We are exploring the process of folding via a series of structural moves for a simple model of two dimensional linkages consisting of hydrophobic (H) and hydrophilic (P) nodes. Most extant work of this type works by restricting the node positions to be on a lattice. This makes more manageable the task of selecting potential node moves, but at the same time restricts each node to move only to adjacent lattice positions. Our off-lattice model allows instead free-form and potentially much larger node motions, by exploiting a pseudo-triangulation construction that was previously investigated by Ilena Streinu. This construction allows coordinated motions of all the nodes in such a way that the motion is guaranteed to be collision free. We still follow Dill's HP model of proteins as chains with an energy function that assigns -1 to every HH non-local interaction. As previously shown by Streinu, a pseudo-triangulation is a minimally rigid graph; removal of any convex hull edge gives rise to a one degree of freedom mechanism. This mechanism, can be moved until it causes a self intersection: the end point of this motion can be found by solving a set of simultaneous quadratic equations. For a Monte Carlo folding simulation we choose a random step by starting with a random pseudo-triangulation of the chain, along with a random hull edge to be removed, and then move the resulting mechanism until it first comes into self-contact. In a standard Monte Carlo fashion, we exploit conformational space by taking random steps and evaluating their energy. The project involves the development of a software for simulating the motion of pseudo-triangulation linkages based upon the pseudo-triangulation workbench developed by Lutz Kettner.

We plan to explore the behavior over time of the system's energy and radius of gyration. We will also investigate how the percentage of rigid linkage edges in the pseudo-triangulation affects this behavior. We hope to be able to fold long chains because of the large motions taken at every step. Perhaps also these off-lattice structures may give insight to the formation of secondary structures. An additional side-benefit is the development of improved tools for two-dimensional linkage motions. In particular, in joint with Jack Snoeyink, we have observed that the maximal rigid components of a pseudo-triangulation can be detected in linear time.