Random walks in the space of conformations of toy proteins

Monte Carlo dynamics of the lattice toy protein of 48 monomers is interpreted as a random walk in an abstract (discrete) space of conformations. To test the geometry of this space, we examine the return probability P(T), which is the probability to find the polymer in the native state after T Monte Carlo steps, provided that it starts from the native state at the initial moment. Comparing computational data with the theoretical expressions for P(T) for random walks in a variety of different spaces, we show that conformation spaces of polymer loops may have nontrivial dimensions and exhibit negative curvature characteristics of Lobachevskii (hyperbolic) geometry.