Influence of Ligand Spatial Organization on Competitive Electrostatic

J. Phys. Chem. 1996, 100, 4305-4313

4305

Influence of Ligand Spatial Organization on Competitive Electrostatic Binding to DNA 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 influence of ligand size on electrostatic binding to DNA in a background of competing counterions is analyzed with the planar Poisson-Boltzmann (P-B) equation, which is analytically integrated in two screening layers near the polyion surface: one sterically accessible only to the smaller counterions and the other accessible to both species. We obtain an explicit expression for the dependence of the electrostatic binding constant on the difference in counterion radii ∆δ. This dependence is approximately exponential with decay length λ/z2, where λ is the thickness of the screening layer due to the smaller species and z2 is the ligand valence. λ is determined by the surface charge density of the polyion and the Bjerrum length, and is a few angstroms for double-helical DNA. This approach agrees well with detailed P-B and Monte Carlo calculations for mixtures of hard sphere counterions, and accounts for NMR results on the competitive binding of monovalent and divalent cations. The distribution of charges on the ligand is shown to affect the power S through which the salt concentration enters the ligand binding constant. If the counterion species have similar radii (∆δ < λ), their competition is well described by a point charge model, which gives the conventional S ) z2/z1. If the ligand is much larger than the salt counterions (∆δ . λ), it does not participate in screening at all, so S ≈ 0. An intermediate difference in counterion radii (∆δ ≈ λ) results in an effective ligand charge zeff different from z2. If the ligand is larger than the salt counterion, then zeff < z2. The opposite situation is also possible depending on ligand structure. We propose a simple method for the approximate prediction of the effective charge in competitive electrostatic binding of a ligand with distributed charges.

Introduction The strong salt dependence of multivalent ligand binding to DNA reflects its predominantly electrostatic nature.1 The main features of such binding can be understood by a model in which two point counterion species of different valences compete in screening the highly charged macroion surface.2 Extensive NMR studies of small ligand binding to DNA, summarized in the review by Anderson and Record,3 show that the binding constant in most cases decreases with the concentration of competing salt to the S ) z2/z1 power, as predicted by all basic approaches.1,2,4 The salt and ligand valences are z1 and z2, respectively. The shape of the binding isotherm is also similar to that expected for point charge counterions. However, for some ligands S is a little smaller than z2/z1, and the magnitudes of the nonspecific binding constants for different ligands of the same charge are often different. In this paper we show that such behavior is a consequence of the difference in counterion sizes, or more precisely in the distances of closest approach of the ligand charges to the DNA surface. Our treatment applies primarily to the competitive binding of relatively small organic and inorganic cations, like metal ions, polyamines, and oligolysines, and to larger ligands with low surface charge density. The results are valid for solutions of less than about 1 M salt, including the physiological range of 0.1-0.2 M ionic strength. These limits are imposed by our desire to separate the effect of ligand size from that of screening of the ligand itself by small solute ions. The latter becomes important if the ligand is highly charged and has a large local radius of curvature, so that some of its regions are in the nonlinear (non-Debye-Hu¨ckel) regime. Such screening generally results in lowering of the effective charge of the ligand. The importance of ligand screening depends on its specific shape * To whom correspondence should be addressed. E-mail: victor@ molbio.cbs.umn.edu. Telephone: (612) 625-2268. FAX: (612) 625-6775. X Abstract published in AdVance ACS Abstracts, February 15, 1996.

0022-3654/96/20100-4305$12.00/0

and charge distribution, and can only be quantified numerically. Such calculations recently performed5,6 for λ cI repressor, EcoRI endonuclease, and some minor groove binding antibiotics show the significance of the ligand screening contribution for these large molecules. However, there is always another more direct and stronger effect of the ligand being larger than the competing salt cation. We use the analytical solution of the nonlinear PoissonBoltzmann (P-B) equation developed in the accompanying paper2 (paper 1) with boundary conditions reflecting the main feature of the effect: different distances of closest approach of the two competing cation species. We arrive at an explicit expression for the dependence of the binding constant on the counterion sizes, DNA charge, and solution conditions. Our conclusions agree well with numerical results from P-B and hypernetted chain (HNC) calculations and Monte Carlo (MC) simulations.7-9 Our treatment attributes the 40-fold difference in the binding constants of eight monovalent counterions to DNA to the 6-fold difference in their sizes. For comparison to experiment, the effective radii of the counterions were assumed equal to their hydrated radii determined from conductance measurements. This assumption does not work so well for divalent counterions, except for diamines. This may indicate the importance of water structuring by the strong electric fields around divalent cations upon their binding to DNA. 2. Qualitative Picture of the Influence of Size on Ligand Binding a. Applicability of the P-B Description. The mean-field P-B approach can be applied to size-dependent ligand binding problems because ion sizes, in many cases, enter only through the distance of closest approach to the surface. Studies of solutions containing a single counterion9-12 have shown that the distribution of ions with finite size is very similar to that for point counterions, if counted from the distance of closest © 1996 American Chemical Society

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Rouzina and Bloomfield

approach to the surface. Physically, this comes from the fortuitous cancellation of two approximations of the simplest P-B treatment: ignoring the ions’ excluded volume is balanced by considering their Coulombic repulsion only in the meanfield approximation. Therefore, the distribution is independent of ion size until the size is close to the average distance between ions in the screening cloud. The smallest possible value for that distance,

h ) (zqe/σ)1/2

(2-1)

would be reached if all screening ions were spread uniformly over the macroion surface. For monovalent counterions on the DNA surface h is about 10 Å. σ is the surface charge density of the cylinder:

σ ) qe/2πab

(2-2)

which for B-DNA with radius a ) 10 Å and length per unit charge along the cylindrical axis b ) 1.7 Å equals qe/106.8 Å-2. Therefore, the distribution of counterions with radii up to several angstroms is not significantly different from that predicted from simple P-B theory. The effects of the inaccuracy of the P-B description for larger ions will be discussed in section 3d. b. Simple Picture of the Counterion Distribution near a Highly Charged Surface. Our analysis in paper 1 of the P-B equation for a macroion showed that the majority of the counterions of valence z neutralizing the highly charged surface in a relatively low ionic strength solution reside within a thin surface layer of thickness O(λz)

λz ) qe/4πσlbz

product is proportional to (1 - 1/zξ).2 This is the fraction of cylindrical polyion charge neutralized by z-valent counterions,4 compared to 1 for a plane. This represents a novel point of view on the traditional problem of the counterion distribution around a charged cylinder.3 Indeed, the counterion distribution described above was obtained in numerous numerical studies, but the universality of the λz and ns characteristics and their dependence on the parameters of the system were not understood. The most important idea is that the decay length λz is the fundamental length scale of the short range counterion distribution. It is defined purely by the electrostatic field at the surface E0 ) 4πσ/ , the temperature, and the counterion valence. It has the physical significance of the average distance from the surface in that field of a z-valent ion with energy kBT: zλzqeE0 ) kBT. This decay length characterizes the counterion distribution independent of the presence of other ionic species, ionic strength or ion size, or nonelectrostatic contributions to the binding. c. Consequences for Competitive Electrostatic Binding. The fact that the majority of screening counterions reside within a thin surface layer a few λz in thickness, which does not depend on the ionic strength or composition of the bulk solution, allows the simple analytical calculation of the amount of each of several counterion species electrostatically bound to DNA at any point of the titration.2 By extension of (2-7), the fractional charge neutralization ziΘi from each species i is proportional to the thickness of its screening layer λzi and the concentration in this layer nsi relative to the total surface charge density σ:

(2-3) ziΘi(∆) )

at a concentration close to the surface concentration ns

ns ) 2π(σ/qe)2lb

(2-4)

where

lb )

qe2 kBT

(2-5)

is the Bjerrum length, combining the thermal energy kBT, dielectric constant , and electron charge qe. In water at 20 °C, lb ) 7.14 Å. Then for B-DNA, λ ) λz)1 ) 1.2 Å and ns ) 6.65 M. λ and ns are the key parameters for the short range counterion profile at any solution ionic strength up to ns:

I e ns

(2-6)

The universal counterion distribution in units normalized by ns and λz is shown in Figure 1 of paper 1. At distances r e λz this profile is close to the simple function exp(-r/λz). Its further decay is slowed by the extensive screening of the surface, but it is still independent of the bulk solution concentration up to a distance on the order of the Debye length rd . λz. The majority of screening counterions reside within a few λz layers from the surface (Figure 1b of paper 1). This is also reflected in the fact that the total screening charge per unit surface area is close to unity:

zqens2λz zθ ≈ )1 (2-7) σ The short range counterion profile of the highly charged cylindrical surface (zξ . 1) is very similar to that of the plane. ξ ) lb/b is the linear charge density on the cylinder in lb units, equal to 4.2 for B-DNA under typical conditions. The only difference is that the surface concentration ns is multiplied by (1 - 1/zξ)2 and the thickness λz by 1/(1 - 1/zξ), so that their

ziqensi2λzi nsi g(∆) ) g(∆) σ ns

(2-8)

Here ziΘi(∆) is the fractional charge neutralization within the distance ri ) ∆λzi from the surface. g(∆) is the universal function presented in Figure 1b of paper 1, which saturates at unity within a few ∆. The only quantities dependent on the surface competition are the excess surface concentrations nsi

nsi ) nbi(eziψs - 1)

(2-9)

which always sum to ns

ns1 + ns2 ) ns

(2-10)

In the case of nonlinear screening we can neglect 1 in (2-9) with respect to eziψs since 1/ eziψs ≈ I/ns , 1. Here ψs is the reduced potential ψ ) -qeφ/kBT at the surface, which can be eliminated from the above equations to yield the surface concentrations explicitly in terms of the bulk concentrations:

ns2 (1 - ns2)

z2/z1

nb2

)

nb1

z2/z1-1

n z2/z1 s

(2-11)

and

ns1 ) ns - ns2

(2-12)

Equations (2-11) and (2-12) together with (2-8) completely solve the problem of finding ziΘi(∆) in terms of the bulk composition (nb1,nb2), surface characteristics (ns), and ion charges (z1,z2). In particular, the ligand binding isotherm will have a form

K2 )

(

)

Θ2 z2Θ2 ) Kobs 1 nb2 g(∆)

z2/z1

(2-13)

where the apparent binding constant in the limit Θ2 f 0, Kobs, is

Ion Size Effect on Electrostatic Binding

Kobs )

nsz2/z1-1 z2nb1z2/z1

g(∆)

J. Phys. Chem., Vol. 100, No. 10, 1996 4307

(2-14)

This result is accurate to about 10% in z2Θ2 even for infinite cutoff distance ∆, and rapidly improves with decrease in ∆. Thus it should be a very good approximation for the interpretation of NMR titration data, as we shall do below. Note the conventional dependence of Kobs on the salt concentration to the -z2/z1 power, and the novel expression for its dependence on ns. d. Effect of Differences in the Sizes of Counterion Species. The above results are relevant for the competition of point charges, or for species of the same size, since within the P-B approximation the ionic radii δ1 and δ2 enter only as distances of closest approach to the surface. The situation with different sized ions is presented in Figure 1. There are two distinct regions of screening. The first is the layer between δ1 and δ2 away from the surface (we assume δ1 < δ2 for concreteness). This layer is inaccessible to the larger species, and the only screening ions are the small ones. The second region is that beyond δ2, where both species participate in screening. The most important factor in competition between species is the extent to which smaller ions screen the surface before the larger ones can come into play. This factor is determined by the relation between the thickness ∆δ ) (δ2 - δ1) of the first region and the characteristic thickness of the monovalent species distribution: λ ) λz)1. If ∆δ , λ, then the situation is very similar to the competition of point charges. Binding of the ligand permits its total charge to play a role in electrostatic competition. The main effect of the counterion size difference then appears as a factor exp(z2∆δ/λ) in the effective ligand bulk concentration nb2, or in the effective ns′:

ns′(z2/z1)-1 ) ns(z2/z1)-1e-z2∆δ/λ

(2-15)

With this modification we can still use (2-8) for the amount of ligand bound. On the other hand, if ∆δ . λ, then the ligand ions never play an important role in screening. The electrostatic ligand binding is weak (with free energy less than kBT per ion) and almost independent of the concentration of competing salt. In the intermediate case, when ∆δ ≈ λ, the effective charge zeff of the ligand is smaller than its total charge, since only that part of the ligand which is closer to the surface than δ1 + λ actually participates in electrostatic competition. Then in our point charge description of the competition, both the effective ns and z2 will be altered. They can be found, for example, by fitting the experimental Scatchard plot to (2-13) with respect to z2 and Kobs. The resultant values of these quantities should be reasonably smaller than the total ligand charge z2 and Kobs predicted by (2-14). All of these features are derived from the P-B equation in the Appendix. Below we use another experimental quantity, the binding ratio D

D)

( )( ) pb2 pf1

pf2 pb1

z2/z1

(2-16)

introduced by Record and co-authors (ref 3 and references therein), to test our analytical prediction. pbi and pfi are the bound and the free fractions of the ith species. In our terms the bound fractions can be expressed as pbi ) Θi[P]/nbi (where [P] is the DNA phosphate concentration), while pfi ≈ 1. It

Figure 1. Simplest model of ligand size influence on binding. The sizes of the competing counterion species δ1, δ2 and their distributions with the decay lengths λz are depicted.

follows from (2-8), (2-11), and (2-12) that D should be constant (independent of bulk composition) throughout the titration

D)

(

ns

)

z2/z1-1

g(∆)[P]

[(

) ]

Rz1 ∆δ z1z2/z1 exp - z2 z2 2 λ

(2-17)

determined mainly by the surface charge through ns, and weakly changing with the cutoff distance (g(∆) e 1). The last exponential term contains the dependence on the size difference of the ions, which is tested below by comparison with the numerical P-B and MC results as well as the NMR data. R is a factor of order unity, defined in (A-11). 3. Comparison of Analytical prediction to Numerical Results The main observation of the numerical P-B and MC studies8 is that the logarithm of the binding ratio D varies linearly with the ratio of the ion sizes. This agrees with the result in (2-17), which can be put in the form

( )

δ1 δ2 -1 λ δ1

ln D ) -(z2 - R/2)

(

ln(z2) + (z2 - l) ln

ns

)

g(∆)[P]

(3-1)

Here we used z1 ) 1 to simplify the general expression, since that is the case in all situations below. Fitting of the numerical and NMR results to this expression should provide information on the decay length λ and surface concentration ns in the screening cloud. Comparison of these quantities with their analytically predicted values, (2-3) and (2-4), will test the consistency of our description. a. Binding of Counterions with Similar Sizes but Different Valences. In this simplest case, similar to that for point charges, (3-1) reduces to

(

ln D ) -ln(z2) + (z2 - 1) ln

ns

)

g(∆)[P]

(3-2)

which means that the binding ratio for the ligand should increase strongly with its charge. In Figure 3 this expression is fitted to

4308 J. Phys. Chem., Vol. 100, No. 10, 1996

Rouzina and Bloomfield

Figure 2. Logarithm of the binding ratio D for monovalentmultivalent competition of similar size ions as a function of the valence z2 of the higher valent species. (0) P-B and (O) MC results from ref 8. Dashed lines, theoretical dependence from (3-2) with ns ) 3.4 M for P-B and ns ) 13.2 M for MC, [P] ) 0.02 M, and g(∆) ) 1. The prediction for ns from (2-4), with aeff ) a + δ1 ) 12.76 Å, is 4.1 M.

Figure 3. Dependence of logarithm of binding ratio D on ratio of species sizes δ2/δ1 for monovalent-monovalent ion competition. (0) P-B and (O) MC results from ref 8. Dashed lines, theoretical dependence from (3-4) with λexp ≈ 1.3 Å, δ1 ) 2.76 Å, and R ) 0.88. The analytical estimate for λ from (2-3) is 1.5 Å.

the results of P-B and MC calculations.8 In this simulation, hard spheres with a 2.76 Å radius equal to that of the sodium ion were used. D was indeed remarkably independent of the concentration of DNA phosphate as well as of both bulk counterion concentrations. The plot of ln D vs z2 was linear as predicted by (3-2), with the surface concentration ns ) 3.4 M from the numerical P-B calculations and ns ) 13.2 M from the MC calculations (here we assumed g(∆) ) 1). Our prediction for ns from (2-3) with the surface charge density σ corrected for the effective radius of the cylinder,

aeff ) a + δ1 ) 10 + 2.76 ) 12.76 Å

(3-3)

rather than a ) 10 Å, is 4.1 M, only a little larger than its P-B value. Thus the two P-B approaches yield similar results. The reason for their discrepancy with the MC value, and their relation to the real physical situation, will be discussed below. b. Competitive Binding of Two Monovalent Counterion Species of Different Sizes. If the species have equal valences, for instance z1 ) z2 ) 1, then (3-1) takes the form

( )

δ1 δ2 -1 ln D ) -(1 - R/2) λ δ1

(3-4)

where 0 < R < 1, (A-11). Comparison of this expression to the numerical results from Paulsen et al.8 in Figure 3, shows

Figure 4. Same as in Figure 3 but for monovalent-divalent ion competition. Fitting the P-B dependence with (3-5) yields λexp ≈ 2.0 Å. (2) Experimental data for Mg2+ and Ca2+;13-16 (O) experimental values of ln D for the series of divalent polyamines and methylated divalent polyamines19 positioned on the MC curve in order to estimate the effective distance of closest approach of the charges to DNA (see text).

that ln D indeed depends linearly on δ2/δ1, and that the ligand binding ratio decreases dramatically with increasing size. The intercept ln D ) 0 at δ2 ) δ1 indicates the equal binding ability of equally charged ions of equal sizes. Given the radius of the reference ion (δ1 ) 2.76 Å), and the average slope of the numerical P-B results (-1.37), we can estimate the decay length of ionic distribution to be λ ) (1 - R/2)δ1/slope ) 1-2 Å, depending on the value of R. We estimated R ) 0.88 from (A-11) for the particular bulk counterion concentrations used in the calculation. This yields λ ) 1.3 Å, which agrees quite well with the value of 1.5 Å calculated from (2-3) with σ ) qe/2πaeffb. In general, the decay length of a few angstroms for the dependence of the binding ratio on the ratio of ion sizes indicates the relevance of our approach. (3-4) was applied here in a range of ionic sizes slightly larger than allowed by the condition |δ2 - δ1| < λ, which for δ1 ) 2.76 Å and λ ≈ 1.5 Å means that the size of the competitor should be between 1.3 and 4.3 Å, or the ratio δ2/δ1 between 0.5 and 1.5. At distances larger than a few λ from the surface, the decay of the smaller species distribution decreases, resulting in a slower decay of the apparent binding ratio with δ2/δ1. Thus applying (3-4) in the larger range of sizes may result in a slight increase of the apparent λ. c. Competition of Monovalent and Divalent Counterions. If the counterions have the same size, (3-1) yields

(

ln D(δ2)δ1) ) ln

ns

)

2g(∆)[P]

(3-5)

for z2 ) 2 and z1 ) 1. In the calculation by Paulsen et al.,8 [P] was 0.02 M. Therefore, using the estimate for ns of 3.4 M that we obtained from the slope of the P-B line in Figure 3, we obtain ln D(δ2)δ1) ) 4.5, consistent with its value in Figure 4. Equation (3-1) for z2 ) 2 predicts the slope of the ln D vs δ2/δ1 dependence to be about twice that for monovalent ion competition. The slope of the P-B line in Figure 4 is 1.87, about 1.4 times larger than the slope in Figure 3. The same ratio holds for the slopes calculated by Monte Carlo. The ratio lower than 2 may result from the narrower range of ion sizes in Figure 4. Thus our interpretation explains the higher sensitivity of the binding ratio to the ionic sizes for higher valent ligands. It comes from the reduction of the potential of the polyion at the distance of closest approach of the ligand by the factor

Ion Size Effect on Electrostatic Binding

J. Phys. Chem., Vol. 100, No. 10, 1996 4309

exp[-∆δ/λ], which causes a decrease of surface concentration by the factor exp[-z2∆δ/λ]. d. P-B, MC, and the Actual Size Dependence of Ligand Binding. In Figures 2-4 we have shown that the analytical and numerical P-B results are consistent with each other and have the same qualitative behavior as the MC calculations. This suggests that the simplest P-B approximation for the uniformly charged cylinder provides a reasonable model for the size dependence of electrostatic ligand binding. The deviation of the P-B from the MC results then gives an estimate of the accuracy of the simple analytical P-B theory in describing the actual binding behavior. The MC model considered by Paulsen et al.8 contains two major features which may affect electrostatic binding. There are two helical lines of point charges mimicking the DNA phosphates, instead of a uniform charge on the cylinder, and electrostatic correlations in the counterion cloud are included by the very nature of the MC model. Both effects qualitatively favor stronger binding of the ligand with increase of its charge (Figure 2), and weaker binding of the ligand with the increase of its size (Figures 3 and 4) relative to the P-B prediction. Several numerical studies7,9-12 and our own analytical estimates indicate that neglect of correlations is the main source of inaccuracy of the P-B theory in describing the size dependence of binding, while the nonuniform distribution of charges on the DNA makes only a minor contribution. Correlations, giving rise to inhomogeneities in the thin surface layer due to electrostatic repulsion, worsen the screening, and thus promote additional binding. This effect is strongest for point counterions and is suppressed by excluded volume effects for larger ions. Therefore, correlations introduce an additional size dependence of ligand binding. The numerical results in Figures 3 and 4 show that for mono- and divalent counterions this dependence introduces a relatively small correction to the P-B results. Thus our analytical P-B picture is relevant to the actual binding, but its error will become larger for counterions of higher valences. e. Free Energy of Electrostatic Ligand Binding. The enhancement of the relative binding ability of the ligand with increasing charge is equivalent to an increase in its (negative) free energy of electrostatic binding. The electrostatic free energy per ion (relative to infinite separation of ligand and surface) in units of kBT, in the limit of the zero ligand binding, is

-Ge/kBT ) ln Kobs ≈ z2 z2 ln[ns exp(-z1∆δ/λ)] - ln nb1 - ln ns (3-6) z1 z1 The first term in this expression is the attractive electrostatic energy of the surface weakened by the difference in counterion sizes; the second and third terms arise from ideal entropy of mixing. Thus a larger ligand interacts with a macroion surface of effectively decreased charge density resulting in the effective surface concentration ns′ instead of ns, (2-15). The condition of relatively high free energy of electrostatic binding imposes a limit on the solution ionic strength:

I ≈ nb1 < ns′ ≈ ns exp(-z1∆δ/λ)

(3-7)

This replaces the condition defining the nonlinear screening regime for point charges, (2-6). For DNA with ns ≈ 4.1 M, it requires I e 1 M. At higher salt, electrostatic binding is always weak and its salt dependence negligible. Below about 1 M the salt dependence of binding is strong and with power proportional to the ligand charge.

4. Comparison with NMR Binding Data a. Surface Electrostatics from the NMR Binding Data. A major accomplishment of the study by Paulsen et al.8 was to demonstrate that MC calculations account for NMR data on binding competition between monovalent ligands, if the hard sphere ion radius is taken equal to the hydrodynamic radius. Thus in Figure 3 we see that the 6-fold variation in the ion size reasonably accounts for the 40-fold variation in the binding ability of eight monovalent counterions. That suggests that binding of the monovalent cations studied is almost purely electrostatic, with the possible exception of Li+. The slope of the experimental ln D vs δ2/δ1 plot is 1.7 ( 0.3, leading to an experimental value of the characteristic length λ between 1 and 2 Å, according to (3-4). This is close to our predicted 1.5 Å, indicating reasonable values of the electrostatic parameters used. That is, factors like the difference of the dielectric constant of water near the surface from its bulk value, the presence of a dielectric discontinuity at the boundary, and the specific distribution of phosphate charges on DNA alter the actual surface electrostatic field by not more than twofold. b. NMR Data on the Binding of Divalent Cations. Multivalent cations sometimes have a strong affinity for particular base sequences on the double helix. In most cases, e.g., with Mg2+, Ca2+, and Co(NH3)63+, these constitute only a small fraction (less than 10%) of the total electrostatic binding sites.13-17 The behavior observed over a wide range of ligand binding is still predominantly electrostatic. This follows from the fact that the effective charge of the ligand that yields the most constant binding ratio D in (2-17), or that is determined from the power law dependence of Kobs on the monovalent salt concentration, is close to the nominal charge z2. In Figure 4 we have combined data from several NMR binding studies on divalent cations by the Record group.8,18,19 Mg2+ and Ca2+ bind more strongly than expected on the basis of their solution hydrodynamic radii: 5.3 and 4.7 Å, respectively. Since D was determined from measurements made over a range of ion binding >10%, where the binding is nonspecific, this indicates that the extended water structure induced by the strong fields around these compact ions affects their binding to DNA. However, the change in free energy of ligand binding due to this putative water structure effect is much smaller than the electrostatic free energy of binding. The additional binding energy, in units of kBT per ion, is equal to the difference between the experimental and theoretical values of ln D in Figure 4: ∼2.5 for Mg2+ and ∼1.5 for Ca2+. The electrostatic energy in the same units is considerably larger, ∼6.9 for Mg2+ and 7.4 for Ca2+, estimated from Ge/kBT ≈ ln D + (z2 - 1) ln([P]/nb1) + ln z2, where [P] ) 0.02 M and nb1 ) 0.01 M were the values in the NMR experiment. This behavior is quite different from that of the divalent transition metals like Mn2+, Cu2+, and Zn2+, which coordinate strongly to the DNA bases. Their free energy of binding is almost independent of salt, and is much larger than Ge. Polyamines, and especially methylated polyamines, are multivalent cations known to have little if any specific interaction with B-DNA. Competitive binding in sodium buffer of a series of divalent polyamines H3N+-(CH2)m-+NH3 and their methylated analogs (H3C)3N+-(CH2)m-+N(CH3)3 (m ) 3, 4, 5, or 6) was measured by NMR.18,19 The binding ratio D with respect to sodium ions calculated from the two-state model depends to some extend on the value of the model parameter r0Na. But the main result, the two-10-fold reduction of the binding of the polyamines after methylation, is obvious in Figure 4. It is very similar to the 10-fold reduction of D after methylation of the simple ammonium ion NH4+.20 If we use the experimental

4310 J. Phys. Chem., Vol. 100, No. 10, 1996

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values of D together with the ln D vs δ2/δ1 MC curve to estimate the effective size for each cation, δ2 for the nonmethylated ions appears to be much closer to the hydrodynamic radius of NH4+ (∼2.5 Å) than of the entire diammine ion (∼5.5 Å). This is consistent with the idea that the distance of closest approach of the charges, rather than the actual ion size, determines the binding. The change in ln D with methylation for the two first amines in the series (m ) 3 and 4) is of the same order as for the methylation of NH4+, and according to Figure 4 corresponds to ∼2.3 Å increase in the distance of closest approach. For the ions with longer chains (m ) 5 and 6) the apparent increase is smaller (∼1 Å), perhaps due to their flexibility,21 whose effect will be briefly discussed in the next section. 5. Large Ligands with Distributed Charges a. Analysis of Experimental Data. Many natural DNA ligands are large globular proteins with distributed charges, which bind to DNA in a solution containing buffer ions. The situation when the buffer ion is small compared to the ligand is schematically represented in Figure 5. It is clear from the above discussion that only those charges of the ligand which can come closer to DNA than

δcr ) δ1 + λ

(5-1)

will participate in competitive screening. The macroion surface will be essentially completely screened to ligand charges beyond δcr by the buffer cations. For a sodium ion buffer, δcr ) 2.8 Å + 1.5 Å ) 4.3 Å. Thus we can get an idea of the ligand’s effective charge by calculating the sum of all its charges zi which can come closer than δcr, each weighted by the factor exp[∆δ/λ]

zeff ) max

∑

δi z2. Thus we predict that a small ligand, in a solution with larger buffer cations, will bind electrostatically to DNA with an effective charge higher than its nominal value. An example of this is given by the minor groove-binding antibiotics discussed in section c below. b. Effects of Ligand Flexibility. The binding of flexible ligands like polyamines14 and oligolysines22-24 to DNA and other polyions25 raises some additional complications. The ligand flexibility on the one hand allows closer contact of all its charges with the DNA surface, but on the other hand introduces additional degrees of freedom which, if frozen out upon binding, reduce the entropy and thereby the binding affinity. Stigter and Dill21 examined this situation, considering the charges to be subjected to the P-B potential of the uniformly charged rod as well as to connectivity constraints, and integrating the bound charge numerically for each solution condition. The ligands were modeled as chains of freely jointed segments with charges placed at the centers of the segments. The distance of closest approach of segmental charge to the rod was used as a fitting parameter for comparison with experiment. Varying this distance within a few angstroms did not affect the slope S, but changed the apparent binding constant by up to 3 orders of magnitude. These features are consistent with the picture of

Ion Size Effect on Electrostatic Binding competitive binding of ligands of slightly different sizes that we have developed here. It is hard to clearly separate the effect of chain flexibility from that of other parameters in the Stigter-Dill theory. Its effect on S ) zeff/z1 for moderate chain lengths is not very large. The calculated effective charge for di-, tri- and tetravalent oligolysines is at least 80% of the nominal ligand charge. However, the relative reduction of the effective charge (1 zeff/z) increases with degree of oligomerization such that for the 10-mer it reaches almost 0.5. The authors consider this effect to be an artifact of the P-B approximation which does not work very well for highly charged ions. However, correlation between the ions results in their tighter binding to the polyion,26-28 which would increase, rather than decrease, their effective charge. Whatever the explanation, it appears that flexibility most affects the effective charge of longer oligocations. c. Comparison with Detailed Numerical P-B Calculations on Binding of Antibiotics and Globular Proteins to DNA. Recently the results of very detailed P-B calculations of the salt dependence for several ligands binding to DNA became available.5,6 Both DNA and ligand were represented with considerable detail and treated as macromolecules immersed in 1:1 salt. For both separated and bound configurations the free energy of interaction with small (2 Å radius) salt ions was calculated. The dependence of the free energy of binding ∆∆Gs on the salt concentration was analyzed in terms of the effective charge of the ligand zeff ) -∂∆∆Gs/∂ ln [M+] ) -∂ ln Kobs/∂ ln [M+], which appeared to be in reasonable agreement with experiment. Two groups of ligands were studied: small minor groove binding mono- and divalent antibiotics, and large highly-charged globular proteins. From the electrostatic point of view the main difference between these ligands is that molecules from the first group are weakly screened by the salt in the Debye-Hu¨ckel regime, while those from the second group can accumulate high concentrations of counterions close to them, or, in other words, be screened in the nonlinear regime. Since our approach neglects correlations within the DNA screening environment, it should work better for the first group. The error in our estimate of the free energy of binding then should be close to the free energy of the isolated ligand molecule relative to its uncharged state, which should be less than kBT per ligand molecule for the first group, and up to several kBT for the second. That is in fact what was obtained in the numerical calculations as seen in Figure 3 of ref 5 and Figure 2 of ref 6. Therefore our simple approach should better estimate zeff for antibiotics than for proteins. Indeed, the results for binding of divalent DAPI and netropsin as well as monovalent Hoechst 33258 to the various sequences in Tables 1 and 3 of ref 5, show that the calculated zeff for these ligands is generally a little larger then the actual charge z. We attribute this to the closer approach of the antibiotic charges to the DNA surface in the bound state relative to the 2 Å distance for the salt counterions (mimicking Na+). The only exception is complex II of Hoechst 33258, which is consistent with our reasoning since it is the only complex in which the charge of the antibiotic molecule is not embedded in the groove but reaches out into solution. Here we again emphasize that it is not the distance of the ligand charge to the nearest phosphate on the DNA, but rather the distance to the cylinder with a 10 Å radius representing the average DNA surface, which should be used. In fact the study of Misra et al.5 showed that using the average cylinder instead of the detailed B-DNA model has only a minor effect on the calculated zeff.

J. Phys. Chem., Vol. 100, No. 10, 1996 4311 For the proteins two major additional effectssthe change in the DNA conformation upon binding and extensive screening of the protein molecule by the saltscan introduce a further salt dependence of binding. Despite this complexity there are still effects which can be understood in terms of simple geometric factors. Substitution in the λcI repressor of Glu34-1 by Lys+1 makes a change of about 3.5 in the calculated and experimental zeff for binding to both the operator and nonspecific DNA. This change in zeff, larger than the +2 change in actual protein charge, occurs because the substitution is at the protein-DNA interface. Therefore in the bound state the highly charged surfaces approach each other more closely than the layers of 2 Å radius Na+ ions which screen them in the unbound state. The wild type EcoRI endonuclease and its three deletion mutations in Table 1 of ref 6 provide an opposite example. The deletions of the first 4, 12, and 29 amino acid residues each decrease the total protein charge by 2, but produce a smaller reduction of zeff. This occurs because the deletions are made in the N-terminal arm of the protein, far from the contact surface. The authors6 state that protein charges farther from the surface than 10-15 Å do not affect the salt dependence of binding. This distance is much smaller than the Debye screening length at the lower end of the studied ionic strength range (∼10-4 M). It appears due to the nonlinear screening of the DNA by the salt counterions. However, this screening is already very effective at the smaller distance δcr ) δsalt + λ, (2-2), which in this case is 2 + 1.5 ) 3.5 Å. Thus we predict that protein charges farther than 3.5 Å from the DNA surface (not from the individual phosphate charges) will have minor effects on zeff. These considerations cannot substitute for detailed numerical calculations, but can help to interpret and sometimes even qualitatively predict their results. 6. Conclusions We can now summarize the effects of the spatial organization of the ligand on its competitive electrostatic properties. The effective charge of the ligand zeff, defined by the power S ) zeff/z1 of the dependence of binding constant on the salt concentration, can be equal to, smaller than, or even larger than its total charge z2. zeff is determined by the relative magnitudes of the difference in the distances of closest approach of the two competing species, ∆δ ) δ2 - δ1, and the screening length λ. If ∆δ/λ < 1, then we expect that zeff ≈ z2. For B-DNA under typical experimental conditions λ ≈ 1.5 Å according to (2-3), and the size of the salt counterion is usually in the range δ1 ≈ 2-3 Å. Therefore ligands with effective radii (or distance of closest approach of the charges to the DNA surface) up to δcr ) δ1 + λ ) 3.5-4.5 Å can reveal their total charge in the electrostatic competition. Examples of this sort include monovalent, hydrated metal ions whose hydrodynamic radius defines their distance of closest approach; more complex but compact and symmetric cations such as hexaamminecobalt(III) with effective radius 3.5 Å;29 and linear, flexible polycations such as homologous series of diamines,19 polyamines,14 and polylysines22 whose individual charges can approach as close to the DNA as the Na+ or K+ buffer ions. Although the slope S remains unaltered for such ions, the binding constant Kobs and binding ratio D are still expected to vary severalfold in proportion to the factor exp(-z2∆δ/λ). With ligand size ∆δ as the variable, the characteristic decay length is λ/z2, so the size effect will be more pronounced for larger ligand charge. However, because λ is small the size dependence is strong even for monovalent ligands, as illustrated by the 40fold variation in binding constant among the eight monovalent counterions shown in Figure 3.

4312 J. Phys. Chem., Vol. 100, No. 10, 1996

Rouzina and Bloomfield

In the other limiting case, where the competing counterions have very different sizes and ∆δ/λ . 1, the larger ligand ions will not significantly bind to polyions through electrostatic interactions, until the ligand concentration is on the order of the ionic strength. The electrostatic free energy of such binding is < kBT per ion, and its salt dependence is negligible, leading to zeff < 1. For example, the binding of globular proteins whose charges are far away from the specific binding site is practically unaffected by the buffer concentration. In the intermediate range of ligand sizes, where ∆δ/λ ≈ 1, zeff of a multivalent ligand will be >1, causing strong salt dependence of the ligand binding, but xd (region II) yields the reduced electrostatic field f(x) ) -∂ψ/∂x in each region:

1 1 (fI)2 ) (∑ nbieziψ + C1) I i*2

(A-3a)

1 II (fII)2 ) (∑ nbieziψ + C2) I i

(A-3b)

The constants of integration C1 and C2 can be determined from the boundary conditions:

fII(xf∞) ) 0

(A-4a)

ψI(xd) ) ψII(xd) ) ψd

(A-4b)

fI(xd) ) fII(xd) ) fd

(A-4c)

fs ) f(x)0) )

()

4πσqe rd ns ) ) kBTrd λ I

1/2

Here λ ) λ(z)1) and ns are given by (2-3) and (2-4). Then C2 ) 0, C1 ) nb2ez2ψd, and

1 (fs)2 ) (∑ nbieziψs + nb2ez2ψd) I i*2

Appendix. Analytical Solution of the P-B Equation for Competitive Screening of a Highly Charged Plane by Two Counterion Species of Different Sizes To understand the influence of ion size on competition, it is sufficient to solve the planar form of the P-B equation:

∂2ψ ∂x2

)

1 2I

∑i zinbi exp(ziψ)

(A-1)

where ψ ) -φqe/kBT and x ) r/rd are the reduced electrostatic potential and distance from the plane, rd and I are the Debye screening length and ionic strength, rd2 ) kBT/8πqe2I and I ) 1/ ∑ z 2n , respectively, and n and z are the bulk concentrations 2 i i bi bi i and valences of each species. Screening starts at δ1, the distance of closest approach of the smaller counterion. At δ2 the larger species starts to participate in the screening. Thus it is convenient to measure distances starting at δ1 from the surface, so that x ) (r - δ1)/rd, and the boundary between screening areas is

xd ) (δ2 - δ1)/rd ) ∆δ/rd

(A-2)

(A-5)

where ψs ) ψ(x)0). Equations (A-3) can be further integrated, yielding the desired dependence of the potential on distance in the form

dψI s f(ψI)

x(ψ) ) ∫ψ

ψ

dψII d f(ψII)

x(ψ) - xd ) ∫ψ

ψ

Acknowledgment. This research was supported by NIH Research Grant GM28093. We are grateful to Drs. Timothy Lohman and David Mascotti for sharing their binding data on oligolysine, and to Drs. Dirk Stigter and Ken Dill for a preprint of their work on binding of flexible charged ligands.

(A-4d)

(A-6a)

(A-6b)

At x ) xd these equations give the connection between ψs, ψd, and xd:

xd ) I1/2∫ψ dψ (∑ nbieziψ + nb2ez2ψd)-1/2 ψs d

(A-7)

i*2

This equation, together with (A-4) and (A-5), allows determination of the potentials ψs and ψd in terms of xd (the counterion sizes) and fs (the surface charge). In this paper we are concerned with a highly charged surface screened in the nonlinear regime, so that the electrostatic ligand binding to DNA is strong (free energy more than kBT per ion). Then fs . 1, and different regimes of ligand-salt counterion competition arise depending on the value of the parameter xdfs ) ∆δ/λ. If the ion sizes are only slightly different, so that ∆δ/λ , 1 and xdfs , 1, (A-5) and (A-7) yield

ψd ≈ ψs - xdfs

(A-8a)

ns ) nb1ez1ψs + nb2ez2ψd ) ns1 + ns2

(A-8b)

This shows that the sum of the surface concentrations

ns1 ) nb1ez1ψs

(A-9a)

ns2 ) nb2 exp(z2ψd) ) nb2 exp(z2ψs) exp(-z2∆δ/λ) (A-9b)

Ion Size Effect on Electrostatic Binding

J. Phys. Chem., Vol. 100, No. 10, 1996 4313

is always equal to ns. The only difference with the analogous result for point charges, (2-10), is the factor exp[-z2∆δ/λ] for the ligand. As with (2-11) we want to calculate the amount of each species bound as a product of its surface concentration and the decay length. For the first species the decay, being determined by fs, (A-4), is still given by (2-4) independent of competition. For the second species the decay is governed by the field fd rather than fs. Therefore, the decay length of the second species distribution λ2 different by the factor fs/fd:

λ2′ ) λ2(fs/fd)

(A-10a)

(fs/fd)2 ) 1 + Rz1fsxd ) 1 + Rz1∆δ/λ

(A-10b)

∆ns2 ≈ nb2(ez1ψd - 1) ≈ nb2z1ψd ≈ nb2e-∆δ/rd

(A-15)

and the Debye length rd. Both quantities depend weakly on the concentration of buffer through rd ∼ nb11/2, so that the ligand binding constant also varies slowly with nb1. In other words, if the difference in sizes of the ligand and buffer ions is larger than a few λ, the larger species does not participate in counterion competition, it behaves as if its charge were very small (zeff < 1), and its free energy of electrostatic binding is less than kBT per charge. For intermediate values of the size difference, ∆δ ≈ λ, the larger ions bind electrostatically as particles with effective charge lower than their total charge: zeff < z2. References and Notes

where

R ) (1 + nb2e

z2ψd

z1ψd -1

/nb1e

)

(A-11)

is less than unity and depends on the bulk ion concentrations. Binding measurements, as well as P-B and MC calculations, are usually performed for bulk concentrations that provide comparable amounts of the two species at the surface. Then R ≈ 1/2, so that fs/fd ≈ 1 + z1∆δ/4λ. Thus the two fields fs and fd are only slightly different. The fractional charge neutralization by the bound ligand is then

ziΘi(∆) ≈

ziqensi2λzi ∆δ R ns2 z - z ≈ exp σ λ 2 2 1 ns

(

(

))

(A-12)

which differs from (2-8) only by the exponential size-dependent factor. The same factor will appear in the binding constant, (2-14), and the binding ratio D. However, the rest of the binding isotherm (2-13) will not be altered, so that the effective ligand charge will still equal the total charge: zeff ) z2. If the ion sizes are very different, ∆δ/λ . 1 and xdfs . 1, then the potential at the boundary is much smaller than at the surface: ψd , ψs. Then (A-5) and (A-7) take the form

fs2 )

nb1 z1ψs e I

xd ) I1/2∫ψ dψ (∑ nbieziψ)-1/2 ψs d

(A-13a) (A-13b)

i*2

Integration of the latter expression with the condition ψd , ψb gives

xd ) ln(4/z1ψd)

(A-14a)

z1ψd ) 4 exp(-xd) ) 4 exp[-∆δ/rd]

(A-14b)

Thus the maximum potential experienced by the larger species, ψd, decays exponentially with the size difference ∆δ and with the characteristic decay length rd, as might be expected for a surface almost completely screened by the smaller species. The remainder of the charge is screened by the larger ions in the linear (Debye-Hu¨ckel) regime. The amount of bound ligand is then proportional to the product of its excess concentration at the boundary:

(1) Record, M. T., Jr.; Anderson, C. F.; Lohman, T. M. Q. ReV. Biophys. 1978, 11, 103. (2) Rouzina, I.; Bloomfield, V. A. J. Phys. Chem. 1996, 100, 4292. (3) Anderson, C. F.; Record, M. T., Jr. Annu. ReV. Biophys. Biophys. Chem. 1990, 19, 423. (4) Manning, G. S. Q. ReV. Biophys. 1978, 11, 179. (5) Misra, V. K.; Hecht, J. L.; Sharp, K. A.; Friedman, R. A.; Honig, B. J. Mol. Biol. 1994, 238, 264. (6) Misra, V. K.; Sharp, K. A.; Friedman, R. A.; Honig, B. J. Mol. Biol. 1994, 238, 245. (7) Paulsen, M. D.; Richey, B.; Anderson, C. F.; Record, M. T., Jr. Chem. Phys. Lett. 1987, 139, 448. (8) Paulsen, M. D.; Anderson, C. F.; Record, M. T. J. Biopolymers 1988, 27, 1249. (9) Bacquet, R. J.; Rossky, P. J. J. Phys. Chem. 1988, 92, 3604. (10) Le Bret, M.; Zimm, B. H. Biopolymers 1984, 23, 271. (11) Mills, P.; Anderson, C. F.; Record, M. T., Jr. J. Phys. Chem. 1985, 89, 3984. (12) Mills, P.; Anderson, C. F.; Record, M. T., Jr. J. Phys. Chem. 1986, 90, 6541. (13) Braunlin, W. H.; Xu, Q. Biopolymers 1992, 32, 1703. (14) Braunlin, W. H.; Strick, T. J.; Record, M. T., Jr. Biopolymers 1982, 21, 1301. (15) Braunlin, W. H.; Nordenskio¨ld, L.; Drakenberg, T. Biopolymers 1991, 31, 1343. (16) Braunlin, W. H.; Drakenberg, T.; Nordenskio¨ld, L. J. Biomol. Struct. Dyn. 1992, 10, 333. (17) Xu, Q.; Jampani, S. R. B.; Braunlin, W. H. Biochemistry 1993, 32, 11754. (18) Padmanabhan, S.; Richey, B.; Anderson, C. F.; Record, M. T., Jr. Biochemistry 1988, 27, 4367. (19) Padmanabhan, S.; Brushaber, V. M.; Anderson, C. F.; Record, M. T., Jr. Biochemistry 1991, 30, 7550. (20) Bleam, M. L.; Anderson, C. F.; Record, M. T., Jr. Proc. Natl. Acad. Sci. U.S.A. 1980, 77, 3085. (21) Stigter, D.; Dill, K. A. Personal communication. (22) Mascotti, D. P.; Lohman, T. M. Proc. Natl. Acad. Sci. U.S.A. 1990, 87, 3142. (23) Mascotti, D. P.; Lohman, T. M. Biochemistry 1992, 31, 8932. (24) Mascotti, D. P.; Lohman, T. M. Biochemistry 1993, 32, 10568. (25) Mascotti, D. P.; Lohman, T. M. Biochemistry 1995, 34, 2908. (26) Le Bret, M.; Zimm, B. H. Biopolymers 1984, 23, 271. (27) Jonsson, B.; Wennerstrom, H.; Halle, B. J. Phys. Chem. 1980, 84, 2179. (28) Svensson, B.; Jonsson, B.; Woodward, C. E. J. Phys. Chem. 1990, 94, 2105. (29) Gessner, R. V.; Quigley, G. J.; Wang, A. H.-J.; van der Marel, G. A.; van Boom, J. H.; Rich, A. Biochemistry 1985, 24, 237. (30) Sinden, R. R. DNA Structure and Function; Academic Press: San Diego, 1994. (31) Ha, J.-H.; Spolar, R. S.; Record, M. T., Jr. J. Mol. Biol. 1989, 209, 801. (32) Livingstone, J. R.; Spolar, R. S.; Record, M. T., Jr. Biochemistry 1991, 30, 4237. (33) Spolar, R.; Record, M. T., Jr. Science 1994, 263, 777.

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