Amazing results in the mathematics of proteins

This report describes the two-hour mini-course given by myself at the Pretty Structures 2011 workshop (http://www.lix.polytechnique.fr/~liberti/pretty_structures). This work is due to a research team also including Carlile Lavor, Jon Lee, Benoıt Masson and Antonio Mucherino. The proofs and mistakes in this presentations are, however, entirely mine. The problem I treat here is the Distance Geometry Problem (DGP): we are given a simple undirected graph G = (V,E) with a real-valued edge weight function d : E → R≥0 and we look for an embedding x : V → R of the vertices of G which preserves the Euclidean distances assigned to each edge, i.e. ∀{u, v} ∈ E ‖xu − xv‖ = duv. I shall limit the problem to consist of a set of instances describing protein graphs. Past computational experience showed us that protein graphs always had a set of solutions of cardinality either empty or a power of two; but we could manually construct counterexamples to this conjecture. I find the results contained here amazing because they show that essentially the conjecture holds notwithstanding the counterexamples! Furthermore, an amazing consequence (not discussed here) of this “power of two” property is that in we can find protein conformations in polynomial time.