Random Conical Tilt Reconstruction without Particle Picking in Cryo-electron Microscopy
11 Random Conical Tilt Reconstruction without ParticlePicking in Cryo-electron Microscopy
Ti-Yen Lan, a * Nicolas Boumal b and Amit Singer a,ca Program in Applied and Computational Mathematics, Princeton University,Princeton, NJ 08544, USA, b Institute of Mathematics, EPFL, CH-1015 Lausanne,Switzerland , and c Department of Mathematics, Princeton University, Princeton, NJ08544, USA. E-mail: [email protected]
Cryo-EM ; random conical tilt ; autocorrelation analysis ; structure reconstruction Abstract
We propose a method to reconstruct the 3-D molecular structure from micrographscollected at just one sample tilt angle in the random conical tilt scheme in cryo-electronmicroscopy. Our method uses autocorrelation analysis on the micrographs to estimatefeatures of the molecule which are invariant under certain nuisance parameters suchas the positions of molecular projections in the micrographs. This enables us to recon-struct the molecular structure directly from micrographs, completely circumventingthe need for particle picking. We demonstrate reconstructions with simulated dataand investigate the effect of the missing-cone region. These results show promise toreduce the size limit for single particle reconstruction in cryo-electron microscopy.
PREPRINT:
A Journal of the International Union of Crystallography a r X i v : . [ phy s i c s . d a t a - a n ] J a n
1. Introduction
Random conical tilt (RCT) (Radermacher et al. , 1987; Radermacher, 1988) is animportant technique in single-particle cryo-electron microscopy (cryo-EM) to gener-ate a de novo ◦ , which makes a considerablefraction of information about the molecular structure inaccessible to the technique:this is the so-called “missing-cone” problem. Another limitation is the need to collectdata from the same field of view at two different sample tilt angles. For each of thetwo tilt angles, the signal-to-noise ratio (SNR) must be high enough so that it ispossible to reliably locate the molecular projections (that is, pick particles) in thenoisy micrographs. This essentially doubles the required electron dose on the sample.Meanwhile, the molecule must be large enough so that the irreversible structuraldamage caused by incident electrons is limited enough to allow for particle picking.Indeed, this has led to the common belief that small biological molecules are out ofthe reach for cryo-EM (Henderson, 1995). IUCr macros version 2.1.11: 2020/04/29
In this study, we develop an approach to reconstruct the 3-D molecular structurefrom data collected at just one large sample tilt angle, as depicted in Figure 2(a). Moreimportantly, our approach circumvents the need for particle picking to reconstruct themolecular structure directly from the micrographs. The main idea is to first estimatefeatures of the molecule that are invariant to the 2-D positions of molecular projectionsin the micrographs. The estimation is done through a variant of Kam’s autocorrelationanalysis (Kam, 1980). We subsequently determine the molecular structure by fittingthe estimated invariants through an optimization problem. We address the problemof missing information by adding a regularizer in the optimization. Assuming whitenoise, this approach can in principle handle cases of arbitrarily low SNR as longas sufficiently many micrographs are used to estimate the invariants. Figure 2(b)shows one such noisy micrograph where particle picking becomes challenging. Thisobservation notably suggests that the feasibility of particle picking does not limit thesmallest usable molecule size in single-particle cryo-EM.Kam’s autocorrelation analysis was also applied for analyzing X-ray single particleimaging data (Kam, 1977; Saldin et al. , 2010; Donatelli et al. , 2015; von Ardenne et al. ,2018). In particular, Saldin et al. (2010) considered the problem of reconstructingthe top-down projection of molecules randomly oriented about a single axis, which issimilar to the case of no tilt in RCT. Subsequently, Elser (2011) designed an algorithmto reconstruct the 3-D structure of such partially oriented molecules from a tilt series.Kam’s method was recently demonstrated with actual data collected from randomlyoriented virus particles (Kurta et al. , 2017).This work belongs to a methodical program to develop algorithms to reconstructmolecular structures without the need for particle picking, which was first proposedin Bendory et al. (2018). The development started with the studies of a simplified 1-Dmodel, where multiple copies of a target signal occur at unknown locations in a noisy
IUCr macros version 2.1.11: 2020/04/29 long measurement (Bendory et al. , 2018; Bendory et al. , 2019; Lan et al. , 2020). Theextension to the 2-D case, where multiple copies of a target image are randomly rotatedand translated in a large noisy measurement image, was later studied in Marshall et al. (2020) and Bendory et al. (2021). These results can be used to reconstruct thetop-down molecular projection from the micrographs collected at no tilt in the RCTscheme.We organize the rest of the paper as follows. We describe the data simulation pro-cedure in Sections 2.1 to 2.3. The details of our approach are discussed in Sections 2.4and 2.5. In Section 3, we study the effect of the missing-cone region on the quality ofreconstruction and present the reconstructions of two molecular structures from sim-ulated noisy micrographs. The computational details are described in the appendix.
2. Methods
In the cryo-EM imaging process, the incident electrons are scattered by the 3-DCoulomb potential of the sample f s ( x, y, z ). We define the coordinate system for datacollection S by the orthogonal x - and y -axes along the edges of the detector and thenormally incident electron beam, as the z -axis. Under the weak-phase object approx-imation, the micrograph recorded by an m × m pixelated detector can be modeledas M ( x i , y i ) = ( h ∗ P f s ) ( x i /ξ, y i /ξ ) + ε ( x i , y i ) , (1)where i ∈ { , . . . , m } , ( x i , y i ) ∈ {−(cid:98) m/ (cid:99) , . . . , (cid:100) m/ − (cid:101)} is the 2-D coordinate ofthe i th pixel, and ξ denotes the pixel sampling rate. The operator P generates thetomographic projection of f s along the z -axis by( P f s )( x, y ) = (cid:90) ∞−∞ f s ( x, y, z ) dz. (2) IUCr macros version 2.1.11: 2020/04/29
The 2-D function h ( x, y ) represents the point spread function of the imaging system,and the operator ∗ denotes the 2-D convolution, where( h ∗ g )( x, y ) = (cid:90) ∞−∞ (cid:90) ∞−∞ h ( u, v ) g ( x − u, y − v ) du dv (3)for any 2-D function g ( x, y ). Finally, the measurement noise is modeled by the additiverandom variable ε ( x i , y i ).In this work, we consider the simplified scenario where we ignore the effect of thepoint spread function by making the idealistic assumption that it is a 2-D Dirac deltafunction, namely, ( h ∗ g )( x, y ) = g ( x, y ). Moreover, we assume that the random noise ε is drawn from an i.i.d. Gaussian distribution with zero mean and variance σ . Thearising challenges beyond these assumptions will be discussed in Section 4. The sample used in RCT consists of multiple copies of partially oriented molecules.Specifically, the molecules adsorb to a 2-D substrate such that a particular axis withinthe molecules aligns with the substrate normal. The molecular orientations are limitedto rotations about the particular body axis by angles uniformly drawn from [0 , π ).Let S (cid:48)(cid:48) be the body frame of one particular molecule, where the z (cid:48)(cid:48) -axis coincides withits body rotation axis. We further define another reference frame S (cid:48) fixed on the 2-Dsubstrate such that the x (cid:48) -axis coincides with the tilt axis of the substrate and the z (cid:48) -axis aligns with the substrate normal. In the following, we also assume that the x -axisof the lab frame is parallel to the x (cid:48) -axis. After specifying these reference frames, wedefine the substrate tilt angle θ as the angle between the z - and z (cid:48) -axes. The rotationangle α of the particular molecule with respect to its body rotation axis is defined asthe angle between the x (cid:48) - and x (cid:48)(cid:48) -axes. The relationships between the reference framesare shown in Figure 3.Let f ( x (cid:48)(cid:48) , y (cid:48)(cid:48) , z (cid:48)(cid:48) ) be the 3-D Coulomb potential of the particular molecule in its own IUCr macros version 2.1.11: 2020/04/29 body frame S (cid:48)(cid:48) . Hereafter, we refer to f as the structure of the molecule. From thegeometries shown in Figure 3, the coordinate transformation between S and S (cid:48)(cid:48) isgiven by r = cos α − sin α θ sin α cos θ cos α − sin θ sin θ sin α sin θ cos α cos θ r (cid:48)(cid:48) + t = R θα r (cid:48)(cid:48) + t , (4)where r = [ x, y, z ] T , r (cid:48)(cid:48) = [ x (cid:48)(cid:48) , y (cid:48)(cid:48) , z (cid:48)(cid:48) ] T , R θα is the rotation matrix that aligns the axesof S with the axes of S (cid:48)(cid:48) , and t = [ t x , t y , t z ] T is the vector pointing from the originof S to the origin of S (cid:48)(cid:48) . We can therefore express the molecular structure in the labframe S by f (( R θα ) T ( r − t )), and its tomographic projection along the z -axis is givenby I θα ( x − t x , y − t y ), where I θα ( x, y ) = (cid:90) ∞−∞ f (( R θα ) T r ) dz. (5)Taking the 2-D Fourier transform on both sides of (5), with the Fourier slice theo-rem, we get ˆ I θα ( k x , k y ) = ˆ f (( R θα ) T [ k x , k y , T ) , (6)where ˆ f ( k x , k y , k z ) denotes the 3-D Fourier transform of f ( x, y, z ). As a result, aprojection image contains the same information as the central slice of the 3-D Fouriertransform that is perpendicular to the direction of projection. Since the molecularorientations are limited to in-plane rotations on the 2-D substrate, which is itselftilted by an angle θ , the corresponding Fourier slices fill the whole 3-D Fourier spaceexcept for the region within a double cone, whose axis coincides with the body rotationaxis of the molecules. The double cone has an opening angle 2 θ and the region withinthe missing cone represents the inaccessible information of the molecular structure inthe setting of RCT. IUCr macros version 2.1.11: 2020/04/29
Before discussing our model for simulating micrographs, we first consider the compu-tation of the molecular projection images. Let F be the discretization of the molecularstructure f that is defined on a cubic grid ( x, y, z ) ∈ {− r, . . . , r } by F ( x, y, z ) = f ( x/ξ, y/ξ, z/ξ ) . (7)The integer r represents the radius of a spherical support such that F ( x, y, z ) is negli-gible for ( x + y + z ) / ≥ r . In addition, we define the discretization of the molecularprojection I θα by I θα ( x, y ) = I θα ( x/ξ, y/ξ ) , (8)where ( x, y ) ∈ {− r, . . . , r } , and it immediately follows that I θα has a circular sup-port of radius r . From the Fourier slice theorem, we can compute the discrete Fouriertransform (DFT) of I θα from the DFT of F byˆ I θα ( k x , k y ) = ˆ F (( R θα ) T [ k x , k y , T ) , (9)where ( k x , k y ) ∈ {− r, . . . , r } . To reduce the interpolation error, we use the FIN-UFFT package (Barnett et al. , 2019; Barnett, 2020) to evaluate ˆ F on the non-uniformgrid points. Finally, we obtain the molecular projections I θα by the inverse DFT of ˆ I θα .We simulate the micrographs measured in a RCT experiment at the substrate tiltangle θ by M ( x i , y i ) = n p (cid:88) j =1 I θα j ( x i − t x j , y i − t y j ) + ε ( x i , y i ) , (10)where ( x i , y i ) ∈ {−(cid:98) m/ (cid:99) , . . . , (cid:100) m/ − (cid:101)} , n p is the number of molecular projectionsin the micrograph, α j is the in-plane rotation of the j th molecule that is uniformlydrawn from [0 , π ), ( t x j , t y j ) ∈ {−(cid:98) m/ (cid:99) + r, . . . , (cid:100) m/ − (cid:101) − r } is the center of thetomographic projection of the j th molecule, and ε ( x i , y i ) is i.i.d. Gaussian noise withzero mean and variance σ . For a reason that will be clear in Section 2.4, we further IUCr macros version 2.1.11: 2020/04/29 assume that (( t x j − t x k ) + ( t y j − t y k ) ) / > r for j (cid:54) = k (11)such that the molecular projections are well separated in the micrographs. Figure 4shows a sample micrograph with SNR = 1. We define SNR as the ratio of the meansquared pixel values of molecular projections to the noise variance. Specifically,SNR = 12 π (cid:90) π dα πr (cid:88) x i + y i 3. Results In this section, we explore the effect of the missing-cone region on the quality ofreconstruction by considering micrographs measured at different substrate tilt angles θ = 60 ◦ , ◦ and 10 ◦ . The molecule used in our simulation is Bovine Pancreatic TrypsinInhibitor (BPTI), which has size of 35 ˚A and weight of 6 . F from the PDB entry 1QLQ (Czapinska et al. , 2000) using the UCSF Chimera software (Pettersen et al. , 2004) at a resolutionof 5 ˚A. The resulting contrast has a spherical support of radius r = 15 voxels, and IUCr macros version 2.1.11: 2020/04/29 F , wesimulate the micrographs as described in Section 2.3. To obtain the baseline resultson the effect of the missing cone region, we consider the idealistic scenario that thein-plane rotation of the j th molecule is given by α j = 2 πj/n p , j ∈ { , . . . , n p } , andthe noise variance σ = 0. By setting the micrograph length m = 4096 pixels and thenumber of molecules n p = 400, we only simulate one micrograph at each given valueof the substrate tilt angle.From the simulated micrographs, we compute the rotationally averaged autocorre-lations of molecular projections and the values of ˜ s F ( k , k ) and τ | q | . Figure 5 showsthe comparison of the mean intensities (cid:104)| ˆ F ( q ) | (cid:105) and the scale parameters τ | q | and | q | − /β for θ = 60 ◦ . We first see that τ | q | provides a good estimate for (cid:104)| ˆ F ( q ) | (cid:105) up tothe Nyquist frequency. On the other hand, the scale parameter | q | − /β is substantiallygreater than (cid:104)| ˆ F ( q ) | (cid:105) outside the Nyquist frequency, which may inevitably preservesome high-resolution noise in the reconstruction.We use the BFGS algorithm in the tensorflow software library (Abadi et al. , 2015) tominimize the cost function (21) over a set of regularization parameters λ = 10 − , − , . . . , .In order to speed up the optimization, we initialize u from the discretization of a 3-D Gaussian profile, whose variance is set as r . We further rescale u such that thesum of the corresponding F is equal to the estimate for (4 r + 1) (cid:104) a I θα (cid:105) α from data.From the converged solutions, we choose the optimal value of λ using the L-curvemethod (Hansen, 1992). Our reconstructed structures are the estimates for F withthese optimal values of λ .Figure 6(a) shows the comparison of the reconstructed BPTI structures with theground truth used to simulate the micrographs. As expected, the visual quality ofthe reconstructions degrades when the sample tilt angle θ decreases, which resultsin a larger missing-data region. To assess the reconstructions in more detail, we plot IUCr macros version 2.1.11: 2020/04/29 θ =35 ◦ correlates to the ground truth worse than the one at θ = 60 ◦ , both of them havethe same resolution as the ground truth (5 ˚A) according to the FSC = 0.5 criterion.Using the same criterion, the resolution of the reconstruction at θ = 10 ◦ is 8.3 ˚A. After having the baseline results for reconstructions from noiseless micrographs, weturn to test our approach on noisy micrographs. At the sample tilt angle θ = 60 ◦ ,we simulate 500 micrographs of size m = 4096 using the same discrete contrast F forBPTI. We adjust the noise level such that the micrographs have SNR = 1. By max-imizing the density of molecular projections while still preserving the requirement ofwell separation (11), the resulting micrographs contain 1 . × molecular projectionsin total.From the noisy micrographs, we compute the estimates for the rotationally aver-aged autocorrelations of molecular projections. Figure 7(a) shows the reconstructionfrom these estimates along with the ground truth. The negative effect of noise on thequality of the reconstruction can best be seen by comparing this reconstruction withits counterpart in Figure 6(a). As plotted in Figure 7(b), we determine the resolutionof this reconstructed structure to be 6.5 ˚A using the FSC = 0.5 criterion.To demonstrate that our approach applies to other biological molecules, we testour approach on another dataset simulated from the myoglobin molecule, which hassize of 40 ˚A and weight of 17.8 kDa. We generate the discrete molecular structure F for myoglobin from the PDB entry 1MBN (Watson, 1969) using the UCSF Chimerasoftware at a resolution of 5 ˚A. The resulting contrast has a spherical support ofradius r = 16 voxels, and is further zero-padded to be a cubic grid of size 65. At the IUCr macros version 2.1.11: 2020/04/29 θ = 60 ◦ , we generate 500 micrographs of size m = 4096 from F . Thenumber of molecular projections in these micrographs totals 1 . × , and we also setSNR = 1 for the micrographs. The reconstructed myoglobin structure from the noisymicrographs is shown in Figure 8(a) along with the ground truth. We can see that ourreconstruction recovers most of the main features of the ground truth. We plot theFSC of our reconstruction with the ground truth in Figure 8(b), and we determinethe resolution of the reconstruction to be 7.0 ˚A according to the FSC = 0.5 criterion. 4. Discussion In this paper, we present a method to reconstruct the 3-D molecular structure fromdata collected at just one sample tilt angle in RCT. Our method reduces data toquantities that are invariant to the 2-D positions of molecular projections in themicrographs, which removes the need for particle picking when analyzing data. Inorder to address the missing data in the double-cone region of the molecule’s Fouriertransform, we design a regularized optimization problem to reconstruct the molecularstructure by fitting the autocorrelations estimated from micrographs. Our numericalstudies illustrate the effect of the missing-cone region on the quality of reconstruction.In addition, we demonstrate structure reconstruction from the autocorrelations com-puted from noisy micrographs. Since the accuracy of the autocorrelation estimates canbe improved by averaging many more micrographs, our results show promise of apply-ing autocorrelation analysis to reconstruct the structures of small biological moleculesin the setting of RCT.A few issues still stand in the way of applying our approach to real RCT data. InSection 2.1, we make the assumption that the point spread function is a 2-D Diracdelta function to ignore its effect. In reality, however, we may have to consider avarying point spread function with respect to the locations on the detector because IUCr macros version 2.1.11: 2020/04/29 σ .Furthermore, structure heterogeneity of the target molecule will be another test forour approach.Additionally, we assume that the molecular projections are well separated in themicrographs. This assumption enables us to directly relate the micrograph autocor-relations to the autocorrelations of molecular projections. However, it is preferable inpractice to have the molecular projections densely packed in micrographs to maximizethe available structural information within limited data collection time. We expect toremove this assumption by considering the cross correlations between neighboringmolecular projections. A similar idea was recently demonstrated in Lan et al. (2020)for the simplified 1-D model.Another practical concern is the amount of required data. As a proof of concept,we reconstruct the molecular structures from simulated micrographs with SNR = 1.For small biological molecules that challenges particle picking, we expect the SNR ofthe micrographs to be much lower. Since our approach uses autocorrelations up to the3 rd order, the sample complexity would scale as SNR − . This means that we will need10 times more molecules to estimate the autocorrelations with similar accuracy whenthe SNR drops from 1 to 0.1. Although densely packing the molecular projections inmicrographs helps improve the SNR of the estimated autocorrelations, it would bebeneficial to investigate methods to denoise the autocorrelations.In the long run, we would like to extend the approach described here to real cryo-EM data to reconstruct high-resolution structures directly from micrographs, withoutbeing restricted to molecules which have a preferred orientation on their substrate. IUCr macros version 2.1.11: 2020/04/29 5. Acknowledgements TYL and AS were supported in part by AFOSR Awards FA9550-17-1-0291 andFA9550-20-1-0266, the Simons Foundation Math+X Investigator Award, the MooreFoundation Data-Driven Discovery Investigator Award, NSF BIGDATA Award IIS-1837992, NSF Award DMS-2009753, and NIH/NIGMS Award R01GM136780-01. Wewould like to thank Tamir Bendory, Joe Kileel, Eitan Levin and Nicholas Marshall forproductive discussions. Appendix AComputational Details The data simulation and structure reconstruction were performed on an NvidiaTesla P100 GPU, which has 16 GB RAM. The computation of the micrograph auto-correlations for relevant step sizes took 1 . × seconds on average for a 4096 × λ to converge. Therefore, if oneknows the correct λ for some setting, it may be advantageous to use the same λ in a similar case. The code is publicly available at https://github.com/tl578/RCT-without-detection. References Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G. S., Davis,A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M.,Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Man´e, D., Monga, R.,Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I.,Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Vi´egas, F., Vinyals, O., Warden,P., Wattenberg, M., Wicke, M., Yu, Y. & Zheng, X., (2015). Software available fromtensorflow.org.von Ardenne, B., Mechelke, M. & Grubm¨uller, H. (2018). Nature communications , , 2375.Barnett, A. H. (2020). arXiv preprint arXiv:2001.09405 .Barnett, A. H., Magland, J. & af Klinteberg, L. (2019). SIAM Journal on Scientific Computing , (5), C479–C504. IUCr macros version 2.1.11: 2020/04/29 Bendory, T., Boumal, N., Leeb, W., Levin, E. & Singer, A. (2018). arXiv preprintarXiv:1810.00226 .Bendory, T., Boumal, N., Leeb, W., Levin, E. & Singer, A. (2019). Inverse Problems , (10),104003.Bendory, T., Boumal, N., Ma, C., Zhao, Z. & Singer, A. (2017). IEEE Transactions on SignalProcessing , (4), 1037–1050.Bendory, T., Lan, T.-Y., Marshall, N., Rukshin, I. & Singer, A. (2021). To appear on arXiv .Czapinska, H., Otlewski, J., Krzywda, S., Sheldrick, G. & Jask´olski, J. (2000). Journal ofMolecular Biology , (5), 1237–1249.Donatelli, J. J., Zwart, P. H. & Sethian, J. A. (2015). Proceedings of the National Academy ofSciences , (33), 10286–10291.Elser, V. (2011). New Journal of Physics , , 123014.French, S. & Wilson, K. (1978). Acta Crystallographica Section A , (4), 517–525.Hansen, P. C. (1992). SIAM Review , (4), 561–580.Harauz, G. & Van Heel, M. (1986). Optik (Stuttgart) , (4), 146–156.Henderson, R. (1995). Quarterly Reviews of Biophysics , (2), 171–193.Kam, Z. (1977). Macromolecules , (5), 927–934.Kam, Z. (1980). Journal of Theoretical Biology , (1), 15–39.Kurta, R. P., Donatelli, J. J., Yoon, C. H., Berntsen, P., Bielecki, J., Daurer, B. J., DeMirci,H., Fromme, P., Hantke, M. F., Maia, F. R. N. C., Munke, A., Nettelblad, C., Pande, K.,Reddy, H. K. N., Sellberg, J. A., Sierra, R. G., Svenda, M., van der Schot, G., Vartanyants,I. A., Williams, G. J., Xavier, P. L., Aquila, A., Zwart, P. H. & Mancuso, A. P. (2017). Physical Review Letter , , 158102.Lan, T.-Y., Bendory, T., Boumal, N. & Singer, A. (2020). IEEE Transactions on Signal Pro-cessing , , 1589–1601.Marshall, N., Lan, T.-Y., Bendory, T. & Singer, A. (2020). In ICASSP 2020 - 2020 IEEE Inter-national Conference on Acoustics, Speech and Signal Processing (ICASSP) , pp. 5780–5784.Pettersen, E. F., Goddard, T. D., Huang, C. C., Couch, G. S., Greenblatt, D. M., Meng, E. C.& Ferrin, T. E. (2004). Journal of Computational Chemistry , (13), 1605–1612.Radermacher, M. (1988). Journal of Electron Microscopy Technique , (4), 359–394.Radermacher, M., Wagenknecht, T., Verschoor, A. & Frank, J. (1987). Journal of Microscopy , (2), 113–136.Sadler, B. M. & Giannakis, G. B. (1992). Journal of the Optical Society of America A , (1),57–69.Saldin, D. K., Poon, H. C., Shneerson, V. L., Howells, M., Chapman, H. N., Kirian, R. A.,Schmidt, K. E. & Spence, J. C. H. (2010). Physical Review B , , 174105.Scheres, S. (2012). Journal of Molecular Biology , (2), 406–418.Tukey, J. (1953). Reprinted in The Collected Works of John W. Tukey , , 165–184.Watson, H. (1969). Progress in Stereochemistry , , 299–333.Wilson, A. (1949). Acta Crystallographica , (5), 318–321. IUCr macros version 2.1.11: 2020/04/29 (a)(b) (c) Fig. 1. The micrographs of the same field of view collected at (a) one large sampletilt angle and (b) no tilt. (c) The Fourier transforms of the molecular projectionsrecorded in (a), which are assembled in Fourier space with respect to their corre-sponding orientations according to the Fourier slice theorem discussed in Section 2.2. detectorincident electrons substrate (a) (b) Fig. 2. (a) The data collection scheme of RCT with just one sample tilt angle. (b) Amicrograph that is so noisy that picking particles is challenging. IUCr macros version 2.1.11: 2020/04/29 detectorincident electrons substrate substrate Fig. 3. The relationships between the lab frame S , the frame fixed on the 2-D substrate S (cid:48) and the body frame of one particular molecule S (cid:48)(cid:48) .Fig. 4. A sample micrograph with SNR = 1. IUCr macros version 2.1.11: 2020/04/29 |q| −2 −4 −6 −8 ⟨| ̂F⟨q̂| ⟩τ |q| −2 ⟩β Fig. 5. The comparison of the mean intensities (cid:104)| ˆ F ( q ) | (cid:105) and the scale parameters τ | q | and | q | − /β for the BPTI molecule at the substate tilt angle θ = 60 ◦ . (b)(a) Fig. 6. (a) The reconstructed BPTI structures from noiseless micrographs at differentsample tilt angles: θ = 60 ◦ (yellow), θ = 35 ◦ (cyan) and θ = 10 ◦ (purple). Thegrey one is the ground truth used to simulate the micrographs. (b) The FSC of thereconstructed structures with the ground truth. IUCr macros version 2.1.11: 2020/04/29 (a) (b) Fig. 7. (a) The reconstructed BPTI structure (yellow) from noisy micrographs withSNR = 1 at the sample tilt angle θ = 60 ◦ . The ground truth is rendered in grey.(b) The FSC of the reconstructed structure with the ground truth. (a) (b) Fig. 8. (a) The reconstructed myoglobin structure (yellow) from noisy micrographswith SNR = 1 at the sample tilt angle θ = 60 ◦ . The ground truth is rendered ingrey. (b) The FSC of the reconstructed structure with the ground truth. Synopsis