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Please use this identifier to cite or link to this item: http://hdl.handle.net/1942/8761

Title: Statistical mechanics of RNA folding: a lattice approach
Authors: Leoni, Peter
Advisors: Vanderzande, Carlo
Issue Date: 2003
Publisher: UHasselt Diepenbeek
Abstract: In the statistical mechanics of polymers one studies the large-length behaviour of polymer chains. These large molecules are built of small units or monomers. In the most simple examples the polymer consists of a sequence of identical atomic groups, joined together into a chain by covalent bonds. One example of such a homopolymer is the well-known polyethylene where the monomers are ethylene CH2 molecules. In other molecules such as proteins, DNA and RNA, the building stones are not identical and these polymers are called heteropolymers. At a microscopic level the binding angle between two neighbouring monomers in the chain is usually a fixed number, called the valence angle. This leaves only one rotational degree of freedom and the flexibility of the chain is concentrated at individual points. However it is also known that the correlation between the directions of two monomers in the chain decays exponentially with the number of monomers between them. There exists a characteristic length , called the persistence length above which the directions are uncorrelated. This length depends on the monomers in question and can vary from 5 monomers for a simple flexible polymer to 150 for the double helix DNA. Since we are mainly interested in large scale properties, it is convenient to take groups of molecules of length of the order of the persistence length. We will refer to these groups as monomers again. This will allow us to take the binding angle between successive monomers as independent. The most simple model for a. polymer is then the ide;.11 polymer chain in which each polymer is modelled by a random walk in JR3 . One knows that at large length scales the proper ties of the random walk are the same as the properties of a random walk on a lattice Z3 . This gives us the most simple lattice model for a polymer. Below we will explain that most often the random walk description is not appropriate. Polymer physics have a long history and we do not wish to discuss all the aspects of its history, but a thesis using polymer models cannot leave out some fundamental landmarks in the history. A first break-through occurred when Flory [Flo49] provided a simple argument for the swelling of flexible polymer chains. The resulting prediction for the critical exponents is remarkably accurate in all dimensions. Today, the reason for understanding why this simple argument work:, so will, is still lacking. In 1972 is was shown by P. G. de Gennes [dG72, Gen79] that the study of polymer physics can be related to the study of critical phenomena. This has led to a major advancement in understanding polymers since the complete machinery that is available for studying magentic systems could now be applied to polymer models. In this thesis we will always investigate polymers in a dilute regime, which means that we can study isolated polymers. Another assumption that is often made is that the polymer is immersed in a good solvent. This means that there is an advantage in energy for the monomers to be surrounded by molecules from the solvent rather than by other monomers of the chain . As a consequence, if a certain monomer is located somewhere in space, there is some small volume element surrounding it in which it is very unlikely to find another monomer. This effect is called the excluded volume effect and it leads to a completely different structure on a large scale as we will illustrate in the section 1.3. A very simple model that includes this excluded volume effect is the self-avoiding walk, which is a lattice walk in which two monomers cannot share the same location in space (we will give a more formal definition in section 1.3). By now this model has been well studied and there is little doubt that the self-avoiding walk model is not only a very good model, but in fact, provides a perfect model for some properties of linear polymers in a good solvent . However there exist some situations in which long polymers actually behave ideally. We mention as an example the B-point [Van98] in three dimensions where attractive forces between monomers exactly balance t he excluded volume repulsion. The fact that a lattice model can capture the basic properties of a real molecule can be understood from universality. It is known that critical systems can be divided into a number of universality classes. Systems belonging to the same universality class share the same critical behaviour. These classes are only determined by rather general properties such as dimension and symmetry. As a simple example we already mentioned that the random walk in space and the random walk on the lattice share the same asymptotic behaviour. From this point of view it becomes clear that one should take a model that is as simple as possible since it will capture the most important properties of any other model within the same universality class.
URI: http://hdl.handle.net/1942/8761
Category: T1
Type: Theses and Dissertations
Appears in Collections: PhD theses
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