Action Concertée Incitative: Nouvelles Interfaces des Mathématiques
Project funded by the CNRS
Nonlinear localisation and applications to the physics of biological molecules
2004-2007
Teams:
Members:
Project coordinator : Guillaume James
Institut de Mathématiques de Toulouse : Guillaume James (MCF, now professor at LJK Grenoble), Cynthia Ferreira (PhD student), Yannick Sire (PhD INSA Toulouse, now MCF at LATP Marseille).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)
Aims of the project and results :
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.
I - Discrete breathers and relative periodic orbits in Euclidean-invariant Hamiltonian systems :
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 :
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 :
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.
II - Potential energy barrier for base-pairs reclosing in the Peyrard-Bishop model :
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 y of the bond connecting the two bases. We consider a homopolymer (i.e. all base pairs are identical) in order to avoid any influence of disorder, which could introduce another source of localisation. The model takes the form of a Hamiltonian chain of coupled oscillators representing the base pairs.
The effective potential V describes the interaction between the two bases in a pair. The original model of Peyrard and Bishop uses a Morse potential represented by a dotted curve in the following graph (the plateau of the potential corresponds to the open state of a base pair). The interaction potential W is chosen anharmonic in the most elaborate versions of the model, but it is sometimes assumed harmonic (with stiffness constant K) to simplify analytical studies.
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 :
we show how the model can be modified to be quantitatively correct. We consider a new type of on-site potential V (continuous curve in the graph given above), which admits a plateau after a hump corresponding to a barrier for reclosing. Indeed, since open bases can fluctuate a lot, their motions include rotations which may hinder the reclosing of the pairs (inducing an energetic barrier for closing in the effective potential V).
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 K goes to 0.
In addition, we have proved analytically the existence of a spatially localized equilibrium of the lattice which ``bifurcates from infinity'' as K-->0. We also numerically obtain a new class of discrete breathers that do not oscillate around a ground state of the system but around this static localized equilibrium, and correspond to a Lyapunov family of periodic orbits in the limit of vanishing amplitude. An analytical proof of their existence should be available soon.
III - Effect of spatial inhomogeneities on the bifurcations of discrete breathers in the Peyrard-Bishop model :
Beyond spatially periodic systems, it is a fundamental and challenging problem to understand breather properties in nonlinear and inhomogeneous media, such as non-periodic or disordered crystals, amorphous solids and biological macromolecules. In particular, the modelling of thermal denaturation of DNA and the analysis of its local fluctuational openings represents a problem where heterogeneity is important. For parameters corresponding to real DNA sequences, Langevin molecular dynamics of the spatially inhomogeneous Peyrard-Bishop model have shown that some locations of discrete breathers heavily depend on the sequence and coincide with functional sites in DNA (Kalosakas et al, 2004).
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 K. These solutions are nonlinear ``continuations'' of a given Anderson mode from the limit of zero amplitude, for frequencies belonging to fat Cantor sets. In addition, the existence of breathers in such systems was proved by Sepulchre and MacKay (1998) in the limit K-->0, following a method initiated by MacKay and Aubry (1994).
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 K, particle masses and on-site potentials can have small variations along the chain. We look for small amplitude and time-periodic solutions, and in particular spatially localized ones (discrete breathers). The problem is reformulated as a nonautonomous recurrence in a space of time-periodic functions, where the dynamics is considered along the discrete spatial coordinate (this process is known as spatial dynamics in the context of PDE, and here this dynamics becomes discrete). Generalizing to nonautonomous maps a centre manifold theorem previously obtained by one of us for infinite-dimensional autonomous maps (James, 2003), we show that small amplitude oscillations are determined by finite-dimensional nonautonomous mappings, whose dimension depends on the solutions frequency. We consider the case of two-dimensional reduced mappings, which occurs for frequencies close to the minimal or maximal normal mode frequency (computed for the unperturbed homogeneous chain). For an homogeneous chain, the reduced map is autonomous and reversible, and bifurcations of reversible homoclinics or heteroclinic solutions are found for appropriate parameter values. These orbits correspond respectively to discrete breathers for the infinite chain, or ``dark solitons'' superposed on a spatially extended standing wave. Breather existence is shown in some cases for any value of the coupling constant K, which generalizes (for small amplitude solutions) an existence result obtained by MacKay and Aubry (1994) at small coupling.
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 image of the unstable manifold under a linear transformation T depending on the sequence intersects the stable manifold. This provides a new geometrical understanding of tangent bifurcations of discrete breathers commonly observed in classes of systems with impurities as defect strengths are varied. Moreover, by computing the principal part of the linear transformation (in the limit of small defect strength and near-critical breather frequencies), we show that the effect of the parameter sequence on the set of small amplitude breather solutions should mainly depend on weighted averages of the defects values.
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 m0 at the site n=0 (the particle mass is 1+m0), and the bifurcation occurs as m0 is increased (the breathers energies are plotted versus m0). For m0 close to 0, the upper branch represents a two-site breather centered between sites n=0 and n=1. The lower branch represents a one-site breather centered at n=1. The breathers profiles (i.e., for a DNA chain, the stretching y of base pairs as a function of n) are plotted in the central panels at the time of maximal amplitude (the corresponding value of m0 is marked with a dashed line in the left panel). The tangent bifurcation shown in the left panel has in fact a hidden geometrical explanation. The right panel shows the first intersection points between the stable and unstable manifolds of the origin for the reduced recurrence relation (continuous curve, case m0=0), and the dashed line depicts the image of the unstable manifold by the linear transformation T (in that case a linear shear) depending on m0, for some positive value of m0. Two homoclinic intersection points collapse when m0 is sufficiently increased, yielding a tangent bifurcation of breather solutions for the infinite lattice.