Reconstruction of the real 3D shape of the SARS-CoV-2 virus

The photographs of the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) virus taken by electron transmission microscopy and cryoelectron microscopy provide only a 2D silhouette. The viruses appear to look like distorted circles. The present paper questions the real shape of the SARS-CoV-2 virus and makes an attempt to give an answer. Is this a general ellipsoid, a spheroid with rotational symmetry, a sphere, or something else? The answer requires the application of tools from three different disciplines: structural mechanics, microbiology, and statistics. A total of 590 virus photographs taken from 22 recently published papers were examined. From this experimental data pool, the histogram of diameter ratios was built from the 283 measurements where the virus images could be approximated as ellipses. The curve peaks at the diameter ratio of 1.22. The transformation equation for the spatial shape to the planar shade was derived for a fixed light source of the microscope. This equation involves an unknown orientation of the viruses with respect to the microscope. Two sets of models were developed, one with a uniform distribution of the virus orientation and the other with the orientation defined by the normalized beta distribution. In both sets of models, the unknown diameter ratio of the spheroidal virus was regarded as a random realization from translated gamma distributions. The parameters of the distribution of the kernel functions were determined by minimizing the mean square difference between the predicted and measured 2D histograms. The information included in the measured histograms was found to be insufficient to find an unknown distribution of the virus's orientation. Simply too many unknown parameters render the solution physically unrealistic. The minimization procedure with a uniform probability of virus orientation predicted the peak of the aspect ratio of the 3D spheroid at 1.32. Based on this result, models of the virus will be developed in the continuation of this research for a full dynamic analysis.