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Publication# On the statistical physics of chains and rods, with application to multi-scale sequence-dependent DNA modelling

Résumé

The complex mechanisms involved in cellular processes have been increasingly understood this past century and the central role of the DNA molecule has been recognized. The base pair sequence along a DNA fragment is observed not only to encode the genomic information, but also to induce locally very specific physical properties, such as significantly bent or stiff regions. These variations in the molecule constitution are for instance believed to be involved in DNA-protein recognition and in nucleosomes positioning. Modelling the sequence dependent DNA mechanical properties is consequently an important step towards understanding many biological functions. However, in a cell, vastly different length scales are involved, ranging from a few base pairs to several thousands, which makes difficult the definition of \textit{one} appropriate model. A promising strategy seems then to be given by the multi-scale modeling of sequence dependent DNA mechanics. In this framework, the sequence dependent rigid base and rigid base pair models have been proposed. In these coarse grain models either each base pair or each base is described as a rigid body configuration, which leads to either a chain or a bichain representation of the DNA molecule. A sequence dependent configurational distribution has then been parametrized, either from experimental data or directly from atomistic molecular dynamic simulations, and provides an efficient and realistic description at the scale of hundreds of base pairs. Important questions that can be studied in these models are for instance the influence of the sequence on the probability of contact of two sites, which are distant along the molecule length, or on the expectation of the relative configuration of these two sites. In this thesis, we propose to approach these physical situations both from the discrete and the continuum modeling point of view, and then to discuss in which sense they actually constitute only one multi-scale point of view. In the first part, we discuss mechanical properties of heterogeneous rigid body chains and bichains, as well as continuum rod and birods, in classical statics and in equilibrium statistical physics. Equilibirum conditions, variational principles and configurational distributions are studied for single chains and rods, and then extended to bichains and birods. We have introduced in particular an original coordinate free Hamiltonian formulation in arc-length of the birod equilibrium conditions, and the notion of the persistence matrix for the configurational moment for chains and rods. We then present deterministic and stochastic exponential Cauchy-Born rules allowing to bridge the scales between the discrete and continuum representations. In the second part, we present applications of the proposed multi-scale mechanical theory for chains and rods to sequence dependent DNA modelling. We discuss the approximation using the birod model of most probable bichain configurations satisfying prescribed end conditions. Similarly, we then present the computation of the sequence dependent frame correlation matrix and the Flory persistence vector for chains using a continuum rod model. In addition, a homogenization method is proposed. These results are believed to constitute a substantial improvement in the multi-scale modeling of DNA mechanics.

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Principe fondamen

Sequence dependent mechanics of DNA is believed to play a central role in the functioning of the cell through the expression of genetic information. Nucleosome positioning, gene regulation, DNA looping and packaging within the cell are only some of the processes that are believed to be at least partially governed by mechanical laws. Therefore it is important to understand how the sequence of DNA affects its mechanical properties. For exploring the mechanical properties of DNA, various discrete and continuum models have been, and continue to be, developed. A large family of these models, including the model considered in this work, assume that bases or base pairs of DNA are rigid bodies. The most standard are rigid base pair models, with parameters either obtained directly from experimental data or from Molecular Dynamics (MD) simulations. The drawback of current experimental data, such as crystal structures, is that only small ensembles of configurations are available for a small number of sequences. In contrast, MD simulations allow a much more detailed view of a larger number of DNA sequences. However, the drawback is that the results of these simulations depend on the choice of the simulation protocol and force field parameters. MD simulations also have sequence length limitations and are currently too intensive for (linear) molecules longer than a few tens of base pairs. The only way to simulate longer sequences is to construct a coarse-grain model. The goal of this work is to construct a small parameter set that can model a sequence- dependent equilibrium probability distribution for rigid base configurations of a DNA oligomer with any given sequence of any length. The model parameter sets previously available were for rigid base pair models ignoring all the couplings beyond nearest neighbour interactions. However it was shown in previous work, that this standard model of rigid base pair nearest neighbour interactions is inconsistent with a (then) large scale MD simulation of a single oligomer [36]. In contrast we here show that a rigid base nearest neighbour, dimer sequence dependent model is a quite good fit to many MD simulations of different duration and se- quence. In fact a hierarchy of rigid base models with different interaction range and length of sequence-dependence is discussed, and it is concluded that the nearest neighbour, dimer based model is a good compromise between accuracy and complexity of the model. A full parameter set for this model is estimated. An interesting feature is that despite the dimer dependence of the parameter set, due to the phenomenon of frustration, our model predicts non local changes in the oligomer shape as a function of local changes in the sequence, down to the level of a point mutation.