Strategies for 3D reconstruction of macromolecules existing as single particles

The aim of the "single-particle" image processing with SPIDER is to obtain a 3D reconstruction of a macromolecule from a large set of particle images (that are obtained with the electron microscope), based on the premise that each of these particle images shows the same structure. Since the macromolecule is single, without a structural context that would stabilize its orientation, it occurs in many different orientations. Thus, the electron micrograph normally displays a wide range of particle views. But it is unknown, in the absence of prior knowledge, how these views are related to one another. Thus the 3D reconstruction procedure must deal with two separate issues:

• How to find the relative orientations (each given in terms of 3 Eulerian angles and two translational parameters ) of the particle projections;
• Provided these orientations are known, how to reconstruct the macromolecule from the projections.

The following is a guide on how to proceed, using established methods of orientation search and 3D reconstruction.

Let's say the structure is entirely unknown. In that case, you have to start from scratch:

• Orientation determination using the random-conical data collection method. This method uses a defined geometry in the data collection, and is able to find the handedness of the structure unambiguously. Each specimen field is imaged twice, once tilted, once untilted. Particles are selected simultaneously from both untilted- and tilted-specimen fields, using a special interactive particle-selection program that is able to "predict" the location of a particle in the tilted-specimen field when its counterpart has been selected in the untilted field. This program is part of WEB. From the untilted-specimen particle data set, all particles are selected that exhibit the same view. This can be done by using alignment followed by classification. The corresponding tilted-specimen data subset can be used to compute a reconstruction: the orientations of the tilted-particle projections lie on a cone with fixed angle (the tilt angle) and random azimuths (the in-plane angles found in the alignment of the untilted particle set).

• Orientation determination using common lines (a.k.a. " angular reconstitution "). This method is based on the fact that in Fourier space any two projections intersect along a central line ("the common line"). Hence, in principle, the relative orientations between three projections can be determined - except that the handedness of the constellation is ambiguous. Because of the low signal-to-noise ratio of raw particle images, averages of projections falling into roughly the same orientation must be used. In the SPIDER implementation (operation OP), any number of views can be used in a simultaneous optimization scheme. Since the procedure leads to solutions presenting local minima, it must be repeated several times to find solutions that form a cluster, presumably around the global minimum. Such clustering of solutions can be detected by multivariate statistical analysis of the resulting 3D maps. Two clusters are expected, one for each enantiomorph. After an initial structure is obtained, it should be further refined using 3D projection matching strategy described next.

• Orientation determination by 3D projection matching. Here the existing 3D map is projected from many orientations on a regular angular grid, and the resulting projections that are kept in computer memory are compared, one by one, with each of the experimental projections. This comparison (by cross-correlation ) yields a refined set of Eulerian angles , with which a refined reconstruction can be computed using one of the reconstruction techniques listed below. This procedure then goes on in several cycles, until the angles for each projection stabilize.

  When you have a set of projections whose orientations are known, you can use either one of the following two reconstruction techniques. Note that the second of these is much slower but gives better results:

  • Weighted back-projection using general weighting functions ( BP 3D ). The back-projection operation is the reverse of the projection operation: each projection image is translated in the direction of the projection, to yield a "back-projection body." All these back-projection bodies are superimposed and summed in their correct orientations, yielding an approximation to the reconstruction. Filtration by a weighting function is necessary since the transition from the polar to the Cartesian coordinate system leads to an imbalance in the representation of spatial frequencies. This weighting is very simple in the case of equal angular increments, but becomes very complicated (General weighting) when an arbitrary distribution of angles must be accommodated, as in our case. Since general weighting entails some approximation, and since weighted "BP 3D" methods do not allow the use of constraints, the results are not as good as with SIRT, described below.

  • Simultaneous Iterative Reconstruction Technique (SIRT) ( BP RP and BP CG ). The 3D reconstruction algorithm called Simultaneous Algebraic Reconstruction Technique (SIRT) seeks to minimize the discrepancy between the 2D projection data and 2D projections of the structure. The reconstruction process begins with setting to zero the elements of the initial volume and proceeds iteratively by using the 2D error between projection data and projected current approximation of the structure as current corrections. Although the results of this algorithm are of very high quality, its rate of convergence is slow and the recommended number of iterations is 100. In addition, the BP RP implementation of SIRT has a provision for reconstruction of symmetric structures. The list of symmetries has to be provided in a form of angular document file. The Eulerian angles have to be given in the order (psi, theta, phi) and the null transformation (psi,theta,phi)=(0,0,0) has to included on the list.