Abstract

A theory is developed for estimating the three-dimensional (3-D) variance of a 3-D image reconstructed from projections by weighted backprojection. The theory is applicable for any data-collection schemes that produce partially redundant sampling of the angular space. The particular data collection considered here, the single-exposure random-conical scheme, is used for the reconstruction of macromolecules in electron microscopy. In this context, the purpose of the 3-D variance estimation is to detect and localize the conformational variability, to assess the significance of structural differences between two experimentally related 3-D images, and to assess the significance of local features in a 3-D image. The 3-D variance estimate of each reconstruction voxel is obtained by (i) the comparison of closest points on Fourier sections associated with difference projections, (ii) the comparison of neighbor projections in real space, or (iii) the comparison of projections with reprojections of the reconstruction.

© 1995 Optical Society of America

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References

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  1. J. Radon, “Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten. Berichte über die Verhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften zu Leipzig,” Math. Phys. Klasse 69, 262–277 (1917).
  2. A. M. Cormack, “Representation of a function by its line integrals, with some radiological applications. I,” J. Appl. Phys. 35, 2908–2912 (1964).
    [Crossref]
  3. D. J. DeRosier and A. Klug, “Reconstruction of three-dimensional structures from electron micrographs,” Nature (London) 217, 130–134 (1968).
    [Crossref]
  4. P. W. Hawkes, “The electron microscope as a structure projector,” in Electron Tomography, Three-Dimensional Imaging with the Transmission Electron Microscope, J. Frank, ed. (Plenum, New York, 1992), pp. 17–38.
  5. R. A. Crowther, D. J. DeRosier, and A. Klug, “The reconstruction of a three-dimensional structure from its projections and its application to electron microscopy,” Proc. R. Soc. London A 317, 319–340 (1970).
    [Crossref]
  6. R. Henderson and P. N. T. Unwin, “Three-dimensional model of purple membrane obtained by electron microscopy,” Nature (London) 257, 28–32 (1975).
    [Crossref]
  7. J. Frank, B. F. McEwen, M. Radermacher, J. N. Turner, and C. L. Rieder, “Three-dimensional tomographic reconstruction in high voltage electron microscopy,” J. Electron Tech. Microsc. 6, 193–205 (1987).
    [Crossref]
  8. W. Hoppe, J. Gassmann, N. Hunsmann, H. J. Schramm, and M. Sturm, “Three-dimensional reconstruction of individual negatively stained yeast fatty-acid synthetase molecules from tilt series in the electron microscopy,” Hoppe-Seyler’s Z. Physiol. Chem. 335, 1483–1487 (1974).
  9. J. Frank and W. Goldfarb, “Methods for averaging of single molecules and lattice-fragments,” in Electron Microscopy at Molecular Dimensions, State of the Art and Strategies for the Future, W. Baumeister and W. Vogell, eds. (Springer-Verlag, New York, 1980), pp. 261–269.
    [Crossref]
  10. M. Radermacher, T. Wagenknecht, A. Verschoor, and J. Frank, “Three-dimensional reconstruction from a single-exposure random conical tilt series applied to the 50S ribosomal subunit of Escherichia coli,” J. Microsc. (Oxford) 146, 131–136 (1987).
    [Crossref]
  11. M. Radermacher, T. Wagenknecht, A. Verschoor, and J. Frank, “Three-dimensional structure of the large subunit from Escherichia coli,” EMBO J. 6, 1107–1114 (1987).
    [PubMed]
  12. N. Boisset, J. C. Taveau, J. Lamy, T. Wagenknecht, M. Radermacher, and J. Frank, “Three-dimensional reconstruction of native Androctonus australishemocyanin,” J. Mol. Biol. 216, 743–760 (1990).
    [Crossref] [PubMed]
  13. T. Wagenknecht, R. Grassucci, J. Frank, A. Saito, M. Inui, and S. Fleisher, “Three-dimensional architecture of the calcium channel/foot structure of sarcoplasmic reticulum,” Nature (London) 338, 167–170 (1989).
    [Crossref]
  14. J. Frank, P. Penczek, R. Grassucci, and S. Srivastava, “Three-dimensional reconstruction of the 70S Escherichia coliribosome in ice: the distribution of ribosomal RNA,” J. Cell Biol. 115, 597–605 (1991).
    [Crossref] [PubMed]
  15. P. Penczek, M. Radermacher, and J. Frank, “Three-dimensional reconstruction of single particles embedded in ice,” Ultramicroscopy 40, 33–53 (1992).
    [Crossref] [PubMed]
  16. S. Srivastava, A. Verschoor, and J. Frank, “Eukaryotic initiation factor 3 does not prevent association through physical blockage of the ribosomal subunit–subunit interface,” J. Mol. Biol. 226, 301–304 (1992).
    [Crossref] [PubMed]
  17. N. Boisset, M. Radermacher, R. Grassucci, J.-C. Taveau, W. Liu, J. Lamy, J. Frank, U. C. Taveau, and J. N. Lamy, “Three-dimensional immunoelectron microscopy of scorpion hemocyanin labeled with a monoclonal Fab fragment,” J. Struct. Biol. 111, 234–244 (1993).
    [Crossref] [PubMed]
  18. N. Boisset, J.-C. Taveau, P. Penczek, J. N. Lamy, J. Frank, and J. Lamy, “Three-dimensional reconstruction of Androctonus australishemocyanin labeled with a monoclonal Fab fragment,” J. Struct. Biol. (to be published).
  19. W. Liu, “3-D variance of weighted back-projection reconstruction and its application to the detection of 3-D particle conformational changes,” in Proceedings of the Forty-Ninth Annual Meeting of the Electron Microscopy Society of America (San Francisco Press, San Francisco, 1991), pp. 542–543.
  20. W. Liu, J. Frank, and N. Boisset, “An application protocol of 3-D variance estimation theory for significance assessment of reconstructions and detection of 3-D conformational changes,” in Proceedings of the Fiftieth Annual Meeting of the Electron Microscopy Society of America (San Francisco Press, San Francisco, 1991), pp. 1064–1065.
  21. M. Radermacher, “Three-dimensional reconstruction of single particles from random and nonrandom tilt series,” J. Electron Microsc. Tech. 9, 359–394 (1998).
    [Crossref]
  22. J. Frank, “Classification of macromolecular assemblies studied as ‘single particles’,” Q. Rev. Biophys. 23, 281–329 (1990).
    [Crossref] [PubMed]
  23. J. L. Smith, W. A. Hendrickson, R. B. Honzatko, and S. Sheriff, “Structural heterogeneity in protein crystals,” Biochemistry 25, 5018–5027 (1986).
    [Crossref] [PubMed]
  24. E. J. Hoffman and M. E. Phelps, “Positron emission tomography: principles and quantitation,” in Positron Emission Tomography and Autoradiography: Principles and Applications for the Brain and Heart, M. Phelps and H. Schelbert, eds. (Raven, New York, 1986), pp. 237–286.
  25. K. Wuthrich, NMR of Proteins and Nucleic Acids (Wiley, New York, 1986).
  26. R. Ernst, G. Bodenhausen, and A. Wokaun, Principles of Nuclear Magnetic Resonance in One and Two Dimensions (Oxford Scientific, Oxford, 1990).
  27. R. Hegerl and W. Hoppe, “Influence of electron noise on three-dimensional image reconstruction,” Z. Naturforsch. 31a, 1717–1721 (1976).
  28. M. Radermacher, T. Wagenknecht, A. Verschoor, and J. Frank, “A new 3-D reconstruction scheme applied to the 50S ribosomal subunit of E. coli,” J. Microsc. 141, PR1–RP2 (1986).
    [Crossref]
  29. G. Harauz and M. van Heel, “Exact filters for general geometry three dimensional reconstruction,” Optik 73, 146–156 (1986).
  30. W. Liu, N. Boisset, and J. Frank, “Estimation of variance distribution in three-dimensional reconstruction. II. Applications,” J. Opt. Soc. Am. A 12, 2628–2637 (1995).
    [Crossref]
  31. M. Schatz, E. V. Orlova, P. Dube, J. Jäger, and M. van Meel, “Structure of Lumbricus temestrishemoglobin at 30 Å resolution determined using angular reconstitution,” J. Struct. Biol. 114, 28–40 (1995).
    [Crossref] [PubMed]
  32. J. Frank, J. Zhu, P. Penzek, Y. Li, S. Srivastava, A. Ver-schoor, M. Radermacher, R. Grassucci, R. K. Lata, and R. K. Agrawal, “A model of protein synthesis based on cryo-electron microscopy of the E. coliribosome,” Nature (London) 376, 441–444 (1995).
    [Crossref]
  33. M. Radermacher and W. Hoppe, “Properties of 3-D reconstructions from projections by conical tilting compared to single axis tilting,” in Proceedings of the Seventh European Congress on Electron Microscopy (Seventh European Congress on Electron Microscopy Foundation, Leiden, The Netherlands, 1980), Vol. 1, pp. 132–133.

1998 (1)

M. Radermacher, “Three-dimensional reconstruction of single particles from random and nonrandom tilt series,” J. Electron Microsc. Tech. 9, 359–394 (1998).
[Crossref]

1995 (3)

M. Schatz, E. V. Orlova, P. Dube, J. Jäger, and M. van Meel, “Structure of Lumbricus temestrishemoglobin at 30 Å resolution determined using angular reconstitution,” J. Struct. Biol. 114, 28–40 (1995).
[Crossref] [PubMed]

J. Frank, J. Zhu, P. Penzek, Y. Li, S. Srivastava, A. Ver-schoor, M. Radermacher, R. Grassucci, R. K. Lata, and R. K. Agrawal, “A model of protein synthesis based on cryo-electron microscopy of the E. coliribosome,” Nature (London) 376, 441–444 (1995).
[Crossref]

W. Liu, N. Boisset, and J. Frank, “Estimation of variance distribution in three-dimensional reconstruction. II. Applications,” J. Opt. Soc. Am. A 12, 2628–2637 (1995).
[Crossref]

1993 (1)

N. Boisset, M. Radermacher, R. Grassucci, J.-C. Taveau, W. Liu, J. Lamy, J. Frank, U. C. Taveau, and J. N. Lamy, “Three-dimensional immunoelectron microscopy of scorpion hemocyanin labeled with a monoclonal Fab fragment,” J. Struct. Biol. 111, 234–244 (1993).
[Crossref] [PubMed]

1992 (2)

P. Penczek, M. Radermacher, and J. Frank, “Three-dimensional reconstruction of single particles embedded in ice,” Ultramicroscopy 40, 33–53 (1992).
[Crossref] [PubMed]

S. Srivastava, A. Verschoor, and J. Frank, “Eukaryotic initiation factor 3 does not prevent association through physical blockage of the ribosomal subunit–subunit interface,” J. Mol. Biol. 226, 301–304 (1992).
[Crossref] [PubMed]

1991 (1)

J. Frank, P. Penczek, R. Grassucci, and S. Srivastava, “Three-dimensional reconstruction of the 70S Escherichia coliribosome in ice: the distribution of ribosomal RNA,” J. Cell Biol. 115, 597–605 (1991).
[Crossref] [PubMed]

1990 (2)

N. Boisset, J. C. Taveau, J. Lamy, T. Wagenknecht, M. Radermacher, and J. Frank, “Three-dimensional reconstruction of native Androctonus australishemocyanin,” J. Mol. Biol. 216, 743–760 (1990).
[Crossref] [PubMed]

J. Frank, “Classification of macromolecular assemblies studied as ‘single particles’,” Q. Rev. Biophys. 23, 281–329 (1990).
[Crossref] [PubMed]

1989 (1)

T. Wagenknecht, R. Grassucci, J. Frank, A. Saito, M. Inui, and S. Fleisher, “Three-dimensional architecture of the calcium channel/foot structure of sarcoplasmic reticulum,” Nature (London) 338, 167–170 (1989).
[Crossref]

1987 (3)

J. Frank, B. F. McEwen, M. Radermacher, J. N. Turner, and C. L. Rieder, “Three-dimensional tomographic reconstruction in high voltage electron microscopy,” J. Electron Tech. Microsc. 6, 193–205 (1987).
[Crossref]

M. Radermacher, T. Wagenknecht, A. Verschoor, and J. Frank, “Three-dimensional reconstruction from a single-exposure random conical tilt series applied to the 50S ribosomal subunit of Escherichia coli,” J. Microsc. (Oxford) 146, 131–136 (1987).
[Crossref]

M. Radermacher, T. Wagenknecht, A. Verschoor, and J. Frank, “Three-dimensional structure of the large subunit from Escherichia coli,” EMBO J. 6, 1107–1114 (1987).
[PubMed]

1986 (3)

J. L. Smith, W. A. Hendrickson, R. B. Honzatko, and S. Sheriff, “Structural heterogeneity in protein crystals,” Biochemistry 25, 5018–5027 (1986).
[Crossref] [PubMed]

M. Radermacher, T. Wagenknecht, A. Verschoor, and J. Frank, “A new 3-D reconstruction scheme applied to the 50S ribosomal subunit of E. coli,” J. Microsc. 141, PR1–RP2 (1986).
[Crossref]

G. Harauz and M. van Heel, “Exact filters for general geometry three dimensional reconstruction,” Optik 73, 146–156 (1986).

1976 (1)

R. Hegerl and W. Hoppe, “Influence of electron noise on three-dimensional image reconstruction,” Z. Naturforsch. 31a, 1717–1721 (1976).

1975 (1)

R. Henderson and P. N. T. Unwin, “Three-dimensional model of purple membrane obtained by electron microscopy,” Nature (London) 257, 28–32 (1975).
[Crossref]

1974 (1)

W. Hoppe, J. Gassmann, N. Hunsmann, H. J. Schramm, and M. Sturm, “Three-dimensional reconstruction of individual negatively stained yeast fatty-acid synthetase molecules from tilt series in the electron microscopy,” Hoppe-Seyler’s Z. Physiol. Chem. 335, 1483–1487 (1974).

1970 (1)

R. A. Crowther, D. J. DeRosier, and A. Klug, “The reconstruction of a three-dimensional structure from its projections and its application to electron microscopy,” Proc. R. Soc. London A 317, 319–340 (1970).
[Crossref]

1968 (1)

D. J. DeRosier and A. Klug, “Reconstruction of three-dimensional structures from electron micrographs,” Nature (London) 217, 130–134 (1968).
[Crossref]

1964 (1)

A. M. Cormack, “Representation of a function by its line integrals, with some radiological applications. I,” J. Appl. Phys. 35, 2908–2912 (1964).
[Crossref]

1917 (1)

J. Radon, “Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten. Berichte über die Verhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften zu Leipzig,” Math. Phys. Klasse 69, 262–277 (1917).

Agrawal, R. K.

J. Frank, J. Zhu, P. Penzek, Y. Li, S. Srivastava, A. Ver-schoor, M. Radermacher, R. Grassucci, R. K. Lata, and R. K. Agrawal, “A model of protein synthesis based on cryo-electron microscopy of the E. coliribosome,” Nature (London) 376, 441–444 (1995).
[Crossref]

Bodenhausen, G.

R. Ernst, G. Bodenhausen, and A. Wokaun, Principles of Nuclear Magnetic Resonance in One and Two Dimensions (Oxford Scientific, Oxford, 1990).

Boisset, N.

W. Liu, N. Boisset, and J. Frank, “Estimation of variance distribution in three-dimensional reconstruction. II. Applications,” J. Opt. Soc. Am. A 12, 2628–2637 (1995).
[Crossref]

N. Boisset, M. Radermacher, R. Grassucci, J.-C. Taveau, W. Liu, J. Lamy, J. Frank, U. C. Taveau, and J. N. Lamy, “Three-dimensional immunoelectron microscopy of scorpion hemocyanin labeled with a monoclonal Fab fragment,” J. Struct. Biol. 111, 234–244 (1993).
[Crossref] [PubMed]

N. Boisset, J. C. Taveau, J. Lamy, T. Wagenknecht, M. Radermacher, and J. Frank, “Three-dimensional reconstruction of native Androctonus australishemocyanin,” J. Mol. Biol. 216, 743–760 (1990).
[Crossref] [PubMed]

W. Liu, J. Frank, and N. Boisset, “An application protocol of 3-D variance estimation theory for significance assessment of reconstructions and detection of 3-D conformational changes,” in Proceedings of the Fiftieth Annual Meeting of the Electron Microscopy Society of America (San Francisco Press, San Francisco, 1991), pp. 1064–1065.

N. Boisset, J.-C. Taveau, P. Penczek, J. N. Lamy, J. Frank, and J. Lamy, “Three-dimensional reconstruction of Androctonus australishemocyanin labeled with a monoclonal Fab fragment,” J. Struct. Biol. (to be published).

Cormack, A. M.

A. M. Cormack, “Representation of a function by its line integrals, with some radiological applications. I,” J. Appl. Phys. 35, 2908–2912 (1964).
[Crossref]

Crowther, R. A.

R. A. Crowther, D. J. DeRosier, and A. Klug, “The reconstruction of a three-dimensional structure from its projections and its application to electron microscopy,” Proc. R. Soc. London A 317, 319–340 (1970).
[Crossref]

DeRosier, D. J.

R. A. Crowther, D. J. DeRosier, and A. Klug, “The reconstruction of a three-dimensional structure from its projections and its application to electron microscopy,” Proc. R. Soc. London A 317, 319–340 (1970).
[Crossref]

D. J. DeRosier and A. Klug, “Reconstruction of three-dimensional structures from electron micrographs,” Nature (London) 217, 130–134 (1968).
[Crossref]

Dube, P.

M. Schatz, E. V. Orlova, P. Dube, J. Jäger, and M. van Meel, “Structure of Lumbricus temestrishemoglobin at 30 Å resolution determined using angular reconstitution,” J. Struct. Biol. 114, 28–40 (1995).
[Crossref] [PubMed]

Ernst, R.

R. Ernst, G. Bodenhausen, and A. Wokaun, Principles of Nuclear Magnetic Resonance in One and Two Dimensions (Oxford Scientific, Oxford, 1990).

Fleisher, S.

T. Wagenknecht, R. Grassucci, J. Frank, A. Saito, M. Inui, and S. Fleisher, “Three-dimensional architecture of the calcium channel/foot structure of sarcoplasmic reticulum,” Nature (London) 338, 167–170 (1989).
[Crossref]

Frank, J.

W. Liu, N. Boisset, and J. Frank, “Estimation of variance distribution in three-dimensional reconstruction. II. Applications,” J. Opt. Soc. Am. A 12, 2628–2637 (1995).
[Crossref]

J. Frank, J. Zhu, P. Penzek, Y. Li, S. Srivastava, A. Ver-schoor, M. Radermacher, R. Grassucci, R. K. Lata, and R. K. Agrawal, “A model of protein synthesis based on cryo-electron microscopy of the E. coliribosome,” Nature (London) 376, 441–444 (1995).
[Crossref]

N. Boisset, M. Radermacher, R. Grassucci, J.-C. Taveau, W. Liu, J. Lamy, J. Frank, U. C. Taveau, and J. N. Lamy, “Three-dimensional immunoelectron microscopy of scorpion hemocyanin labeled with a monoclonal Fab fragment,” J. Struct. Biol. 111, 234–244 (1993).
[Crossref] [PubMed]

S. Srivastava, A. Verschoor, and J. Frank, “Eukaryotic initiation factor 3 does not prevent association through physical blockage of the ribosomal subunit–subunit interface,” J. Mol. Biol. 226, 301–304 (1992).
[Crossref] [PubMed]

P. Penczek, M. Radermacher, and J. Frank, “Three-dimensional reconstruction of single particles embedded in ice,” Ultramicroscopy 40, 33–53 (1992).
[Crossref] [PubMed]

J. Frank, P. Penczek, R. Grassucci, and S. Srivastava, “Three-dimensional reconstruction of the 70S Escherichia coliribosome in ice: the distribution of ribosomal RNA,” J. Cell Biol. 115, 597–605 (1991).
[Crossref] [PubMed]

N. Boisset, J. C. Taveau, J. Lamy, T. Wagenknecht, M. Radermacher, and J. Frank, “Three-dimensional reconstruction of native Androctonus australishemocyanin,” J. Mol. Biol. 216, 743–760 (1990).
[Crossref] [PubMed]

J. Frank, “Classification of macromolecular assemblies studied as ‘single particles’,” Q. Rev. Biophys. 23, 281–329 (1990).
[Crossref] [PubMed]

T. Wagenknecht, R. Grassucci, J. Frank, A. Saito, M. Inui, and S. Fleisher, “Three-dimensional architecture of the calcium channel/foot structure of sarcoplasmic reticulum,” Nature (London) 338, 167–170 (1989).
[Crossref]

J. Frank, B. F. McEwen, M. Radermacher, J. N. Turner, and C. L. Rieder, “Three-dimensional tomographic reconstruction in high voltage electron microscopy,” J. Electron Tech. Microsc. 6, 193–205 (1987).
[Crossref]

M. Radermacher, T. Wagenknecht, A. Verschoor, and J. Frank, “Three-dimensional structure of the large subunit from Escherichia coli,” EMBO J. 6, 1107–1114 (1987).
[PubMed]

M. Radermacher, T. Wagenknecht, A. Verschoor, and J. Frank, “Three-dimensional reconstruction from a single-exposure random conical tilt series applied to the 50S ribosomal subunit of Escherichia coli,” J. Microsc. (Oxford) 146, 131–136 (1987).
[Crossref]

M. Radermacher, T. Wagenknecht, A. Verschoor, and J. Frank, “A new 3-D reconstruction scheme applied to the 50S ribosomal subunit of E. coli,” J. Microsc. 141, PR1–RP2 (1986).
[Crossref]

J. Frank and W. Goldfarb, “Methods for averaging of single molecules and lattice-fragments,” in Electron Microscopy at Molecular Dimensions, State of the Art and Strategies for the Future, W. Baumeister and W. Vogell, eds. (Springer-Verlag, New York, 1980), pp. 261–269.
[Crossref]

W. Liu, J. Frank, and N. Boisset, “An application protocol of 3-D variance estimation theory for significance assessment of reconstructions and detection of 3-D conformational changes,” in Proceedings of the Fiftieth Annual Meeting of the Electron Microscopy Society of America (San Francisco Press, San Francisco, 1991), pp. 1064–1065.

N. Boisset, J.-C. Taveau, P. Penczek, J. N. Lamy, J. Frank, and J. Lamy, “Three-dimensional reconstruction of Androctonus australishemocyanin labeled with a monoclonal Fab fragment,” J. Struct. Biol. (to be published).

Gassmann, J.

W. Hoppe, J. Gassmann, N. Hunsmann, H. J. Schramm, and M. Sturm, “Three-dimensional reconstruction of individual negatively stained yeast fatty-acid synthetase molecules from tilt series in the electron microscopy,” Hoppe-Seyler’s Z. Physiol. Chem. 335, 1483–1487 (1974).

Goldfarb, W.

J. Frank and W. Goldfarb, “Methods for averaging of single molecules and lattice-fragments,” in Electron Microscopy at Molecular Dimensions, State of the Art and Strategies for the Future, W. Baumeister and W. Vogell, eds. (Springer-Verlag, New York, 1980), pp. 261–269.
[Crossref]

Grassucci, R.

J. Frank, J. Zhu, P. Penzek, Y. Li, S. Srivastava, A. Ver-schoor, M. Radermacher, R. Grassucci, R. K. Lata, and R. K. Agrawal, “A model of protein synthesis based on cryo-electron microscopy of the E. coliribosome,” Nature (London) 376, 441–444 (1995).
[Crossref]

N. Boisset, M. Radermacher, R. Grassucci, J.-C. Taveau, W. Liu, J. Lamy, J. Frank, U. C. Taveau, and J. N. Lamy, “Three-dimensional immunoelectron microscopy of scorpion hemocyanin labeled with a monoclonal Fab fragment,” J. Struct. Biol. 111, 234–244 (1993).
[Crossref] [PubMed]

J. Frank, P. Penczek, R. Grassucci, and S. Srivastava, “Three-dimensional reconstruction of the 70S Escherichia coliribosome in ice: the distribution of ribosomal RNA,” J. Cell Biol. 115, 597–605 (1991).
[Crossref] [PubMed]

T. Wagenknecht, R. Grassucci, J. Frank, A. Saito, M. Inui, and S. Fleisher, “Three-dimensional architecture of the calcium channel/foot structure of sarcoplasmic reticulum,” Nature (London) 338, 167–170 (1989).
[Crossref]

Harauz, G.

G. Harauz and M. van Heel, “Exact filters for general geometry three dimensional reconstruction,” Optik 73, 146–156 (1986).

Hawkes, P. W.

P. W. Hawkes, “The electron microscope as a structure projector,” in Electron Tomography, Three-Dimensional Imaging with the Transmission Electron Microscope, J. Frank, ed. (Plenum, New York, 1992), pp. 17–38.

Hegerl, R.

R. Hegerl and W. Hoppe, “Influence of electron noise on three-dimensional image reconstruction,” Z. Naturforsch. 31a, 1717–1721 (1976).

Henderson, R.

R. Henderson and P. N. T. Unwin, “Three-dimensional model of purple membrane obtained by electron microscopy,” Nature (London) 257, 28–32 (1975).
[Crossref]

Hendrickson, W. A.

J. L. Smith, W. A. Hendrickson, R. B. Honzatko, and S. Sheriff, “Structural heterogeneity in protein crystals,” Biochemistry 25, 5018–5027 (1986).
[Crossref] [PubMed]

Hoffman, E. J.

E. J. Hoffman and M. E. Phelps, “Positron emission tomography: principles and quantitation,” in Positron Emission Tomography and Autoradiography: Principles and Applications for the Brain and Heart, M. Phelps and H. Schelbert, eds. (Raven, New York, 1986), pp. 237–286.

Honzatko, R. B.

J. L. Smith, W. A. Hendrickson, R. B. Honzatko, and S. Sheriff, “Structural heterogeneity in protein crystals,” Biochemistry 25, 5018–5027 (1986).
[Crossref] [PubMed]

Hoppe, W.

R. Hegerl and W. Hoppe, “Influence of electron noise on three-dimensional image reconstruction,” Z. Naturforsch. 31a, 1717–1721 (1976).

W. Hoppe, J. Gassmann, N. Hunsmann, H. J. Schramm, and M. Sturm, “Three-dimensional reconstruction of individual negatively stained yeast fatty-acid synthetase molecules from tilt series in the electron microscopy,” Hoppe-Seyler’s Z. Physiol. Chem. 335, 1483–1487 (1974).

M. Radermacher and W. Hoppe, “Properties of 3-D reconstructions from projections by conical tilting compared to single axis tilting,” in Proceedings of the Seventh European Congress on Electron Microscopy (Seventh European Congress on Electron Microscopy Foundation, Leiden, The Netherlands, 1980), Vol. 1, pp. 132–133.

Hunsmann, N.

W. Hoppe, J. Gassmann, N. Hunsmann, H. J. Schramm, and M. Sturm, “Three-dimensional reconstruction of individual negatively stained yeast fatty-acid synthetase molecules from tilt series in the electron microscopy,” Hoppe-Seyler’s Z. Physiol. Chem. 335, 1483–1487 (1974).

Inui, M.

T. Wagenknecht, R. Grassucci, J. Frank, A. Saito, M. Inui, and S. Fleisher, “Three-dimensional architecture of the calcium channel/foot structure of sarcoplasmic reticulum,” Nature (London) 338, 167–170 (1989).
[Crossref]

Jäger, J.

M. Schatz, E. V. Orlova, P. Dube, J. Jäger, and M. van Meel, “Structure of Lumbricus temestrishemoglobin at 30 Å resolution determined using angular reconstitution,” J. Struct. Biol. 114, 28–40 (1995).
[Crossref] [PubMed]

Klug, A.

R. A. Crowther, D. J. DeRosier, and A. Klug, “The reconstruction of a three-dimensional structure from its projections and its application to electron microscopy,” Proc. R. Soc. London A 317, 319–340 (1970).
[Crossref]

D. J. DeRosier and A. Klug, “Reconstruction of three-dimensional structures from electron micrographs,” Nature (London) 217, 130–134 (1968).
[Crossref]

Lamy, J.

N. Boisset, M. Radermacher, R. Grassucci, J.-C. Taveau, W. Liu, J. Lamy, J. Frank, U. C. Taveau, and J. N. Lamy, “Three-dimensional immunoelectron microscopy of scorpion hemocyanin labeled with a monoclonal Fab fragment,” J. Struct. Biol. 111, 234–244 (1993).
[Crossref] [PubMed]

N. Boisset, J. C. Taveau, J. Lamy, T. Wagenknecht, M. Radermacher, and J. Frank, “Three-dimensional reconstruction of native Androctonus australishemocyanin,” J. Mol. Biol. 216, 743–760 (1990).
[Crossref] [PubMed]

N. Boisset, J.-C. Taveau, P. Penczek, J. N. Lamy, J. Frank, and J. Lamy, “Three-dimensional reconstruction of Androctonus australishemocyanin labeled with a monoclonal Fab fragment,” J. Struct. Biol. (to be published).

Lamy, J. N.

N. Boisset, M. Radermacher, R. Grassucci, J.-C. Taveau, W. Liu, J. Lamy, J. Frank, U. C. Taveau, and J. N. Lamy, “Three-dimensional immunoelectron microscopy of scorpion hemocyanin labeled with a monoclonal Fab fragment,” J. Struct. Biol. 111, 234–244 (1993).
[Crossref] [PubMed]

N. Boisset, J.-C. Taveau, P. Penczek, J. N. Lamy, J. Frank, and J. Lamy, “Three-dimensional reconstruction of Androctonus australishemocyanin labeled with a monoclonal Fab fragment,” J. Struct. Biol. (to be published).

Lata, R. K.

J. Frank, J. Zhu, P. Penzek, Y. Li, S. Srivastava, A. Ver-schoor, M. Radermacher, R. Grassucci, R. K. Lata, and R. K. Agrawal, “A model of protein synthesis based on cryo-electron microscopy of the E. coliribosome,” Nature (London) 376, 441–444 (1995).
[Crossref]

Li, Y.

J. Frank, J. Zhu, P. Penzek, Y. Li, S. Srivastava, A. Ver-schoor, M. Radermacher, R. Grassucci, R. K. Lata, and R. K. Agrawal, “A model of protein synthesis based on cryo-electron microscopy of the E. coliribosome,” Nature (London) 376, 441–444 (1995).
[Crossref]

Liu, W.

W. Liu, N. Boisset, and J. Frank, “Estimation of variance distribution in three-dimensional reconstruction. II. Applications,” J. Opt. Soc. Am. A 12, 2628–2637 (1995).
[Crossref]

N. Boisset, M. Radermacher, R. Grassucci, J.-C. Taveau, W. Liu, J. Lamy, J. Frank, U. C. Taveau, and J. N. Lamy, “Three-dimensional immunoelectron microscopy of scorpion hemocyanin labeled with a monoclonal Fab fragment,” J. Struct. Biol. 111, 234–244 (1993).
[Crossref] [PubMed]

W. Liu, J. Frank, and N. Boisset, “An application protocol of 3-D variance estimation theory for significance assessment of reconstructions and detection of 3-D conformational changes,” in Proceedings of the Fiftieth Annual Meeting of the Electron Microscopy Society of America (San Francisco Press, San Francisco, 1991), pp. 1064–1065.

W. Liu, “3-D variance of weighted back-projection reconstruction and its application to the detection of 3-D particle conformational changes,” in Proceedings of the Forty-Ninth Annual Meeting of the Electron Microscopy Society of America (San Francisco Press, San Francisco, 1991), pp. 542–543.

McEwen, B. F.

J. Frank, B. F. McEwen, M. Radermacher, J. N. Turner, and C. L. Rieder, “Three-dimensional tomographic reconstruction in high voltage electron microscopy,” J. Electron Tech. Microsc. 6, 193–205 (1987).
[Crossref]

Orlova, E. V.

M. Schatz, E. V. Orlova, P. Dube, J. Jäger, and M. van Meel, “Structure of Lumbricus temestrishemoglobin at 30 Å resolution determined using angular reconstitution,” J. Struct. Biol. 114, 28–40 (1995).
[Crossref] [PubMed]

Penczek, P.

P. Penczek, M. Radermacher, and J. Frank, “Three-dimensional reconstruction of single particles embedded in ice,” Ultramicroscopy 40, 33–53 (1992).
[Crossref] [PubMed]

J. Frank, P. Penczek, R. Grassucci, and S. Srivastava, “Three-dimensional reconstruction of the 70S Escherichia coliribosome in ice: the distribution of ribosomal RNA,” J. Cell Biol. 115, 597–605 (1991).
[Crossref] [PubMed]

N. Boisset, J.-C. Taveau, P. Penczek, J. N. Lamy, J. Frank, and J. Lamy, “Three-dimensional reconstruction of Androctonus australishemocyanin labeled with a monoclonal Fab fragment,” J. Struct. Biol. (to be published).

Penzek, P.

J. Frank, J. Zhu, P. Penzek, Y. Li, S. Srivastava, A. Ver-schoor, M. Radermacher, R. Grassucci, R. K. Lata, and R. K. Agrawal, “A model of protein synthesis based on cryo-electron microscopy of the E. coliribosome,” Nature (London) 376, 441–444 (1995).
[Crossref]

Phelps, M. E.

E. J. Hoffman and M. E. Phelps, “Positron emission tomography: principles and quantitation,” in Positron Emission Tomography and Autoradiography: Principles and Applications for the Brain and Heart, M. Phelps and H. Schelbert, eds. (Raven, New York, 1986), pp. 237–286.

Radermacher, M.

M. Radermacher, “Three-dimensional reconstruction of single particles from random and nonrandom tilt series,” J. Electron Microsc. Tech. 9, 359–394 (1998).
[Crossref]

J. Frank, J. Zhu, P. Penzek, Y. Li, S. Srivastava, A. Ver-schoor, M. Radermacher, R. Grassucci, R. K. Lata, and R. K. Agrawal, “A model of protein synthesis based on cryo-electron microscopy of the E. coliribosome,” Nature (London) 376, 441–444 (1995).
[Crossref]

N. Boisset, M. Radermacher, R. Grassucci, J.-C. Taveau, W. Liu, J. Lamy, J. Frank, U. C. Taveau, and J. N. Lamy, “Three-dimensional immunoelectron microscopy of scorpion hemocyanin labeled with a monoclonal Fab fragment,” J. Struct. Biol. 111, 234–244 (1993).
[Crossref] [PubMed]

P. Penczek, M. Radermacher, and J. Frank, “Three-dimensional reconstruction of single particles embedded in ice,” Ultramicroscopy 40, 33–53 (1992).
[Crossref] [PubMed]

N. Boisset, J. C. Taveau, J. Lamy, T. Wagenknecht, M. Radermacher, and J. Frank, “Three-dimensional reconstruction of native Androctonus australishemocyanin,” J. Mol. Biol. 216, 743–760 (1990).
[Crossref] [PubMed]

J. Frank, B. F. McEwen, M. Radermacher, J. N. Turner, and C. L. Rieder, “Three-dimensional tomographic reconstruction in high voltage electron microscopy,” J. Electron Tech. Microsc. 6, 193–205 (1987).
[Crossref]

M. Radermacher, T. Wagenknecht, A. Verschoor, and J. Frank, “Three-dimensional reconstruction from a single-exposure random conical tilt series applied to the 50S ribosomal subunit of Escherichia coli,” J. Microsc. (Oxford) 146, 131–136 (1987).
[Crossref]

M. Radermacher, T. Wagenknecht, A. Verschoor, and J. Frank, “Three-dimensional structure of the large subunit from Escherichia coli,” EMBO J. 6, 1107–1114 (1987).
[PubMed]

M. Radermacher, T. Wagenknecht, A. Verschoor, and J. Frank, “A new 3-D reconstruction scheme applied to the 50S ribosomal subunit of E. coli,” J. Microsc. 141, PR1–RP2 (1986).
[Crossref]

M. Radermacher and W. Hoppe, “Properties of 3-D reconstructions from projections by conical tilting compared to single axis tilting,” in Proceedings of the Seventh European Congress on Electron Microscopy (Seventh European Congress on Electron Microscopy Foundation, Leiden, The Netherlands, 1980), Vol. 1, pp. 132–133.

Radon, J.

J. Radon, “Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten. Berichte über die Verhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften zu Leipzig,” Math. Phys. Klasse 69, 262–277 (1917).

Rieder, C. L.

J. Frank, B. F. McEwen, M. Radermacher, J. N. Turner, and C. L. Rieder, “Three-dimensional tomographic reconstruction in high voltage electron microscopy,” J. Electron Tech. Microsc. 6, 193–205 (1987).
[Crossref]

Saito, A.

T. Wagenknecht, R. Grassucci, J. Frank, A. Saito, M. Inui, and S. Fleisher, “Three-dimensional architecture of the calcium channel/foot structure of sarcoplasmic reticulum,” Nature (London) 338, 167–170 (1989).
[Crossref]

Schatz, M.

M. Schatz, E. V. Orlova, P. Dube, J. Jäger, and M. van Meel, “Structure of Lumbricus temestrishemoglobin at 30 Å resolution determined using angular reconstitution,” J. Struct. Biol. 114, 28–40 (1995).
[Crossref] [PubMed]

Schramm, H. J.

W. Hoppe, J. Gassmann, N. Hunsmann, H. J. Schramm, and M. Sturm, “Three-dimensional reconstruction of individual negatively stained yeast fatty-acid synthetase molecules from tilt series in the electron microscopy,” Hoppe-Seyler’s Z. Physiol. Chem. 335, 1483–1487 (1974).

Sheriff, S.

J. L. Smith, W. A. Hendrickson, R. B. Honzatko, and S. Sheriff, “Structural heterogeneity in protein crystals,” Biochemistry 25, 5018–5027 (1986).
[Crossref] [PubMed]

Smith, J. L.

J. L. Smith, W. A. Hendrickson, R. B. Honzatko, and S. Sheriff, “Structural heterogeneity in protein crystals,” Biochemistry 25, 5018–5027 (1986).
[Crossref] [PubMed]

Srivastava, S.

J. Frank, J. Zhu, P. Penzek, Y. Li, S. Srivastava, A. Ver-schoor, M. Radermacher, R. Grassucci, R. K. Lata, and R. K. Agrawal, “A model of protein synthesis based on cryo-electron microscopy of the E. coliribosome,” Nature (London) 376, 441–444 (1995).
[Crossref]

S. Srivastava, A. Verschoor, and J. Frank, “Eukaryotic initiation factor 3 does not prevent association through physical blockage of the ribosomal subunit–subunit interface,” J. Mol. Biol. 226, 301–304 (1992).
[Crossref] [PubMed]

J. Frank, P. Penczek, R. Grassucci, and S. Srivastava, “Three-dimensional reconstruction of the 70S Escherichia coliribosome in ice: the distribution of ribosomal RNA,” J. Cell Biol. 115, 597–605 (1991).
[Crossref] [PubMed]

Sturm, M.

W. Hoppe, J. Gassmann, N. Hunsmann, H. J. Schramm, and M. Sturm, “Three-dimensional reconstruction of individual negatively stained yeast fatty-acid synthetase molecules from tilt series in the electron microscopy,” Hoppe-Seyler’s Z. Physiol. Chem. 335, 1483–1487 (1974).

Taveau, J. C.

N. Boisset, J. C. Taveau, J. Lamy, T. Wagenknecht, M. Radermacher, and J. Frank, “Three-dimensional reconstruction of native Androctonus australishemocyanin,” J. Mol. Biol. 216, 743–760 (1990).
[Crossref] [PubMed]

Taveau, J.-C.

N. Boisset, M. Radermacher, R. Grassucci, J.-C. Taveau, W. Liu, J. Lamy, J. Frank, U. C. Taveau, and J. N. Lamy, “Three-dimensional immunoelectron microscopy of scorpion hemocyanin labeled with a monoclonal Fab fragment,” J. Struct. Biol. 111, 234–244 (1993).
[Crossref] [PubMed]

N. Boisset, J.-C. Taveau, P. Penczek, J. N. Lamy, J. Frank, and J. Lamy, “Three-dimensional reconstruction of Androctonus australishemocyanin labeled with a monoclonal Fab fragment,” J. Struct. Biol. (to be published).

Taveau, U. C.

N. Boisset, M. Radermacher, R. Grassucci, J.-C. Taveau, W. Liu, J. Lamy, J. Frank, U. C. Taveau, and J. N. Lamy, “Three-dimensional immunoelectron microscopy of scorpion hemocyanin labeled with a monoclonal Fab fragment,” J. Struct. Biol. 111, 234–244 (1993).
[Crossref] [PubMed]

Turner, J. N.

J. Frank, B. F. McEwen, M. Radermacher, J. N. Turner, and C. L. Rieder, “Three-dimensional tomographic reconstruction in high voltage electron microscopy,” J. Electron Tech. Microsc. 6, 193–205 (1987).
[Crossref]

Unwin, P. N. T.

R. Henderson and P. N. T. Unwin, “Three-dimensional model of purple membrane obtained by electron microscopy,” Nature (London) 257, 28–32 (1975).
[Crossref]

van Heel, M.

G. Harauz and M. van Heel, “Exact filters for general geometry three dimensional reconstruction,” Optik 73, 146–156 (1986).

van Meel, M.

M. Schatz, E. V. Orlova, P. Dube, J. Jäger, and M. van Meel, “Structure of Lumbricus temestrishemoglobin at 30 Å resolution determined using angular reconstitution,” J. Struct. Biol. 114, 28–40 (1995).
[Crossref] [PubMed]

Verschoor, A.

S. Srivastava, A. Verschoor, and J. Frank, “Eukaryotic initiation factor 3 does not prevent association through physical blockage of the ribosomal subunit–subunit interface,” J. Mol. Biol. 226, 301–304 (1992).
[Crossref] [PubMed]

M. Radermacher, T. Wagenknecht, A. Verschoor, and J. Frank, “Three-dimensional reconstruction from a single-exposure random conical tilt series applied to the 50S ribosomal subunit of Escherichia coli,” J. Microsc. (Oxford) 146, 131–136 (1987).
[Crossref]

M. Radermacher, T. Wagenknecht, A. Verschoor, and J. Frank, “Three-dimensional structure of the large subunit from Escherichia coli,” EMBO J. 6, 1107–1114 (1987).
[PubMed]

M. Radermacher, T. Wagenknecht, A. Verschoor, and J. Frank, “A new 3-D reconstruction scheme applied to the 50S ribosomal subunit of E. coli,” J. Microsc. 141, PR1–RP2 (1986).
[Crossref]

Ver-schoor, A.

J. Frank, J. Zhu, P. Penzek, Y. Li, S. Srivastava, A. Ver-schoor, M. Radermacher, R. Grassucci, R. K. Lata, and R. K. Agrawal, “A model of protein synthesis based on cryo-electron microscopy of the E. coliribosome,” Nature (London) 376, 441–444 (1995).
[Crossref]

Wagenknecht, T.

N. Boisset, J. C. Taveau, J. Lamy, T. Wagenknecht, M. Radermacher, and J. Frank, “Three-dimensional reconstruction of native Androctonus australishemocyanin,” J. Mol. Biol. 216, 743–760 (1990).
[Crossref] [PubMed]

T. Wagenknecht, R. Grassucci, J. Frank, A. Saito, M. Inui, and S. Fleisher, “Three-dimensional architecture of the calcium channel/foot structure of sarcoplasmic reticulum,” Nature (London) 338, 167–170 (1989).
[Crossref]

M. Radermacher, T. Wagenknecht, A. Verschoor, and J. Frank, “Three-dimensional structure of the large subunit from Escherichia coli,” EMBO J. 6, 1107–1114 (1987).
[PubMed]

M. Radermacher, T. Wagenknecht, A. Verschoor, and J. Frank, “Three-dimensional reconstruction from a single-exposure random conical tilt series applied to the 50S ribosomal subunit of Escherichia coli,” J. Microsc. (Oxford) 146, 131–136 (1987).
[Crossref]

M. Radermacher, T. Wagenknecht, A. Verschoor, and J. Frank, “A new 3-D reconstruction scheme applied to the 50S ribosomal subunit of E. coli,” J. Microsc. 141, PR1–RP2 (1986).
[Crossref]

Wokaun, A.

R. Ernst, G. Bodenhausen, and A. Wokaun, Principles of Nuclear Magnetic Resonance in One and Two Dimensions (Oxford Scientific, Oxford, 1990).

Wuthrich, K.

K. Wuthrich, NMR of Proteins and Nucleic Acids (Wiley, New York, 1986).

Zhu, J.

J. Frank, J. Zhu, P. Penzek, Y. Li, S. Srivastava, A. Ver-schoor, M. Radermacher, R. Grassucci, R. K. Lata, and R. K. Agrawal, “A model of protein synthesis based on cryo-electron microscopy of the E. coliribosome,” Nature (London) 376, 441–444 (1995).
[Crossref]

Biochemistry (1)

J. L. Smith, W. A. Hendrickson, R. B. Honzatko, and S. Sheriff, “Structural heterogeneity in protein crystals,” Biochemistry 25, 5018–5027 (1986).
[Crossref] [PubMed]

EMBO J. (1)

M. Radermacher, T. Wagenknecht, A. Verschoor, and J. Frank, “Three-dimensional structure of the large subunit from Escherichia coli,” EMBO J. 6, 1107–1114 (1987).
[PubMed]

Hoppe-Seyler’s Z. Physiol. Chem. (1)

W. Hoppe, J. Gassmann, N. Hunsmann, H. J. Schramm, and M. Sturm, “Three-dimensional reconstruction of individual negatively stained yeast fatty-acid synthetase molecules from tilt series in the electron microscopy,” Hoppe-Seyler’s Z. Physiol. Chem. 335, 1483–1487 (1974).

J. Appl. Phys. (1)

A. M. Cormack, “Representation of a function by its line integrals, with some radiological applications. I,” J. Appl. Phys. 35, 2908–2912 (1964).
[Crossref]

J. Cell Biol. (1)

J. Frank, P. Penczek, R. Grassucci, and S. Srivastava, “Three-dimensional reconstruction of the 70S Escherichia coliribosome in ice: the distribution of ribosomal RNA,” J. Cell Biol. 115, 597–605 (1991).
[Crossref] [PubMed]

J. Electron Microsc. Tech. (1)

M. Radermacher, “Three-dimensional reconstruction of single particles from random and nonrandom tilt series,” J. Electron Microsc. Tech. 9, 359–394 (1998).
[Crossref]

J. Electron Tech. Microsc. (1)

J. Frank, B. F. McEwen, M. Radermacher, J. N. Turner, and C. L. Rieder, “Three-dimensional tomographic reconstruction in high voltage electron microscopy,” J. Electron Tech. Microsc. 6, 193–205 (1987).
[Crossref]

J. Microsc. (1)

M. Radermacher, T. Wagenknecht, A. Verschoor, and J. Frank, “A new 3-D reconstruction scheme applied to the 50S ribosomal subunit of E. coli,” J. Microsc. 141, PR1–RP2 (1986).
[Crossref]

J. Microsc. (Oxford) (1)

M. Radermacher, T. Wagenknecht, A. Verschoor, and J. Frank, “Three-dimensional reconstruction from a single-exposure random conical tilt series applied to the 50S ribosomal subunit of Escherichia coli,” J. Microsc. (Oxford) 146, 131–136 (1987).
[Crossref]

J. Mol. Biol. (2)

N. Boisset, J. C. Taveau, J. Lamy, T. Wagenknecht, M. Radermacher, and J. Frank, “Three-dimensional reconstruction of native Androctonus australishemocyanin,” J. Mol. Biol. 216, 743–760 (1990).
[Crossref] [PubMed]

S. Srivastava, A. Verschoor, and J. Frank, “Eukaryotic initiation factor 3 does not prevent association through physical blockage of the ribosomal subunit–subunit interface,” J. Mol. Biol. 226, 301–304 (1992).
[Crossref] [PubMed]

J. Opt. Soc. Am. A (1)

J. Struct. Biol. (2)

M. Schatz, E. V. Orlova, P. Dube, J. Jäger, and M. van Meel, “Structure of Lumbricus temestrishemoglobin at 30 Å resolution determined using angular reconstitution,” J. Struct. Biol. 114, 28–40 (1995).
[Crossref] [PubMed]

N. Boisset, M. Radermacher, R. Grassucci, J.-C. Taveau, W. Liu, J. Lamy, J. Frank, U. C. Taveau, and J. N. Lamy, “Three-dimensional immunoelectron microscopy of scorpion hemocyanin labeled with a monoclonal Fab fragment,” J. Struct. Biol. 111, 234–244 (1993).
[Crossref] [PubMed]

Math. Phys. Klasse (1)

J. Radon, “Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten. Berichte über die Verhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften zu Leipzig,” Math. Phys. Klasse 69, 262–277 (1917).

Nature (London) (4)

D. J. DeRosier and A. Klug, “Reconstruction of three-dimensional structures from electron micrographs,” Nature (London) 217, 130–134 (1968).
[Crossref]

R. Henderson and P. N. T. Unwin, “Three-dimensional model of purple membrane obtained by electron microscopy,” Nature (London) 257, 28–32 (1975).
[Crossref]

T. Wagenknecht, R. Grassucci, J. Frank, A. Saito, M. Inui, and S. Fleisher, “Three-dimensional architecture of the calcium channel/foot structure of sarcoplasmic reticulum,” Nature (London) 338, 167–170 (1989).
[Crossref]

J. Frank, J. Zhu, P. Penzek, Y. Li, S. Srivastava, A. Ver-schoor, M. Radermacher, R. Grassucci, R. K. Lata, and R. K. Agrawal, “A model of protein synthesis based on cryo-electron microscopy of the E. coliribosome,” Nature (London) 376, 441–444 (1995).
[Crossref]

Optik (1)

G. Harauz and M. van Heel, “Exact filters for general geometry three dimensional reconstruction,” Optik 73, 146–156 (1986).

Proc. R. Soc. London A (1)

R. A. Crowther, D. J. DeRosier, and A. Klug, “The reconstruction of a three-dimensional structure from its projections and its application to electron microscopy,” Proc. R. Soc. London A 317, 319–340 (1970).
[Crossref]

Q. Rev. Biophys. (1)

J. Frank, “Classification of macromolecular assemblies studied as ‘single particles’,” Q. Rev. Biophys. 23, 281–329 (1990).
[Crossref] [PubMed]

Ultramicroscopy (1)

P. Penczek, M. Radermacher, and J. Frank, “Three-dimensional reconstruction of single particles embedded in ice,” Ultramicroscopy 40, 33–53 (1992).
[Crossref] [PubMed]

Z. Naturforsch. (1)

R. Hegerl and W. Hoppe, “Influence of electron noise on three-dimensional image reconstruction,” Z. Naturforsch. 31a, 1717–1721 (1976).

Other (9)

E. J. Hoffman and M. E. Phelps, “Positron emission tomography: principles and quantitation,” in Positron Emission Tomography and Autoradiography: Principles and Applications for the Brain and Heart, M. Phelps and H. Schelbert, eds. (Raven, New York, 1986), pp. 237–286.

K. Wuthrich, NMR of Proteins and Nucleic Acids (Wiley, New York, 1986).

R. Ernst, G. Bodenhausen, and A. Wokaun, Principles of Nuclear Magnetic Resonance in One and Two Dimensions (Oxford Scientific, Oxford, 1990).

M. Radermacher and W. Hoppe, “Properties of 3-D reconstructions from projections by conical tilting compared to single axis tilting,” in Proceedings of the Seventh European Congress on Electron Microscopy (Seventh European Congress on Electron Microscopy Foundation, Leiden, The Netherlands, 1980), Vol. 1, pp. 132–133.

N. Boisset, J.-C. Taveau, P. Penczek, J. N. Lamy, J. Frank, and J. Lamy, “Three-dimensional reconstruction of Androctonus australishemocyanin labeled with a monoclonal Fab fragment,” J. Struct. Biol. (to be published).

W. Liu, “3-D variance of weighted back-projection reconstruction and its application to the detection of 3-D particle conformational changes,” in Proceedings of the Forty-Ninth Annual Meeting of the Electron Microscopy Society of America (San Francisco Press, San Francisco, 1991), pp. 542–543.

W. Liu, J. Frank, and N. Boisset, “An application protocol of 3-D variance estimation theory for significance assessment of reconstructions and detection of 3-D conformational changes,” in Proceedings of the Fiftieth Annual Meeting of the Electron Microscopy Society of America (San Francisco Press, San Francisco, 1991), pp. 1064–1065.

P. W. Hawkes, “The electron microscope as a structure projector,” in Electron Tomography, Three-Dimensional Imaging with the Transmission Electron Microscope, J. Frank, ed. (Plenum, New York, 1992), pp. 17–38.

J. Frank and W. Goldfarb, “Methods for averaging of single molecules and lattice-fragments,” in Electron Microscopy at Molecular Dimensions, State of the Art and Strategies for the Future, W. Baumeister and W. Vogell, eds. (Springer-Verlag, New York, 1980), pp. 261–269.
[Crossref]

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Figures (4)

Fig. 1
Fig. 1

Data-collection scheme of a single-exposure random-conical tilt series: (a) specimen with particles in the same in-plane orientation and with random azimuthal angles; (b) the same specimen tilted by a high angle, 45°–60°. First an image of tilted specimen (b) is recorded. An image of untilted specimen (a) is taken subsequently. With the azimuthal angle determined from the untilted view and the tilt angle derived by the use of the particles as markers, each particle selected from tilted image (b) can be arranged in its appropriate place in a conical tilt series (c). Three features of this data-collection scheme are worth noting: (i) the azimuthal angles of the projections in a data set are randomly distributed over 360°, (ii) each view angle has only one projection, and (iii) experimentally we can collect as many projections as we desire; hence most projections have at least one neighbor with a closely neighbored view angle. (Reproduced from Ref. 10 with permission by Blackwell Scientific Publishers.)

Fig. 2
Fig. 2

Flow diagrams of reconstruction and corresponding variance. (a) Weighted backprojection reconstruction. The weighting and filtering operations are applied to projections first, and the reconstruction is obtained by summation of the weighted and filtered projections. (b) Variance estimation based on close neighbor comparisons. In the case shown, δ = 1.

Fig. 3
Fig. 3

Fourier projection geometry of a regular conical tilt series (N = 9) as indicated by two Uz slices. For a Fourier point A that is located in the central section plane of projection 6 and on the sphere surface where |U| = Uf, the closest projections are 9 and 1 on the Uz ≠ 0 slice, whereas they are 2 and 1 on the Uz = 0 slice. This shows that the projection density at a given Fourier point can be contributed by directionally remote projections. Also, note that the projection density is m = 4 and the neighbor projection density is m′ = 2 at point A, which illustrates the relationship of m = 2m′ for a regular conical tilt series.

Fig. 4
Fig. 4

Neighbor signal component difference when the object consists of a point at the periphery of a sphere with radius D/2. For explanation, see text.

Equations (78)

Equations on this page are rendered with MathJax. Learn more.

p ( i ) ( k , l ) ¯ lim M [ 1 M m = 1 M p m ( i ) ( k , l ) ] ,
b ( i ) ( R ) ¯ lim M [ 1 M m = 1 M b m ( i ) ( R ) ] .
p ( i ) ( r ) p ( i ) ( k , l ) i ( r ) ,
i ( r ) = { ( 1 | x | ) ( 1 | y | ) when | x | 1 and | y | 1 . 0 when | x | > 1 or | y | > 1
b ( R ) { i = 1 N BP [ p ( i ) ( r ) ] } w ( R ) f ( R ) = i = 1 N BP [ p w f ( i ) ( r ) ] .
p w f ( i ) p ( i ) ( r ) w ( i ) ( r ) f ( i ) ( r ) p ( i ) ( k , l ) h i w f ( i ) ( r ) ,
h i w f ( i ) ( r ) i ( r ) w ( i ) ( r ) f ( i ) ( r ) .
n ( i ) ( r ) p ( i ) ( r ) p ( i ) ( r ) ¯ .
n w f ( i ) ( r ) p w f ( i ) ( r ) p w f ( i ) ( r ) ¯ n ( i ) ( k , l ) h i w f ( i ) ( r ) .
ν ( R ) [ b ( R ) b ( R ) ¯ ] 2 ¯ = i = 1 N { BP [ n w f ( i ) ( r ) ] } 2 ¯ + i , j = 1 i j N BP [ n w f ( i ) ( r ) ] BP [ n w f ( j ) ( r ) ] ¯ = i = 1 N BP [ n w f ( i ) ( r ) ] 2 ¯ + i , j = 1 i j N BP [ n w f ( i ) ( r ) n w f ( j ) ( r ) ] ¯ = i = 1 N BP [ V w f ( i ) ( r ) ] ,
ν w f ( i ) ( r ) [ n w f ( i ) ( r ) ] 2 ¯ .
n w f ( i ) ( r ) = N w f ( i ) ( u ) exp ( i 2 π ru ) d 2 u .
ν w f ( i ) ( r ) [ n w f ( i ) ( r ) ] 2 ¯ = C w f ( i ) ( u , u ) exp [ i 2 π r ( u , u ) ] d 2 u d 2 u ,
C w f ( i ) ( u , u ) = C ( i ) ( u , u ) H i w f ( i ) ( u ) H i w f ( i ) ( u ) ,
C ( i ) ( u , u ) N ( i ) ( u ) N ( i ) C ( i ) ( u ) ¯
N j ( i ) ( u ) = [ P ( i ) ( u ) P ( j ) ( u s ) ] / 2 [ N ( i ) ( u ) N ( j ) ( u s ) ] / 2 .
C ( i ) ( u , u ) = λ j k N j ( i ) ( u ) N k ( i ) ( u ) ,
λ j k { 1 when j = k 2 when j k .
ν ( R ) = i = 1 N BP [ ν w f ( i ) ( r ) ] ,
ν w f ( i ) ( r ) = C ( i ) ( u , u ) H i w f ( i ) ( u ) H i w f ( i ) ( u ) × exp [ i 2 π r · ( u + u ) ] d 2 u d 2 u
p w f ( i ) ( r ) = j 1 j i N p w f j ( i ) ( r ) ,
p w f j ( i ) ( r ) = u S j ( i ) P w f ( i ) ( u ) exp ( i 2 π r · u ) d 2 u .
ν w f ( i ) ( r ) = j 1 j i N k 1 k i N c w f j k ( i ) ( r ) ,
c w f j k ( i ) ( r ) n w f j ( i ) ( r ) n w f k ( i ) ( r ) . ¯
c w f j k ( i ) ( r ) = λ j k n w f j ( i ) ( r ) n w f k ( i ) ( r ) ,
n w f j ( i ) ( r ) = u S j ( i ) N ( i ) ( u ) H w f j ( i ) ( u ) exp ( i 2 π r · u ) d 2 u
ν w f ( i ) ( r ) = j = 1 j i N k = 1 k i N c w f j k ( i ) ( r ) = 2 [ n w f j ( i ) ( r ) ] 2 j = 1 j 1 N [ n w f j ( i ) ( r ) ] 2 ,
n w f ( i ) ( r ) = n ( i ) ( r ) h i w f ( i ) ( r ) .
s ( i ) ( k , l ) = 1 2 δ + 1 j = i δ i + δ p ( i ) ( k , l ) ,
n ( i ) ( k , l ) = ( 2 δ + 1 2 δ ) 1 / 2 [ p ( i ) ( k , l ) s ( i ) ( k , l ) ] .
ν w f ( i ) ( r ) = [ n w f j ( i ) ( r ) ] 2 = [ n ( i ) ( k , l ) h i w f ( i ) ( r ) ] 2 .
b m ( R ) = { i = 1 N BP [ p w f ( i ) ( r ) ] } M ( R ) .
b ( i ) ( r ) P J i [ b m ( R ) ] .
n w f ( i ) ( r ) = p w f ( i ) ( r ) b ( i ) ( r ) .
η = var [ ν 0 ( R ) ] var [ ν ( R ) ] .
η | δ = 1 / 2 = 2 3 ,
η | δ = 1 = 0.514 ,
c ( R 1 , R 2 ) [ b ( R 1 ) b ( R 1 ) ¯ ] [ b ( R 2 ) b ( R 2 ) ¯ ] ¯ = { i = 1 N BP [ n w f ( i ) ( r 1 ) ] } { i = 1 N BP [ n w f ( i ) ( r 2 ) ] } = j = 1 N BP [ n w f ( i ) ( r 1 ) ] j = 1 N BP [ n w f ( i ) ( r 2 ) ] = i = 1 N j = 1 N BP [ c w f ( i ) ( r 1 , r 2 ) ] ,
c w f ( i ) ( r 1 , r 2 ) n w f ( i ) ( r 1 ) n w f ( i ) ( r 2 ) ¯ .
h ( R ) = i = 1 N BP [ h i w f ( i ) ( r ) ] .
= ( 3 sin 2 ϑ 0 2 sin 2 ϑ 0 ) 1 / 2 .
S ( U ) = B ( U ) ¯ = i = 1 N { FT [ p ( i ) ( k , l ) ] ¯ } H i w f ( i ) ( u ) = i = 1 N O ( i ) ( u ) H i w f ( i ) ( u ) = O ( U ) i = 1 N H i w f ( i ) ( u ) = O ( U ) H ( U ) .
s ( R ) = o ( R ) h ( R ) ;
H ( U ) { 1 outside missing cone and | R | < U f 0 within missing cone and | R | > U f { F ( U ) outside missing cone 0 within missing cone F 0 ( U ) .
c ( R , R 0 ) = i = 1 N BP [ n w f ( i ) ( r ) n w f ( i ) ( r 0 ) ] ¯ = i = 1 N BP { k , l [ n ( i ) ( r k l ) 2 ] ¯ h i w f ( i ) ( r r k l ) × h i w f ( i ) ( r 0 r k l ) } .
c ( R , R 0 ) = ν 0 i = 1 N BP [ k , l h i w f ( i ) ( r r k l ) h i w f ( i ) ( r r k l ) ] ν 0 i = 1 N BP [ h i w f ( i ) ( r r ) h i w f ( i ) ( r 0 r ) d 2 r d 2 r ] = ν 0 i = 1 N BP [ h i w f ( i ) ( r 1 ) h i w f ( i ) ( r 1 ) ] | r 1 = r r 0 ,
C ( U 1 ) ν 0 i = 1 N H i w f ( i ) ( u 1 ) 2 = ν 0 [ i = 1 N H i w f ( i ) ( u 1 ) ] [ i = 1 N H i w f ( i ) ( u 1 ) ] = ν 0 H 2 ( U 1 ) ν 0 H ( U 1 ) ,
p ( i ) ( r ) = P J i [ o ( i ) ( R ) ] .
var [ o ( R ) ] [ o ( R ) o ( R ) ¯ ] 2 ¯ = cov [ O ( U 1 ) , O ( U 2 ) ] × exp [ [ i 2 π R · ( U 1 + U 2 ) ] d 3 U 1 d 3 U 2 .
cov [ O ( U 1 ) , O ( U 2 ) ] = C ( i ) ( U 1 , U 2 ) ,
var ( n A ) = var ( n B ) = ν c ,
cov [ n A , n B ] = γ ν c ,
n ( i ) ( r ) = n ( i ) ( r ) + n A δ [ r ( i ) r A ( i ) ] + n B δ [ r ( i ) r B ( i ) ] ,
ν ( R ) = i = 1 N BP [ n w f ( i ) 2 ( r ) ] ¯ = ν ( R ) + ν c 2 [ VPSF ( R R A ) + VPSF ( R R A ) ] + 2 γ ν c 2 ( i = 1 N BP { h i w f ( i ) [ r r A ( i ) ] h i w f ( i ) [ r r A ( i ) ] } ) ,
ν ( R ) = i = 1 N BP [ n w f ( i ) 2 ( r ) ] ¯ ,
VPSF ( R ) = i = 1 N BP [ h i w f ( i ) 2 ( r ) ] ,
p ( i ) ( k , l ) = s ( i ) ( k , l ) + n ( i ) ( k , l ) ,
n w f ( i ) ( r ) = ( 2 δ + 1 2 δ ) 1 / 2 [ s w f ( i ) ( r ) 1 2 δ + 1 j = i δ i + δ s w f ( i ) ( r ) ] + ( 2 δ + 1 2 δ ) 1 / 2 [ n w f ( i ) ( r ) 1 2 δ + 1 j = i δ i + δ n w f ( i ) ( r ) ] = n w f n ( i ) ( r ) + n w f s ( i ) ( r ) .
var [ n w f ( i ) ( r ) ] = var [ n w f n ( i ) ( r ) ] + var [ n w f s ( i ) ( r ) ] .
m = 1 2 m .
o ( R ) = S 0 δ [ R ( D 2 , 0 , 0 ) ] .
s ( i ) ( r ) = P J i [ o ( R ) ] = S 0 d ( i ) [ r r 0 ( i ) ] ,
r 0 ( i ) = [ 0 , D 2 sin ( ϕ i ) ] .
S w f ( i ) ( u ) = S ( i ) ( u ) W ( i ) ( u ) F ( i ) ( u ) = S 0 exp [ i 2 π u · r 0 ( i ) ] W ( i ) ( u ) F ( i ) ( u ) .
F ( i ) ( u ) = [ 1 when | u | U f 0 when | u | > U f ,
W ( i ) ( u ) = | u y | .
U f Δ ϕ = 1 m D ,
D 2 sin ( ϕ j ) = D 2 sin ( j Δ ϕ ) D 2 j Δ ϕ = j 2 m U f .
N w f s ( 0 ) ( u ) = 3 2 [ S w f ( 0 ) ( u ) 1 3 j = 1 1 S w f ( j ) ( u ) ] = 2 3 S 0 [ 1 cos ( π u y m U f ) ] | u y | F ( i ) ( u ) 0 .
n w f s ( 0 ) ( r ) = N w f s ( 0 ) ( u ) exp ( i 2 π u · r ) d 2 u = | N w f s ( 0 ) ( u ) exp ( i 2 π u · r ) | d 2 u = n w f s ( 0 ) ( 0 ) .
max [ n w f s ( 0 ) ( r ) ] = n w f s ( 0 ) ( 0 ) = 2 3 S 0 [ 1 cos ( π u y m U f ) ] | u y | F ( i ) ( u ) d 2 u = 2 3 S 0 4 0 U f [ 1 cos ( π u y m U f ) ] u y ( U f 2 u y 2 ) 1 / 2 d u y ν = u y / U f = 4 2 3 S 0 U f 3 0 1 [ 1 cos ( π m ν ) ] ν ( 1 ν 2 ) 1 / 2 d ν = 4 2 3 S 0 U f 3 0 1 [ 1 2 ( π m ν ) 2 1 24 ( π m ν ) 4 + . . . ] ν ( 1 ν 2 ) 1 / 2 d ν 4 15 2 3 S 0 U f 3 ( π m ) 2 [ 1 1 24 ( π m ) 2 ] .
max [ s w f ( 0 ) ( r ) ] = s w f ( 0 ) ( 0 ) = S w f ( 0 ) ( u ) d 2 u = 4 S 0 U f 3 0 1 ν ( 1 ν 2 ) 1 / 2 d ν = 4 3 S 0 U f 3 ,
max [ n w f s ( 0 ) ( r ) ] max [ s w f ( 0 ) ( r ) ] 1 5 2 3 S 0 U f 3 ( π m ) 2 [ 1 1 21 ( π m ) 2 ] .
max [ s w f ( i ) ( r ) ] = max [ s w f ( 0 ) ( r ) ] ,
max [ n w f s ( i ) ( r ) ] max [ n w f s ( 0 ) ( r ) ] .
var [ n w f s ( i ) ( r ) ] var [ n w f n ( i ) ( r ) ] max [ n w f s ( 0 ) 2 ( r ) ] max [ s w f ( i ) 2 ( r ) ] max [ s w f ( i ) 2 ( r ) ] var [ n w f n ( i ) ( r ) ] 2 75 π 4 m 4 [ 1 1 21 ( π m ) 2 ] 2 R s n 2.60 m 4 R s n ,
R s n max [ S w f ( i ) 2 ( r ) ] var [ n w f n ( i ) ( r ) ] ,
N 0 = 2 π D U f sin ϑ 0 for N 0 even .

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