Structural variability and the incoherent addition of scattered intensities in single-particle diffraction

X-ray lasers may allow structural studies on single particles and biomolecules without crystalline periodicity in the samples. We examine here the effect of sample dynamics as a source of structural heterogeneity on the resolution of the reconstructed image of a small protein molecule. Structures from molecular-dynamics simulations of lysozyme were sampled and aligned. These structures were then used to calculate diffraction patterns corresponding to different dynamic states. The patterns were incoherently summed and the resulting data set was phased using the oversampling method. Reconstructed images of hydrated and dehydrated lysozyme gave resolutions of $3.7\text{Å}$ and $7.6\text{Å}$, respectively. These are significantly worse than the root-mean-square deviation of the hydrated ($2.7\text{Å}$ for all atoms and $1.45\text{Å}$ for $\text{C-}\alpha$ positions) or dehydrated ($3.7\text{Å}$ for all atoms and $2.5\text{Å}$ for $\text{C-}\alpha$ positions) structures. The noise introduced by structural dynamics and incoherent addition of dissimilar structures restricts the maximum resolution to be expected from direct image reconstruction of dynamic systems. A way of potentially reducing this effect is by grouping dynamic structures into distinct structural substates and solving them separately.

Figure 1

(Color online) Modulation of the diffraction intensity on a 1D bond of length $l=1$ due to vibration of pointlike atoms as a function of the scattering vector $q$. (a) The dashed line represents the instantaneous diffraction intensity $\left(I_{inst}\right)$ for ${ϵ}_1={ϵ}_2=0$. The solid line represents the incoherently added intensities $\left(I_{inc}\right)$ for an atom vibration $\sigma =0.1$. (b) The solid line corresponds to the same as in (a) while the dashed line shows the intensities resulting from the coherently added amplitudes using the same parameters. The envelopes of the intensities are drawn as dotted lines.

Figure 2

(Color online) (a) Slice through the origin of the incoherently averaged diffraction intensities of the dehydrated system. (b) Slice through the origin of the diffraction intensities resulting from coherently averaging the amplitudes of the dehydrated system. (c) Relative standard deviation of the intensities $\left({\sigma }_{rel}\right)$ as a function of the magnitude of the scattering vector $\left(q\right)$, for the intensities resulting from coherently averaged amplitudes $\left(I_{coh}\right)$, instantaneous intensities $\left(I_{inst}\right)$, and incoherent averaged intensities $\left(I_{inc}\right)$ of the dehydrated system, plus for incoherent averaged intensities of the hydrated system $\left(I_{hyd}\right)$.

Figure 3

(Color online) PRTF and mean phase error in radians of the averaged reconstructed image. The PRTF drops below $1/e$ at $q=0.132{\text{Å}}^{-1}$ and $q=0.271{\text{Å}}^{-1}$ resulting in a resolution of $7.6\text{Å}$ and $3.7\text{Å}$ for the dehydrated and hydrated systems, respectively. The PRTF and the mean phase flatten out at the same resolution supporting the use of the PRTF as a good indicator of the information content of the reconstruction.

Figure 4

(Color online) Left: isosurface of the reconstructed electron density map of ${\rho }_{rec}$ for hydrated lysozyme assuming perfect image orientation, perfect signal, and no radiation damage to the sample. The map was plotted at $3\sigma$. Right: cartoon representation of the secondary structure of 1AKI [28], the PDB entry used in the molecular-dynamics simulations. The isosurface on the left follows the PDB structure, but details are smoothened out as a result of the lower resolution.