Institute for Biochemical Physics, Russian Academy of Sciences, ul. Kosygina 4, Moscow, 119334, Russian Federation
A single polymer macromolecule is considered with disorder types such as branches, knots, and heterogeneous sequences of chemical units. In all cases, simple theoretical approaches are employed to gain useful physical insights. For branched polymers, a simple Flory-type theory is described by means of which the difference between the universality classes for molecules with quenched and annealed branches is demonstrated. For knots, another Flory-type theory is suggested to describe the swelling and/or collapse of a quenched topology ring or the size distribution for the annealed case. To consider heteropolymers, the Random Energy Model borrowed from the spin glass theory is systematically employed. This allows a simple yet rigorous description of both the freezing transition of a random sequence globule and the use of the canonical ensemble for designing sequences with energy-optimized ground state conformation. Along with the analytical theory, computer tests for the freezing and design processes are discussed. The sequence design scheme is shown to yield a specific prediction concerning the character of correlations in protein sequences. Statistical tests confirming this prediction are described.