Nonlinear effects in the torsional adjustment of interacting DNA

DNA molecules in solution, having negatively charged phosphates and countercations readsorbed on its surface, possess a distinct charge separation motif to interact electrostatically. If their double-helical structure were ideal, duplexes in parallel juxtaposition could choose azimuthal alignment providing attraction, or at least a reduction of repulsion, between them. But duplexes are not perfect staircases and the distortions of their helical structure correlate with their base pair texts. If the patterns of distortions on the opposing molecules are uncorrelated, the mismatch will accumulate as a random walk and attraction vanishes. Based on this idea, a model of recognition of homologous sequences has been proposed [A. A. Kornyshev and S. Leikin, Phys. Rev. Lett. 86, 3666 (2001)]. But DNA has torsional elasticity. How will this help to relax a mismatch between the charge distributions on two nonhomologous DNA's? In the same work, the solution of this problem has been mapped onto a frustrated sine Gordon equation in a nonlocal random field (where the latter represents a pattern of twist angle distortions on the opposing molecules), but the results had been obtained in the limit of torsionally rigid molecules. In the present paper, by solving this equation numerically, we find a strongly nonlinear relaxation mechanism which utilizes static kink-soliton modes triggered by the "random field." In the range of parameters where the solitons do not emerge, we find good agreement with the results of a variational study [A. G. Cherstvy, A. A. Kornyshev, and S. Leikin, J. Phys. Chem. B (to be published)]. We reproduce the first-order transitions in the interaxial separation dependence, but detect also second-order or weak first-order transitions for shorter duplexes. The recognition energy between two nonhomologous DNA sequences is calculated as a function of interaxial separation and the length of juxtaposition. The soliton-caused kinky length dependence is discussed in connection with plots of recombination frequency as a function of the length of homology.