Asymptotic solution of the cylindrical nonlinear Poisson–Boltzmann equation at low salt concentration: Analytic expressions for surface potential and preferential interaction coefficient

Abstract

The analytic solution to the nonlinear Poisson–Boltzmann equation describing the ion distributions surrounding a nucleic acid or other cylindrical polyions as a function of polyion structural quantities and salt concentration ([salt]) has been sought for more than 80 years to predict the effect of these quantities on the thermodynamics of polyion processes. Here we report an accurate asymptotic solution of the cylindrical nonlinear Poisson–Boltzmann equation at low to moderate concentration of a symmetrical electrolyte (≤0.1 M 1:1 salt). The approximate solution for the potential is derived as an asymptotic series in the small parameter ɛ⁻¹, where ɛ ≡ κ⁻¹/a, the ratio of the Debye length (κ⁻¹) to the polyion radius (a). From the potential at the polyion surface, we obtain the coulombic contribution to the salt–polyelectrolyte preferential interaction (Donnan) coefficient (Γ) per polyion charge at any reduced axial charge density ξ. Γ is the sum of the previously recognized low-salt limiting value and a salt-dependent contribution, analytically derived here in the range of low-salt concentrations. As an example of the application of this solution, we obtain an analytic expression for the derivative of the midpoint temperature of a nucleic acid conformational transition with respect to the logarithm of salt concentration (dT_m/d ln[salt]) in terms of [salt] and nucleic acid structural quantities. This expression explains the experimental observation that this derivative is relatively independent of salt concentration but deviates significantly from its low-salt limiting value in the range 0.01–0.1 M.

The cylindrical nonlinear Poisson–Boltzmann (NLPB or PB) equation is widely used for calculating electrostatic potential around rod-like charged objects surrounded by mobile ions both in the theory of polyelectrolyte solutions (1, 2), in applications in plasma physics (3) and in colloid and surface sciences (4). In particular, the surface coulombic potential (and/or its distance dependence) of a charged polyion in an electrolyte solution is required for calculations of such thermodynamic properties as electrostatic free energy (5, 6), polyion-ligand binding constant (7, 8), and especially the fundamental thermodynamic quantity Γ, the coulombic contribution to the preferential interaction coefficient (equivalent to the experimentally observable Donnan coefficient). This coefficient is required for interpretation of thermodynamic experiments on salt–polyion interactions and on effects of salt concentration ([salt], C_b) on polyion processes (1, 9). Numerical calculations of NLPB solution for the model of a long periodically charged polyion as an infinite charged cylinder characterized by only two structural quantities, reduced charge density ξ§ and radius a, are known to successfully describe experimentally measured thermodynamic properties of polyelectrolyte solutions (1). Monte Carlo simulations confirm the accuracy of the cylindrical PB equation in the presence of added univalent salt up to 0.1 M (10, 11). However, despite numerous studies devoted to solving the NLPB equation (2, 3), no sufficiently accurate analytic solution for the cylindrical NLPB equation was known at low- to moderate-salt concentration.

Solution of cylindrical NLPB equation is more challenging at low-salt (LS) concentration than at high [salt], where several useful accurate approximations for the potential, electrostatic free energy and preferential interaction coefficient are known (4, 12–14). In the absence of added salt, the exact analytic solution of this equation exists in the form of elementary functions (“salt-free” solution) for the cell model (15). For an infinite space, Ramanathan (16) derived asymptotic approximation for NLPB potential, and MacGillivray and Winkelman (17) obtained matched asymptotic solution for a weakly charged polyion (ξ < 1). Other solutions of the cylindrical NLPB equation (18, 19) provide analytic expressions for the potential but require numerical calculations for constants. Although the NLPB potential in the presence of salt does not differ greatly from the “salt-free” potential (16, 20), the salt–polyelectrolyte preferential interaction coefficient Γ depends exponentially on the coulombic potential at the polyion surface (21) and is very sensitive to small errors in the latter. We therefore sought a highly accurate analytic expression for the NLPB surface potential of a highly charged polyion accurate at low-to-moderate [salt] and dilute polyion concentration, expressible in elementary functions and useful in the analytical treatment of thermodynamic properties of polyelectrolyte–salt solutions.

Salt–nucleic acid preferential interaction coefficients, which characterize the net thermodynamic consequences of cation accumulation and anion exclusion in the vicinity of a nucleic acid polyion, have been measured by dialysis (22). Differences between preferential interaction coefficients of reactant(s) and product(s) of a nucleic acid process are the fundamental thermodynamic determinant of the strong dependence of the thermodynamics of that process on [salt] (1). Experimentally well characterized examples include the approximately linear dependences on logarithm of [salt] of the melting temperature T_m of nucleic acid conformational transitions (23) and of standard free energy change ΔG = −RT ln Κ_obs (where Κ_obs is the binding constant) for binding of oligocations and other charged ligands to nucleic acids (24) (for more references, see ref. 1). Numerical calculations of preferential interaction coefficients, Donnan coefficient, or related quantities using the NLPB equation and Monte Carlo simulations have been reported in refs. 9–11 and 25–28 and refs. therein. For extremely low but excess [salt], Gross and Strauss (26) deduced from the numerical NLPB solution and Manning (29) subsequently derived from the counterion condensation hypothesis the limiting law (LL) analytic expression for Γ, Γ = −(4ξ)⁻¹ at ξ ≥ 1 and Γ = −1/2 + ξ/4 at ξ < 1. More recently, analytic expressions for Γ at high [salt] have been derived (6, 13). For single-stranded (ss) and double-stranded (ds) nucleic acids, significant deviations from the LL expressions were observed even at submicromolar [salt] (26, 30), but no analytic expression for Γ was available at salt concentration from 10⁻⁶ to 0.1 M.

Previous numerical and approximate analytical solutions of the cylindrical NLPB equation show that the potential at the cylinder surface, y_a, is very large when the surface charge density is high (i.e., ξ ≫ 1), especially at low [salt]. Benham (31) explained this behavior by showing the existence of a singular point of the Painleve equation, to which the cylindrical NLPB equation reduces when written in terms of coion concentration. The exact solution for the Painleve equation is available in the form of a power series in the vicinity of the singular point (31). This solution contains two unknown constants; analytic determination of these constants is required for the expansion at the singular point to have more than a theoretical significance and to serve as a basis for evaluation of the electrostatic potential around the polyion.

Here we present a LS asymptotic expansion of the exact solution of the cylindrical NLPB equation in the form of an asymptotic series in the small parameter ɛ⁻¹, where ɛ⁻¹ is the ratio of the polyion radius (a) to Debye length (κ⁻¹), κ² ≡ 2n_bz²e²/DD₀k_BT, where n_b is the bulk concentration (at r → ∞) of either cation or anion (in SI units), and z is the cation valency. The functional form of the zeroth order term in this expansion is analogous to the “salt-free” potential but with the modified Bessel function of the second kind, Κ₀(κr), replacing the logarithm of the radial coordinate r. Two integration constants in this term are calculated from two boundary conditions. These constants, which depend on ɛ and ξ, determine the salt dependence of the preferential interaction coefficient, Γ. We obtain an analytic expression for Γ as the sum of the LL (low but excess salt) value, Γ, and a [salt]-dependent term. Although results of this work are obtained for any symmetrical electrolyte, we discuss applications of the results to 1:1 salt, because neglecting ion correlation effects in the PB approach reduces its accuracy for multivalent ions (11, 32). The PB solution for multivalent ions may prove useful in extensions of the PB equation (e.g., ref. 33), where the potential is described by the cylindrical NLPB equation everywhere except in a layer near a polyion surface.

The LS asymptotic expansion allows one to calculate the position of the singular point of the PB solution. For the highly charged cylinder, (ξ ≫ 1), we show that the singular point is located in the polyion interior at a distance on the order of a/ξ from the cylinder surface.

The NLPB Potential and Its Relation to Γ.

The NLPB equation describing the electric potential around a uniformly charged cylindrical polyion immersed in a symmetrical (z:z) electrolyte at infinite dilution of the polyion is

1

2

3

where y ≡ zeψ/k_BT, σ* ≡ σeza/DD₀k_BT. Here, ψ and y are reduced and actual potentials, respectively; y′ = dy/dr; r is the radial coordinate; and σ* and σ are the reduced and actual surface charge densities of the polyion. For a cylindrical polyion, the surface charge density is σ = e/(2πab), where b is the axial charge separation (length per elementary charge). Then the reduced surface charge density, σ*, is related to the reduced axial charge density of the cylinder, ξ, as σ* = 2ξz. Setting the derivative of the potential equal to zero at infinity, Eq. 3, guarantees the electroneutrality of the entire space (3). Then y is the potential relative to infinity [y(∞) = 0].¶

The preferential interaction coefficient is calculated either as the integral of the local deficit in coion concentration over volume surrounding the polyion (11) or as the [salt] derivative of electrostatic free energy (6, 9). If the cylindrical NLPB potential is used, the integral of coion deficit can be evaluated in closed form, yielding an expression for the preferential interaction coefficient per polyion charge in terms of ξ and ɛ, and the surface potential, y_a (21)

4

Eq. 4 provides the most direct route to calculating Γ, because it requires knowledge only of the surface potential y_a.

Results

We obtain the solution of the NLPB equation in the form of a zeroth order term of an asymptotic series in the small parameter ɛ⁻¹, y = y₀ + O(f(ɛ)) [for the detailed derivation and an estimation of the remainder of the asymptotic series, O(f(ɛ)), see Appendix A, which is published as supporting information on the PNAS web site, www.pnas.org], where the symbol O(f(ɛ)) means that the remainder of the asymptotic series is on the order of f(ɛ) as f(ɛ) → 0. The zeroth order term, y₀, is analogous to the “salt-free” solution but with the modified Bessel function of the second kind, Κ₀(κr) = Κ₀(xɛ⁻¹), instead of the logarithm of the radial coordinate (ln r):

5

where x = r/a is the reduced radial coordinate, γ = 0.5772 is the Euler–Mascheroni constant, and β and C are integration constants. In the limit ɛ⁻¹ → 0, S(v) = sin v when ξ ≥ 1 and S(v) = sinh v when ξ < 1. For nonvanishing values of ɛ⁻¹ > 0, corresponding conditions on ξ and ɛ⁻¹ are presented in Table 1 in terms of ξ and the product β(ξ − 1)⁻¹, which arises from the boundary condition at the cylinder surface. The two integration constants, β and C, are determined from electroneutrality and boundary conditions, Eqs. 23, and 24 (Appendix A, which is published as supporting information) and depend on ξ and ɛ. Eqs. 23 and 24 yield transcendental equations for constants β and C, which we obtain approximately for various ranges of ξ and β|ξ − 1|⁻¹ specified in Table 1.

6a

6b

7a

7b

7c

Here α = e^γ + ln 2 − γ = 1.897. We express remainders in Eqs. 6a, 7a, and 7b in terms of β and ξ for convenience; alternatively, they may be expressed using ɛ and ξ, because β = O((ln ɛ)⁻¹) (Eqs. 7a and 7b).

Table 1.

Functional forms of low salt approximate analytic solution of cylindrical NLPB equation at different ranges of parameters ξ and β

ξ	>1	>1	1	<1	<1
β|ξ − 1|−1	≪1	≫1		≫1	≅1
S(ν)	sin ν	sin ν	sin ν	sin ν	sinh ν
Eq. for C	6a	6a	6a	6a	6b
Eq. for β	7a	7b	7b	7b	7c
Eq. for Γ	10	11	11	11	12

The value of the surface potential can be expressed from Eq. 5 with the use of the boundary condition at the surface, Eq. 21, as

8

where β is given by Eqs. 7a–7c, “+” refers to the solution with S(v) = sin v, and “−” refers to the case S(v) = sinh v. For a highly charged polyion in the limit of low [salt] (ɛ⁻¹ → 0), β → 0 and the surface potential tends to its limiting value, y_0,a,LL = ln[4ɛ²(ξ − 1)²], which was previously deduced from the PB LL numerical value of Γ and Eq. 4 as a low-salt approximation for y_a (21), and also from the asymptotic solution (16). When ξ < 1, β → 1 − ξ as ɛ⁻¹ → 0, and the limiting value for the surface potential is y_0,a,LL = ln[16ɛ²(ξ − 1)²] − 2(1 − ξ)(ln 2ɛ + C − γ). Accuracy of Eq. 8 and comparison to previously obtained expressions for the surface potential (16, 17) are discussed in Appendix A, which is published as supporting information.

Thermodynamic Applications.

Calculation of the preferential interaction coefficient Γby using the surface potential.

We use y_0,a, Eq. 8, as an approximation for the surface potential, y_a, in Eq. 4 (see Appendix A, which is published in supporting information), which yields the LS expression for Γ

9

where “−” and “+” correspond to the solutions with S(v) = sin v and S(v) = sinh v, respectively. Because for a highly charged polyion β → 0 as [salt] decreases (ɛ⁻¹ → 0), the first term in Eq. 9 [−(4ξ)⁻¹] is the LS limiting value of Γ, and −β²(4ξ)⁻¹ is the salt-dependent term. When ξ < 1, the limiting value of β is (1 − ξ), which produces the LL value for the preferential interaction coefficient, Γ = −1/2 + ξ/4, and the salt-dependent term, given below in terms of parameters ξ and ɛ

10

11

12

In addition to conditions on ξ specified here, we list conditions on ɛ⁻¹ for Eqs. 10–12 in Table 1.

Accuracy of the analytic expression for the preferential interaction coefficient Γ.

For the model of an infinite uniformly charged cylindrical polyion, Γ is a function of two parameters, ξ and ɛ. Fig. 1 shows a comparison of Γ obtained from the LS asymptotic expansion, Eq. 10, and the numerical NLPB solution for dsRNA (A) and ssRNA (B) with structural parameters listed in Table 2. Fig. 1 A and B Inserts present the same comparison, but for (−4ξΓ − 1)^−0.5 plotted as a function of log C_b, for which Eq. 10 predicts a linear dependence. As one can see from Fig. 1 A and B Inserts, the plots are linear for ɛ > 1. Therefore, Eq. 10 describes the salt dependence of the preferential interaction coefficient in the entire range ɛ > 1 with less than 7% error for both ss- and dsRNA. For both ss- and dsRNA, the error monotonically decreases as ɛ⁻¹ → 0 for ɛ > 2.5. For 1 < ɛ < 2.5, the error is not monotonic, which indicates that the asymptotic result is on the margin of its applicability (ɛ ≅ 1), but the approximation is still acceptable for comparison with experimental data, for which the uncertainty is typically ±10%. At a given C_b, the LS asymptotic expression for Γ (Eqs. 10 and 11) becomes more accurate as ξ decreases toward unity. (At C_b = 0.01 M the error for dsRNA (a = 10 Å and b = 1.44 Å) is 7%, whereas for the case of ξ = 1 for a polyion with structural parameters a = 10 Å and b = 7.14 Å, the error is 2%.) Accuracy of the analytic expressions for y_a and Γ for the case ξ ≤ 1 is summarized in Table 3 and Appendix A, which are published as Supporting Information on the PNAS web site.

Figure 1.

Comparison of Γ from the LS approximation, Eq. 10 (solid line), the HS approximation, Eq. 14 (dotted line), and numerical results (dots) for dsRNA (A) and ssRNA (B). The error bars for the numerical values are not shown when they are within the symbols. Dashed line represents both LL value, Γ, and preferential interaction coefficient calculated from the surface potential of ref. 16. (for description of Insets, see text.)

Table 2.

Structural parameters and Γ for models of ss-, ds-, and tsRNA.

RNA	a, Å	b, Å	ξ	Γ	Cb(M) δΓ = δΓ	δΓ max (%)	δΓ max (%)
ss	7.5	3.2	2.23	Γ = −0.112 − 4.42 (3.62 − ln Cb)−2	0.2	7	4
ds	11.8	1.44	4.96	Γ = −0.0504 − 1.99 (1.59 − ln Cb)−2	0.05	7	4
ts	13	1.12	6.38	Γ = − 0.0392 − 1.55 (1.27 − ln Cb)−2	0.04	6	4

Discussion

Low [Salt] Limiting Value and Salt Dependence of Γ.

Eqs. 10–12 provide expressions for Γ as the sum of the LL value (Γ = −(4ξ)⁻¹, ξ ≥ 1, or Γ = −1/2 + ξ/4, ξ < 1), which depends only on the polyion reduced axial (structural) charge density ξ, and a [salt]-dependent term. Previously, Anderson and Record (34) deduced from Eq. 4 and PB LL numerical results (26) that at sufficiently low salt concentration, the preferential interaction coefficient is represented as Γ = Γ(1 + σ_Γ), where σ_Γ is a salt-dependent correction to the LL value, vanishing as [salt] approaches zero. Our asymptotic expansion of PB potential rigorously proves this fact and provides an explicit expression for σ_Γ as a function of ɛ and ξ. For a highly charged polyion at very low [salt], the [salt]-dependent term σ_Γ decreases as (ln ɛ)⁻².

Eqs. 10–12 provide expressions for Γ for the conditions on ξ and β listed in Table 1 arising from the expansion of the boundary condition at the cylinder surface. Because the solution, Eq. 5, is an asymptotic expansion at small ɛ⁻¹, another condition for its application is ɛ ≫ 1. This condition is equivalent to the requirement that β be small for the solution with S(v) =sin v and β be close to (1 − ξ) for the solution with S(v) = sinh v. In practice, for a polyion with reduced charge density distinct from 1 (ξ < 0.5 or ξ > 2), the condition on β|ξ − 1|⁻¹ is automatically satisfied if ɛ > 1, and for a polyion with ξ ≅ 1, Eq. 11 is applicable at not very low [salt] (see, for example, comparison of Γ at ξ = 0.5 and ξ = 0.9 in Table 3, which is published as supporting information). Because nucleic acids have relatively large axial charge density (ξ > 2), Eq. 10 can be used as an accurate analytic expression for their preferential interaction coefficients in the entire [salt] range where ɛ > 1.

To reveal the dependence of Γ on salt concentration and polyion structural parameters for a highly charged polyion, we replace ɛ in denominator in Eq. 10 by using the identity ɛ⁻² ≡ κ²a² ≡ 8C_bξV_u.

13

Here and below, C_b is in molar units (M), and the polyion cylindrical volume per charge is V_u in M⁻¹ units. Substitution of structural parameters of ss-, ds-, and triple-stranded (ts) RNA in Eq. 10 yields expressions for Γ, which are given in Table 2 together with corresponding RNA structural parameters.

Range of Applicability of Low [Salt] Expression for Γ.

The solution derived in this paper is obtained as an asymptotic expansion in small parameter ɛ⁻¹, and its accuracy increases as ɛ increases. The high-salt (HS) approximate expression for Γ derived by Shkel et al. (13) is also an asymptotic expansion result valid in the opposite limit, ɛ → 0.

14

where p = ɛξ, q̃ =. Thus for every polyion, there is a [salt], which separates LS (ɛ > 1) and HS (ɛ < 1) regions, and where accuracy of both approximations is expected to decline. The value of [salt] corresponding to ɛ = 1 is different for polyions of different radii. Both expressions have a comparable error in the crossover range of salt concentrations, ɛ ≅ 1, and both errors increase with increasing ξ. Fig. 1 shows that the error for the HS expression increases more slowly, therefore the HS approximation can be used at salt concentrations where ɛ is slightly larger than 1. For the cylindrical models of ss-, ds-, and tsRNA, Table 2 presents the salt concentrations where δΓ = δΓ and maximum errors for both approximations in regions separated by this [salt]. The two approximations together describe Γ with an uncertainty of <7% for ss-, ds-, and tsRNA.

Analytic Expression for ΔΓ for RNA Transitions at Low [Salt].

Preferential interaction coefficients are the fundamental determinant of the effect of salt concentration on the thermodynamics of nucleic acid processes. We discuss experimental studies of RNA conformational transitions (23) as an example of application of Eq. 10 to thermodynamic calculations, because these reactions involve a wide range of RNA axial charge densities (ξ ≅ 2 − 6.5, b ≅ 1 − 3.2 Å) and radii (a = 7.5 − 13 Å) and independently measured enthalpies of the transitions. Extensive numerical analysis of these transitions with NLPB cylindrical model is available for comparison (27). For the processes of step-wise denaturation of ts- and dsRNA polymers to the ss state, tsRNA → dsRNA + ssRNA and dsRNA → ssRNA, in the range from 0.01 to 0.1 M [salt] (23), the derivative of the melting temperature, T_m, with respect to the logarithm of [salt] is determined by the ratio ΔΓ/ΔH⁰, where ΔΓ is the stoichiometrically weighted difference in preferential interaction coefficients [Δ₁Γ_u = (2/3)Γ_u,ds + (1/3)Γ_u,ss − Γ_u,ts for the first reaction and Δ₂Γ_u = Γ_u,ss − Γ_u,ds for the second reaction], and ΔH⁰ is the transition enthalpy (both expressed per mol of nucleotide monomers). For proper comparison with experiment, one should take into account the excluded volume contributions to ΔΓ_u, ΔΓ_u = ΔΓ + ΔΓ, where ΔΓ = −6.022⋅10⁻⁴C_bπΔ(a²b) and the values of a and b (in Å) are taken from Bond et al. (27).

The salt concentration range in experiments (23) (0.01 M < C_b < 0.1 M) coincides with the crossover region ɛ ≅ 1 for the two approximations: at 0.01 M, the LS approximation Eq. 10 is more accurate, and at 0.1 M the HS approximation Eq. 14 is more accurate.

We derive expressions for ΔΓ and its [salt] derivative in Appendix B, which is published as supporting information, for any polyion conformational transition in terms of changes in reduced charge density Δξ and volume per charge ΔV_u for this process with coefficients depending only on [salt] and average values ξ_av and V_u,av for the process.

15

16

where G_ξ = π²ξf(1 + 2f′_avξ_avf), G_V = 2π²ξVf, G′_ξ = 2π²ξf(1 + 3f′_avξ_avf), G′_V = 6π²ξVf, f(ξ, V_u, C_b) = 1.715 −ln C_b − ln ξ −ln V_u + 2(ξ − 1)⁻¹, f_av = f(ξ_av, V_u,av, C_b), and f′_av = (∂f/∂ξ)|_ξ=ξav. Average values of ξ and V_u are determined by their minimal and maximal values in the reaction ξ_av = (ξ_min + ξ_max)/2, V_u,av = (V_u,max + V_u,min)/2. For the reaction tsRNA → dsRNA + ssRNA, ξ_min = 2.23, ξ_max = 6.38, ξ_av,1 = 4.3, for the reaction dsRNA → ssRNA, ξ_min = 2.23, ξ_max = 4.96, ξ_av,2 = 3.6. The largest, the smallest, and the average volumes for both reactions are V_u,min = 0.339 M⁻¹, V_u,max = 0.378 M⁻¹, and V_u,av = 0.359 M⁻¹. Eqs. 15 and 16 are more accurate at low [salt], because neglected terms are proportional to some negative power of f_av and f_av increases with decreasing [salt]. At 0.01 M, Eq. 15 yields Δ₁Γ = −0.0455, which differs only by 5% from the value obtained from the original Eq. 13 (−0.0434). For the second transition, Eq. 15 yields Δ₂Γ = −0.077, which differs only by 3% from the value obtained from Eq. 13 (−0.0749).

In Fig. 2, we show Δ₁Γ_u and Δ₂Γ_u calculated from the LS and the HS approximations and from numerical NLPB solution. The experimental values of ΔΓ_u (27) are shown in the experimental range of salt concentration, 0.01–0.1 M, with the error of 15%. These error estimates assume average errors of 10% in ΔH⁰ (23) and in dT_m/d log C_b.

Figure 2.

Difference in preferential interaction coefficients; comparison of the LS approximation (solid line), the HS approximation (dotted line), and numerical results (dots) for two RNA transitions: ts → ds + ss (lower curves) and ds → ss (upper curves). The errors of numerical values of ΔΓ_u are less than 2%.

For Δ₁Γ, the error of the LS approximation is less that 11% below C_b = 0.03 M, and the error of the HS approximation is less that 11% above C_b = 0.03 M. For Δ₂Γ, the error of the LS approximation is less that 12% at C_b < 0.02 M. Above C_b = 0.02 M the HS expression, Eq. 14 is a better approximation for Δ₂Γ, accurate within 13%.

Deviation of ΔΓ from ΔΓ.

Eq. 15 allows one to estimate the relative deviation of ΔΓ from its LL value. At low [salt], the leading term in ΔΓ − ΔΓ is π²ξfΔξ (because f_av is large). This term predicts an increase in magnitude of ΔΓ− ΔΓ (Δξ < 0) with increasing salt concentration.

For the transitions with ξ_av ≥ 3.6, the deviation (ΔΓ − ΔΓ)/ΔΓ is less than 10% when f_av > 17.5, i.e., at [salt] lower than 3⋅10⁻⁷ M. For both transitions considered above, Eq. 15 predicts that (ΔΓ − ΔΓ)/ΔΓ exceeds 25% when f_av < 9, i.e., at [salt] higher than 0.001 M. Thus, in range of [salt] used in experimental measurements in ref. 23, the deviation of ΔΓ from the LL value is significant for both transitions and cannot be neglected in comparing with experiments.

Eq. 16 predicts that the derivative of ΔΓ becomes zero at the salt concentration determined by equation f_av = 3ξ_av(Δξ)⁻¹ΔV_uV− 3f′_avξ_av. For reactions for which the first term can be neglected (ΔV_u ≅ 0), the relationship for corresponding [salt] is ln C_b = −0.26 −lnξ_av + 2(ξ_av − 1)⁻¹ − 6ξ_av(ξ_av − 1)⁻². Because the difference in ξ_av for different transitions is not large, the salt concentration where this occurs varies between 0.01 and 0.05 M (for ξ_av from 3.2 to 6.5). This [salt] is in the experimental range for both transitions analyzed here. The broad maximum of ΔΓ at this [salt] explains the experimentally observed linearity of T_m as a function of ln C_b.

Singular Points of the Solution.

The solution of the cylindrical NLPB equation has one or two singular points. The point x = 0 (r = 0) is always a singular point of the solution, because the term 2Κ₀(xɛ⁻¹) becomes infinite. For ξ ≥ 1, the position of the second singular point is determined by the relationship Κ₀(xɛ⁻¹) = Κ₀(ɛ⁻¹) + (ξ − 1)⁻¹. At ɛ⁻¹ → 0, this equation yields r = ae^{−1/(ξ−1)} + O(ɛ⁻¹ln ɛ). This point lies between the point r = 0 and the cylinder surface r = a and approaches the surface as ξ increases. As ξ → 1, this singular point approaches the coordinate origin (r = 0), and at ξ = 1 and ɛ⁻¹ → 0, the two singular points coincide. Thus, at ɛ⁻¹ = 0, ξ = 1 is a bifurcation point for the cylindrical NLPB equation [similar to the cell model solution (35)], which results in different functional forms of the (approximate) solution Eq. 5.

Benham (31) presented the expansion of the exact solution of the NLPB equation for coion concentration at the singular point, which for the potential is rewritten as

17

where, as above, x = r/a.

Series 17 satisfies the NLPB equation exactly, which can be verified by direct substitution of Eq. 17 into the NLPB equation. All coefficients A_n are uniquely expressed through two constants, x₀ and A₂, by induction (31). Constant x₀ determines the location of the singular point. These two constants have to be determined from two boundary conditions. Because the expansion given by Eq. 17 is not valid at infinity, the boundary condition at infinity cannot be applied directly to Eq. 17.

The LS asymptotic solution of the NLPB equation, Eq. 5, provides approximate expressions for A₂ and x₀ when y₀ is expanded at small (x − x₀) and compared with series 17. In the leading order with respect to ɛ⁻¹, these expressions are:

18

19

If ξ ≫ 1, then the singular point is located at the distance on the order of aξ⁻¹ from the cylinder surface and the order of magnitude of every term in the expansion Eq. 17 differs by a factor of ξ⁻¹ from the previous one. Although the expansion Eq. 17 does not converge sufficiently rapidly for calculation of y, it provides a clear illustration of the behavior of the solution near the singular point. When r approaches the singular point, the potential increases as the logarithm of the distance to the singular point. Because at high polyion charge density (ξ ≫ 1) the singular point is close to the cylinder surface, the potential at the polyion surface is very large.

Other Solutions for the Surface Potential and Γ at Low- to Moderate-Salt Concentration.

For a long time, the LL value (value at very low [salt]) of Γ[Γ = −(4ξ)⁻¹, ξ ≥ 1 and Γ = −1/2 + ξ/4, ξ < 1] obtained from numerical NLPB analysis (26) and independently from the counterion condensation hypothesis has been the only existing analytic expression for Γ at low [salt]. The attempt to introduce a deviation from the LL value of Γ due to exclusion of coions from the volume of the condensed layer by Manning (36) was found less satisfactory in explaining experimental data than numerical NLPB calculations (27). Several approximate NLPB analytic solutions for the potential at low salt concentrations are known, but they are unsatisfactory for the calculation of Γ. Previous solutions in the presence of the salt (18 and 19), although analytic in functional form, require extensive numerical calculations to evaluate constants for each [salt] and polyion structural quantity. The leading-order approximation with respect to [salt] to the cylindrical NLPB potential in an infinite space in the presence of salt (16) is not sufficiently accurate for calculation of Γ. It yields the LL value for the surface potential, and hence for Γ, but does not produce the correct [salt] dependence of Γ, as shown in Fig. 1 and discussed in Appendix A, which is published as supporting information. Another solution (17), derived by the matched asymptotic expansions method, was obtained for a weakly charged polyion only (ξ < 1). (See Table 3 and Appendix A, which are published as supporting information, for comparison of this solution (17) with Eq. 8 of the present work.) The LS approximate solution of the cylindrical NLPB equation (Eq. 5) presented here is obtained for any value of polyion charge density ξ and exists in the entire range of the radial coordinate (r ≥ a).

Conclusion

For a stiff polyion of either high or low charge density, Eq. 8 and the accompanying expressions for β (Eqs. 7a–7c) provide analytic expressions for the [salt] dependence of the polyion surface potential y_a, which are sufficiently accurate to calculate the coulombic contribution to the preferential interaction coefficient Γ at all reduced polyion charge densities ξ and over a wide range of low-to-moderate [salt]. Eqs. 10–12 are, to our knowledge, the first rigorously derived analytic expressions for the preferential interaction coefficient Γ at low [salt]. These expressions show deviation of Γ from its LL value with increasing [salt] and depend explicitly on structural quantities of the polyion (radius a, and charge separation b). Different functional forms of y_a and Γ are obtained depending on the value of ξ relative to unity and on salt concentration via β (Table 1). Previously, these behaviors were described analytically only under LL conditions, where the functional forms of y_a and Γ depend on the value of ξ relative to unity, a result attributed to counterion condensation but derived more generally from the NLPB equation without the hypothesis of counterion condensation (16, 35).

Eqs. 10–12 are obtained in the LS limit (ɛ⁻¹ → 0), complementary to the HS limit (ɛ → 0) considered in Shkel et al. (13). The two approximations provide explicit analytic expressions for the preferential interaction coefficient Γ in the entire range of salt concentration in the context of NLPB approach. The accuracy of both solutions is sufficient to explain an experimentally measured derivative of the melting temperature for RNA transitions with respect to the logarithm of [salt].

Analytic expressions Eqs. 8, 10–12 generalize numerical data for PB surface potential and preferential interaction coefficients in the form of explicit, accurate, easy-to-use relationships and eliminate the need for numerical calculations for the cylindrical NLPB model in the future. The calculations of other thermodynamic properties of electrolyte solutions (electrostatic free energy, activity coefficient, osmotic coefficient, etc.) now can be readily performed. The solution reported here also provides a basis for the incorporation of local details at the polyion surface to examine thermodynamic consequences of ion correlations (33) or DNA structural details (9).

Abbreviations

NLPB

nonlinear Poisson-Boltzmann

LL

limiting law

[salt]

salt concentration

ss

single-stranded

ds

double-stranded

LS

low salt

HS

high salt

ts

triple stranded