b) λz* and ns*, respectively. Counterion and coion have the same charge z. Line types indicate the ratio of bulk concentration to surface concentration, nb/ns. Solid line, limiting nonlinear screening profile for nb/ns , 1; dotted line, 0.1; short dash, 0.5; medium dash, 1; long dash, 10.
Integrating the tail of the distribution (r > rd), (3-14), we find that the fraction of the neutralizing charge residing in it is on the order of (I/ns)1/2 , 1, and tends to zero upon infinite dilution (rd f ∞). The excess counterion profile then tends to the limiting profile independent of the bulk, with the two decay ranges given by (3-12) and (3-13), while the fractional charge neutralized tends to the universal function g(r/λz), Figure 1b, such that g(∞) ) 1. This proves (3-7) in the most general way, showing that the planar surface keeps all its neutralizing charge even at infinite dilution. Figure 2a illustrates changes in the counterion distribution accompanying variation of the ionic strength of the pure salt. The strongly salt-dependent D-H distribution in high salt (nb/ ns > 1) gradually transforms into a salt-independent form at nb/ns , 1. Competitive Binding in the Nonlinear Screening Regime. In the case of several counterion species their total surface concentration will sum to ns, (3-2), with the contribution of each depending on bulk ionic composition. The decay length λzi, (2-4), of each ion’s distribution near the surface depends only on the surface properties and the ionic charge, but is independent of bulk composition, (3-12). Therefore the shape of the short
range distribution of each counterion species changes very little upon titration by the other, but the entire profile decreases proportionately to its surface concentration nsi(nb1,nb2) as the species is displaced by its competitor. This is strictly true only within the range of the exponential decay of the distributions where the expansion (3-12) is valid, but due to the continuity of each distribution it appears to continue to hold for some distance beyond λz. As follows from the above discussion of the single counterion profile, the great majority of the screening ions reside within that distance (see Figure 1b). Thus the amount of each species bound also changes proportionally to its surface concentration in the course of titration. As was already mentioned, (3-2) and (3-3) define the surface concentration without solving the P-B equation. This allows us to obtain a closed form for the binding isotherm, as well as an exact expression for the binding constant. Specializing to a two-counterion system, we develop in detail the consequences of competitive binding near the planar surface. The same features will appear for the cylindrical surface, which constitutes the main interest of this paper. Figure 3 shows the numerical results for trivalent (a) and monovalent (b) counterion distributions at five different bulk concentrations of trivalent species from 10-2 to 10-10 M, while the monovalent salt is kept constant at 10-2 M. Plotted vs distance in λz units and normalized to their surface values, which differ more than fourfold, all profiles are very similar to each other and to the single ion profile, Figure 1a. Therefore, it is natural to suggest that the amount of each species bound, ziΘi, being the integral of the distribution, is essentially proportional to the surface
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J. Phys. Chem., Vol. 100, No. 10, 1996 4297
concentration nsi(nb1,nb2). Taking into account that the screening species completely neutralize the surface (3-7), and using (316), we can write for the charge bound within the distance r
ziΘi(r) ≈ n˜ si(nb1,nb2) g(r/λz)
(3-17)
where we have introduced the reduced surface concentration of the ith species
n˜ si(nb1,nb2) ) nsi/ns
(3-18)
This is a dimensionless function of the bulk counterion concentrations, which changes from 0 to 1 as the species replaces its competitor at the surface. g(r/λz) is the same universal function as for the single species, defined by (3-15) and plotted in Figure 1b. Then the total amount bound (g(∞) ) 1) should be
ziΘi ≈ n˜ si(nb1,nb2)
(3-19)
Simple though it is, (3-17) can be regarded as the major result of this study. It reduces the problem of finding the amount of each species bound, ziΘi, to the calculation of its surface concentration. As was already mentioned, the latter can be found from the boundary conditions (3-2) and (3-3), which for two counterion species in the nonlinear regime (∆nsi ≈ nbi exp(ziψs)) take the forms
n˜ s2 n˜ s1z2/z1
)
n˜ b2
(3-20)
n˜ b1z2/z1
and
n˜ s1 + n˜ s2 ) 1
(3-21)
Hence both reduced surface concentrations depend on the system variables through the single parameter Y
Y)
n˜ b2 n˜ b1
z2/z1
)
nb2
z2/z1-1
nb1
n z2/z1 s
(3-22)
The latter equation gives the values of the reduced surface concentrations n˜ s1 and n˜ s2 with relative accuracy exp(-ziψs) ≈ nbi/ns , 1. Thus for a surface as highly charged as B-DNA with ns ) 6.65 M, n˜ s1 and n˜ s2 are simple universal functions of Y, (3-20) and (3-21), in solutions up to 1 M ionic strength. Therefore to estimate the amount of species bound involves simply calculating Y, (3-22), for the particular solution composition, and then reading off the corresponding n˜ s1 and n˜ s2 values on the universal plot given by the solid lines in Figure 4 (see below). The accuracy of approximation (3-17) needs examination. The discrepancy between ziΘi and n˜ si is due to the change in the shape of the counterion distribution in the course of titration. Figure 3 shows that this change becomes more pronounced for the lower valent species at its lowest degrees of binding. Each distribution maintains its decay length λzi at the surface, so the higher valent distribution is more compact. Therefore, increasing the amount of higher valent species at the surface effectively slows down the further decay of the lower valent profile in the λz < r < rd range. This modest increase in the lower valent concentration, integrated out to a large radial distance, can amount to a noticeable increase in z1Θ1 with respect to n˜ s1. We can estimate this increase in the limit when z1Θ1 f 0, assuming that the potential is that for unperturbed screening by the higher valence ion only
ψ)
( )
ns4λz22 1 ln z2 nb2r2
(3-23)
from (3-13). Here we used the potential within the range λz < r < rd, since at closer distances the profile shapes do not change, while only a very small fraction nbi/ns of the screening charge resides beyond rd. Then the lower valent profile n1 ) nb1ez1ψ is also known, and can be integrated:
z1Θ1 )
( )
z1qe rd ns dr n1(r) ∼ n˜ s1 ∫ λz 1 σ nb1
(1/2)-(z1/z2)
(3-24)
This can be much larger than n˜ s1 if z2/z1 g 2 and ns/nb1 . 1 because of the integration up to the large upper limit rd/λz ≈ ns/(nb1)1/2. Thus the accuracy of approximation (3-17) is better for higher ionic strength. Specifically, we found that z1Θ1 is always larger than n˜ s1 by an amount smaller than 0.03(ns/nb1)(1/2)-(z1/z2), with the more pronounced deviations for z1Θ1 < 0.5. The same accuracy pertains to z2Θ2 ≈ n˜ s2, since z2Θ2 ) 1 - z1Θ1. Such behavior is illustrated by Figure 4. However, the whole picture of competitive nonlinear binding holds only when I < ns (see titration curves for I ) 10 M in Figure 4). Therefore approximation (3-19) is good only within a range of solution ionic strengths which depends on the ion valencies: 1 < ns/nb1 < 301/(1/2-(z1/z2). Thus in a 0.01 M monovalent solution with trivalent counterions added, the actual amount of trivalent charge bound can be smaller than n˜ s2 by 0.08, while for pentavalent ions this quantity is 0.2. For competitive binding of mono- and trivalent cations to the B-DNA double helix, the accuracy of (3-19) is better than 0.2 even for 10-4 M ionic strength. Each experimental method of monitoring titration implies its own definition of “bound” and “free” counterions. To estimate the amount of a species bound within a certain distance from the surface, ziΘi(r), one can take the value of the universal function n˜ si, (3-20) and (3-21), for Y corresponding to the experimental bulk concentrations, (3-22), and multiply it by the corresponding value of g(r/λz) from Figure 1b. The accuracy of this estimate, (3-17), improves rapidly with shorter cutoffs, since the change in shape of the counterion profile becomes less important. 4. Binding to the Cylinder If one of the two radii of curvature of the surface is not the largest length in the problem, the P-B equation in cylindrical coordinates must be solved. Table 1 shows that we may expect four qualitatively different regimes of ligand binding. The first (case 3) is peculiar to the highly charged cylinder (zξ ) a/2λz > 1) and will be shown to closely resemble the nonlinear planar regime (case 2). The other cases pertain to the slightly charged cylinder (zξ < 1). Two (cases 4 and 5) are very similar to the linear planar case 1, while case 6 is specific to the cylinder. The classification of Table 1 illuminates the similarity in screening of highly charged surfaces of various curvatures in finite salt.22 If the Debye length rd is smaller than the radius of surface curvature a, then the latter is not important and screening is reduced to the nonlinear planar case 2. A major point of the current paper, however, is that even in the zero salt limit the highly charged cylinder is very similar in counterion distribution to the plane of the same charge density. This commonality is illustrated by the comparison of Figure 2. Distributions corresponding to the three highest ionic strengths are indistinguishable in the two geometries, while the low salt limiting distributions are just a little different. This strong
4298 J. Phys. Chem., Vol. 100, No. 10, 1996
Rouzina and Bloomfield
Figure 4. Charge neutralization ziΘi by the ith species, approximated with the reduced surface concentrations n˜ si(nb1,nb2) (solid lines), and their numerical values at the bulk monovalent concentrations nb1 ) 10-4, 10-2, 10-1, and 10 M (dashed lines). Lines with the longer dashes correspond to the lower ionic strengths.
similarity is nontrivial: the curves in Figure 2a come from the analytically integrated P-B equations for the plane, while the curves Figure 2b are from numerical integration of the cylindrical P-B equation with parameters specialized for B-DNA. This behavior will be justified analytically below. Case 3: Highly Charged Cylinder in Low Salt: λz < a < rd. The single species screening in low salt can be treated analytically using the solution of the P-B equation within the cylindrical cell model.20,21 To mimic the excess low salt limit, the cell radius R should be such that n(R) ≈ nb < ns/(zξ)2.7 Then we find the surface concentration
ns* ) ns(1 - 1/zξ)2
(4-1)
and the complete profile -2
(ar) [(zξ - 1) ln(ar) + 1]
n(r) ) ns*
2
(4-2)
where we adopt the convention of designating quantities pertaining to cylinders by asterisks, to differentiate them from the analogous planar quantities. Expansion of (4-2) close to the cylinder: (r - a)/a e λz/a ) 1/2zξ yields
(
n(r) ) ns* 1 - 2zξ
r-a + ... a
)
(4-3)
which demonstrates the standard decay length λz ) a/2zξ, (29). This common decay length results from the common surface electrostatic field in both geometries. Comparison of the distribution (4-3) in ns* units to the planar profile (from Figure 1a) in Figure 5 shows that the two decays are also very similar beyond the layer of thickness λz; the cylindrical profile decays just a little faster by a factor of order ln(a/r). We now take advantage of the very small thickness of the counterion layer, λz , a, and express the fraction of neutralizing charge near the cylinder through its planar analog, (3-15),
g*(r/λz) ≈
zns*(2λz*) g(r/λz) σ/qe
(4-4)
where λz* is the thickness of the quasiplanar layer:
λz* )
λz 1 - 1/zξ
(4-5)
Figure 5. Comparison of the counterion distributions near a plane (solid line) and a cylinder (long dashes) in the low ionic strength limit. Concentrations are in units of ns for the plane, and ns* ) ns(1 -1/zξ)2 for the cylinder. For r/λz e 1 both are close to the simple exponential e-r/lz (short dashes).
as can be found by expansion of the expression in square brackets in (4-2). Recalling the definitions of ns and λz, (2-6) and (2-4), we conclude that the situation resembles nonlinear screening of the plane with an effective surface charge density lower for the cylinder by the factor (1 -1/zξ):
σ* ) σ(1 - 1/zξ)
(4-6)
The main physical difference between the two geometries is the total fraction of charge neutralized. This is unity for the plane, (3-7), and
zΘ ) 1 - 1/zξ
(4-7)
for the cylinder, the well-known result of counterion condensation theory.5 This result can be obtained either by direct integration zΘ ) z(2πb)∫∞a dr rn(r) with (4-2), or by substituting expressions (4-1) and (4-5) into (4-4): 2λ* zzqens*/σ ) 1 zξ. In the intermediate ionic strength range, such that ns/(zξ)2 < n(R) ≈ nb < ns, transition to the nonlinear planar regime occurs. The surface concentration changes from ns(1 -1/zξ)2 to ns, the effective decay length from λz/(1 -1/zξ) to λz, and the total fraction of macroion charge neutralized from 1 -1/zξ to 1. At the highest ionic strengths, such that n(R) ≈ nb g ns, the cell model no longer gives a good approximation for the excess salt situation. However, at about the same ionic strength, the excess surface concentration for the cylinder becomes indistinguishable from that for the plane of the same surface charge density. This transition from nonlinear cylindrical screening at very low salt (I , I′′ ) ns/ξ2), through nonlinear planar at low salt (I , I′ ) ns), to linear in high salt (I g ns), is manifested for B-DNA by the weak growth of the counterion concentration at the DNA surface from ns(1 -1/zξ)2 ) 3.9 M below ∼0.1 M to ns ) 6.6 M in ∼1 M salt, with further growth proportional to the bulk ionic strength. Decreasing the ionic strength below a few tenths molar has no effect on the surface counterion concentration or its short range decay length. Such behavior is confirmed by all available numerical studies, but has not until now been recognized as general. Competitive Binding in the Regime of Nonlinear Cylindrical Screening. The similarity between the single ion distributions in planar and cylindrical geometries suggests that corresponding similarities should be observed in competitive binding. This is confirmed by detailed numerical calculations (not shown). The short range decay length λz for each ion remains independent of the relative amount of the competing species.
Competitive Binding of Charged Ligands
J. Phys. Chem., Vol. 100, No. 10, 1996 4299
When normalized to its maximum surface value, the distribution of each species changes very little in the course of titration out to 10-20 λz layers from the surface, which includes the range where the majority of screening ions reside. Therefore the amounts of bound counterions change proportionally to the corresponding surface concentrations in the course of titration, and by analogy with (3-17) we can write
ziΘi(r) ≈ n˜ si*(nb1,nb1)(1 - 1/ziξ)g(r)
(4-8)
Here n˜ si*(nb1,nb2) is the ratio
n˜ si* ) )
nsi* nsi*max
(4-9)
nsi*
(4-10)
ns(1 - 1/ziξ)2
of the actual value nsi*(nb1,nb2) of the excess surface concentration of the ith species at the particular bulk concentration to its maximum value nsi*max ) ns(1 -1/ziξ)2. The fraction of the surface counterion concentration due to the ith species n˜ si*(nb1,nb2) lies between 0 and 1. In (4-8) we also took into account that for the short range cylindrical distribution gi*(r) ≈ (1 -1/ziξ) gi(r), (4-4). As in the planar case, this reduces the problem of determining the amount of each species bound within the arbitrary distance from macroion to the calculation of the surface concentrations. (3-3) is valid in any geometry, so that similarly to (3-20) for the cylinder we have
n˜ s2*/(n˜ s1*)z2/z2 ) Y* where
Y* )
nb2 nb1z2z1
[
nsz2/z1-1
(4-11)
]
(1 - 1/z1ξ)z2/z1 1 - 1/z2ξ
2
(4-12)
However, the cylindrical P-B equation cannot be integrated to provide the boundary condition (3-2), and therefore we cannot obtain the cylindrical analog of (3-21) by analytical methods. Nevertheless, extensive numerical studies for mixtures of monovalent-trivalent, divalent-trivalent, and monovalentpentavalent counterions over a wide range of ionic conditions pertinent to B-DNA show that with high accuracy (∼nbi/ns) a relationship similar to (3-21)
n˜ s1* + n˜ s2* ) 1
(4-13)
holds. This should be even more accurate for higher ion valences and surface charge densities. Together (4-11) and (413) completely determine both surface concentrations n˜ s1* and n˜ s2* and imply that
n˜ si*(Y*) ) n˜ si(Y)
(4-14)
so that our problem of finding ziΘi with (4-8) seems to be solved with the same accuracy as in the planar case. There are, however, two important points to be noted. First, the cylindrical equations (4-8)-(4-13) are to be used for calculating ziΘi only if I , ns/ξ2, or I , 0.4 M for B-DNA. At higher ionic strength the planar equations (3-17), (3-20), and (3-21) should be used instead. Second, one cannot safely apply expressions (4-8)-(4-13) to estimate the total amounts of each species bound, ziΘi(∞). To be sure, (4-8) gives for the maximum values of these quantities 1 -1/ziξ, but this is true only for infinite dilution. At any finite ionic strength the charge is completely neutralized due to the electroneutrality of the
Figure 6. Amounts of each species bound within the r/λzi ) 8 cutoff from the B-DNA surface calculated from (4-8), (dashed lines), and their numerical values from the solution of P-B equation with nb1 ) 10-4 M vs log Y*, (4-12). Charge was integrated up to 3 Å for the trivalent and 9 Å for the monovalent species. The value g(r/λzi)8) ) 0.77 was taken from Figure 1b.
system. Contrary to CC theory, P-B theory shows that the cylindrical radius RM containing the fraction 1 -1/zξ is not constant and small, but diverges as rd1/2 with lowering of the ionic strength.7 The radius containing any fraction of charge higher than 1 -1/zξ diverges even faster. In other words, the residual screening from 1 -1/zξ to 1 is performed beyond rd in the D-H regime, primarily by the lower valent species dominating in the solution. Therefore, if we are interested in the amount of charge bound within a distance shorter than rd, then (4-8) can be used. The ziΘi quantities are essentially equal to those for the plane, multiplied by 1 -1/ziξ as a function of Y* rather than Y. The condition for the cutoff to be smaller than rd is essentially the same as that for the potential energy of the counterions to be lower than kBT. This has been used28 to demonstrate the constancy of the fractional charge bound with ionic strength and its dependence on zξ. The above considerations provide a rationale for that choice. Many experimental measurements (e.g., electrophoretic mobility, equilibrium dialysis, NMR) are sensitive to ions bound with energy kBT or more. Figure 6 compares amounts of each species bound within r/λzi ) 8 from the B-DNA surface, as calculated from (4-8), to their values from numerical solution of the P-B equation with nb1 ) 10-4 M. The charge was integrated up to 3 Å for the trivalent and 9 Å for the monovalent species. The value g(r/ λzi)8) ) 0.77 was taken from Figure 1b. We see that the accuracy of (4-8) in predicting ziΘi is better than 0.05 even at very low ionic strength. To obtain the proper dependence on ziξ, the cutoff for the higher valence ion should be chosen z1/z2 times smaller than for the lower valence (i.e., the same g(r/λzi) value). This is a reasonable choice since the potential energies of the species at a given distance are in that ratio. The total amounts bound, ziΘi(∞), can be reasonably estimated as the sum of the amounts within rd, (4-8), and the residual charge beyond rd partitioned between the species in accord with their share in the ionic strength zi2nbi/I. Case 4: Linear Pseudoplanar Cylindrical Screening: rd < a < λz. We consider now the situation in which the cylinder has a low charge density (a < 2λz, or zξ < 1) and in addition the ionic strength is so high that the Debye length is smaller than the cylinder radius (rd < a or I > ns/ξ2). Both of these conditions are necessary for the potential to be small everywhere. Then expanding the general cylindrical P-B equation, we find that the fraction of charge neutralized by the ith species is
4300 J. Phys. Chem., Vol. 100, No. 10, 1996
ziΘi ) zi∫∆ni(r) d3r )
Rouzina and Bloomfield
zi2n3bi 1 ∫∞ xK (x) dx ) 2I rd/aK1(rd/a) rd/a 1
zi2nbi (1 - O(rd/a)) (4-15) 2I Comparison with (3-2) shows that the binding behavior in this case is essentially the same as for planar geometry. The curvature of the cylinder is felt only in that the minimal ionic strength for linear behavior to apply is somewhat higher: not just I > ns, but I > ns/ξ2, which is significant if ξ < 1. Case 5: Linear Cylindrical Screening: a < rd < λz. This case differs from the previous one only in the relative values of a and rd. The excess surface counterion concentration then is ∆ns ) nb[exp(zψ0) - 1] ≈ nb(2zξ) ln(rd/a)
(4-16)
which, in the case of a single counterion species, behaves like nb ln(1/nb). The thickness of the counterion layer varies as rd ∼ nb-1/2, so the fraction Θ of bound counterions, being proportional to the product of these two factors, decreases with dilution as nb1/2 ln(1/nb). This is the main difference with the previous planar and pseudoplanar cases, where all of the screening charge remains associated with the polyion at any dilution of the salt. It is the result of faster decay of the potential in cylindrical geometry, which also shows up in the smaller than planar fraction of charge neutralized by each species, (32), by the factor [(a/rd) ln(rd/a)2] < 1. The apparent binding constant is smaller by the same factor:
( ( ))
Θ2 z2 a rd ≈ ln K2 ) nb2 2I rd a
2
∼ I-1/2 ln(I)
generally of most interest in analyzing titration data. The apparent binding constant Kobs is defined as the limit of K2, (3-8), as Θ2 f 0. In this case we can find the exact relationship between z2Θ2 and n˜ s2: z2Θ2 ) n˜ s2/(2 - z1/z2), using the same method as was used to obtain (3-22), but in the opposite limit of z1Θ1 f 0. In general, correction to (3-19) can be made by writing
z2Θ2 ) ν2n˜ s2
where the coefficient ν2 has the limiting values 1/(2 - z1/z2) at z2Θ2 ) 0 and 1 at z2Θ2 ) 1. At high degrees of ligand binding, ν2 can become rather large for very low ionic strengths (
