Coulombic free energy of polymeric nucleic acid: low- and high-salt analytical approximations for the cylindrical Poisson-Boltzmann model.

An accurate analytical expression for the Coulombic free energy of DNA as a function of salt concentration ([salt]) is essential in applications to nucleic acid (NA) processes. The cylindrical model of DNA and the nonlinear Poisson-Boltzmann (NLPB) equation for ions in solution are among the simplest approaches capable of describing Coulombic interactions of NA and salt ions and of providing analytical expressions for thermodynamic quantities. Three approximations for Coulombic free energy G(u,infinity)(coul) of a polymeric nucleic acid are derived and compared with the numerical solution in a wide experimental range of 1:1 [salt] from 0.01 to 2 M. Two are obtained from the two asymptotic solutions of the cylindrical NLPB equation in the high-[salt] and low-[salt] limits: these are sufficient to determine G(u,infinity)(coul) of double-stranded (ds) DNA with 1% and of single-stranded (ss) DNA with 3% accuracy at any [salt]. The third approximation is experimentally motivated Taylor series up to the quadratic term in ln[salt] in the vicinity of the reference [salt] 0.15 M. This expression with three numerical coefficients (Coulombic free energy and its first and second derivatives at 0.15 M) predicts dependence of G(u,infinity)(coul) on [salt] within 2% of the numerical solution from 0.01 to 1 M for ss (a = 7 A, b = 3.4 A) and ds (a = 10 A, b = 1.7 A) DNA. Comparison of cylindrical free energy with that calculated for the all-atom structural model of linear B-DNA shows that the cylindrical model is completely sufficient above 0.01 M of 1:1 [salt]. The choice of two cylindrical parameters, the distance of closest approach of ion to cylinder axis (radius) a and the average axial charge separation b, is discussed in application to all-atom numerical calculations and analysis of experiment. Further development of analytical expression for Coulombic free energy with thermodynamic approaches accounting for ionic correlations and specific effects is suggested.