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A method for three-dimensional (3D) reconstruction of macromolecule assembles, that is, octahedral symmetrical adapted functions (OSAFs) method, was introduced in this paper and a series of formulations for reconstruction by OSAF method were derived. To verify the feasibility and advantages of the method, two octahedral symmetrical macromolecules, that is, heat shock protein

The determination of three-dimensional (3D) structures of macromolecular assemblies plays a key role in understanding their functions and properties. In the course of structure reconstruction of single particles during the last several decades, the Cryo-electron microscopy (referred to as “CryoEM”) has been successively used to solve 3D structures up to subnanometer resolution [_{24} [

In this paper, we would like to concentrate our attention to the octahedral symmetry-adapted function (OSAF) method. And a series of formulations for 3D reconstruction of octahedral symmetry macromolecules had been derived. To verify the feasibility and the advantages of this approach, two octahedral symmetrical simulated data such as heat shock protein Degp_{24} (3Cs0.pdb) [

For readily understanding the OSAF method for 3D reconstruction, it is necessary to describe the principle of the OSAF method briefly.

Due to SAFs being the linear combinations of the spherical harmonics

The multiplicity

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | |

1 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 1 | 0 |

To calculate OSAFs, at first, one should calculate the normalized spherical harmonics by the formula

where the

According to (

The coefficients of the OSAF for

Normalized constant | ||||
---|---|---|---|---|

4 | 6 | |||

6 | 4 | |||

9 | 12 |

For example,

Due to the properties of the normalized spherical harmonic functions, the

So that

According to (

The OSAFs of all

The combination coefficients of real OSAFs | |||
---|---|---|---|

0 | 1 | ||

4 | 1 | ||

6 | 1 | ||

8 | 1 | ||

9 | 1 | ||

10 | 1 | ||

12 | 1 | ||

12 | 2 |

Figures

Density contour maps of the OSAFs looking down along a fourfold axis. (a)

It should be pointed out that the OSAFs (

It is well known that the structure of macromolecular complexes can be described as its potential functions which are determined by Fourier inversion transformation of the structure factors

In contrast to Crowther’s Fourier-Bessel method [

According to (

Substituting (

The reconstruction by OSAF can be carried out in the following procedure.

Calculate the OSAFs by (

Construct two linear equation groups with experimental determined structure factors

Find the fitting coefficients

Determine 3D structures of octahedral symmetrical objects according to (

To verify the feasibility and advantages of the OSAF method for reconstruction of macromolecules with octahedral symmetry, two biological objects with octahedral symmetry, heat shock protein Degp_{24} and the Red-cell L Ferritin, were taken as examples. The atomic structures were downloaded from PDB (3cs0.pdb and 1rcc.pdb). The both 3D structures with 4.0 Å resolution were generated as standard structure models (SSMs) by the EMAN’s pdb2mrc procedure. Then two thousand random projections of these two proteins with predefined orientations and centers were created using real-space projection. Then random noise was added to each projection at 3 different signal to noise ratios (

where

Finally, 3D structures of two models by making use of all the 2000 projections (_{24} and the Red-cell L Ferritin proteins with _{24} and 4-fold axis of Red-cell L Ferritin proteins, respectively. From the two-dimensional section maps, one may find that some noises still present in the final structures. Usually a perfect reconstruction is impossible in the case of added noise. However it can be seen intuitively from Figures _{24} protein is approximately equal to 5.8 Å, 4.6 Å, and 4.3 Å and those of Red-cell L Ferritin protein, are approximately equal to 5.6 Å, 4.7 Å, and 4.2 Å with

The Relative Errors with different ^{-1}).

Fourier radius | Relative errors | |
---|---|---|

1/14.4 Å^{-1} | 7.96% | |

1/14.4 Å^{-1} | 7.62% | |

1/14.4 Å^{-1} | 7.60% | |

1/14.4 Å^{-1} | 7.59% |

Three representative projection images of the DegP_{24} model generated from PDB data. (a) Without noise, (b) with

Three representative projection images of the erythrocyte L Ferritin model generated from PDB data. (a) Without noise, (b) with

Maps of two-dimensional sections perpendicular to _{24} model and reconstruction results viewing along the 3-fold axis by the OSAF method: (a) the standard model, (b) the reconstructed results with

Maps of two-dimensional sections perpendicular to

3D structures reconstructed by the OSAF method viewing along the 4-fold axis of DegP_{24} protein with

3D structure viewing along the 4-fold axis of erythrocyte L Ferritin model with

The relative error between the erythrocyte L Ferritin model and the results reconstructed by the OSAF method.

Since the

The algorithm time of the OSAF method for reconstructing DegP_{24} model at low and high resolution (S means second).

Particle numbers | Time (^{-1}) | Time (^{-1}) |
---|---|---|

500 | 17 (S) | 165 (S) |

2000 | 37 (S) | 186 (S) |

5000 | 88 (S) | 230 (S) |

A set of formulations for 3D reconstruction of macromolecular assemblies with octahedral symmetry by the OSAF method has been established.

The OSAF method is feasible and efficiently suppresses the influence of the noise because of its sufficiently utilizing the symmetry of the objects.

The calculation can be greatly speed up by dealing with the reconstruction in an asymmetrical unit of the octahedral symmetry group.

It should be pointed out that in the simulation, one may use projections with predetermined centers and orientations to reconstruct structures; however in practice, one should reconstruct based on experimented measured data with unknown centers and orientations. In this case, one should first determine the center and orientation of a projection. At this stage, we did not write a program to determine the center and orientation by the OSAF method itself yet, So far we should use the other program such as EMAN [

where

The OSAF method for reconstruction is just at beginning stage, there is a plenty of space for optimizing the program. We believe that this method has a prospective future. A reconstruction with experimental data is proceeding based on the principle mentioned above and will be reported later on.

This work is supported by the National Natural Scientific Foundation of China (no. 10874144), National Basic Research Program of China (Grant no. 2010CB912400) and the Scientific Foundation of Education Department of Hunan Province (no. 08C885).