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For most of the important processes in DNA metabolism, a protein has to reach a specific binding site on the DNA. The specific binding site may consist of just a few base-pairs while the DNA is usually several millions of base-pairs long. How does the protein search for the target site? What is the most efficient mechanism for a successful search? Motivated by these fundamental questions on intracellular biological processes, we have developed a model for searching a specific site on a model DNA by a single protein. We have made a comparative quantitative study on the efficiencies of sliding, inter-segmental hoppings and detachment/re-attachments of the particle during its search for the specific site on the DNA. We also introduce some new quantitative measures of efficiency of a search process by defining a relevant quantity, which can be measured in in-vitro experiments.

Mathematical and numerical models for studying the electrophoresis of topologically nontrivial molecules in two and three dimensions are presented. The molecules are modeled as polygons residing on a square lattice and a cubic lattice whereas the electrophoretic media of obstacle network are simulated by removing vertices from the lattices at random. The dynamics of the polymeric molecules are modeled by configurational readjustments of segments of the polygons. Configurational readjustments arise from thermal fluctuations and they correspond to piecewise reptation in the simulations. A Metropolis algorithm is introduced to simulate these dynamics, and the algorithms are proven to be reversible and ergodic. Monte Carlo simulations of steady field random obstacle electrophoresis are performed and the results are presented.

We describe a simple, discrete model of deterministic chiral motion on a square lattice. The model is based on rotating walkers with trailing tails spanning L lattice bonds. These tail segments cannot overlap and their leading A segments cannot be crossed. As prescribed by their chirality, walkers must turn if possible, or go straight, or else correct earlier steps recursively. The resulting motion traces unbound trajectories and complex periodic orbits with various symmetries. Periods tend to decrease with increasing L and vary between L and L². Interacting walkers can form intricate pair states. Some orbits match pinned spiral tip trajectories observed experimentally in excitable systems.

We prove recurrence of self-repelling and self-attracting walks on the pre-Sierpiński gasket and ℤ. We also determine a set of parameters for which this walk defines infinite, σ-finite and ergodic measure preserving dynamical systems.

The scaling behavior of linear polymers in disordered media, modelled by self-avoiding walks (SAWs) on the backbone of percolation clusters in two, three and four dimensions is studied by numerical simulations. We apply the pruned-enriched Rosenbluth chain-growth method (PERM). Our numerical results bring about the estimates of critical exponents, governing the scaling laws of disorder averages of the configurational properties of SAWs.

We discuss possible discretizations of complex analysis and some of their applications to probability and mathematical physics, following our recent work with Dmitry Chelkak, Hugo Duminil-Copin and Clément Hongler.

We outline a proof, by a rigorous renormalisation group method, that the critical two-point function for continuous-time weakly self-avoiding walk on ℤ^d decays as |x|^-(d-2) in the critical dimension d = 4, and also for all d > 4.