Extensions of counterion condensation theory. I. Alternative geometries and finite salt concentration

The counterion condensation theory originally proposed by Manning is extended to take account of both finite counterion concentration (m(C)) and the actual structure of the array of discrete changes. Counterion condensation is treated here as a binding isotherm problem, in which the unknown free volume is replaced by an unknown local binding constant beta', which is expected to vary with m(C) and polyion structure. The relation between the condensed fraction of counterion charge, r, beta' and m(C) is obtained from the relevant grand partition function via the maximum term method. In the case of the single polyion in a large salt reservoir, the result is practically identical to Manning's equation. In order to determine the values of beta' and r at arbitrary m(C), a second relation between r, beta' and m(C) is required. We propose an alternative auxiliary relation that is equivalent to previous assumptions near m(C) = 0, but which yields qualitatively correct and quantitatively useful results at finite m(C). Simple expressions for r vs. m(C) and beta' vs. m(C) are obtained by simultaneously solving the binding isotherm and auxiliary equations. Then r and beta' are evaluated for five different linear arrays of infinite extent with different geometries: (1) a straight line of charges with uniform axial spacing; (2) two parallel lines of in-phase uniformly spaced charges; (3) a single-helix of discrete charges with uniform axial spacing; (4) a double-helix of discrete charges with uniform axial spacing of pairs of charges; (5) a cylindrical array of many parallel charged lines, chosen to simulate a uniformly charged cylinder. In all cases, the computed binding isotherms exhibit qualitatively correct behavior. As m(C) approaches zero, r approaches the Manning limit, r = 1-1/(L(B)/b) where b is the average axial spacing of electronic charges in the array and L(B) is the Bjerrum length. However, beta' varies with polyion geometry, even in the zero salt limit, and matches the Manning value only in the case of a single straight charged line. With increasing m(C), r declines significantly below its limiting value whenever lambda(b) > or approximately equal 0.3, where lambda is the Debye screening parameter. In the case of cylindrical arrays containing either 2 or 100 parallel charged lines, r also decreases, whenever lambda(d) > or approximately equal 2.0, where d is the diameter of the array. In the case of two parallel charged lines, each with axial charge spacing b=3.4 A, which are separated by d = 200 A, r exhibits a plateau value, 0.76, characteristic of the two combined lines, when lambda(d)<<2.0, and declines with increasing m(C) to a shelf value, 0.52, characteristic of either single line, when lambda(d) > or approximately equal 2.0 and the lines become effectively screened from one another. beta' behaves in a roughly similar fashion. In the case of a cylindrical array of charged lines with the diameter and linear charge density of DNA, the r-values predicted by the present theory agree fairly well with those predicted by non-linear Poisson-Boltzmann theory up to 0.15 M uni-univalent salt.