Phases of Polymers and Biopolymers

In this thesis we develop coarse grained models aiming at understanding physical problems arising from phase transitions which occur at the single molecule level. The thesis will consist of two parts, grossly related to and motivated by the two subjects dealt with above. In the first half, we will focus on critical phenomena in stretching experiments, namely in DNA unzipping and polymer stretching in a bad solvent. In the second part, we will develop a model of thick polymers, with the goal of understanding the origin of the protein folds and the physics underlying the folding ‘transition’, as well as with the hope of shedding some light on some of the fundamental questions highlighted in this Introduction. In the first part of the thesis we will introduce a simple model of self-avoiding walks for DNA unzipping. In this way we can map out the phase diagram in the force vs. temperature plane. This reveals the present of an interesting cold unzipping transition. We then go on to study the dynamics of this coarse grained model. The main result which we will discuss is that the unzipping dynamics below the melting temperature obeys different scaling laws with respect to the opening above thermal denaturation, which is governed by temperature induced fluctuating bubbles. Motivated by this and by recent results from other theoretical groups, we move on to study the relation to DNA unzipping of the stretching of a homopolymer below the theta point. Though also in this case a cold unzipping is present in the phase diagram, this situation is richer from the theoretical point of view because the physics depends crucially on dimension: the underlying phase transition indeed is second order in two dimensions and first order in three. This is shown to be intimately linked to the failure of mean field in this phenomena, unlike for DNA unzipping. In particular, the globule unfolds via a series (hierarchy) of minima. In two dimensions they survive in the thermodynamic limit whereas if the dimension, d, is greater than 2, there is a crossover and for very long polymers the intermediate minima disappear. We deem it intriguing that an intermediate step in this minima hierarchy for polymers of finite length in the three-dimensional case is a regular mathematical helix, followed by a zig-zag structure. This is found to be general and almost independent of the interaction potential details. It suggests that a helix, one of the well-known protein secondary structure, is a natural choice for the ground state of a hydrophobic protein which has to withstand an effective pulling force. In the second part, we will follow the inverse route and ask for a minimal model which is able to account for the basic aspects of folding. By this, we mean a model which contains a suitable potential which has as its ground state a protein-like structure and which can account for the known thermodynamical properties of the folding transition. The existing potential which are able to do that[32] are usually constructed ‘ad hoc’ from knowledge of the native state. We stress that our procedure here is completely different and the model which we propose should be built up starting from minimal assumptions. Our main result is the following. If we throw away the usual view of a polymer as a sequence of hard spheres tethered together by a chain (see also Chapter 1) and substitute it with the notion of a flexible tube with a given thickness, then upon compaction our ’thick polymer’ or ’tube’ will display a rich secondary structure with protein-like helices and sheets, in sharp contrast with the degenerate and messy crumpled collapsed phase which is found with a conventional bead-and-link or bead-and-spring homopolymer model. Sheets and helices show up as the polymer gets thinner and passes from the swollen to the compact phase. In this sense the most interesting regime is a ‘twilight’ zone which consists of tubes which are at the edge of the compact phase, and we thus identify them as ‘marginally compact strucures’. Note the analogy with the result on stretching, in which the helices were in the same way the ‘last compact’ structures or the ‘first extended’ ones when the polymer is being unwinded by a force. After this property of ground states is discussed, we proceed to characterize the thermodynamics of a flexible thick polymer with attraction. The resulting phase diagram is shown to have many of the properties which are usually required from protein effective models, namely for thin polymers there is a second order collapse transition (O collapse) followed, as the temperature is lowered, by a first order transition to a semicrystalline phase where the compact phase orders forming long strands all aligned preferentially along some direction. For thicker polymers the transition to this latter phase occurs directly from the swollen phase, upon lowering T, through a first order transition resembling the folding transition of short proteins.