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

The counterion condensation theory originally proposed by Manning is extended to take account of both finite counterion concentration (mC) 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 β′, which is expected to vary with mC and polyion structure. The relation between the condensed fraction of counterion charge, r, β′ and mC 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 Manningʹs equation. In order to determine the values of β′ and r at arbitrary mC, a second relation between r, β′ and mC is required. We propose an alternative auxiliary relation that is equivalent to previous assumptions near mC=0, but which yields qualitatively correct and quantitatively useful results at finite mC. Simple expressions for r vs. mC and β′ vs. mC are obtained by simultaneously solving the binding isotherm and auxiliary equations. Then r and β′ 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 mC approaches zero, r approaches the Manning limit, r=1−1/(LB/b) where b is the average axial spacing of electronic charges in the array and LB is the Bjerrum length. However, β′ 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 mC, r declines significantly below its limiting value whenever λb≳0.3, where λ is the Debye screening parameter. In the case of cylindrical arrays containing either 2 or 100 parallel charged lines, r also decreases, whenever λd≳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 Å, which are separated by d=200 Å, r exhibits a plateau value, 0.76, characteristic of the two combined lines, when λd≪2.0, and declines with increasing mC to a shelf value, 0.52, characteristic of either single line, when λd≳2.0 and the lines become effectively screened from one another. β′ 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.