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 $P\left(T\right)$, 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 $P\left(T\right)$ 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.