Bayesian orientation estimate and structure information from sparse single-molecule x-ray diffraction images

We developed a Bayesian method to extract macromolecular structure information from sparse single-molecule x-ray free-electron laser diffraction images. The method addresses two possible scenarios. First, using a “seed” structural model, the molecular orientation is determined for each of the provided diffraction images, which are then averaged in three-dimensional reciprocal space. Subsequently, the real space electron density is determined using a relaxed averaged alternating reflections algorithm. In the second approach, the probability that the “seed” model fits to the given set of diffraction images as a whole is determined and used to distinguish between proposed structures. We show that for a given x-ray intensity, unexpectedly, the achievable resolution increases with molecular mass such that structure determination should be more challenging for small molecules than for larger ones. For a sufficiently large number of recorded photons $(>200)$ per diffraction image an $M^{1/6}$ scaling is seen. Using synthetic diffraction data for a small glutathione molecule as a challenging test case, successful determination of electron density was demonstrated for $20000$ diffraction patterns with random orientations and an average of 82 elastically scattered and recorded photons per image, also in the presence of up to 50% background noise. The second scenario is exemplified and assessed for three biomolecules of different sizes. In all cases, determining the probability of a structure given set of diffraction patterns allowed successful discrimination between different conformations of the test molecules. A structure model of the glutathione tripeptide was refined in a Monte Carlo simulation from a random starting conformation. Further, effective distinguishing between three differently arranged immunoglobulin domains of a titin molecule and also different states of a ribosome in a tRNA translocation process was demonstrated. These results show that the proposed method is robust and enables structure determination from sparse and noisy x-ray diffraction images of single molecules spanning a wide range of molecular masses.

Figure 1

Two approaches to structure determination from single-molecule diffraction experiments. For each recorded image, photon arrival positions are shown in red (dark gray) dots. (a) The spatial orientation $\left(\theta ,\psi ,\phi \right)$ of the irradiated molecule, here a glutathione, is determined for each of the images separately, and the molecular transform is derived by averaging the images in reciprocal space. (b) A structure that fits best to the entire set of photon arrival positions is determined.

Figure 2

Examples of posterior probability surfaces $\pi \left(\mathit{\Theta }\right|\mathbf{X})$. Shown are cuts through the three-dimensional angular probability landscapes obtained for a glutathione molecule with actual orientation $\theta ={73}^{\circ },\psi ={52}^{\circ },\phi ={34}^{\circ }$ using diffraction images with shot noise only (left) and with additional 50% background noise (right). Slices at the $\Theta$ value corresponding to the posterior maximum (${\theta }_{\max }={75}^{\circ }$ for shot noise only, ${\theta }_{\max }={63}^{\circ }$ for background noise) are depicted at a logarithmic scale (top row). The insets (bottom row) show the maximum peaks at a linear scale.

Figure 3

Quality of determined molecular transforms. Shown are cuts along the $k_{\mathrm{x}}$ axis of the calculated molecular transform (blue solid lines) compared to the reference (red dashed). Below each panel, the difference [green (light gray)] between the calculated and the reference molecular transform is shown. The top row shows differences in the high-resolution regime between the two tested methods, the maximum likelihood and Bayesian; the bottom row shows the effect of different levels of background noise for the Bayesian method.

Figure 4

Quality of calculated electron densities. Top: Reference density; middle: densities obtained by maximum likelihood (left) and Bayesian (right) methods. The bottom row demonstrates the robustness of the Bayesian method to the background noise (BN) levels of 10% (left) and 50% (right).

Figure 5

Achievable resolution for different molecular masses and incident beam intensities. Solid lines (dots) represent a signal with shot noise only and dashed lines (diamonds) a signal with additional 50% background noise. Line colors encode incident beam intensities $(\mathrm{photons}/\AA {}^2)$. The resolution achievable with x-ray crystallography $(\sim 1–5 \AA )$ is highlighted in green (light gray). The molecular masses of the test molecules used in this study are indicated on the $x$ axis; glutathione (GTT), titin, and ribosome. The dot-dash line shows the expected slope of $-1/6$.

Figure 6

Monte Carlo structure refinement of glutathione. The logarithm of normalized probability that the accepted structure simultaneously agrees with 200 synthetic diffraction images containing 76 photons was plotted for 12 independent MC runs (color lines) starting from different random structures. Two examples of starting structures are shown in green (mid-left) and pink (bottom left) boxes. After about 500 steps, the most probable structure is found. A comparison of the refined [orange (dark gray)], i.e., the most probable, and the reference structure [yellow (light gray)] is shown in the bottom right corner.

Figure 7

Ability to identify the correct titin structure. Two hundred synthetic images with 376 photons per picture (inset: sample image) were generated for the reference structure (a). For 290 different conformations, the normalized posterior probability (blue symbols) is plotted as a function of the RMSD with respect to the reference structure. Insets show three intermediate structures, (b), (c), and (d).

Figure 8

Ability to identify the correct ribosomal translocation state. Two hundred synthetic diffraction images with $1.075\times {10}^5$ photons per picture were generated from the reference structure (pre1 state). For seven different translocation states, the logarithm of the normalized posterior probability and the RMSD with respect to the reference (pre1 state) was calculated. The inset shows an overlay of the pre1 [red (dark gray)] and the pre5 [yellow (light gray)] states, in particular, structural differences within subunits (surface) and the tRNA chains (cartoon) are highlighted. Boxes next to each of the points depict the location of the tRNA chains in the ribosome at corresponding translocation states.

Figure 9

Tracing local structural changes during tRNA translocation. For each translocation state model, obtained by embedding the native tRNA chains conformation within the subunit structure of the pre1 state, normalized posterior probability was calculated ($y$ axis) using 200 synthetic diffraction images with $1.075\times {10}^5$ photons per picture generated from the pre1 state. Bottom $x$ axis corresponds to the RMSDs of whole structures, while the upper axis shows the RMSDs of the tRNA chains alone. The difference between the translocated [yellow (light gray)] and the reference chains [red (dark gray)] is shown for each of the states. The inset demonstrates the differences between the tRNA chains locations from the pre1 and the pre5 states, embedded in the ribosomal structure of the pre1 state (shaded surface).

Figure 10

Tracing local structural changes during tRNA translocation using an inaccurate structure model. For each translocation state model, obtained by embedding the native tRNA chains conformation within the subunit structure of the pre2 state (inaccuracy of the model), normalized posterior probability was calculated ($y$ axis) using 200 synthetic diffraction images with $1.075\times {10}^5$ photons per picture generated from the pre1 state. The bottom $x$ axis corresponds to the RMSDs of whole structures, while the upper axis shows the RMSDs of the tRNA chains alone. The difference between the translocated [yellow (light gray)] and the reference chains [red (dark gray)] is shown for each of the states. The inset demonstrates the difference between the tRNA chains locations from the pre1 and the pre5 state embedded in the ribosomal structure of the pre2 state (shaded surface).