  Single molecule X-ray scattering experiments using free electron lasers hold
the potential to resolve both single structures and structural ensembles of
biomolecules. However, molecular electron density determination has so far not
been achieved due to low photon counts, high noise levels and low hit rates.
Most analysis approaches therefore focus on large specimen like entire viruses,
which scatter substantially more photons per image, such that it becomes
possible to determine the molecular orientation for each image. In contrast,
for small specimen like proteins, the molecular orientation cannot be
determined for each image, and must be considered random and unknown.
  Here we developed and tested a rigorous Bayesian approach to overcome these
limitations, and also taking into account intensity fluctuations, beam
polarization, irregular detector shapes, incoherent scattering and background
scattering. We demonstrate using synthetic scattering images that it is
possible to determine electron densities of small proteins in this extreme high
noise Poisson regime. Tests on published experimental data from the coliphage
PR772 achieved the detector-limited resolution of $9\,\mathrm{nm}$, using only
$0.01\,\%$ of the available photons per image.
