A dichotomy theorem for minimizers of monotone recurrence relations

Variational monotone recurrence relations arise in solid state physics as generalizations of the Frenkel-Kontorova model for a ferromagnetic crystal. For such problems, Aubry-Mather theory establishes the existence of "ground states" or "global minimizers" of arbitrary rotation number. A nearest neighbor crystal model is equivalent to a Hamiltonian twist map. In this case, the global minimizers have a special property: they can only cross once. As a nontrivial consequence, every one of them has the Birkhoff property. In crystals with a larger range of interaction and for higher order recurrence relations, the single crossing property does not hold and there can exist global minimizers that are not Birkhoff. In this paper we investigate the crossings of global minimizers. Under a strong twist condition, we prove the following dichotomy: they are either Birkhoff, and thus very regular, or extremely irregular and nonphysical: they then grow exponentially and oscillate. For Birkhoff minimizers, we also prove certain strong ordering properties that are well known for twist maps

Scalable parallel debugging with statistical assertions

Traditional debuggers are of limited value for modern scientific codes that manipulate large complex data structures. This paper discusses a novel debug-time assertion, called a "Statistical Assertion", that allows a user to reason about large data structures, and the primitives are parallelised to provide an efficient solution. We present the design and implementation of statistical assertions, and illustrate the debugging technique with a molecular dynamics simulation. We evaluate the performance of the tool on a 12,000 cores Cray XE6

Analyzing the components of the free-energy landscape in a calcium selective ion channel by Widom’s particle insertion method

The selectivity filter of the L-type calcium channel works as a Ca2 + binding site with a very large affinity for Ca2 + versus Na+. Ca2 + replaces half of the Na+ ions in the filter even when these ions are present in 1 μM and 30 mM concentrations in the bath, respectively. The energetics of this strong selectivity is analyzed in this paper. We use Widom’s particle insertion method to compute the space-dependent profiles of excess chemical potential in our grand canonical Monte Carlo simulations. These profiles define the free-energy landscape for the various ions. Following Gillespie [Biophys. J. 94, 1169 (2008)], the difference of the excess chemical potentials for the two competing ions defines the advantage that one of the ions has over the other in the competition for space in the crowded selectivity filter. These advantages depend on ionic bath concentrations: the ion that is present in the bath in larger quantity (Na+) has the “number” advantage which is balanced by the free-energy advantage of the other ion (Ca2 +). The excess chemical potentials are decomposed into hard sphere exclusion and electrostatic components. The electrostatic terms correspond to interactions with the mean electric field produced by ions and induced charges as well to ionic correlations beyond the mean field description. Dielectrics are needed to produce micromolar Ca2 + versus Na+ selectivity in the L-type channel. We study the behavior of these terms with changes in bath concentrations of ions, charges, and diameters of ions, as well as geometrical parameters such as radius of the pore and the dielectric constant of the protein. Ion selectivity in calcium binding proteins probably has a similar mechanism

Improved HIV-1 neutralization breadth and potency of V2-apex antibodies by in silico design

Summary: Broadly neutralizing antibodies (bNAbs) against HIV can reduce viral transmission in humans, but an effective therapeutic will require unusually high breadth and potency of neutralization. We employ the OSPREY computational protein design software to engineer variants of two apex-directed bNAbs, PGT145 and PG9RSH, resulting in increases in potency of over 100-fold against some viruses. The top designed variants improve neutralization breadth from 39% to 54% at clinically relevant concentrations (IC80 < 1 μg/mL) and improve median potency (IC80) by up to 4-fold over a cross-clade panel of 208 strains. To investigate the mechanisms of improvement, we determine cryoelectron microscopy structures of each variant in complex with the HIV envelope trimer. Surprisingly, we find the largest increases in breadth to be a result of optimizing side-chain interactions with highly variable epitope residues. These results provide insight into mechanisms of neutralization breadth and inform strategies for antibody design and improvement