Long range correlations and folding angle with applications to α-helical proteins.

The conformational complexity of chain-like macromolecules such as proteins and other linear polymers is much larger than that of point-like atoms and molecules. Unlike particles, chains can bend, twist, and even become knotted. Thus chains might also display a much richer phase structure. Unfortunately, it is not very easy to characterize the phase of a long chain. Essentially, the only known attribute is the radius of gyration. The way how it changes when the degree of polymerization becomes different, and how it evolves when the ambient temperature and solvent properties change, is commonly used to disclose the phase. But in any finite length chain there are corrections to scaling that complicate the detailed analysis of the phase structure. Here we introduce a quantity that we call the folding angle to identify and scrutinize the phase structure, as a complement to the radius of gyration. We argue for a mean-field level relationship between the folding angle and the scaling exponent in the radius of gyration. We then estimate the value of the folding angle in the case of crystallographic α-helical protein structures in the Protein Data Bank. We also show how the experimental value of the folding angle can be obtained computationally, using a semiclassical Born-Oppenheimer description of α-helical chiral chains.

Arnol'd cat map lattices.

We construct Arnol'd cat map lattice field theories in phase space and configuration space. In phase space we impose that the evolution operator of the linearly coupled maps be an element of the symplectic group, in direct generalization of the case of one map. To this end we exploit the correspondence between the cat map and the Fibonacci sequence. The chaotic properties of these systems also can be understood from the equations of motion in configuration space. These describe inverted harmonic oscillators, where the runaway behavior of the potential competes with the toroidal compactification of the phase space. We highlight the spatiotemporal chaotic properties of these systems using standard benchmarks for probing deterministic chaos of dynamical systems, namely, the complete dense set of unstable periodic orbits, which, for long periods, lead to ergodicity and mixing. The spectrum of the periods exhibits a strong dependence on the strength and the range of the interaction.