Laboratoire de Physique de l'ENS Lyon : Michel Peyrard (professor), Santiago Cuesta Lopez (postdoc).

Institut Camille Jordan : Pascal Noble (MCF).

External collaborators : Bernardo Sànchez-Rey, Jesus Cuevas (nonlinear physics group, univ. Sevilla)

Biological macromolecules are highly dynamical objects whose functions generally require large conformational changes. For example, DNA must be locally open by breaking the pairing between its bases in order to allow access to the genetic code embedded inside the double helix. Moreover, even in the absence of enzymes involved in the reading or duplication of the code, DNA undergoes large amplitude fluctuations known by biologists as the ``breathing of DNA''. Such large amplitude motions induce important nonlinear effects whose understanding constitutes a challenging problem. Moreover, the spatial structures of biological molecules and their nonuniform charge distributions induce important discreteness effects, which have to be taken into account to understand their dynamical properties.

Fundamental effects obtained by combining nonlinearity and spatial discreteness can be studied theoretically by considering lattices of interacting particles subject to anharmonic potentials. In particular, such models typically allow for the existence of time-periodic and spatially localized excitations known as ``discrete breathers'' or ``intrinsic localized modes'' (see e.g. the reviews Flach and Willis, 1998; Flach and Gorbach, 2007) This picture is consistent with experimental studies of the DNA breathing, which have indeed shown that fluctuational openings of DNA are highly localized in space.

The mathematical theory of localized waves in nonlinear lattices has known important developments in the last 15 years, but the models considered up to now missed important ingredients in order to be applied to the context of biomolecules. Our project has selected three directions of improvements, which raised interesting mathematical questions and were at the same time related with experiments.

The first direction was to generalize an existence theorem of discrete breathers in networks of weakly coupled particles (MacKay and Aubry, 1994) to the case of Euclidean-invariant Hamiltonian systems in more than one dimension (the 1D case is much simpler and was solved by Livi, Spicci and MacKay in 1997). Such systems are relevant in the context of chemical physics since all molecular classical Hamiltonians fall within this category. We have obtained a proof for the 2D case in reference :

Original publication available at www.springerlink.com. pdf file

Our existence theorem is valid for an arbitrary (finite) number of particles. It is based on the continuation of breather solutions with respect to a parameter measuring the mass ratio between certain groups of atoms. Discrete breathers are proved to exist in the limit of large mass ratio, which turns out to be consistent with experimental observations of local modes in three-dimensional tetraedric molecules (Halonen et al, 1998).

We have generalized this approach to relative periodic orbits (i.e. time-periodic orbits in a frame rotating at constant velocity) in reference :

Original publication available at www.sciencedirect.com. pdf file

where we consider relative periodic orbits in a class of triatomic Euclidean-invariant (planar) Hamiltonian systems. These systems consist of two identical heavy atoms and one light atom, and their atomic mass ratio is treated as above as a continuation parameter. Here we do not address the problem of spatial localisation (since the model consists in a small triatomic molecule), but instead revisit the theory of relative periodic orbits in a class of molecular Hamiltonians.

Indeed, under some nondegeneracy conditions, we show that a given family of relative periodic orbits existing at infinite mass ratio (and parametrized by phase, rotational degree of freedom and period) persists at sufficiently large mass ratio and for nearby angular velocities (this result is valid for small angular velocities). Our results provide several types of relative periodic orbits, which extend from small amplitude relative normal modes (as in the general results by Ortega, 2003) up to large amplitude solutions which are not restrained to a small neighborhood of a stable relative equilibrium. In particular, we show the existence of large amplitude motions of inversion, where the light atom periodically crosses the segment between heavy atoms.

The extension of these theories to the three-dimensional case appears to be mainly technical. Two challenging theoretical problems raise up after these works : 1-the jump from local continuation results to global ones, and 2-the persistence of the discrete breathers and relative periodic orbits we have obtained under the addition of quantum effects.

This part of the project leaves the atomic scale for a mesoscopic scale. We consider a model of DNA at the scale of a base pair originally introduced by Peyrard and Bishop (1989) to study the thermal denaturation of DNA. It considers a single variable for each base pair, the stretching

The effective potential

The Peyrard-Bishop model reproduces quite accurately denaturation curves of short DNA segments obtained in experiments and, from a dynamical point of view, it is able to qualitatively describe DNA breathing. However this description is quantitatively not satisfactory. The problem concerns the lifetime of the open states of the base pairs which appear as breathing modes: excited base pairs open and close with a period of the order of a picosecond, whereas the lifetimes of open states measured in experiments are in the nanosecond range. Thus, although the model gives statistical averages which can be fitted to experimental results for the melting curves of short DNA segments, its dynamics is not consistent with the observations. In reference :

Original publication available at www.springerlink.com. pdf file

we show how the model can be modified to be

In addition, the new model is interesting by itself from a mathematical point of view because it admits a new class of discrete breather which bifurcate from infinity as the coupling constant

In addition, we have proved analytically the existence of a spatially localized equilibrium of the lattice which ``bifurcates from infinity'' as

Original publication available at www.tandfonline.com. pdf file

Original publication available at www.aimsciences.org. Preprint available on arXiv: pdf file.

Beyond spatially periodic systems, it is a fundamental and challenging problem to understand breather properties in

From a mathematical point of view, Albanese and Fröhlich have proved in 1988 the existence of breathers for a class of random Hamiltonian systems consisting of an infinite array of nonlinear oscillators (with randomly chosen linear frequencies), coupled by harmonic springs of stiffness

Although providing existence results in two different asymptotic limits, these methods do not allow to analyze the bifurcations of discrete breathers when defect strengths varied. From a physical point of view it is however quite important to understand how a local change in the lattice parameters modifies the set of spatially localized solutions for the Peyrard-Bishop model or similar systems. For example, this could help to explain how a mutation at a specific location of a DNA sequence will modify the structure of fluctuational openings (Kalosakas et al, 2004).

We have developed a new analytical tool in this direction in reference :

In this work we consider an infinite chain of particles linearly coupled to their nearest neighbours and subject to an anharmonic local potential. The chain is assumed weakly inhomogeneous, i.e. coupling constants

For an inhomogeneous chain the study of the nonautonomous reduced map is in general far more involved. Here this problem is considered when the chain presents a finite number of defects. For the principal part of the reduced recurrence, using the assumption of weak inhomogeneity, we show that homoclinics to 0 exist when

The case of a mass impurity has been studied in detail, and our geometrical analysis could be successfully compared with direct numerical simulations.

A simple example is illustrated in the above figure. The left plot shows a tangent bifurcation between two breather solutions (having the same frequency) numerically computed in the oscillator chain. The chain presents a mass defect