We address the problem of protein folding by studying a model of random copolymers with a Gaussian distribution of quenched disorder in the monomers' hydrophobicity. We develop the Gaussian self-consistent method in kinetics with the effective potentials depending on the disorder variables, which are further integrated out perturbatively. As an alternative, at equilibrium, we also suggest a version of a variational approach in the replica space. These methods allow us to achieve a unified description of the extended coil, liquid-like, frozen and folded globular states, as well as of the kinetic transformations between them. We discuss the role of the finite-size effects and the optimization of the disorder distribution for improving the folding properties of random sequences. We believe that the final resolution of these issues may shed light on the protein folding puzzle.

We consider a simple model of a random heteropolymer chain that is known to undergo the freezing transition. We investigate the change in the elastic properties of the chain upon freezing. The elastic constants are evaluated and it is shown that they acquire a non-trivial contribution in the frozen phase.

We apply a replica variational approach to investigate the equilibrium properties of a random amphiphilic copolymer as a model for protein folding. We discuss the properties of replica symmetric and replica symmetry-broken solutions in the glassy phase. We show that the freezing transition has a scale dependence due to the connectivity of the copolymer chain. This reflects the existence of connectivity-induced frustration, which is the simplest case of the more general class of topology-induced frustration.

We present the results of our study of the glassy phase of a random copolymer. We use a simple Ansatz to develop a qualitative theory, which enables us to recover the main computational results without considerable effort.

We study the phase transitions of a random copolymer chain with quenched disorder. We calculate the average over the quenched disorder in replica space and apply a Gaussian variational approach based on a generic quadratic trial Hamiltonian in terms of the correlation functions of monomer Fourier coordinates. This has the advantage that it allows us to incorporate fluctuations of the density, determined self-consistently, and to study collapse, phase separation transitions and the onset of the freezing transition within the same mean field theory. The effective free energy of the system is derived analytically and analyzed numerically in the one-step Parisi scheme. Such quantities as the radius of gyration, end-to-end distance or the average value of the overlap between different replicas are treated as observables and evaluated by introducing appropriate external fields to the Hamiltonian. As a result we obtain the phase diagram in terms of model parameters, scaling for the freezing transition and the dependence of correlation functions on the chain index.

We present the results of our study of the freezing transition of an amphiphilic random copolymer. We here confirm that a replica variational approach predicts a ''scale''-dependent freezing transition due to the connectivity of the chain. In addition we suggest that two systems, a random copolymer and an Ising spin-glass, can be directly related to each other on the mean-field level in the vicinity of the freezing transition. Both systems have the same type of effective free energy. The properties of replica symmetrical (RS) and replica symmetry broken (RSB) solutions are discussed. The latter has larger radius of gyration and effective free energy, and is less phase separated. It might be related to a globule with more than one hydrophobic core.

The Blume–Emery–Griffiths three-spin state lattice model is applied to describe three component mixtures of water, polymer, and amphiphile. A simple mean-field theory similar to the Flory–Huggins approach is used to analyze the lattice model. The shape of the demixation curve is derived analytically in the mean-field approximation for small concentrations of amphiphile. We analyze the system in the vicinity of the critical point. In addition we consider mixtures of two polymers in solvent, and discuss the demixation curve for such systems.

We use the mean field replica theory to analyze the dependence of the freezing transition temperature on the composition and coupling constants for two letter random copolymers. Gaussian disorder is considered.

We study the phase transitions of a random copolymer chain with quenched disorder. We apply a replica variational approach based on a Gaussian trial Hamiltonian in terms of the correlation functions of monomer Fourier coordinates. This allows us to study collapse, phase separation and freezing transitions within the same mean field theory. The effective free energy of the system is derived analytically and analysed numerically. Such quantities as the radius of gyration or the average value of the overlap between different replicas are treated as observables and evaluated by introducing appropriate external fields to the Hamiltonian. We obtain the phase diagram and show that this system exhibits a scale dependent freezing transition. The correlations between replicas appear at different length scales as the temperature decreases. This indicates the existence of the topological frustration.