Dinamica Del Adn

Journal of Nonlinear Mathematical Physics

Volume 15, Number 2 (2008), 186–204

Article

Solitary waves in helicoidal models of DNA dynamics
Giuseppe GAETA and Laura VENIER Dipartimento di Matematica, Universit` di Milano, via Saldini 50, 20133 Milano (Italy) a E-mail: gaeta@mat.unimi.it and laura venier@infinito.it
Received November 6, 2007; Accepted January 3, 2008 Abstract We analyze travellingsolitary wave solutions in the Barbi-Cocco-Peyrard and in a simplified version of the Cocco-Monasson models of nonlinear DNA dynamics. We identify conditions to be satisfied by parameters for such solutions to exist, and provide first order ODEs whose solutions give the required solitary waves; these are not solvable in analytical terms, but are easily integrated numerically.

Introduction
It isconjectured since a long time [13] that soliton-like excitations could be present, and play a functional role, in DNA. They could be relevant both for denaturation (here the relevant excitations would be breathers [17, 20]) and for DNA transcription (here the relevant excitations would be travelling kink solitons [15, 24]). When trying to model DNA dynamics, one is faced with an extremely complexmolecule; the nearly-regular structure (a regular backbone with attached bases, themselves of four possible types with similar properties) suggests to start from modelling a regular (i.e. homogeneous) chain. In any case, the number of classical degrees of freedom per nucleotide is extremely high. On the other hand, the dynamics of molecules is governed by quantum mechanics, so that a number ofdegrees of freedom could (and actually will) be frozen at body temperature; one is thus led to identify the softer degrees of freedom and focus attention on these. Such an analysis leads to consider two degrees of freedom, whose activation energies are similar and lower than those of other ones [15]. One of these corresponds essentially to a radial opening of the double helix (that is, the bases in thepairs and hence the two helices separate moving away from the axis of the double helix); the other one corresponds to rotations of the bases (around the sugarphosphate backbone) in a plane roughly orthogonal to the double helix axis. Actually, it is quite lucky that in DNA denaturation it is essentially the first of these which is at play, while in DNA transcription the opening of the double chain(to let the RNA Polymerase access the genetic sequence) happens essentially via rotation of the bases so not to disrupt the double helix. It was thus entirely natural to elaborate models taking

Copyright c 2008 by G Gaeta and L. Venier

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into account either one of these two degrees of freedom, depending on the process one was aiming atmodelling. In fact, models for DNA denaturation – in particular, the Peyrard-Bishop (PB) model [18] and improvements thereof [19, 20] – consider a radial degree of freedom; while model dealing with the DNA structure modification met in the transcription process1 – like the Yakushevich (Y) model [22] and improvements thereof [23, 24] – consider angular degrees of freedom. As mentioned above, dependingon the biological process at hand one is also interested in different nonlinear excitations. In the first case (PB model), one is mainly interested in breathers; while in the second one (Y model) one looks for travelling (topological [12]) solitons.2 Simple DNA models were able to support relevant nonlinear excitations, and quite successful in describing several experimentally measurable quantitiesassociated with the dynamics of the DNA molecule [17, 20, 24]. By the end of the nineties, however, one was able to perform more refined experiments – able to carefully study the dynamics of a single DNA molecule subject to exactly controlled external forces. This called for more detailed models, and an obvious improvement was to consider both the softer degrees of freedom, i.e. the “radial” and...