NONUNIFORM FOURIER TRANSFORMS FOR RIGID-BODY AND MULTI-DIMENSIONAL ROTATIONAL CORRELATIONS

On the construction of general cubature formula by flat extensions

We describe a new method to compute general cubature formulae. The problem is initially transformed into the computation of truncated Hankel operators with flat extensions. We then analyze the algebraic properties associated to flat extensions and show how to recover the cubature points and weights from the truncated Hankel operator. We next present an algorithm to test the flat extension property and to additionally compute the decomposition. To generate cubature formulae with a minimal number of points, we propose a new relaxation hierarchy of convex optimization problems minimizing the nuclear norm of the Hankel operators. For a suitably high order of convex relaxation, the minimizer of the optimization problem corresponds to a cubature formula. Furthermore cubature formulae with a minimal number of points are associated to faces of the convex sets. We illustrate our method on some examples, and for each we obtain a new minimal cubature formula.

Computational Refinement and Validation Protocol for Proteins with Large Variable Regions Applied to Model HIV Env Spike in CD4 and 17b Bound State

Envelope glycoprotein gp120 of HIV-1 possesses several variable regions; their precise structure has been difficult to establish. We report a new model of gp120, in complex with antibodies CD4 and 17b, complete with its variable regions. The model was produced by a computational protocol that uses cryo-electron microscopy (EM) maps, atomic-resolution structures of the core, and information on binding interactions. Our model has excellent fit with EMD: 5020, is stereochemically and energetically favorable, and has the expected binding interfaces. Comparison of the ternary arrangement of the loops in this model with those bound to PGT122 and PGV04 suggested a possible motion of the V1V2 away from the CCR5 binding site and toward CD4. Our study also revealed that the CD4-bound state of the V1V2 loop is not optimal for gp120 bound with several neutralizing antibodies.

X-ray, Cryo-EM, and computationally predicted protein structures used in integrative modeling of HIV Env glycoprotein gp120 in complex with CD4 and 17b

We present the data used for an integrative approach to computational modeling of proteins with large variable domains, specifically applied in this context to model HIV Env glycoprotein gp120 in its CD4 and 17b bound state. The initial data involved X-ray structure PDBID:1GC1 and electron microscopy image EMD:5020. Other existing X-ray structures were used as controls to validate and hierarchically refine partial and complete computational models. A summary of the experiment protocol and data was published (Rasheed et al., 2015) [26], along with detailed analysis of the final model (PDBID:3J70) and its implications.

PF2fit: Polar Fast Fourier Matched Alignment of Atomistic Structures with 3D Electron Microscopy Maps

There continue to be increasing occurrences of both atomistic structure models in the PDB (possibly reconstructed from X-ray diffraction or NMR data), and 3D reconstructed cryo-electron microscopy (3D EM) maps (albeit at coarser resolution) of the same or homologous molecule or molecular assembly, deposited in the EMDB. To obtain the best possible structural model of the molecule at the best achievable resolution, and without any missing gaps, one typically aligns (match and fits) the atomistic structure model with the 3D EM map. We discuss a new algorithm and generalized framework, named PF(2) fit (Polar Fast Fourier Fitting) for the best possible structural alignment of atomistic structures with 3D EM. While PF(2) fit enables only a rigid, six dimensional (6D) alignment method, it augments prior work on 6D X-ray structure and 3D EM alignment in multiple ways: Scoring. PF(2) fit includes a new scoring scheme that, in addition to rewarding overlaps between the volumes occupied by the atomistic structure and 3D EM map, rewards overlaps between the volumes complementary to them. We quantitatively demonstrate how this new complementary scoring scheme improves upon existing approaches. PF(2) fit also includes two scoring functions, the non-uniform exterior penalty and the skeleton-secondary structure score, and implements the scattering potential score as an alternative to traditional Gaussian blurring. Search. PF(2) fit utilizes a fast polar Fourier search scheme, whose main advantage is the ability to search over uniformly and adaptively sampled subsets of the space of rigid-body motions. PF(2) fit also implements a new reranking search and scoring methodology that considerably improves alignment metrics in results obtained from the initial search.