The first consideration before starting the 3D reconstruction is whether there are actually enough particles to achieve a reconstruction at the desired resolution. The Crowther equation gives a good approximation.

N is the number of particles (or minimum number of unique projections needed), d is the resolution and D is the diameter of the particle. This could also be presented as

Δθ is the angular increment between the orientations. The size of tilt angles between the successive orientations will prescribe how well the 3D reconstruction will be covered. Missing orientations means a less complete 3D Fourier transform and so poorer resolution in those particular regions of the reconstructed map. This is a rough estimate as many other factors influence the resolution.

There are 2 main approaches for the 3D reconstruction

1. Projection-matching alignment. This is based on the Central Section or Fourier Slice Theorem. Basically, the Fourier transform of a single 2D projection of a 3D object is a central section through the 3D Fourier transform of that object perpendicular to the direction of the projection. Images of the object in many different orientations provides lots of different central sections. Combining all of these 2D Fourier transforms together gives a 3D Fourier transform and a 3D inverse Fourier transform of this will produce a reconstruction of the original object. This involves determining the orientations of all the projections in the dataset by comparing against a known model 3D map. The model can come from a low resolution negative stain derived reconstruction, an X-ray model or an EM map of a homolog and should be low resolution to avoid introducing any bias.


Central Section Theorem. The 2D projections from particles in different orientations are converted to 2D Fourier transforms. Combining all the Fourier transforms from all the orientations produces a 3D Fourier transform. Performing an inverse Fourier transform on this will produce a 3D image of the object in real space. Figure courtesy of Lou Brillault, University of Queensland.




2. Maximum likelihood methods. These do not need a model to compare against for the initial orientations but consider all possible orientations of the particle and assign weights to each direction according to their probability. It finds the most likely 3D reconstruction for the dataset by maximising the probability that the observed particle images would be generated, given the cryo-EM reconstruction produced during the refinement. The maximum likelihood approach is best for data that has high levels of noise, like cryo-TEM data.

Both techniques produce new maps that could be of low resolution, between 20 to 60Å, that serve as a new reference structure for the next round of alignment. This process keeps cycling to produce improved higher resolution maps each time. The iterations continue until there is no improvement in reconstruction resolution at what is called convergence.

3D classification of the maps helps again to evaluate the quality of the data, often showing heterogeneity that wasn’t obvious in the 2D classification and leading to the removal of more particles from the dataset. It may also indicate missing orientations or a favoring of specific orientations that could bias the models. This could mean preparation of new samples, trying to avoid the orientation issues.

The signal in the final 3D reconstruction is dampened at high frequencies due to many issues both during imaging in the microscope and during the image processing. The measure of each of these signal losses is an attenuation envelope. There are many envelopes that lead to deterioration of the data and they can include things like optical imperfections in the imaging system, the thickness of the ice and noise-induced errors in averaging alignments. It is possible to determine an aggregate envelope function that encompasses all of these elements. This is described as the “B factor”. The B-factor is really a number estimated from the data that tells you about the rate at which the signal-to-noise ratio deteriorates, relative to resolution (or spatial frequency). The 3D map is sharpened by applying a negative B factor to each Fourier component to counteract these effects. The B factor can help to guide the reconstruction as it can be used to give a more accurate estimate of how the resolution will improve with an increased number of particles. It can be estimated from the slope of a Guinier plot which graphs log F (the log of the amplitudes in the three-dimensional Fourier transform squared) against the inverse of the resolution squared for each refinement. At a very simplistic level, log F can be considered as representing the amount of signal in the data. The B factor can also be measured using the slope of a Rosenthal-Henderson plot which also graphs the inverse of the resolution squared for each refinement but against the logarithm of the number of particles in the corresponding subset.

A Guinier plot of data from a refinement of apoferritin. The slope of a straight line fitted through the data gives the B factor.

Typically, the global resolution of maps is measured by a “gold-standard” procedure. This involves splitting the original data into two halves and independently refining each to produce two reconstructions. A comparison between these two maps is done in 3D Fourier space. A correlation between corresponding Fourier shells (spatial frequencies that are the 3D equivalent of Thon rings) in reciprocal space is produced. This is called the Fourier Shell Correlation (FSC). The average value for all shells are plotted as a curve with the correlation on the Y-axis and spatial frequency on the X-axis. The FSC is really a measure of the SNR of the map, so a correlation of 1 is a perfect correlation with no noise while 0 represents no correlation or just noise at the spatial frequency. The spatial frequencies are measured in the units of Å^-1 which is a direct reciprocal of Å so this can be used for reading the resolution of the maps. The cryo-EM community has now accepted that the value of 0.143 is the threshold for reading the resolution as seen on the example graph.

Gold standard of map refinement. The dataset is divided into 2 half sets and the 3D reconstruction is done on each. A Fourier shell correlation between the 2 shows the resolution of the map at the value of 0.143. Figure courtesy of Lou Brillault, University of Queensland.

3D refinement of models is a very complex process and many different strategies have been adopted to address specific issues. For one thing the resolution of different regions of the model can be quite different and so specific refinement of the region is necessary. One way to address this is to apply a soft 3D mask on the specific part, so the classification will be focused solely in that area.

FSC graph for an apoferritin map showing the resolution for the map with no masks used, a loose mask and a tight mask during refinement. The resolution is read at the 0.143 correlation.

Angular distribution plot showing the particle orientations contributing to the final map (red: high occurrence, blue: low occurrence). From such a plot it is possible to tell if certain orientations are under-represented in the map. This is from a reconstruction of the pore-forming A subunit from the Yersinia entomophaga ABC toxin and is overlaid on a map of this. Image courtesy of Michael Landsberg, University of Queensland.

Map of GroEL rendered according to local resolution. Blue indicates the highest resolution regions and red the lowest resolution regions.