Quantifying the local resolution of cryo-em density maps

Transcription

1 BRIEF COMMUNICATIONS npg 2014 Nature America, Inc. All rights reserved. Quantifying the local resolution of cryo-em density maps Alp Kucukelbir 1, Fred J Sigworth 1,2 & Hemant D Tagare 1,3 We propose a definition of local resolution for three-dimensional electron cryo-microscopy (cryo-em) density maps that uses local sinusoidal features. Our algorithm has no free parameters and is applicable to other imaging modalities, including tomography. By evaluating the local resolution of singleparticle reconstructions and subtomogram averages for four example data sets, we report variable resolution across a 4- to 40-Å range. Various resolution measures for cryo-em have been proposed in the past three decades 1. Unlike the classical Rayleigh resolution that characterizes instruments, these measures characterize features present in the data. A commonly used cryo-em resolution measure is the Fourier shell correlation (FSC) procedure. It quantifies the strength, relative to noise, of sinusoidal features across the entire density map. FSC produces a single resolution for the entire density map. FSC cannot assess locally varying resolution, which may be caused by sample heterogeneity and image processing errors 2. Our goal was to overcome this limitation of FSC by presenting a definition of local resolution that can assess variable resolution across the density map. As a resolution measure, FSC has other limitations. FSC uses split data sets, that is, the images are grouped into two halves each contributing to a separate density map. The resolution is measured using both density maps. The computational stage at which the data are split can affect the FSC resolution 3. Further, calculating the resolution from FSC requires a threshold, whose value and interpretation has been debated 1. Alternative approaches 4,5 address some of these shortcomings but do not define local resolution. Recent structural studies 6,7 have used windowed FSC for local resolution 8. Windowed FSC masks the split data set density maps with a window and calculates FSC resolutions as the window moves through the map. This requires a window size parameter, whose value is often arbitrary. Although this approach implicitly conducts multiple tests on the density map, it does not control the false discovery rate (FDR) in the thresholding of the FSC. FDR control is critical because local resolution tests are repeated at many points in the volume. Additionally, there is data dependency between neighboring points that windowed FSC does not account for. We propose a mathematical theory and an efficient algorithm for measuring local resolution that address all of the above limitations. The theory (Online Methods) is based on the following idea: a -Å feature exists at a point in the volume if a threedimensional (3D) local sinusoid of wavelength is statistically detectable above noise at that point. A likelihood-ratio hypothesis test of the local sinusoid versus noise can detect this feature at a given P value (typically P = 0.05). We define the local resolution at a point as the smallest at which the local sinusoid is detectable, and we account for multiple testing with an FDR procedure. Our algorithm, named ResMap, implements this theory. ResMap begins by initializing a local-sinusoid model at = 2, where is the voxel spacing in Å. Likelihood-ratio tests are conducted at all voxels in the volume, with explicit FDR control that accounts for data dependency. Voxels that pass the test are assigned resolution, whereas those that fail are tested at a larger. The algorithm produces a local-resolution map with a number assigned to every voxel in the density map (Fig. 1a). There are no algorithm parameters to tune, and local resolution may be defined unambiguously at the given P value. In ResMap, local sinusoids of wavelength are approximated by a set of functions called H2. This set is derived from Gaussian windowed second-order Hermite polynomials 9,10, with window size proportional to the wavelength (Fig. 1b and Online Methods). ResMap results with H2 are specifically denoted as ResMap-H2. H2 functions are steerable, so their linear combination can locally model any arbitrarily oriented local sinusoid in three dimensions (Supplementary Note 1). At a fixed wavelength, the standard likelihood-ratio test 11 can detect whether a local sinusoid is present in the steerable function approximation. The test requires an estimate of the noise variance, which we obtain from the region surrounding the particle. The likelihood-ratio test does not depend on how this variance is estimated. Other noise estimates, such as those obtained by analyzing split data set density maps, can also be used. The smallest at which the likelihood-ratio test passes at a given P value defines the resolution. We control for false discoveries using a method that takes into account the dependencies between tests 12 (Online Methods). We first evaluated ResMap using a simulated density map of a radially symmetrical chirp signal whose wavelength decreased with radius. We added white and non-white noise with two different variance levels (Supplementary Fig. 1). ResMap-H2 estimates show an intuitive relation to the underlying signal features (Fig. 1c). 1 Department of Biomedical Engineering, Yale University, New Haven, Connecticut, USA. 2 Department of Cellular and Molecular Physiology, Yale University, New Haven, Connecticut, USA. 3 Department of Diagnostic Radiology, Yale University, New Haven, Connecticut, USA. Correspondence should be addressed to H.D.T. (hemant.tagare@yale.edu). RECEIVED 30 MAY; ACCEPTED 2 OCTOBER; PUBLISHED ONLINE 10 NOVEMBER 2013; DOI: /NMETH.2727 NATURE METHODS VOL.11 NO.1 JANUARY

2 npg 2014 Nature America, Inc. All rights reserved. BRIEF COMMUNICATIONS Figure 1 Local resolution. (a) The ResMap algorithm. Wavelength is initialized to twice the voxel spacing. Likelihood-ratio tests decide whether the local-sinusoid model is detectable at each voxel. Voxels that pass the test are controlled for false discoveries. Voxels that fail the test are tested again after is increased (Online Methods). (b) Cosine- and sine-like H2 functions oriented along an axis. White and black indicate negative and positive parts, respectively (Supplementary Note 1). (c) Left, slice through noisy simulated density maps with voxel spacing of 1 Å. Right, radial plots. ResMap-H2 resolution estimates show a steady improvement as the simulated signal becomes more finely varying. Bottom, corresponding results for 1/f noise display robustness against non-white noise (Supplementary Fig. 1). 2, variance. Further, increasing the noise worsens the resolution at every point. ResMap-H2 results for this simulation exhibit a ripple in the transitions between the peaks and valleys of the signal. This is because transitions have more energy in the higher frequencies and are thus detectable with local sinusoids of smaller scale. We then tested ResMap with four different density maps ranging from near-atomic single-particle reconstructions (~4 Å) to typical subtomogram averages (~40 Å). All results were obtained with a P value of We compared ResMap-H2 results to regular and gold-standard 3 FSC plots and to windowed FSC maps. First, we analyzed a single-particle 80S ribosome reconstruction (EMDataBank: EMD-2275) 13. The original publication estimated a resolution of 4.5 Å (gold-standard FSC at 0.143) and noted the blurring from the heterogeneity in the 40S subunit (Fig. 2a). Our ResMap-H2 resolution estimates fall between 4.5 and 5.5 Å in the 60S subunit and between 4.5 and 9 Å in the 40S subunit. Some parts of the 40S are just as resolved as the 60S, which ResMap-H2 results show in the portion of 40S adjacent to 60S. The median a b c d Resolution 3D visualization Density map slice(s) ResMap-H2 slice(s) (Å) S 60S a c Initialize. Set voxels inside the particle to have unassigned resolution. Conduct likelihood-ratio tests at unassigned voxels to evaluate the -Å local-sinusoid model. Assign Å to voxels that pass the test with FDR control. White noise, 2 = 2.0 1/f noise, 2 = 2.0 ResMap-H2 (Å) ResMap-H2 (Å) Increase YES More unassigned voxels? Radius (voxels) Signal ResMap-H2 resolution ( 2 = 2.0) ResMap-H2 resolution ( 2 = 0.5) ResMap-H2 resolution ( 2 = 2.0) ResMap-H2 resolution ( 2 = 0.5) Radius (voxels) ResMap-H2 resolutions in the 40S and 60S subunits are 6.5 and 5 Å, respectively, which agree with a map versus atomic model FSC plot (Fig. 3J in the original publication 13 ). Our ResMap-H2 results additionally point to a decrease in resolution near the edges of the particle. This may be due to image alignment errors or the interaction of the ribosome with the solvent. Second, we analyzed a single-particle Tulane virus reconstruction (EMD-5529) 14. The original publication estimated a 6.3-Å (gold-standard FSC at 0.143) resolution for the entire particle and highlighted the considerable flexibility of the protruding domains of the virus. Our ResMap results corroborate these findings, estimating the resolution of the shell as between 6 and 7 Å and the resolution of the protruding domains as between 7 and 9 Å (Fig. 2b). Third, we analyzed a subtomogram average of GroEL (EMD-2221) 15. The original publication reported an 8.4-Å (FSC at 0.5) resolution. Our ResMap-H2 estimates suggest that many -helices are resolved up to 7.5 Å (Fig. 2c). This is evident in the close-up rendering that displays the central part of a helix at 7.5 Å but the end and adjoining loop at ~9.5 Å. These results are corroborated in Supplementary Video 1, in which the central part of Figure 2 ResMap-H2 results using experimental density maps. Color bars apply to both volumes and slices. 3D visualizations are rendered using UCSF Chimera 17. White dotted lines in color bars indicate FSC and 0.5 thresholds from the original publications. (a) 80S ribosome (EMD-2275). ResMap-H2 results indicate a decreased resolution within the 40S subunit and near the edges of the particle. (b) Tulane virus (EMD-5529). ResMap estimates lower resolutions in the protruding domains, whereas the shell appears well resolved. (c) Subtomogram GroEL (uncropped version of EMD-2221). ResMap-H2 results show an -helix with varying levels of resolution (Supplementary Video 1). (d) Subtomogram ATP synthase dimer (uncropped version of EMD-2161). ResMap delineates the central dimer as better resolved than the neighboring dimers and membrane. NO DONE b VOL.11 NO.1 JANUARY 2014 NATURE METHODS

3 npg 2014 Nature America, Inc. All rights reserved. the helix is shown to maintain its tubular structure under a range of surface threshold values. Finally, we analyzed a subtomogram average of ATP synthase dimers (EMD-2161) 16. The original publication estimated a 37-Å (FSC at 0.5) resolution. Our ResMap-H2 resolution estimates are between 30 and 42 Å in the central dimers, which are better resolved than the neighboring dimers and membrane (Fig. 2d). The edges of the central dimers appear to be at a higher resolution than the cores. This is likely due to the strong dark bands surrounding the particle, as is typical in particles reconstructed without contrast transfer function correction. ResMap results are consistent with windowed FSC but differ in some important aspects. Windowed FSC results appear to be sensitive to the fixed size of the user-defined window (Supplementary Fig. 2). Too large a window size may include the solvent in the FSC computation and lead to underestimation of resolution (Supplementary Fig. 3). ResMap does not suffer from this effect because the localization uncertainty in the joint spatial-frequency domain is minimized by the use of the H2 functions. The behavior of ResMap for typical P values is shown in Supplementary Figure 4. Moreover, windowed FSC can be slow, taking anywhere between 25 min and 4 h to compute, depending on the window size; ResMap usually requires a few minutes. For all cases above, ResMap-H2 local resolutions within the reconstructed particle were almost always between the 0.5 and threshold of the FSC in the original publications. This is consistent with the idea that the 0.5 threshold may be too conservative and that the threshold may be too optimistic 3,13. Moreover, ResMap-H2 results agree with published flexibility analyses and also visually match the level of detail in the density maps. ResMap is available as a cross-platform executable package with a simple graphical user interface (Supplementary Fig. 5). The software and test data are publicly accessible (Supplementary Software; Users can also apply ResMap to other fields by choosing features other than local sinusoids. For instance, a 2D Gaussian feature may be appropriate for optical nanoscopy, whereas rotated 2D arcs may be of use in radio telescopy. The anticipated 2 recent increase in heterogeneity studies 6,7,13 highlights the pressing challenge of evaluating the local resolution of cryo-em density maps. We believe that this local-resolution method is both statistically rigorous and practical and therefore BRIEF COMMUNICATIONS provides a critical step in enabling researchers to assess the quality of cryo-em density maps. METHODS Methods and any associated references are available in the online version of the paper. Note: Any Supplementary Information and Source Data files are available in the online version of the paper. ACKNOWLEDGMENTS We are grateful to S. Scheres, G. Lander, A. Bartesaghi and K. Davies for stimulating discussions and sharing their density maps for this study. This work was supported by Natural Sciences and Engineering Research Council of Canada award PGS-D3 (A.K.) and US National Institutes of Health grants R01LM (H.D.T.), R01GM (A.K. and H.D.T.) and R01NS (F.J.S.). AUTHOR CONTRIBUTIONS A.K. and H.D.T. conceived of the theory. A.K. developed the algorithm and performed experiments. A.K., F.J.S. and H.D.T. designed the experiments and wrote the manuscript. COMPETING FINANCIAL INTERESTS The authors declare no competing financial interests. Reprints and permissions information is available online at com/reprints/index.html. 1. Liao, H.Y. & Frank, J. Structure 18, (2010). 2. Leschziner, A.E. & Nogales, E. Annu. Rev. Biophys. 36, (2007). 3. Scheres, S.H. & Chen, S. Nat. Methods 9, (2012). 4. Unser, M. et al. J. Struct. Biol. 149, (2005). 5. Sousa, D. & Grigorieff, N. J. Struct. Biol. 157, (2007). 6. Lander, G.C. et al. Nature 482, (2012). 7. Hashem, Y. et al. Nature 494, (2013). 8. Heymann, J.B. & Belnap, D.M. J. Struct. Biol. 157, 3 18 (2007). 9. Freeman, W.T. & Adelson, E.H. IEEE Trans. Pattern Anal. Mach. Intell. 13, (1991). 10. Derpanis, K.G. & Gryn, J.M. IEEE Int. Conf. Image Processing 3, (2005). 11. Lehmann, E.L. & Romano, J.P. Testing Statistical Hypotheses 3rd edn. (Springer, New York, 2005). 12. Benjamini, Y. & Yekutieli, D. Ann. Stat. 29, (2001). 13. Bai, X., Fernandez, I.S., McMullan, G. & Scheres, S.H. elife 2, e00461 (2013). 14. Yu, G. et al. PLoS ONE 8, e59817 (2013). 15. Bartesaghi, A., Lecumberry, F., Sapiro, G. & Subramaniam, S. Structure 20, (2012). 16. Davies, K.M., Anselmi, C., Wittig, I., Faraldo-Gómez, J.D. & Kühlbrandt, W. Proc. Natl. Acad. Sci. USA 109, (2012). 17. Pettersen, E.F. et al. J. Comput. Chem. 25, (2004). NATURE METHODS VOL.11 NO.1 JANUARY

4 npg 2014 Nature America, Inc. All rights reserved. ONLINE METHODS This section presents mathematical details of the theory and algorithm. The mathematical formulation is in three dimensions, but actual computations are performed on column vectors, where the elements in three dimensions are inserted in order into the vector. To highlight this difference, 3D variables are displayed in ordinary type (A) and their vectorized counterparts in bold type (A). Modeling the signal locally. We first describe how the density map can be approximated locally by any basis. Then we introduce the 3D sinusoid-like feature basis used in our algorithm. The 3D density map S is in a V V V voxel array. The voxels are indexed by discrete-valued coordinates x, y, z. We refer to a voxel in the array as v = (v x, v y, v z ) where v x, v y, v z are its coordinates. The vectorized density map of S is a V 3 1 column vector S. Suppose W v, is a spherically symmetric Gaussian function centered at voxel v with scaling parameter Wv, ( x, y, z) exp( [( x vx) ( y vy) ( z vz) ]) and that k v,, k = 1,, K are basis functions centered at v. We then have D D Wv, S Wv, v, which locally approximates the density map S by basis functions k v,. Here W D v, is a diagonal matrix with W v, along the principal diagonal; v, is a matrix whose columns are the basis functions k v,; is a column vector of the coefficients of the basis functions; and is zero-mean Gaussian noise with variance 2. Note that the weighting function W v, determines the spatial extent of the local model. To fit the local model to S, we minimize the weighted residual sum of squares (WRSS v, ) T D WRSSv, ( Sv, ) Wv, ( S v, ) 2 D Wv, ( S v, ) with respect to. The minimizing coefficient vector is denoted as. ˆ 3D sinusoid-like features. A natural basis for density maps is one containing rotated 3D sinusoids with wavelength Å. Unfortunately, describing all orientations in 3D requires an infinite number of basis functions. A computationally tractable alternative is to use steerable filters 9,10, which we refer to as steerable functions. Steerable functions are a finite set of functions with the property that every 3D rotation of any of the functions is produced by linear combinations of the functions 9 (Supplementary Note 1). The steerable functions we use are the second-order Hermite polynomial and its approximate quadrature, multiplied by a Gaussian function. We call this set H2. The elements of H2 match cosine and sine functions up to their second-order Taylor expansion terms. They can also be scaled such that their spectral peak occurs at any desired wavelength. The H2 steerable functions are constructed from a pair of functions Gv, ( x, y, z) PG v ( x, y, z) Wv x y z,, (,, ) 2 [( 4 ( x vx)) 2] Wv, (, x y, z ) Hv, ( x, y, z) P ( x, y, z) W x y z Hv v (,, ),, 3 [( ( x vx)) 2254(. ( x vx))] Wv, ( x, y, z) (1) (2) (3) (4) where G v, is the cosine-like function and H v, is the sine-like function. The scalar controls the peak frequency and the width of the Gaussian function. Setting = 2/ 2/5 gives a spectral peak for G v, and H v, at wavelength (Supplementary Note 2). The functions in equation (4) are each composed of the Hermite polynomial (P Gv, ) or its quadrature (P Hv, ) multiplied by W v,, the spherically symmetric Gaussian weighting function. Their spectral peak occurs on the x axis of the frequency domain because the functions are oriented along the spatial x axis. Rotating G v, and H v, so that their spectral peaks occur along the vertices and faces of an icosahedron respectively gives D steerable functions. The linear combination of these 16 functions produces all possible rotations of G v, and H v, in three dimensions 10, thereby covering a shell in the 3D Fourier domain (Supplementary Note 1). i We denote the rotated Hermite polynomials as P Gv, i = 1,, 6 j, and P H, j = 1,, 10. These polynomials and the constant function v, 1 are our local-sinusoid signal model. Because the weighting function W v, appears outside of the basis matrix v, in equation (2), the v, matrix need only contain the vectorized polynomials v, 1 P 1 Gv P, G 6 v P, H 1 v P, H 10 v, where the boldface denotes vectorization. Likelihood-ratio testing. Testing whether the data in the neighborhood of voxel v support the local-sinusoid model is equivalent to testing the two hypotheses 0 : The constant term 0 is unconstrained, but : All of 0, 1,, 1 6 are unconstrained (6) where the null hypothesis 0 states that the data do not support the local-sinusoid model (the coefficients of all local-sinusoid terms are 0). The alternate hypothesis 1 allows the coefficients to take on any finite value. The likelihood-ratio test 11 is a standard procedure for comparing such hypotheses. For this test, we calculate the s that maximize the likelihood (probability) under each hypothesis and then use the logarithm of the ratio of the maximized likelihoods. The s that maximize the likelihood are found by minimizing the WRSS from equation (3) under 0 and 1, respectively. Many common statistical tests such as the Pearson 2 test and the F-test are derived from the likelihood-ratio test (Supplementary Note 3). A simple calculation shows that the negative logarithm of the log-likelihood ratio, called the likelihood-ratio statistic (LRS), is given by 2 2 D alrs(;, ) ( ˆ, ) D S v 1 Wv S 1 Wv, ( S v, ˆ) 2 0 where T S ( ) 0 S 2 D D T D 1 T D 0 Wv, Wv, 11 ( Wv, 1) 1 Wv, D D T Wv, Wv, v, ( v, Wv D 1, v, ) v T, Wv D, (5) (7) NATURE METHODS doi: /nmeth.2727

5 npg 2014 Nature America, Inc. All rights reserved. The LRS is a difference of weighted residuals between the null model fit and the local-sinusoid model fit. It takes large values when the local-sinusoid model is a better fit than the null model. The likelihood-ratio test is applied by comparing the LRS to a number c, defined by T S ( 0 ) S Pr 2 c 1 p for some P value p, usually If LRS < c, then the data do not support the model and we accept the null hypothesis. Otherwise, we accept the hypothesis that the local-sinusoidal model fits the data. Calculating the threshold c requires the statistical distribution of the LRS. Unfortunately, because of the weighting function W v,, the LRS does not have a closed-form statistical distribution. However, S T ( 0 )S asymptotically tends to a weighted sum of 2 random variables r r 2, where r are the eigenvalues 18 of 0. Fast and accurate numerical methods are available to compute such distributions 19. The LRS computation requires the value of the noise variance 2. We estimate this variance accurately by taking nonoverlapping cubes of voxels from the region of the density map surrounding the particle. We use the following variance estimator recommended for local-signal modeling 18 ˆ ( ) 2 1 T B C b 0 Cb B b 1 trace( 0 ) (8) (9) where C b is a cube of voxels from the background and B is the number of nonoverlapping cubes that are available in the background. This estimator is robust to non-white noise as it only requires the noise spectrum to be relatively flat within the shell in 3D Fourier space that the local-sinusoid model, implicitly indicated by, is approximating. The noise variance may also be estimated from the difference map between split data set density maps. In this case, the estimator from equation (9) is adjusted to accept cubes of voxels from the region inside the particle. The noise statistics inside and outside the particle are nearly identical (Supplementary Note 4). Both noise-variance estimators are implemented in the accompanying software package (Supplementary Software; sourceforge.net/). Multiple-testing correction. The likelihood-ratio test chooses between two hypotheses at each voxel. Because this test has to be repeated for many voxels in S, some sort of false discovery rate (FDR) control is necessary. The tests in neighboring windows are not independent from each other; therefore, we use the Benjamini-Yekutieli FDR procedure 12 that accounts for dependencies between tests. Summary. ResMap works by applying a hypothesis test at every voxel. The null hypothesis is that the data in the neighborhood of a voxel do not support a local sinusoid. The alternative hypothesis is that the data describe a local sinusoid. These features are modeled by 3D steerable functions. The likelihood-ratio statistic is used to decide between the hypotheses at a given P value. Noise variance is estimated from the area surrounding the particle, and multiple-testing correction is applied to carry out the test at many voxels. 18. Loader, C. Local Regression and Likelihood (Springer, New York, 1999). 19. Farebrother, R.W. J. R. Stat. Soc. Ser. C Appl. Stat. 33, (1984). doi: /nmeth.2727 NATURE METHODS

6 Supplementary Material The Local Resolution of Cryo-EM Density Maps Alp Kucukelbir 1, Fred J. Sigworth 1,2, and Hemant D. Tagare 1,3 1 Department of Biomedical Engineering, Yale University, New Haven, United States 2 Department of Cellular and Molecular Physiology, Yale University, New Haven, United States 3 Department of Diagnostic Radiology, Yale University, New Haven, United States Correspondence should be addressed to HDT (hemant.tagare@yale.edu) Contents Supplementary Figure 1: Simulated data visualization 2 Supplementary Figure 2: Comparison of windowed FSC results 3 Supplementary Figure 3: Comparison of windowed FSC and ResMap-H2 results 4 Supplementary Figure 4: ResMap behavior for typical statistical confidence levels 5 Supplementary Figure 5: Screenshot of ResMap s graphical user interface 6 Supplementary Note 1: Steerable Functions 7 Supplementary Note 2: Scaling of the Basis Functions 8 Supplementary Note 3: Likelihood Ratio Testing Framework 9 Supplementary Note 4: Noise Statistics Inside and Outside the Particle 10 References 15 1

7 Supplementary Figure 1 a b Arbitrary Units white, variance = 0.5 1/f, variance = 0.5 white, variance = 2.0 1/f, variance = Synthetic Signal Signal plus white σ 2 = 0.5 noise 10 Signal plus 1/f σ 2 = 0.5 noise Voxels Supplementary Figure 1 Simulated data visualization. The simulated signal is 3 cos(2 (1/(1 4p R +0.15))) + 3 where R is a radius map with zeros outside radius 38. We added both white and 1/f noise to the data. The purpose of the latter is to demonstrate the robustness of the local resolution estimates to non-white noise, which is typical of cryo-em images 1. (a) Central slices through simulated data corrupted by white and 1/f non-white noise at two variance levels. (b) Plot through central line for the variance level 2 =0.5. 2

8 Supplementary Figure boxsize = 10 boxsize = 18 boxsize = Å Supplementary Figure 2 Comparison of windowed FSC results. S.cerevisiae ribosome 2 (EMD2275, split-dataset maps courtesy of the authors) volumes are rendered and colored with the output of blocres 3 using varying window sizes. Values are reported in A. Results are sensitive to the boxsize parameter and take 25 minutes (boxsize = 10) to four hours (boxsize = 25) to compute on a four-core Fedora 14 workstation. 3

9 Supplementary Figure 3 a b Windowed FSC ResMap-H A B 9.5 Density Map ResMap-H2 Windowed FSC 8.5 Supplementary Figure 3 Comparison of windowed FSC and ResMap-H2 results for the 26S Proteasome 4 (EMD-1992). Windowed FSC results were obtained using blocres 3 with boxsize = 18 (personal communication with Gabriel Lander). Values are reported in Å. (a) Volumes were median filtered in UCSF Chimera for visualization. Left: windowed FSC results. Right: ResMap-H2 results. The windowed FSC results estimate a lower resolution than ResMap-H2 in the area delineated in the black circle. This may be the result of windowed FSC using neighboring voxels in the solvent. (b) Central slice through unfiltered density map. Letters A and B highlight regions where the visual detail level appears to be comparable. ResMap-H2 corroborates this by assigning similar resolutions to both regions, while windowed FSC results appears to underestimate resolution in region A. 7.5 Å 4

10 Supplementary Figure 4 p-value = 0.01 p-value = 0.05 Supplementary Figure 4 ResMap behavior for typical statistical confidence levels. Results shown on the 80S ribosome 2 (EMD-2275). Colorbar values are reported in Å. As the p-value decreases, it is expected that fewer likelihood-ratio tests pass at smaller wavelengths. This is observed in the histograms above (note the di erent y-axis limits). Visualizations and histograms are generated from ResMap (Supplementary Software; 5

11 Supplementary Figure 5 Supplementary Figure 5 Screenshot of ResMap s graphical user interface on Mac OSX. The volume to be analyzed is the only required input. The optional inputs may be used to analyze a specific resolution range and/or reduce computation time by taking larger step sizes. The algorithm will automatically delineate the particle from its surrounding solvent, though the option to override this using a mask volume is also available. Further details are provided within the software package manual (Supplementary Software; 6

12 Supplementary Note 1: Steerable Functions Steerable functions are a finite set of functions with the property that linear combinations of these functions produce all possible rotations of themselves 5. For example, the G v, function defined in the Online Methods section is a steerable function. In three dimensions, six G i v, s oriented along the vertices of the icosahedron are su cient to produce any arbitrary rotation of G 6 v,. The icosahedrally oriented set of functions G i v, for i =1,, 6, are shown in Supplementary Figure 6a-f. Supplementary Figure 6g shows a G v, function oriented along the z-axis. This function is not in the original set G 1 v,,,g 6 v,, but can be obtained as, G v, = G 1 v, G 2 v, 0.25 G 3 v, G 4 v, 0.25 G 5 v, G 6 v,. Similar results follow for H v, with 10 rotated versions, oriented along the faces of the icosahedron. a b c g d e f Supplementary Figure 6: G i v, steerable functions. Red and blue indicate negative and positive parts, respectively. Subfigures (a) through (f) depict G i v, for i =1,, 6. Subfigure (g) shows an arbitrarily rotated G v, that can be obtained as a linear combination of the previous six functions. 7

13 Supplementary Note 2: Scaling of the Basis Functions For simplicity we consider the one-dimensional G v, function centered at v x = 0. Extending the result to 2D and 3D is trivial and the centering does not a ect the spectral peak. The one-dimensional G v, function is defined as G v, (x) = 4( x) 2 ( x) 2 2 exp, 2 and its Fourier transform is F[G v, ](!) = 1! exp 2 2!2. Taking the derivative of the Fourier transform and setting it to zero gives the maximum at r 5! = ± 2. To scale G v, such that its spectrum peaks at! =2 (1/ )weset = 2 r 2 5, where is the desired spatial resolution we seek in Angstroms. We can use the same factor for the quadrature function H v,, since it is a polynomial fit to the Hilbert transform of G v,, which in this case is simply a phase shift 5. 8

14 Supplementary Note 3: Likelihood Ratio Testing Framework The model described in the Online Methods section implies the following likelihood of observing the density map S at voxel v under a local-sinusoid model with scaling parameter, 1 q L(S ; v, ) / exp 2 2 Wv, (S D v, ) 2, where / means proportional to and k k 2 is the sum of voxels squared. The likelihood ratio is then simply the following ratio (S ; v, ) = max{l(s ; v, ) : restrained to H 0} max{l(s ; v, ) : restrained to H 1 } = L(S b 0 ; v, ) L(S b ; v, ) where b 0 and b are the coe cient vectors that minimize the weighted residual sum of squares under the local constant and local-sinusoid models, respectively. To ease computations, it is common to calculate the likelihood ratio statistic (LRS). Below we present the calculation of the LRS: LRS(S ; v, ) = 2ln (S ; v, ) = 2ln L(S b 0 ; v, ) +2ln L(S b ; v, ) = 1 q W 2 v, (S D 1 b 0) 2 q Wv, (S D b v, ) 2 = 1 (S 1 b 2 0) > Wv, (S D 1 b 0) (S b v, ) > Wv, (S D b v, ) = S> 0S S > S 2 = S> ( 0 )S 2 0 = W D v, W D v, 1 1 > W D v, > W D v, =W D v, W D v, v, > v, W D v, v, 1 > v, W D v, 9

15 Supplementary Note 4: Noise Statistics Inside and Outside the Particle The theory presented in the Online Methods section is independent of the method used to estimate the noise variance. We use two noise estimators: one which analyzes the region outside the particle of a single reconstruction, and another which analyzes the di erence between two split-dataset reconstructions in the region of the particle. Both approaches provide similar estimates of the noise variance and the noise statistics inside and outside the particle are nearly identical. The variance estimates agree to better than 20%, and their power spectra are indistinguishable for our purposes. Supplementary Figures 7-9 show the first order statistics of the noise inside and outside the particle region of three particles that span the 4 to 40Å range. Supplementary Figures show the corresponding noise power spectra Slice 120 of the di erence map Standard deviation inside: 8.29e-03 Gaussian with estimated sample mean and variance Histogram Slice 120 of the particle mask Standard deviation outisde: 6.99e-03 Gaussian with estimated sample mean and variance Histogram Supplementary Figure 7: Noise histograms for the S.cerevisiae Ribosome (EMD-2275). The di erence map is computed by subtracting one of the split reconstruction from the other. The particle mask delineates where the protein is located. The histograms of the voxels show that the noise distribution follows a Gaussian shape, both inside and outside the particle. Moreover, the standard deviations of the histograms are very similar. 10

16 0 Slice 120 of the di erence map Standard deviation inside: 6.31e-01 Gaussian with estimated sample mean and variance Histogram Slice 120 of the particle mask Standard deviation outside: 6.72e-01 Gaussian with estimated sample mean and variance Histogram Supplementary Figure 8: Noise histograms for the 26S Proteasome (EMD-1992). The di erence map is computed by subtracting one of the split reconstruction from the other. The particle mask delineates where the protein is located. The histograms of the voxels show that the noise distribution follows a Gaussian shape, both inside and outside the particle. Moreover, the standard deviations of the histograms are very similar. 11

17 Standard deviation inside: 4.30e+00 0 Slice 48 of the di erence map Gaussian with estimated sample mean and variance Histogram Standard deviation outside: 4.18e+00 0 Slice 48 of the particle mask Gaussian with estimated sample mean and variance Histogram Supplementary Figure 9: Noise histograms for the ATP Synthase Dimer (EMD-2161). The di erence map is computed by subtracting one of the split reconstruction from the other. The particle mask delineates where the protein is located. The histograms of the voxels show that the noise distribution follows a Gaussian shape, both inside and outside the particle. Moreover, the standard deviations of the histograms are very similar. 12

18 Radially Averaged Spectral Power ( f 2 ) S.cerevisiae Ribosome 10 4 Inside the protein Outside the protein (background) Frequency Supplementary Figure 10: Noise spectra for the S.cerevisiae Ribosome (EMD-2275). A box of size 40 voxels was placed inside and outside the protein mask and outside the protein. A 3D Fourier transform was computed for each box and then radially/spherically averaged to produce the plot above. 13

19 Radially Averaged Spectral Power ( f 2 ) S Proteasome Inside the protein Outside the protein (background) Frequency Supplementary Figure 11: Noise spectra for the 26S Proteasome (EMD-1992).A box of size 40 voxels was placed inside and outside the protein mask and outside the protein. A 3D Fourier transform was computed for each box and then radially/spherically averaged to produce the plot above. Radially Averaged Spectral Power ( f 2 ) ATP Synthase Dimer Inside the protein Outside the protein (background) Inside the membrane Frequency Supplementary Figure 12: Noise spectra the ATP Synthase Dimer (EMD-2161). A box of size 22 voxels was placed inside the protein (i.e. where there is the ATP Synthase dimer), inside the membrane (i.e. where there is membrane density), and outside both. A 3D Fourier transform was computed for each box and then radially/spherically averaged to produce the plot above. 14

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