Nonuniform Fourier Transforms for Rigid-Body and Multidimensional Rotational Correlations

Bajaj, Chandrajit L.; Bauer, Benedikt; Bettadapura, Radhakrishna; Vollrath, Antje

doi:10.1137/120892386

Cited by 13 publications

(21 citation statements)

References 54 publications

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“…In the third stage, after structural optimization and energy minimization (Chopra et al, 2010) of the complete models a small, diverse, and high-scoring subset is selected. Then Local refinement-based docking (Chowdhury et al, 2013) and fitting (Bettadapura et al, 2012; Bajaj et al, 2013), together with energy minimization is used to improve the binding interactions of chains in the selected models (Figure 1D). …”

Section: Resultsmentioning

confidence: 99%

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

2015

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Summary Envelope glycoprotein gp120 of HIV-1 possesses several variable regions whose 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 which uses cryo-electron microscopy (EM) maps, atomic-resolution structures of the core, and information of binding interactions. Our model has excellent fit with EMD5020 (Liu et al., 2008), 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 (Julien et al., 2013a) and PGV04 (Lyumkis et al., 2013) suggested a possible motion of the V1V2 away from the CCR5 binding site and towards CD4. Our study also revealed that the CD4-bound state of the V1V2 loop is not optimal for gp120 bound with several neutralizing antibodies.

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Section: Resultsmentioning

confidence: 99%

“…Each model was then flexibly fit using PF3Fit (Bettadapura et al, 2012; Bajaj et al, 2013) into EMD5020 (Liu et al, 2008). The fitted models were next clustered based on their fold similarity (measured using TM-score (Xu & Zhang, 2010)), and the best scoring models from each cluster were picked (Figures 1A-B).…”

Section: Methodsmentioning

confidence: 99%

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

2015

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“…Additional applications abound in numerical integration for low dimensional (6–100 dimensions) convolution integrals that appear naturally in computational molecular biology [3,2], as well in truly high dimensional (tens of thousands of dimensions) integrals that occur in finance [32,8]. …”

Section: Cubature Formulamentioning

confidence: 99%

On the construction of general cubature formula by flat extensions

Bucero

Bajaj

Mourrain

2016

Linear Algebra and its Applications

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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.

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“…There are several methods to efficiently evaluate Fourier transforms specifically on SO(3) [ 21 , 26 , 27 ]. There are also works that tackle Fourier transforms on the entire motion group SE(3), [ 28 , 29 ]. The use of fast and efficient algorithms to evaluate the Fourier transform on non-uniformly distributed points, cf.…”

Section: Introductionmentioning

confidence: 99%

PF2 fit: Polar Fast Fourier Matched Alignment of Atomistic Structures with 3D Electron Microscopy Maps

Bettadapura

Rasheed²,

Vollrath

et al. 2015

PLoS Comput Biol

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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 PF2 fit (Polar Fast Fourier Fitting) for the best possible structural alignment of atomistic structures with 3D EM. While PF2 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. PF2 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. PF2 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. PF2 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. PF2 fit also implements a new reranking search and scoring methodology that considerably improves alignment metrics in results obtained from the initial search.

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Nonuniform Fourier Transforms for Rigid-Body and Multidimensional Rotational Correlations

Cited by 13 publications

References 54 publications

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

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

On the construction of general cubature formula by flat extensions

PF2 fit: Polar Fast Fourier Matched Alignment of Atomistic Structures with 3D Electron Microscopy Maps

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