11528
J. Phys. Chem. B 2000, 104, 11528-11540
Transformed Poisson-Boltzmann Relations and Ionic Distributions† Hong Qian‡ Department of Applied Mathematics, UniVersity of Washington, Seattle, Washington 98195
John A. Schellman* Institute of Molecular Biology, UniVersity of Oregon, Eugene, Oregon 97403 ReceiVed: NoVember 23, 1999; In Final Form: February 24, 2000
For many applications, charge distributions around macromolecules in aqueous solution are of greater interest than the electrical potential. We show that it is possible to use the Poisson-Boltzmann (PB) relation to develop differential equations for the ionic distributions. The solutions to these equations are the integral distribution functions whose derivatives give the charge density functions for counterions and coions. In this formalism the salt-free atmosphere of a cylindrical polyelectrolyte is very easily solvable for the counterion. Quantities such as the “condensation radius” (Le Bret, M.; Zimm, B. H. Biopolymers 1984, 23, 287-312) and the “Bjerrum association radius” (Bjerrum, N. Investigations on Association of Ions, I. In Niels Bjerrum Selected Papers; Munksgaard: Copenhagen, 1926; pp 108-19) appear naturally as inflection points in curves of the counterion distribution functions. Moreover, a number of the properties of condensation theory arise as scaling limits of the transformed PB equation. In the presence of added salt separate equations can be derived for the excess charge distributions of counterions and coions. In this case the total excesses of counterion φct and of coion φco are simply related to experiment. Various combinations of these two quantities lead to formulas for (1) the total charge, (2) Donnan exclusion, (3) counterion release (Record, M. T.; Lohman, T. M.; de Haseth, P. J. Mol. Biol. 1976, 107, 145-158), and (4) fraction of “condensed” ions. Bjerrum’s theory of ion assocition and Manning’s theory of counterion condensation are discussed in the context of the transformed Poisson-Boltzmann theory.
Electrostatic interactions play important roles in chemistry and biology. In biochemistry, interactions between macromolecules often involve charged polyelectrolytes such as proteins and nucleic acids. The interactions are intimately related to the ionic distributions around the macromolecular polyelectrolytes in aqueous solution. A quantitative description of the ionic distributions can be derived from the Poisson-Boltzmann (PB) equation. (For applications to a cylindrical geometry see D. Stigter.29) The solution of the PB equation is considered to be a good approximation for the thermoelectrostatic potential around a macroion.9 However, it is the ionic distribution rather than the potential which is of ultimate interest to many molecular physical chemists. On the basis of this realization, we introduce the transformed Poisson-Boltzmann equations (TPB, eqs 9, below), which offer insight and analytical power by being expressed directly in terms of ionic distributions. The PB and TPB equations are two mathematical formulations of the same physical theory. An advantage of the latter is that it makes the ionic distributions intuitively accessible, and many of the properties of most interest to an experimentalist are simply related to them. On the other hand, properties which are explicitly related to the potential such as electrophoresis and the free energy are best viewed from the point of view of the standard PB equation. Most of the results of the present work are general for a polyelectrolyte modeled as a very long † This paper is dedicated to Robert (Buzz) Baldwin in appreciation of years of fruitful interaction and friendship. The results of this paper were first presented in partial form at his retirement celebration in 1998. * To whom correspondence should be addressed. E-mail: john@ jas1.uoregon.edu. ‡ E-mail:
[email protected].
cylinder. However, because of our special interest, the numerical examples are mostly based on molecules with the dimensions and charge distribution of DNA. Among other things this means that our examples are concentrated on cases where the Manning charge density parameter ξ > 1, and in particular on the B-form DNA value of ξ ) 4.22. Most of the results which are of direct interest to experimentalists will be found in the section on “The Excess Functions φct and φco.” The paper is developed as follows: 1. The notation, model and basic electrostatics are described and developed. 2. The charge distribution functions are defined and their differential equations derived. The condensation radius and a connection with the condensation theory as a limit are deduced. 3. Systems with no added salt are presented and explicitly solved for the three cases, ξ < 1, ξ ) 1, ξ > 1. 4. The solutions for the PB equation are compared with the derivatives of the solutions for the TPB equation. 5. Numerical solutions for the case of added salt, the condensation radius and condensation limit for the saltpresent case are obtained. 6. The excess functions φct and φco are defined and derived from the numerical solutions of the TPB equations. Their direct relations with experimental quantities (Donnan exclusion, thermodynamic binding function, condensation) are derived and presented. Quantitative distinctions between the PB and condensation theories are discussed. 7. The dissociation radius, Rd; and the accumulation of counterions and its relation to the Bjerrum theory of ion association are studied. 8. Major conclusions are summarized. 1. Notation and Background The model for a polyelectrolyte will be the usual charged cylinder with length L and radius a. The cylinder is considered
10.1021/jp994168m CCC: $19.00 © 2000 American Chemical Society Published on Web 11/01/2000
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J. Phys. Chem. B, Vol. 104, No. 48, 2000 11529
to be sufficiently long that end effects can be ignored and is treated as a uniformly charged cylinder with uniform linear charge density, z ) Z/L, where Z is the net number of fixed, elementary charges on the macromolecule. There are two classic cases. In the first there is no added salt and therefore no coions. All the counterions are in a finite cylinder with radius R, and the PB equation can be solved analytically.10,1 In the second case salt is present. A somewhat formidable analytical solution has recently been obtained for this case31 but most published work has made use of numerical integration of the differential equation. The situation with the TPB is completely analogous. The salt free case is solvable and numerical solutions will be used for the presence of salt. In the presence of added salt the ionic atmosphere contains both counterions (ct) and coions (co). These subscripts are used rather than “+” or “-” to make the formulas independent of the sign of the charge on the polyion. The salt is assumed to be a 1:1 electrolyte. With the mean-field Boltzmann treatment local concentrations at a distance r from the cylinder axis are approximated as
nct ) noey
and
nco ) noe-y
l and z(r) is linear charge density, specifically the number of elementary charges per unit length (polyion plus the net charge of the ion atmosphere within the cylinder). Introducing the dimensionless potential and radius (y and x)
e2z(x) ∂y x )) 2lBz(x) ∂x 2πkT
(4)
The right side of eq 4 can be interpreted as minus twice the number of elementary charges contained in length lB of the cylinder. In particular, at r ) a, x ) xa
xa
(∂y∂x)
xa
) -2lBz(xa) ) -2ξ
(5)
where ξ ) e2/(4πkTb) is the dimensionless charge density discussed by Manning. z(xa) is often written as 1/b, where b is the average charge separation. From this derivation we see that ξ ) lBz(xa) is the number of elementary charges that are contained in a Bjerrum length of the polyion.
(1) 2. Transformed Poisson-Boltzmann Equations
where y is the reduced potential y(r) ) (e|ψ(r)|)/(kT), ψ(r) is the mean electrostatic potential at r, and e is the protonic charge. y is always positive. no is the number density of the counterions at a distance r sufficiently remote from the polyelectrolyte that the presence of the polyelectrolyte is undetectable, and thus y ) 0. For a 1:1 electrolyte this is same as the salt concentration far removed from polyelectrolyte molecules or the concentration of the salt in dialysis equilibrium with the polyelectrolyte solution. With cylindrical symmetry and coordinates Poisson’s equation is
∇2ψ )
1 d dψ r ) -F/ r dr dr
( )
where F is the net charge density, in units of the elementary charge, resulting from counterions and coions. It is useful to introduce the dimensionless radial distance x ) r/lD where lD ) 1/κ is the Debye length and κ2 ) (2e2no)/(kT). Replacing r by x and ψ by y, Poisson’s equation takes the form29
1 d dy ey - e-y x ) ) sinh y xdx dx 2
( )
(2)
Another characteristic length of importance is the Bjerrum length,
lB ≡ e2/4πkT
(3)
It is the distance at which a pair of singly charged ions have an interaction free energy of kT. It arises automatically in calculations of ion distributions in solution. lB is a function of the dielectric constant and temperature. For water at 25 °C its value is 7.13 Å. We apply Gauss’ law to the surface of a cylinder, coaxial with the polyion, of radius r and length l, with a e r e R,
lz(r)e ∂ψ Q )- )∂r
(2πlr)
Q is the total charge contained in the cylindrical section of height
We will be mainly concerned with the ionic distributions around the polyion rather than the potential and it is therefor desirable to transform the PB equation to a form in which the unknown functions are the ionic distributions. We consider a cylindrical section of the ion atmosphere with a height of lB extending from a to an outer limit R. In the case where there is no salt (no coion) R is the outer limit of the cylindrical volume surrounding the polyelectrolyte. This is the cell model. When salt is present R is chosen to be sufficiently remote that the ionic concentrations at R do not differ appreciably from their values at infinity where y ) 0. This condition will be discussed further when actual calculations are presented. Within the cylindrical section described above we define the cumulative quantity of ions between a variable radius r, a e r e R, and the external boundary at R. For the counterions this is given by
Φct(r) ) 2πlb
∫rRnctr′dr′ ) 2πlbno∫rReyr′dr′
Integrals of this form have appeared in the works of many authors especially for the case where r is equal to the radius of the cylinder to include the entire ionic atmosphere.2,16,15,21,29 From the above definitions of κ2 and lB we have κ2 ) 8πlBno, so that the definition may be written as
Φct(r) ) κ2/4
∫rReyr′dr′ ) 1/4∫xXeyx′dx′
(6a)
where X ) κR. Similarly, for the coion
Φco(r) ) κ2/4
∫rRe-yr′dr′ ) 1/4∫xXe-yx′dx′
(6b)
Note that Φct and Φco are positive irrespective of the charges on the ions. We now find differential equations for these quantities. Φct - Φco is the magnitude of the net charge between the surfaces of r and R. Because of the overall neutrality of the system it is also the magnitude of the charge within r, i.e., the charge of the polyion plus the ionic atmosphere within r. From (4) and
11530 J. Phys. Chem. B, Vol. 104, No. 48, 2000
Qian and Schellman We introduce the variable ln r10, and eq 10 becomes
(6a) we have
x
dy ) -2(Φct - Φco) dx dΦct 1 ) - xey dx 4
(7) (8)
taking the second derivative, (d2Φct)/(dx2) ) -1/4(ey + xey(dy)/ (dx)), and eliminating terms in y with eqs 7 and 8, we obtain the transformed PB equations
d2Φct 2
)
dx
1 - 2(Φct - Φco) dΦct x dx
(9a)
and for Φco
d2Φ dΦ ) 2(1 - Φ) d ln r d ln r2 The first integral of eq 11 can be obtained by inspection
dΦ ) C - (Φ - 1)2 d ln r
dx
2
)
1 + 2(Φct - Φco) dΦco x dx
(9b)
Equations 9 are a pair of coupled, nonautonomous, ordinary differential equations for Φct and Φco. We note that in both cases it is the net charge density, Φct - Φco, which controls the behavior of the functions. The boundary conditions at r ) a and r ) R (or x ) xa and x ) X) are
Φco(xa) ) Ns; Φco(X) ) 0 Φct(xa) ) ξ + Ns; Φct(X) ) 0 Here Ns is the number of co- and counterions contributed by the neutral salt in the volume between a and R and ξ is the total charge on the ionic atmosphere, which balances the charge on the polyelectrolyte. Note that Φct(xa) - Φco(xa) ) ξ. While the PB equation is derived by combining Poisson’s equation (second order) and the Boltzmann law to the local ionic concentrations, the TPB equations arise from Gauss’ law for a cylindrical distribution (first order), and the Boltzmann law to obtain cumulative distributions functions. Poisson’s equation is, of course, the differential form of Gauss’ law. 3. Counterion Distribution with No Added Salt In the absence of additional salt there are no coions and it will be useful to write eqs 6a and 9a with r as a variable rather than x, since it will turn out that the results depend only on the ratios of distances and any scaling factor is automatically canceled. This can be seen by substituting Rx for x in eqs 9a,b and observing the cancellation of R. The subscript for the counterions will be dropped in this section to distinguish it from the case with added salt, Φct ) Φ,
Φ(r) ) (k2ct/4)
∫rReydr
d2Φ 1 - 2Φ dΦ ) r dr dr2
(6a)
(10)
Φ(a) ) ξ; Φ(R) ) 0 When additional salt is not present, kct is defined in terms of the counterion concentration at the outer boundary,16 i.e., as k2ct ) (2e2nct(R)/(kT).
(12)
There are three solutions to eq 12 depending on the sign and magnitude of C
1 -1Φ - 1 tan ) - ln r + c2, c1 c1 Φ-1)
d2Φco
(11)
C ) c21 < 0
1 , C)0 ln r + c2
Φ-1 1 coth-1 ) ln r + c2, C ) c21 > 0 c1 c1
(13c)
These results come from standard integral tables. It should be noted how more direct these solutions are than that for the PB equation itself.1,10 We now discuss three physical situations. High Charge Density, ξ > 1. From its definition dΦ/d ln γ must always be negative. For ξ > 1, the value of Φ -1 passes through zero at a distance which we will call Rc in imitation of the RM of LeBret and Zimm.16 If C is positive, the slope is positive in the neighborhood of Rc where (Φ - 1)2 < C. Consequently, C in eq 12 must be negative to maintain the negative slope and we require solution 13a which can be written as
Φ ) 1 - c1tan(c1(ln r - c2))
(13a′)
Since Φ is a continuous and monotonic function of ln r, its range must lie on a single branch of the tangent function. c1 and c2 can be evaluated by numerically solving the two boundary equations (1/c1) tan-1(ξ - 1)/(c1) ) -ln a + c2 and (1/c1) tan-1(-1/c1) ) -ln rR + c2. This case is illustrated in Figure 1a for ξ ) 4.22 (DNA) and R/a ) 100. The distance Rc at which Φ ) 1 has been called the condensation radius by Zimm and LeBret.32 In our analysis it shows up as an inflection point in the Φ vs ln r curve and will play a prominent role in the analysis of results. This is especially clear if we note that at Φ ) 1, c2 ) ln Rc and
Φ ) 1 - c1tan(c1(ln(r/Rc))
(14)
This is a negative tangent curve centered on the inflection point (ln Rc, 1) with positive curvature within the condensation radius and negative curvature on the outside. These regions correspond to the condensed and dissociable regions of the ion atmosphere as discussed by Zimm and LeBret.32,16 It would be of pedagogic interest to pursue these aspects of the problem with the present formalism, but we will not do so since this area has already been extensively discussed by the cited authors. We do wish to call attention to the fact that the condensation radius and the inner and outer ionic atmospheres are generated simply and automatically with the TPB formalism. In the limit of a f 0, our result can be related to the Manning condensation model.20 In this case we have ln r f -∞ in the neighborhood of the polyelectrolyte. The function Φ is bounded at both limits. For a smooth monotonic function to be bounded
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J. Phys. Chem. B, Vol. 104, No. 48, 2000 11531
Figure 1. (a) The salt-free case. Plot of Φ vs r for ξ ) 4.22, A ) 10 Å, R ) 1000 Å. Rc occurs where the unit line crosses the Φ curve. (b) Plot of Φ vs r for ξ ) 0.5 and the same limits as in a.
at -∞ it must have an asymptote parallel to the abscissa, i.e., dΦ/d ln r ) 0. From eq 11 when dΦ/d ln r ) 0, the second derivative also equals 0. From this we infer that in this limit the inflection point and therefore Rc, the condensation radius, has moved to the origin, or we might prefer to say that it is infinitesimally separated from it. This is a mathematical description of the Manning model. This type of conclusion has been obtained by others,17,12,16 but once again we emphasize the simplicity and directness of the present method. This conclusion is reached without eVen solVing the differential equation. We can look as well at the case for finite a and infinite dilution, i.e., ln R f ∞. Again Φ is bounded, the limiting slope must be zero, and the inflection point Rc moves to infinity. Mathematically, eq 11 is the same if we substitute r′ ) r/R, and a′ ) a/R f 0. In our view these results are a matter of scaling. There are no zero or infinite boundaries. Material charges cannot be confined to a line. When R is fixed and ln a f ∞, we are adopting a macroscopic point of view. With this perspective the radius of the charge and Rc are both infinitesimally close to the z axis and we see only the negative curvature of the outer atmosphere. When a is fixed and ln R f ∞, the dissociable charges are concentrated in a shell at the outer edge of the cylindrical boundary, and on a molecular scale we see only the positive curvature of the condensed atmosphere. These are two viewpoints of the same problem when R/a is extremely large. Low Charge Density, ξ < 1. In this case C can take on positive values provided that C ≡ c21 < (1 - ξ)2. From eq 13c the solution is
Φ ) 1 + c1coth(c1(ln r + c2))
(13c′)
The constants are evaluated from the boundary relations (1/c1) coth-1(ξ - 1)/(c1) ) ln a + c2 and (1/c1) coth-1(-1/c1) ) ln R + c2. The function is monotonic between ln a and ln R, and there is no inflection point or accumulation of condensed ions. This is in agreement with all other studies. This function is shown in Figure 1b for ξ ) 0.5, r/a ) 100. The Special Case, C ) 0. From Eq 13b we have
Φ)1+
1 ln r + c2
(13b′)
Applying the boundary condition Φ ) 0 at r ) R, this becomes
Φ)1-
1 1 + ln(R/r)
(14)
ln(R/r) is always positive so that Φ is always less than 1 except at r f 0 where it becomes unity. We see that this case is appropriate for the special case of a line charge with ξ ) 1. We note also that in the limit of c1 f 0 eq 13c becomes eq 13b′. 4. Solving the PB Equation. There is naturally a great deal of overlap between the above results and those obtained by others using the PB equation. In fact the equations derived in the appendix of ref 16 are identical to the boundary conditions of our differential equation. One can easily obtain one set of solutions from the other. Equation 6a transforms from the potential to the ion distribution functions by an integration. The inverse is performed with a differentiation. From eq 6a
(κr)2ey dΦ )d ln r 4 and
y ) ln
[
-4 dΦ (κr)2d ln r
]
(15)
For ξ > 1 we differentiate eq 14 and substitute the result in eq 15
y ) -2ln
[
]
(κr) cos(c1(ln r + c2) 2c1
(16)
This is the solution to this problem reported by Alfrey et al.1 For ξ < 1 differentiation of eq 13c′ and substitution yields
y ) -2ln
[
]
(κr) sin(c1(ln r + c2)) 2c1
(17)
This is the solution of Fuoss et al.10 for this case. Using the
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Qian and Schellman
TABLE 1: Coions within the Condensation Radiusa C
Rc
Rc - a
Φco(
