Detecting consistent common lines in cryo-EM by voting

Singer, Amit; Coifman, Ronald R.; Sigworth, Fred J.; Chester, David W.; Shkolnisky, Yoel

doi:10.1016/j.jsb.2009.11.003

Cited by 67 publications

(82 citation statements)

References 27 publications

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“…Particles were picked, contrast transfer function correction was performed per Sander et al (34), and further image processing was done in Imagic (35). A starting model was generated by using angular reconstitution facilitated by a voting algorithm (36). Two distinct conformations were detected in the dataset and it was split based on cross correlations.…”

mentioning

confidence: 99%

Structural basis for cooperativity of CRM1 export complex formation

Monecke

Haselbach

Voß

et al. 2012

Proc. Natl. Acad. Sci. U.S.A.

101

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In eukaryotes, the nucleocytoplasmic transport of macromolecules is mainly mediated by soluble nuclear transport receptors of the karyopherin-β superfamily termed importins and exportins. The highly versatile exportin chromosome region maintenance 1 (CRM1) is essential for nuclear depletion of numerous structurally and functionally unrelated protein and ribonucleoprotein cargoes. CRM1 has been shown to adopt a toroidal structure in several functional transport complexes and was thought to maintain this conformation throughout the entire nucleocytoplasmic transport cycle. We solved crystal structures of free CRM1 from the thermophilic eukaryote Chaetomium thermophilum. Surprisingly, unbound CRM1 exhibits an overall extended and pitched superhelical conformation. The two regulatory regions, namely the acidic loop and the C-terminal α-helix, are dramatically repositioned in free CRM1 in comparison with the ternary CRM1-Ran-Snurportin1 export complex. Single-particle EM analysis demonstrates that, in a noncrystalline environment, free CRM1 exists in equilibrium between extended, superhelical and compact, ring-like conformations. Molecular dynamics simulations show that the C-terminal helix plays an important role in regulating the transition from an extended to a compact conformation and reveal how the binding site for nuclear export signals of cargoes is modulated by different CRM1 conformations. Combining these results, we propose a model for the cooperativity of CRM1 export complex assembly involving the long-range allosteric communication between the distant binding sites of GTPbound Ran and cargo.

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mentioning

confidence: 99%

Structural basis for cooperativity of CRM1 export complex formation

Monecke

Haselbach

Voß

et al. 2012

Proc. Natl. Acad. Sci. U.S.A.

101

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“…In other words, a necessary and sufficient condition for common lines data to be valid for N = 3 is that it satisfies the spherical triangle inequality, a fact already observed both by the cryo-EM ( [7], pp. 198-199) and mathematics ( [8], Equations 11 and 12) communities.…”

Section: Methodsmentioning

confidence: 99%

“…Thus, the coordinates on the Grassmannian Gr(3, 2N) are the common line coordinates, together with these 'bad' minors Equation 8. If we consider the projection where we discard the 'bad' minors, we obtain the rational map…”

Section: Plücker Coordinates For Common Linesmentioning

confidence: 99%

Semi-algebraic geometry of common lines

Dynerman

2014

Res Math Sci

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Purpose: Cryo-electron microscopy is a technique in structural biology for determining the 3D structure of macromolecules. A key step in this process is detecting common lines of intersection between unknown embedded image planes. We wish to characterize such common lines in terms of the unembedded geometric data detected in experiments. Methods: We use techniques from spherical geometry, real algebraic geometry, and linear algebra. Results: We show that common lines are the solutions to a system of polynomial equalities and inequalities, i.e., they form a semi-algebraic set. These polynomials are low degree, and we explicitly derive them in this paper. Conclusions:The polynomials we derive provide the desired intrinsic characterization of common lines. We discuss possible applications of these polynomials to reconstruction algorithms that are robust to the high levels of noise present in cryo-electron images.

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“…More fundamentally, it has proved difficult to assign physical meaning to the outcome of graph-theoretic analyses [9], in the sense that the meaning of the dimensions in which the data manifold is embedded is unknown. Strategies to overcome this problem have included exploring the graph-theoretic consequences of specific changes in model systems [10], making restrictive assumptions about the nature of the data, and/or extracting specific information from the data first and subjecting this information to graph-theoretic analysis [9,11].…”

Section: Previous Workmentioning

confidence: 99%

“…However, it has proved difficult to assign physical meaning to the results [9,10]. Using the arsenal of tools developed in differential geometry, general relativity, and quantum mechanics, we elucidate the physical meaning of the outcome of graph-theoretic analysis of scattering data, without the need for restrictive a priori assumptions [11]. Finally, perception can be formulated as learning some functions on the observation manifold.…”

Section: Introductionmentioning

confidence: 99%

The symmetries of image formation by scattering I Theoretical framework

Giannakis¹,

Schwander²,

Ourmazd³

2012

Opt. Express

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We perceive the world through images formed by scattering. The ability to interpret scattering data mathematically has opened to our scrutiny the constituents of matter, the building blocks of life, and the remotest corners of the universe. Here, we deduce for the first time the fundamental symmetries underlying image formation. Intriguingly, these are similar to those of the anisotropic "Taub universe" of general relativity, with eigenfunctions closely related to spinning tops in quantum mechanics. This opens the possibility to apply the powerful arsenal of tools developed in two major branches of physics to new problems. We augment these tools with graph-theoretic means to recover the three-dimensional structure of objects from random snapshots of unknown orientation at four orders of magnitude higher complexity than previously demonstrated. Our theoretical framework offers a potential link to recent observations on face perception in higher primates. In a later paper, we demonstrate the recovery of structure and dynamics from ultralow-signal random sightings of systems with no orientational or timing information.

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Detecting consistent common lines in cryo-EM by voting

Cited by 67 publications

References 27 publications

Structural basis for cooperativity of CRM1 export complex formation

Structural basis for cooperativity of CRM1 export complex formation

Semi-algebraic geometry of common lines

The symmetries of image formation by scattering I Theoretical framework

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