The 20 residue long Trp-cage is the smallest protein known, and thus has been the subject of several in vitro and in silico folding studies. Here, we report the multistate folding scenario of the miniprotein in atomic detail. We detected and characterized different intermediate states by temperature dependent NMR measurements of the (15)N and (13)C/(15)N labeled protein, both at neutral and acidic pH values. We developed a deconvolution technique to characterize the invisible--fully folded, unfolded and intermediate--fast exchanging states. Using nonlinear fitting methods we can obtain both the thermodynamic parameters (ΔH(F-I), T(m)(F-I), ΔC(p)(F-I) and ΔH(I-U), T(m)(I-U), ΔC(p)(I-U)) and the NMR chemical shifts of the conformers of the multistate unfolding process. During the unfolding of Trp-cage distinct intermediates evolve: a fast-exchanging intermediate is present under neutral conditions, whereas a slow-exchanging intermediate-pair emerges at acidic pH. The fast-exchanging intermediate has a native-like structure with a short α-helix in the G(11)-G(15) segment, whereas the slow-exchanging intermediate-pair presents elevated dynamics, with no detectable native-like residue contacts in which the G(11)-P(12) peptide bond has either cis or trans conformation. Heteronuclear relaxation studies combined with MD simulations revealed the source of backbone mobility and the nature of structural rearrangements during these transitions. The ability to detect structural and dynamic information about folding intermediates in vitro provides an excellent opportunity to gain new insights into the energetic aspects of the energy landscape of protein folding. Our new experimental data offer exceptional testing ground for further computational simulations.

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Motivated by the algorithmic study of 3-dimensional manifolds, we explore the structural relationship between the JSJ decomposition of a given 3-manifold and its triangulations. Building on work of Bachman, Derby-Talbot and Sedgwick, we show that a "sufficiently complicated" JSJ decomposition of a 3-manifold enforces a "complicated structure" for all of its triangulations. More concretely, we show that, under certain conditions, the treewidth (resp. pathwidth) of the graph that captures the incidences between the pieces of the JSJ decomposition of an irreducible, closed, orientable 3-manifold M yields a linear lower bound on its treewidth tw(M) (resp. pathwidth pw(M)), defined as the smallest treewidth (resp. pathwidth) of the dual graph of any triangulation of M.We present several applications of this result. We give the first example of an infinite family of bounded-treewidth 3-manifolds with unbounded pathwidth. We construct Haken 3-manifolds with arbitrarily large treewidth-previously the existence of such 3-manifolds was only known in the non-Haken case. We also show that the problem of providing a constant-factor approximation for the treewidth (resp. pathwidth) of bounded-degree graphs efficiently reduces to computing a constant-factor approximation for the treewidth (resp. pathwidth) of 3-manifolds.

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