Abstract

A previously developed theory of three-dimensional (3-D) variance estimation [ J. Opt. Soc. Am. A 12, 2615– 2627 ( 1995)] is applied to the structural study of a hemocyanin-Fab complex with the electron microscope. The precise locations of structurally variable regions of the macromolecule are determined from the 3-D variance maps. The structural differences among different classes of the macromolecular complex are assessed by the use of the statistical t-test, and the 3-D antibody binding sites are revealed. From a model analysis, a rule is demonstrated for visually identifying a 3-D conformational change by the inspection of the 3-D variance map. Our analysis lays the foundation for numerous practical applications of variance estimation in the 3-D imaging of macromolecules.

© 1995 Optical Society of America

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  1. W. Liu, J. Frank, “Estimation of variance distribution in three-dimensional reconstruction. I. Theory,” J. Opt. Soc. Am. A 12, 2615–2627 (1995).
    [Crossref]
  2. J. Frank, A. Verschoor, M. Boublik, “Computer averaging of 40S ribosomal subunits,” Science 214, 1353–1355 (1981).
    [Crossref] [PubMed]
  3. W. O. Saxton, W. Baumeister, “The correlation averaging of a regularly arranged bacterial envelope protein,” J. Microsc. 127, 127–128 (1982).
    [Crossref]
  4. M. Unser, B. L. Trus, A. C. Steven, “A new resolution criterion based on spectral signal-to-noise ratios,” Ultramicroscopy 23, 39–42 (1987).
    [Crossref] [PubMed]
  5. W. Hänicke, J. Frank, H. P. Zingsheim, “Statistical significance of molecule projections by single particle averaging,” J. Microsc. 133, 223–238 (1984).
    [Crossref]
  6. L. Sachs, Applied Statistics, A Handbook of Techniques, 2nd ed. (Springer-Verlag, New York, 1984).
  7. E. J. Dudewicz, S. N. Mishra, Modern Mathematical Statistics, Vol. 19 of Wiley Series in Probability and Mathematical Statistics (Wiley, New York, 1988).
  8. N. Boisset, M. Radermacher, R. Grassucci, J. C. Taveau, W. Liu, J. Lamy, J. Frank, 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]
  9. N. Boisset, J. C. Taveau, J. Lamy, T. Wagenknecht, M. Radermacher, J. Frank, “Three-dimensional reconstruction of native Androctonus australishemocyanin,” J. Mol. Biol. 216, 743–760 (1990).
    [Crossref] [PubMed]
  10. M. Radermacher, “Three-dimensional reconstruction of single particles from random and nonrandom tilt series,” J. Electron Microsc. Tech. 9, 359–394 (1988).
    [Crossref] [PubMed]
  11. R. J. Poljak, L. M. Amzel, H. P. Avey, L. N. Becha, A. Nisonoff, “Structure of Fab′ new at 6 Ångstroems resolution,” Nature/New Biology (London) 235, 137–141 (1972).
  12. J. Frank, “Classification of macromolecular assemblies studied as ‘single particles’,” Q. Rev. Biophys. 23, 281–329 (1990).
    [Crossref] [PubMed]
  13. J. Frank, W. Liu, N. Boisset, “Classification and 3D variance estimation: complementary tools in the 3D reconstruction of macromolecules,” in Proceedings of the 10th European Congress on Electron Microscopy (Secretariado de Publicaciones de la Universidad de Granada, Granada, Spain, 1992), pp. 427–429.

1995 (1)

1993 (1)

N. Boisset, M. Radermacher, R. Grassucci, J. C. Taveau, W. Liu, J. Lamy, J. Frank, 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]

1990 (2)

N. Boisset, J. C. Taveau, J. Lamy, T. Wagenknecht, M. Radermacher, 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]

1988 (1)

M. Radermacher, “Three-dimensional reconstruction of single particles from random and nonrandom tilt series,” J. Electron Microsc. Tech. 9, 359–394 (1988).
[Crossref] [PubMed]

1987 (1)

M. Unser, B. L. Trus, A. C. Steven, “A new resolution criterion based on spectral signal-to-noise ratios,” Ultramicroscopy 23, 39–42 (1987).
[Crossref] [PubMed]

1984 (1)

W. Hänicke, J. Frank, H. P. Zingsheim, “Statistical significance of molecule projections by single particle averaging,” J. Microsc. 133, 223–238 (1984).
[Crossref]

1982 (1)

W. O. Saxton, W. Baumeister, “The correlation averaging of a regularly arranged bacterial envelope protein,” J. Microsc. 127, 127–128 (1982).
[Crossref]

1981 (1)

J. Frank, A. Verschoor, M. Boublik, “Computer averaging of 40S ribosomal subunits,” Science 214, 1353–1355 (1981).
[Crossref] [PubMed]

1972 (1)

R. J. Poljak, L. M. Amzel, H. P. Avey, L. N. Becha, A. Nisonoff, “Structure of Fab′ new at 6 Ångstroems resolution,” Nature/New Biology (London) 235, 137–141 (1972).

Amzel, L. M.

R. J. Poljak, L. M. Amzel, H. P. Avey, L. N. Becha, A. Nisonoff, “Structure of Fab′ new at 6 Ångstroems resolution,” Nature/New Biology (London) 235, 137–141 (1972).

Avey, H. P.

R. J. Poljak, L. M. Amzel, H. P. Avey, L. N. Becha, A. Nisonoff, “Structure of Fab′ new at 6 Ångstroems resolution,” Nature/New Biology (London) 235, 137–141 (1972).

Baumeister, W.

W. O. Saxton, W. Baumeister, “The correlation averaging of a regularly arranged bacterial envelope protein,” J. Microsc. 127, 127–128 (1982).
[Crossref]

Becha, L. N.

R. J. Poljak, L. M. Amzel, H. P. Avey, L. N. Becha, A. Nisonoff, “Structure of Fab′ new at 6 Ångstroems resolution,” Nature/New Biology (London) 235, 137–141 (1972).

Boisset, N.

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

J. Frank, W. Liu, N. Boisset, “Classification and 3D variance estimation: complementary tools in the 3D reconstruction of macromolecules,” in Proceedings of the 10th European Congress on Electron Microscopy (Secretariado de Publicaciones de la Universidad de Granada, Granada, Spain, 1992), pp. 427–429.

Boublik, M.

J. Frank, A. Verschoor, M. Boublik, “Computer averaging of 40S ribosomal subunits,” Science 214, 1353–1355 (1981).
[Crossref] [PubMed]

Dudewicz, E. J.

E. J. Dudewicz, S. N. Mishra, Modern Mathematical Statistics, Vol. 19 of Wiley Series in Probability and Mathematical Statistics (Wiley, New York, 1988).

Frank, J.

W. Liu, J. Frank, “Estimation of variance distribution in three-dimensional reconstruction. I. Theory,” J. Opt. Soc. Am. A 12, 2615–2627 (1995).
[Crossref]

N. Boisset, M. Radermacher, R. Grassucci, J. C. Taveau, W. Liu, J. Lamy, J. Frank, 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, 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]

W. Hänicke, J. Frank, H. P. Zingsheim, “Statistical significance of molecule projections by single particle averaging,” J. Microsc. 133, 223–238 (1984).
[Crossref]

J. Frank, A. Verschoor, M. Boublik, “Computer averaging of 40S ribosomal subunits,” Science 214, 1353–1355 (1981).
[Crossref] [PubMed]

J. Frank, W. Liu, N. Boisset, “Classification and 3D variance estimation: complementary tools in the 3D reconstruction of macromolecules,” in Proceedings of the 10th European Congress on Electron Microscopy (Secretariado de Publicaciones de la Universidad de Granada, Granada, Spain, 1992), pp. 427–429.

Grassucci, R.

N. Boisset, M. Radermacher, R. Grassucci, J. C. Taveau, W. Liu, J. Lamy, J. Frank, 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]

Hänicke, W.

W. Hänicke, J. Frank, H. P. Zingsheim, “Statistical significance of molecule projections by single particle averaging,” J. Microsc. 133, 223–238 (1984).
[Crossref]

Lamy, J.

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

Lamy, J. N.

N. Boisset, M. Radermacher, R. Grassucci, J. C. Taveau, W. Liu, J. Lamy, J. Frank, 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]

Liu, W.

W. Liu, J. Frank, “Estimation of variance distribution in three-dimensional reconstruction. I. Theory,” J. Opt. Soc. Am. A 12, 2615–2627 (1995).
[Crossref]

N. Boisset, M. Radermacher, R. Grassucci, J. C. Taveau, W. Liu, J. Lamy, J. Frank, 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, W. Liu, N. Boisset, “Classification and 3D variance estimation: complementary tools in the 3D reconstruction of macromolecules,” in Proceedings of the 10th European Congress on Electron Microscopy (Secretariado de Publicaciones de la Universidad de Granada, Granada, Spain, 1992), pp. 427–429.

Mishra, S. N.

E. J. Dudewicz, S. N. Mishra, Modern Mathematical Statistics, Vol. 19 of Wiley Series in Probability and Mathematical Statistics (Wiley, New York, 1988).

Nisonoff, A.

R. J. Poljak, L. M. Amzel, H. P. Avey, L. N. Becha, A. Nisonoff, “Structure of Fab′ new at 6 Ångstroems resolution,” Nature/New Biology (London) 235, 137–141 (1972).

Poljak, R. J.

R. J. Poljak, L. M. Amzel, H. P. Avey, L. N. Becha, A. Nisonoff, “Structure of Fab′ new at 6 Ångstroems resolution,” Nature/New Biology (London) 235, 137–141 (1972).

Radermacher, M.

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

M. Radermacher, “Three-dimensional reconstruction of single particles from random and nonrandom tilt series,” J. Electron Microsc. Tech. 9, 359–394 (1988).
[Crossref] [PubMed]

Sachs, L.

L. Sachs, Applied Statistics, A Handbook of Techniques, 2nd ed. (Springer-Verlag, New York, 1984).

Saxton, W. O.

W. O. Saxton, W. Baumeister, “The correlation averaging of a regularly arranged bacterial envelope protein,” J. Microsc. 127, 127–128 (1982).
[Crossref]

Steven, A. C.

M. Unser, B. L. Trus, A. C. Steven, “A new resolution criterion based on spectral signal-to-noise ratios,” Ultramicroscopy 23, 39–42 (1987).
[Crossref] [PubMed]

Taveau, J. C.

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

Trus, B. L.

M. Unser, B. L. Trus, A. C. Steven, “A new resolution criterion based on spectral signal-to-noise ratios,” Ultramicroscopy 23, 39–42 (1987).
[Crossref] [PubMed]

Unser, M.

M. Unser, B. L. Trus, A. C. Steven, “A new resolution criterion based on spectral signal-to-noise ratios,” Ultramicroscopy 23, 39–42 (1987).
[Crossref] [PubMed]

Verschoor, A.

J. Frank, A. Verschoor, M. Boublik, “Computer averaging of 40S ribosomal subunits,” Science 214, 1353–1355 (1981).
[Crossref] [PubMed]

Wagenknecht, T.

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

Zingsheim, H. P.

W. Hänicke, J. Frank, H. P. Zingsheim, “Statistical significance of molecule projections by single particle averaging,” J. Microsc. 133, 223–238 (1984).
[Crossref]

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 (1988).
[Crossref] [PubMed]

J. Microsc. (2)

W. O. Saxton, W. Baumeister, “The correlation averaging of a regularly arranged bacterial envelope protein,” J. Microsc. 127, 127–128 (1982).
[Crossref]

W. Hänicke, J. Frank, H. P. Zingsheim, “Statistical significance of molecule projections by single particle averaging,” J. Microsc. 133, 223–238 (1984).
[Crossref]

J. Mol. Biol. (1)

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

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

J. Struct. Biol. (1)

N. Boisset, M. Radermacher, R. Grassucci, J. C. Taveau, W. Liu, J. Lamy, J. Frank, 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]

Nature/New Biology (London) (1)

R. J. Poljak, L. M. Amzel, H. P. Avey, L. N. Becha, A. Nisonoff, “Structure of Fab′ new at 6 Ångstroems resolution,” Nature/New Biology (London) 235, 137–141 (1972).

Q. Rev. Biophys. (1)

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

Science (1)

J. Frank, A. Verschoor, M. Boublik, “Computer averaging of 40S ribosomal subunits,” Science 214, 1353–1355 (1981).
[Crossref] [PubMed]

Ultramicroscopy (1)

M. Unser, B. L. Trus, A. C. Steven, “A new resolution criterion based on spectral signal-to-noise ratios,” Ultramicroscopy 23, 39–42 (1987).
[Crossref] [PubMed]

Other (3)

L. Sachs, Applied Statistics, A Handbook of Techniques, 2nd ed. (Springer-Verlag, New York, 1984).

E. J. Dudewicz, S. N. Mishra, Modern Mathematical Statistics, Vol. 19 of Wiley Series in Probability and Mathematical Statistics (Wiley, New York, 1988).

J. Frank, W. Liu, N. Boisset, “Classification and 3D variance estimation: complementary tools in the 3D reconstruction of macromolecules,” in Proceedings of the 10th European Congress on Electron Microscopy (Secretariado de Publicaciones de la Universidad de Granada, Granada, Spain, 1992), pp. 427–429.

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Figures (4)

Fig. 1
Fig. 1

Contour representation of the resolution filtration PSF function h(R), with missing cone angle θ0 = 40° and low-pass cutoff frequency of the filter Uf = 1/(45 Å): (a) central XY plane, (b) central XZ plane. Shading indicates regions where h(R) < 0. The dots represent the resampling points with the sampling rates according to Eqs. (2.1) and (2.2). Note that h(R) ≈ 0 in these points. [Contours in (a) are not completely circular because of numerical and sampling inaccuracies.]

Fig. 2
Fig. 2

Model of the quaternary structure of scorpion Androctonus australis hemocyanin from its top view in which the molecule lies on its flip face. A. australis hemocyanin is a copper-containing oxygen carrier that has a 4 ×6-mer structure. Its 24 subunits belong to eight different polypeptide chains with a molecular weight of 75 kDa each. The molecule possesses a twofold axis of symmetry passing through its center. Because of the noncoplanar arrangement of the hexameric building blocks, the molecule is able to rock around an axis that passes through the two diagonally opposed axes on the grid.9 The locations of the IgG 104 antibody epitopes on the four copies of subunit Aa6 are shown by arrows (reprinted from Ref. 9, with permission from Academic Press).

Fig. 3
Fig. 3

Significance assessment of difference between reconstructions of class-II and class-I hemocyanin-Fab complex by the use of the t-test: (a), (b) Z-slice montages of 3-D reconstructions of class II and class I, respectively, with the cutoff frequency of low-pass filter Uf = 1/(31.1 Å) and the number of projections N2 = 184 and N1 = 109; (c) Z-slice montage of the 3-D difference map from the reconstructions of class II and class I as shown in (a) and (b), respectively; (d), (e) Z-slice montages of 3-D variance estimates of class II and class I, respectively; (f) montage of the 3-D t-test map of the differences as shown in (c), with bright areas corresponding to the significance level of 99%, where, t ̂ ( R ) > n 0.01 and bright plus gray areas corresponding to 95%, where. t ̂ ( R ) > n 0.05 The distance between consecutive slices is 15.6 Å for all the montages in Figs. 3 and 4.

Fig. 4
Fig. 4

Z-slice montages of 3-D reconstructions and corresponding 3-D variance estimates of the hemocyanin-Fab complex: (a)–(d) correspond to the total flip-top data set for two frequency cutoffs Uf of the low-pass filter. The majority of the NT = 404 projections have Fab bound to each of the four sites. (a), (b) Reconstruction and 3-D variance estimate for Uf = 1/ (46.7 Å); (c), (d) reconstruction and 3-D variance estimate for Uf = 1/(31.1 Å); (e), (f) reconstruction and 3-D variance estimate of class II for Uf = 1/(46.7 Å).

Equations (26)

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T X Y = 1 / ( 2 U f ) ,
T Z = T X Y ,
m = V 0 / ( T X Y 2 T Z ) .
var [ ν ( R ) ] = 1 N e ν 2 ( R ) .
η = var [ ν 0 ( R ) ] var [ ν ( R ) ] ,
N e = η ν 2 ( R ) var [ ν 0 ( R ) ] = η N 2 ν w f ( i ) 2 ( r ) N ν w f ( i ) 2 ( r ) = η N ,
N e > N / 2 .
F ̂ M b M w ,
b ¯ 1 m k = 1 m b k 1 n R i V 0 b ( R i ) ,
M b 1 m 1 k = 1 m ( b b ¯ ) 2 , m m 1 1 n R i V 0 [ b ( R i ) b ¯ ] 2 ,
M w 1 m k = 1 m ν k 1 n R i V 0 ν ( R i ) ,
Δ b ν b ¯ ν b 0 = 1 m k = 1 m b k b 0 1 n R i V 0 b ( R i ) b 0 .
ν ν var ( Δ b ν ) = 1 m k = 1 m var ( b k ) = 1 m k = 1 m ν k .
ν ν = 1 m k = 1 m ν k = M w .
t ̂ = Δ b ν ν ν = Δ b ν M w ,
var [ ν i ( R ) ] = 1 N e i ν i 2 ( R ) ,
( N e 1 + 1 ) ν 1 2 ( R ) = ( N e 2 + 1 ) ν 2 2 ( R ) .
t ̂ ( R ) = | b 1 ( R ) b 2 ( R ) | [ ν 1 ( R ) + ν 2 ( R ) ] 1 / 2 ,
υ ( R ) = [ ν 1 ( R ) + ν 2 ( R ) ] 2 ν 1 2 ( R ) N e 1 + ν 2 2 ( R ) N e 2 .
t ̂ = | b ¯ 1 b ¯ 2 | ( ν 1 + ν 2 ) 1 / 2 ,
ν L 1 m ν ¯ L 1 m [ 1 V 0 R V 0 b 1 ( R ) d 3 R ] .
Δ ν ( R ) / ν ( R ) = Δ ν ( R ) ν 0 ( R ) + Δ ν ( R ) P c ( 1 P c ) .
ν ( R ) 1 N .
var [ ν ( R ) ] / ν ( R ) = 2 N e 1 N ,
ν out ( R ) / ν ( R ) 1 N .
Δ ν ( R ) / ν ( R ) = const ,

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