Bootstrap resampling for voxel-wise variance analysis of three-dimensional density maps derived by image analysis of two-dimensional crystals

Abstract

Difference density maps are commonly used in structural biology for identifying conformational changes in macromolecular complexes. For interpretation of the results, it is essential to estimate the variance or standard deviation of the difference density and the distribution of errors in space. In order to compare three-dimensional density maps of gap junction channels with and without the C-terminal regulatory domain, we developed a bootstrap resampling method for estimation of the voxel-wise standard deviation. The bootstrap approach has been successfully used for estimating the sampling distribution from a limited data set and for estimating the statistical properties of the derived quantities [Efron, B., 1979. Bootstrap methods: another look at the jackknife. Ann. Stat. 7, 1-26]. In our application, the standard deviation map can be estimated by bootstrapping the images. Our results show that, apart from the symmetry axes and small regions bordering the lumen of the extracellular vestibule, difference maps normalized by the mean of the standard deviation map can be used as a good approximation of the t-test map of the gap junction crystals.

Figure 1

Illustration of the sampling distribution meta experiment (a,b,c) and its correspondence to the bootstrap resampling method (d,e,f) for a population distribution (gray curve) of colored balls. In a, b and c the sample mean is the quantity of interest, represented by the green ellipse. (a) With 7 samples taken from a known population, a mean can be calculated. (b) By repeating the sampling process without changing the population, a mean is calculated for each of the 7 sample sets. The value of the mean changes with each sampling, as shown by the different colors (blue and purple) for the sample ellipses. The sampling distribution shown by the red curve in (c) is then the distribution of these means. Note that this distribution is narrower compared with the population distribution (gray). (d) When the population distribution is not known, bootstrap resampling can be used to estimate the distribution. As in (a), a single value of the quantity of interest can be calculated from the 7 samples. (e) The bootstrap resampling estimate is generated using the 7 sampled balls shown in the gray rectangular box. Samples can be repeatedly drawn with replacement so that the probability of sampling a particular ball is not altered by the resampling process. For each loop, the resampling is repeated 7 times to match the number of the original sampling from the population. Since the resampling follows the distribution of the samples, the many calculated values for the quantity of interest from a large number of loops will distribute, shown in different colors (blue and purple), according to the propagated distribution that originated from the same distribution. This process yields a good estimate for the distribution of the quantity, shown by the red curve in (f).

Figure 2

Application of the bootstrap method to projection images of simulated crystals that contain only structure variation. The unit cells in (a) and (b) contain two possible conformations of an artificial Gaussian molecule. Shown in (c) is the reconstructed projection map of 17 simulated crystals displaying an average of the two conformations. The inset shows a portion of a simulated 2D crystal. (d) Standard error map of the mean for an individual reconstruction, ${\delta }_{\overline{{\tilde{\rho }}_i}}$, calculated from Eqns. 1-4. (e) and (f) show the Bootstrap estimation, ${\sigma }_{{\tilde{\rho }}_q^B}$, and Jackknife estimation, ${\widehat{\delta }}_{\tilde{\rho }}^J$, respectively, of the standard error map, at the same contour level as in (d).

Figure 3

Noise level dependence of the bias of the estimation for reconstructed map mean and standard error from simulated crystals. (a,b) Standard error of the reconstruction, ${\delta }_{\tilde{\rho }}$ (blue), is estimated by ${\delta }_{\overline{{\tilde{\rho }}_i}}$ (red) or ${\sigma }_{{\tilde{\rho }}_q^B}$ (green). (c,d) Reconstructed density, $\tilde{\rho }$, is estimated by $\overline{{\tilde{\rho }}_i}$ or ${\widehat{\tilde{\rho }}}^B$, shown in the same color scheme. The two positions of the unit cell where the plots are indicated on the inset map in (b). Results at position a are shown in (a,c), and those at b are in (b,d).

Figure 4

Surface-shaded views (a-d) of the 3D density map of Cx43-WT contoured at 1.5 ${\mathit{sd}}_{\mathit{sp}}\left[{\tilde{\rho }}_{WT}\right]$. The boundaries shown in the side view (a) indicate the locations of sections for (b-d). The arrows indicate the viewing directions. Shown in (b), (c) and (d), respectively, are the helices within the top connexon, a cross-section showing the continuous wall of protein density that lines the extracellular vestibule, and a cross-section showing the helices within the bottom connexon.. The wedge lines in (b) demarcate a pie slice used for the representations in Figure 5. The crystallographic and one of the non-crystallographic 2-fold symmetry axes are indicated in (c). Shown in (e) is a projection map contoured at ${\mathit{sd}}_{\mathit{sp}}\lfloor {\delta }_{\Delta \tilde{\rho }}\rfloor$. We note that the viewing direction of this map and those shown in Fig. 5 are consistent with the viewing direction of the 3D maps in our (Unger et al., 1999). This is opposite to that of the projection maps we published previously (Unger et al., 1997; Cheng et al., 2003) Scale bars: 20 Å

Figure 5

Maps of the standard error of the mean projection density derived from 2D crystals of Cx43-TR. Seventeen images with tilt angle values less than 4° were used in the analysis. (a) Standard error, ${\delta }_{\overline{{\tilde{\rho }}_i}}$, of the mean of the individually reconstructed map calculated according to Eqns. 1-4. (b) and (c) Estimated standard error of the reconstruction, ${\widehat{\delta }}_{\tilde{\rho }}^B$, from two independent bootstrap resamplings, each consisting of 200 cycles. The red outlines show the approximate boundary of the projected protein density. All maps are scaled and contoured equally, with each contour corresponding to the rms deviation of the pixel values in ${\delta }_{\overline{{\tilde{\rho }}_i}}$. The similarity between (b) and (c) confirms that the number of cycles of bootstrap resampling was adequate.

Figure 6

Simple bootstrap estimation test on a 3D map with large variations in structure. All maps shown are asymmetric unit pie slices around the top connexon as indicated in Figure 4b. (a) and (b) are the mirrored pair of Cx43-WT in yellow and white, respectively. (c) surface-shaded representation of the 3D reconstruction of the combined data set (light purple), contoured at a higher value than the wire-frame rendering (white and yellow) from (a) and (b). (d) 3D reconstruction of the combined data in wire-frame rendering (purple), shown with a line indicating one of the mirror planes. The map has approximate 6/mmm point group non-crystallographic symmetry. (e) and (f) Bootstrap estimated standard error map (green) superimposed on the same density in (d). The line in (f) indicates the location of the mirror plane passing through the middle of the helical bundle. The pseudo mirror symmetry is apparent after all the processing that used only p6 symmetry. Scale bar: 20 Å.

Figure 7

Bootstrap estimated standard error maps of the difference, ${\widehat{\delta }}_{\Delta \tilde{\rho }}^B$ between Cx43-WT and Cx43-TR. The green maps in (a) and (c) show the results for simple boot strapping, surface rendered at 1.5 ${\mathit{sd}}_{\mathit{sp}}\lfloor {\widehat{\delta }}_{\Delta \tilde{\rho }}^B\rfloor +{\text{mean}}_{\mathit{sp}}\lfloor {\widehat{\delta }}_{\Delta \tilde{\rho }}^B\rfloor$. The green maps in (b) and (d) show the results for residual bootstrapping, scaled to the peak on the 6-fold axis in (a) and (c). As a reference, the error map is superimposed on the surface-shaded 3D map of Cx43-WT (gold), rendered at 1.5 ${\mathit{sd}}_{\mathrm{sp}}\left[{\tilde{\rho }}_{WT}\right]$. The top views in (a) and (b) are through a section of the extracellular gap, indicated by the white boxes in (c) and (d). This section contains the largest non-crystallographic error peaks, which border the lumen of the extracellular vestibule. The dashed line in (a) indicates the slicing plane for the side views shown in (c) and (d). Scale Bar: 20 Å.

Figure 8

Similarity of the 3D difference and t- density maps between Cx43-WT and Cx43-TR. Top (a) and side views (c) of the difference map, contoured at 2.9 ${\text{mean}}_{\mathit{sp}}\lfloor {\widehat{\delta }}_{\Delta \tilde{\rho }}^B\rfloor$. The positive difference, i.e., $\Delta \tilde{\rho }={\tilde{\rho }}_{WT}-{\tilde{\rho }}_{TR}>0$, is colored in red and negative difference in sky blue. Top (b) and side views (d) of the t-test map surface rendered at t=± 2.681 or 99% confidence level for 47 degree of freedom. Positive t values are colored in orange and negative values in violet. As a reference, the maps are superimposed on the surface-shaded 3D map of Cx43-WT (gold), rendered at 1.5 ${\mathit{sd}}_{\mathrm{sp}}\left[{\tilde{\rho }}_{WT}\right]$. The sections and viewing direction are identical to that of Figure 7.