Solitons in a double pendulums chain model, and DNA roto-torsional dynamics

It was first suggested by Englander et al to model the nonlinear dynamics of DNA relevant to the transcription process in terms of a chain of coupled pendulums. In a related paper [4] we argued for the advantages of an extension of this approach based on considering a chain of double pendulums with certain characteristics. Here we study a simplified model of this kind, focusing on its general features and nonlinear travelling wave excitations; in particular, we show that some of the degrees of freedom are actually slaved to others, allowing for an effective reduction of the relevant equations. Introduction and motivation In a seminal paper appeared a quarter century ago, Englander, Kallenbach, Heeger, Krumhansl and Litwin [12] suggested that DNA solitons, i.e. nonlinear mechanical excitations of the DNA double helix, could have a key role in DNA functional processes – such as DNA transcription and replication – in that they would provide automatic focusing of energy and synchronization of opening along the chain. They also were the first to suggest a simple mechanical model to illustrate their argument; this consisted of a one-dimensional chain of simple pendulums, each of them coupled to the neighboring ones. In this way travelling solitons were described by the sine-Gordon equation, which is known to support both topological and dynamical solitons. (Note that, contrary to the Davydov soliton in alpha-helices [10] which is intrinsically quantum, in this case we deal with classical mechanics and hence classical solitons.) Following the suggestion of Englander et al., a number of models for the nonlinear dynamics of DNA have been elaborated by different scientists in later years. Research was pursued essentially in two directions: on the one hand, in DNA denaturation the relevant aspect is the separation between the two helices, and one looks mainly at the degree of freedom describing distance between the two bases in a (Watson-Crick) Copyright c © 2006 by M Cadoni, R De Leo and G Gaeta Work supported in part by the Italian MIUR under the program COFIN2004, as part of the PRIN project “Mathematical Models for DNA Dynamics (M ×D)”. 2 M Cadoni, R De Leo and G Gaeta pair. Considerable progress in this direction has been obtained via the model introduced by Peyrard and Bishop [25]; this has then been refined by Dauxois [9] and extended in the BCP model by Barbi, Cocco and Peyrard [1, 2, 7, 23] (related models are discussed in [21, 35]). See e.g. [3, 24] for a discussion of results and recent advances in this direction, and [8, 26, 30] for matters related to (thermodynamic) stability and bubble formation. On the other hand, models for the roto-torsional dynamics of DNA have been considered by other authors; the torsional degrees of freedom are relevant in connection with the opening of the DNA double helix taking place to allow RNA-Polymerase to access the base sequence to transcript genetic information [34]. Several models have been proposed in this direction; references and some detail on these can be found in [19, 34].2 A particularly simple model (Y model) was proposed by Yakushevich [32, 33]; see also [13, 14, 16, 19]. Despite its simplicity, the Y-model succeeds in describing several relevant features of the DNA nonlinear (and linear) dynamics related to the relevant processes [19, 34, 36], and is thus the subject of continuing interest. In recent works it has been shown that all the (sometimes, very crude) approximations introduced by Yakushevich in her model – some of these substantially affecting the dispersion relations for the model [20] – have very little impact on the fully nonlinear dynamics and in particular on solitons’ shape [17, 18]. The Y model should in any case be seen as only a first step in a hierarchy of increasingly accurate models [33], and has some drawbacks which should be removed by considering more detailed versions of the model; these are both quantitative and conceptual. On the quantitative side, we mention the impossibility to fit the observed speed of transversal waves in DNA with physical values of the parameters [36]; and the fact that soliton speed remains – provided it is smaller than a maximal speed – essentially a free parameter [15]. On the conceptual side, solitons are possible in these models thanks to the homogeneous character of the chain; but we know that actual DNA is strongly inhomogeneous, as bases are quite different from each other, and homogeneous DNA would not carry any information.3 The model we consider here describes the DNA double chain via two (rotational) degrees of freedom per nucleotide, hence it will called a “composite Y model”, following the nomenclature introduced in [4]. These degrees of freedom are related separately to rotation of the sugar-phosphate backbone (which can be of any magnitude) on the one hand, and of the nitrogen bases (which are constrained to a limited range) on the other hand. Both rotations are in the plane perpendicular to the double helix axis. This model is a simplified – and somehow a “skeleton” – version of the more realistic (and involved) model considered in [4]: in that paper we triggered our model to the actual It should be stressed that these models deal with the DNA double helix alone, i.e. do not consider its interaction with the environment or other agents as RNA Polymerase; as such, they are not of direct relevance for processes involving other actors, albeit they are definitely a first step in the study of these (see e.g. [24, 34] for a discussion of the possible functional role of nonlinear excitations in DNA). On the other hand, the description they provide of the dynamics of the DNA molecule could nowadays be tested, in principles, by means of single-molecule experiments [22, 27]. The idea behind considering such a model is of course that the homogeneous model can be considered as an “average” version of a more realistic model, which could be studied perturbatively; but in a model like the Y model (and more generally those considered in the literature) a perturbation breaking translational invariance would destroy the soliton solutions. Solitons in a double pendulum chain model 3 DNA geometry and dynamics, while here we consider a simplified model so to focus on the general abstract mechanism of interaction between topological and non topological degrees of freedom. For the same reason we will restrict to motions which are symmetric under the exchange of the two chains (as often done also in the analysis of the Peyrard-Bishop and the BCP models). The model we consider, and more generally the class of composite Y models [4], represents an improvement with respect to the standard Y model both from a quantitative and qualitative point of view. On the quantitative, phenomenological, side, we have a much higher flexibility of the chain described by the model and dramatic consequences on the model ability to provide realistic physical quantities. E.g., with the composite Y model considered in [4], one obtains the experimentally observed transverse phonon speed using interaction energies of the physical order of magnitude4, and a selection of solitons’ speed. On the qualitative side, which is maybe even more interesting (and more widely applicable than merely DNA), the composite Y model is remarkable in that the uniform and the non-uniform parts of the DNA molecule (backbone and bases respectively) are described by separate degrees of freedom. It happens that, as a consequence of the geometry of the DNA molecule, the degree of freedom describing the backbone supports topological – hence strongly stable – solitons, while the one describing the motion of bases performs quite limited excursion due to steric hindrances; as mentioned above, this is taken into account in our model. It is thus quite conceivable that introducing in the model a non-uniformity which affects only the latter degree of freedom, a perturbation approach would allow to obtain solutions in terms of perturbed solutions for the uniform model5; see sections 4 and 5 below. The perturbative approach is also attractive in that by a suitable limiting procedure, see section 3 below (and [4]), the composite Y model reduces to a standard Y model, for which exact solutions can be obtained. Thus, a perturbative description for the uniform composite Y model can be obtained by perturbing the system near the standard Y solutions. A drawback of general composite Y models is that the equations describing its dynamics are too complex to be solved analytically, and even the perturbative expansion around the solitonic Y solution can be very hard to control [4]. On the other hand, most of the nice features of the composite model seem to be quite generic, related mostly to doubling of the degrees of freedom and their different topological features, i.e. largely independent of the model details. Numerical investigations for the “realistic” composite Y model of [4] showed that introducing a number of simplifications into the model does not affect its main features – confirming the observations for the standard Y model [17, 18] – but surely makes it easier to handle it at the analytical level. Motivated by the previous arguments, we study in this paper a simple – possibly the simpler – composite Y model, for which the comparison with the standard Y model is immediate. Our main purpose is indeed to focus on the essential features introduced by As mentioned above, within the standard Y model the same speed can be fitted only using an intrapair interaction energy which is about 6000 times the physical value [36]. It should be mentioned that the BCP model [1, 2, 24] also presents the interaction of topological and non-topological degrees of freedom; however, there the topological degree of freedom is actually a cyclic variable and one has correspondingly a conservation law: this leads to a dynamics less rich topologically. 4 M Cadoni, R De Leo and G Gaeta Figure 1. Notation for each of the double pendulums along the two chains; see text. having two different kinds of degrees of freedom in chain models. For the sake of concreteness we make reference to – and discuss the consequences for – DNA dynamics, but it will be quite clear that most of our discussion is more general.