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Introduction

This page will describe how EMAN2 defines the asymmetric unit, and its associated mirror portion, for each of the symmetries. This is mostly important in the context the the projection tools (EMAN1: project3d, EMAN2: e2project3d.py) which project quasi-uniformly over the asymmetric unit for each of the symmetries, generating the reference images for particle classification.

A symmetric object is often defined by its symmetric axes and how they are arranged with respect to one another. For each symmetry one may define a set of affine transformations (i.e. rotations), which when applied to the symmetric object yield it in exactly equivalent (but unique) orientation. These will be referred to as the set of rotational symmetry operations, the total in number of which is the same as the overall symmetry of the object. For instance the 60-fold symmetric icosahedral symmetry has a total of 60 rotational symmetry operations (including the identity). Similarly the 14-fold symmetric D7 symmetry has 14 and so on.

Similar to the rotational symmetry operations, the total number of asymmetric units will match the overall symmetry of the object. The asymmetric unit of a symmetric 3D object can be defined in many ways. In one regard, it is the 3D subvolume of the entire 3D object that an asymmetric object can exist inside of, in each of the asymmetric units, in equivalent orientation, without destroying the overall symmetry of the object, that is, applying a rotational symmetry must yield the 3D object in indentical conformation.

In single particle reconstruction we must project over this asymmetric unit as accurately as possible. To do this we often think in terms of the unit sphere and imagine it as encapsulating the symmetric object. We isolate a single asymmetric unit and define the surface of the unit sphere we must uniformly project through to ensure the sampling of the single asymmetric unit in all of its unique orientations.

This seems simple enough, but depending on the approach, we must also consider the influence of mirror (or reflective) symmetry which occurs when we generate a projection from the opposite side of the hemisphere. Take this idea one step futher, if it is true that a projection from one side of the hemisphere is the mirror of the projection from the opposite side, then if we apply the correct rotational symmetry operation, the mirror projection must exist in same asymmetric unit that produced the equivalent "un mirrored" projection. In fact it is true that exaclty half of the projections over the asymmetric unit are the mirror projections of the other half.

In EMAN1 it was imperative that the mirror portion of the asymmetric unit was accurately demarcated. This was due to EMAN1's approach which used two rounds of classification for each reconstruction, one using the raw (unmodified) particle data, and one where the particle data was mirrored. The particle was then classified to which projection it exhibited the greatest similarity toward, based on the best results of the two runs. In EMAN2 the equivalent approach of projecting over the entire asymmetric unit (including the mirrored portion) but doing only one round of classification will be available to the user.

This document will therefore describe the way in which EMAN2 demarcates the entire and mirror portions of the asymmetric unit, for each of the symmetries.

C symmetry

C even symmetry is handled this way...

attachment:csym_ai.png

D symmetry

Icosahedral symmetry

Octahedral symmetry

Tetrahedral symmetry