Theoretical models for studying thermodynamic properties of biological macromolecules

The three-dimensional organization of biological macromolecules as well as their superstructural transitions appears to be a complex problem involving static and large scale dynamic effects. However, in many cases some features dominate over the others; the possibility of identifying these determinants allows considerable simplifications of the models, because a lower number of variables have to be considered. In the case of the superstructural transformations of DNA, the dominant effect seems to be the elastic energy involved in the continuous deformations of the structure. The aim of this work is the development of an analytical model to study the transitions of circular DNAs toward supercoiled states, commonly referred as writhe transitions. Although there are several papers in the literature concerning experimental electrophoretic studies of natural DNAs, the current results have been compared with Monte Carlo data by different authors, for a more straightforward comparison. This is because electrophoretic experiments just reveal the DNA behaviour averaged over a conformation ensemble, whereas Monte Carlo data are split into the contributions of the forms with different topological numbers. The comparison of the elastic ground-state energy and entropy of a 468 bp isotropic chain, calculated with the present model, and the ensemble average energy evaluated by Monte Carlo simulations (Gebe & Schurr, 1996) shows a straightforward correlation. The two sets of data differ for a strictly constant quantity, supporting the initial hypothesis of the equivalence of the interactions with the solvent and/or counterions, and the independence of the fluctuations of the writhing number in the whole set of the possible transformations. The application of the model to intrinsically curved DNAs has shown that curvature always seems to stabilize the supercoiled DNA forms, shifting the transitions to lower linking number differences. Also the bent of the DNA axis due to protein binding seems to favour the occurrence of writhe transitions, as in the case of the catabolite gene activator protein (CAP). These findings suggest a new role of the sequence in determining the mechanical behaviour of DNA, either determining its intrinsic curvature or directing the binding of bent-inducing proteins. In the latter half of this work, the same topological approach has been extended to the study of the tertiary organization of globular proteins. In particular, a new algorithm is proposed to identify proteins domains, based on the statistical analysis of the distribution of the local axes of the polypeptide chain, evaluated on topological basis. A polypeptide chain can be modelled as a ribbon and, therefore, can be studied in terms of topological quantities. The complexity of a space curve is conveniently described by its writhing number, which represents the number of self-crossings in the curve projections onto a unit sphere. Each crossing is taken to be positive or negative depending if it is right- or left-handed. Consequently, the summation of the absolute values of the writhing-number contributions, here referred as absolute writhing number, can conveniently describe the three-dimensional organization of a non-directional space curve. It is interesting that the absolute writhing number of 71 crystallographic and NMR structures from the Protein Data Bank (Bernstein et al., 1977) shows a very good correlation (R = 0.996) with the number of amino acid residues. For a more straightforward comparison, the absolute writhing number was calculated for a set of random coils. They show the same dependence of the native proteins with respect to the amino-acids number, but they appear much more spread out (R = 0.953) and the fitting line lies below that relative to the native proteins. The result is consistent with the fact that random coils are generally less compact of native proteins. This finding supports the hypothesis that topological requirements should be satisfied in the process of protein folding and in the final organization of the tertiary structures.