Statistical Mechanics of Polymer Stretching

The interplay between Physics and Biology is certainly one of the most exciting field in modern Science. In particular, the discovery that proteins, DNA and RNA have rather peculiar spatial arrangements [1, 2, 3] has convinced biological physicists that these simple forms may be deduced from an underlying principle. Besides, new experimental techniques have supplied high-quality data, which can be investigated and compared to theoretical models. In particular, in this Thesis, we have focused our attention on the theoretical study of some elastic and thermodynamic properties of polymers and, in particular, biopolymers such as proteins, DNA and RNA [4]. This research work is organized as follows: In Chap. 1, we introduce some basic concept on polymers and biopolymers. In particular, biopolymers has attracted the attention of many research groups. Probably, their most appealing property is that they are organized in simple hierarchical structures [5]. In fact, the primary amino acid sequence of proteins is disposed in some fascinating forms as alpha-helices and beta-strands, which at an outer level form compact structures called domains. Moreover, 50 years ago, Watson and Crick [2] discovered the marvelous double helix of DNA. Furthemore, polymers seem to display many intriguing features, since they can not be described in terms of ordinary solids. This is due to the covalent nature of the bonds between consecutive monomers. Due to temperature fluctuations of these bonds, a polymer can not be viewed as a rigid macromolecule. Since these fluctuations favor many different spatial conformation, a Statistical Mechanics approach has revealed very useful [3]. Then, we focus on the important problem of polymer elasticity and introduce some preliminary concepts as the Kuhn length and the persistence length [4]. In Chap. 2, we focus our attention about some recent experiments on polymer stretching. Firstly, we begin with a brief introduction to some recent experimental techniques. Mainly, we focus on optical tweezers [6], atomic force microscopes [7] and soft microneedles [8]. We also give a short explanation about their technical features, including practical limitations and available force ranges. Besides, we describe in great details many force driven phase transition which occur in real polymers. Then, Statistical Mechanics allows for a rigorous approach to these phenomena. As explained above, we also address the important problem of elasticity in polymers, introducing the freely jointed chain (FJC) model and the worm like chain (WLC) model [3]. In Chap. 3, we describe the stretching behaviour of polymers, with the introduction of some chosen 2d on-lattice models and 3d off-lattice models. In the framework of a simplified approach on a self-interacting directed selfavoiding walk (DSAW) [9], we have discussed the importance of some scaling laws that we think to be of more general validity. Then, we introduce a more realistic model for a self-interacting SAW [9]. In particular we are able to describe its phase diagram. Through the introduction of an off-lattice self-avoiding polymer, we also give a simple explanation of some recent puzzling experimental results described in Chap. 2. In Chap. 4, we shall focus our attention on the stretching behaviour of polymer in a good solvent [10]. Generalizing the WLC approach of Marko and Siggia [11], we obtain a new interpolation formula, which perfectly describes some numerical data, obtained with Monte Carlo simulations. Furthermore, this formula seems to be more powerful than Marko and Siggia’s one. In fact, it fits well some experimental data taken from literature, that the previous approach was not able to describe correctly. Finally, we outline final conclusions and perspectives.