# A local-optimization refinement algorithm in single particle analysis for macromolecular complex with multiple rigid modules

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## ABSTRACT

Single particle analysis, which can be regarded as an average of signals from thousands or even millions of particle projections, is an efficient method to study the three-dimensional structures of biological macromolecules. An intrinsic assumption in single particle analysis is that all the analyzed particles must have identical composition and conformation. Thus specimen heterogeneity in either composition or conformation has raised great challenges for high-resolution analysis. For particles with multiple conformations, inaccurate alignments and orientation parameters will yield an averaged map with diminished resolution and smeared density. Besides extensive classification approaches, here based on the assumption that the macromolecular complex is made up of multiple rigid modules whose relative orientations and positions are in slight fluctuation around equilibriums, we propose a new method called as local optimization refinement to address this conformational heterogeneity for an improved resolution. The key idea is to optimize the orientation and shift parameters of each rigid module and then reconstruct their three-dimensional structures individually. Using simulated data of 80S/70S ribosomes with relative fluctuations between the large (60S/50S) and the small (40S/30S) subunits, we tested this algorithm and found that the resolutions of both subunits are significantly improved. Our method provides a proof-of-principle solution for high-resolution single particle analysis of macromolecular complexes with dynamic conformations.

## KEYWORDS

cryo-electron microscopy single particle analysis conformational heterogeneity rigid module local optimization refinement## INTRODUCTION

Single-particle analysis (SPA) of electron cryo-microscopy (cryo-EM) has become an efficient method to reveal structural information of macromolecular complexes. In theory, it is possible to solve a 3 Å resolution structure when thousands of single particle images are averaged (Henderson, 1995). Nowadays, with improved detectors and image processing techniques, the prediction comes true with not only large, highly symmetrical viruses (Wang et al., 2014; Zhang et al., 2010) or asymmetrical ribosome (Fischer et al., 2015), but also small membrane proteins (Liao et al., 2013) (TRPV1), whose structures were resolved at near-atomic resolutions.

Besides the rapid progress in pushing resolution, however, intrinsic sample heterogeneity in composition or conformation is becoming a threshold stopping us obtaining higher-resolution structure. The current solution to deal with both heterogeneity problems is to divide the data set into different classes with each class corresponding to one homogenous composition/conformation (Leschziner and Nogales, 2007). A couple of classification methods have been developed, such as the normal mode analysis (NMA) method that uses simulated models as references for multi-reference supervised classification (Brink et al., 2004; Jin et al., 2014), 3D multivariate statistical analysis (MSA) that projects a 3D mask of the area with the most variance to a series of 2D images in the same orientation and performs classification focusing on the masked highly varied regions (Penczek et al., 2006a; Penczek et al., 2006b; Zhang et al., 2008), and a Bayesian based 3D classification method (Scheres, 2012b). These classification methods can work well upon the assumption that the heterogeneous sample only contains a finite number of compositions/conformations.

In practice, for many macromolecular complexes that exhibit heterogeneity with dynamic conformations and thereby an infinite conformational states, the above classification methods may not produce a good result. A typical scenario is that the macromolecular complex comprises a number of stable modules (such as domains, subunits, or sub-complexes) that can be treated as rigid bodies, the overall flexibility of the complex is due to the dynamic slight fluctuations of the relative orientations and positions between rigid modules. In these cases, the conventional SPA approach will yield a 3D reconstruction with smeared densities (i.e. a decreased resolution) because SPA approach assumes that all the particles within a class have an identical structure while this is not correct for such flexible complexes and the assigned parameters could be inaccurate for all of the modules. In addition, conventional classification approaches are also not able to classify the infinite continuous conformations of the complex into a finite number of discrete states with enough homogeneity within a class. Examples for these kinds of macromolecular complexes include the ribosome containing two subunits with relative motion (Bai et al., 2013) and the splicesome with substantial flexibility among subunits that is still poorly resolved by cryo-EM SPA approach (Azubel et al., 2004). Besides developing new sample preparation and freezing procedures to reduce the flexibility and heterogeneity, there is a great need of new image-processing algorithms to adequately treat the dynamic conformation problem.

Here we report a new image-processing algorithm that can yield a better resolution by resolving the accurate orientation and shift parameters of each individual structural module respectively. Since the orientation and shift parameters of each module are searched within a local range and only the local area of the particle image is counted, we call this method as the local optimization refinement algorithm (LO-refinement). In a test case, we used the ribosome (80S or 70S) that has two rigid modules (60S/50S or 40S/30S) with the fluctuating relative orientations and positions to prove the concept of this method.

## THEORY AND ALGORITHM

The LO-refinement is based on the assumption that the imaged macromolecular complex comprises a number of rigid modules with slightly varying relative orientations and positions between different modules. The relative orientation and position between any two modules can fluctuate due to module rotation and shift. The goal of the LO-refinement is to resolve a higher resolution structure of each rigid module.

### The rationality of using cross-correlation coefficient

*J*dimension, \( {<}f_{i}\!\!{>}= 1/J\mathop \sum \limits_{j = 1}^{J} \, f_{i} \left( {r_{j} } \right) \); i = 1,2. This formula is composed of a numerator representing the similarity between two images and a denominator for normalization, resulting \( - 1 \le \rho_{12} \le 1 \).

Assuming that a molecular complex comprises two modules A and B, the projection of the whole complex \( f\left( {{r}_{{j}} } \right) \) is the summation of the projections of two modules \( f_{A} \left( {{r}_{{j}} } \right) \) and \( f_{B} \left( {{r}_{{j}} } \right) \).

*ρ*

_{12}is approximately equivalent to maximizing term (5), which is our target. That is to say, although the information of an individual module can not be explicitly separated from that of other modules within one particle projection, maximizing the cross-correlation coefficient between the experimental and simulated whole projections to search the parameters of the target module can, with most probability, yield more precise parameters for the orientation and position of target module in the experimental projection.

### Search range of the target module orientation and position

For the orientation of a rigid module, there are five parameters to be optimized, three orientation angles φ, θ, ψ and two in-plane translations x, y. Here, φ, θ define the projection direction in 3D space, ψ is the in-plane rotation angle and x, y determine the position of the projection in the plane. In an experimental projection, for molecular complexes with limited module motion, the optimal parameters of the target module should be near the preliminarily determined global ones from conventional SPA procedures, resulting in a constrained space for optimizing those five parameters (φ, θ, ψ, x, y) of the target module.

*e*

_{ c }reflecting the final orientation of the camera. The projection direction (φ, θ) can be represented by the unit vector

*e*

_{ r }with the following formula,

_{0}, θ

_{0}). OC is a projection direction (φ

_{i}, θ

_{i}) near OB. For the pre-determined preliminary projection direction

*e*

_{ r0 }and the projection direction

*e*

_{ ri }within the search range, the span angle α by these two directions can be determined with the formula,

_{i}, θ

_{i}) can be confined locally with a pre-defined maximum span angle α

_{0}as follows,

Here, D is a set of evenly distributed projection directions and can be generated by SPIDER command VO NEA.

_{i}should be recomputed at every new projection direction (φ

_{i}, θ

_{i}). As shown in Fig. 2B, the way to move camera from B to C with zero in-plane rotation of camera in the coordinate system of camera itself is to keep its angle with the arc BC unchanged during movement. As a result, the apparent in-plane rotation angle ψ

_{i}in our defined coordinate system is changed and can be determined with the following relation,

By considering every possible situation on a sphere, we obtain:

_{i}− φ

_{0}| ≤π

_{i}− φ

_{0}| >π

_{i}= φ

_{0}+ ψ

_{0}− φ

_{i.}Finally, the search range of ψ for a particular (φ

_{i}, θ

_{i}) is

*d*

_{1}is the search step size and

*t*

_{1}is the number of search steps.

*d*

_{2}is the search step size and

*t*

_{2}is the number of search steps.

Over all, we optimize the orientation of the target module by searching all five parameters (φ, θ, ψ, x, y) in a confined range defined by (11), (15) and (16).

### Procedure of local orientation optimization

With the preliminary parameters for each particle, all of possible parameters defined by (11), (15) and (16) are considered for the target module. The target module is transformed with all possible orientation parameters (φ, θ, ψ, x, y) and then combined together with the rest part of the model to generate a series of simulated projections, which are compared with the experimental image by using CCC defined by (1). The optimized orientation parameters of the target module for each particle are determined according to the highest CCC. All of the operations (rotations and shift/translations) will be combined to minimize interpolation errors. With the optimized parameters for target module, a new and improved 3D reconstruction can be generated using the conventional 3D reconstruction method (e.g. WBP, SIRT or Fourier method).

Here we develop two procedures to search the optimized parameters of the target module. The first one is to perform an exhaustive search of angle parameters (φ, θ, ψ) and shift parameters (x, y) simultaneously, which requires to generate simulating images with every combination of angle and shift parameters to make comparison (Fig. 3A). The second procedure is to separate shift parameter searching from angle parameter searching (Fig. 3B). We randomly choose n sets of angle parameters within the constriction defined by (11) and (15) and then search all possible shift parameters defined by (16). As a result, n sets of optimized shift parameters are obtained, which are averaged to reduce error. Thereafter, an exhaust search of the angle parameters within the defined ranges is performed with the pre-optimized shift parameters.

## RESULTS

### Reconstruction and LO-refinement for the datasets with Gaussian noise

With the conventional SPA procedure, the 3D density map of the 80S ribosome was reconstructed from the datasets with Gaussian noise. The resolution was assessed at FSC = 0.5 by calculating the FSC curve between the reconstructed map and the ground-truth map generated from the PDB file. For the dataset with SNR of 0.25, the conventional SPA procedure produced a final reconstruction with the resolution of 11.5 Å (Fig. 4C). For the dataset with SNR of 0.11, the resolution of the final density map was assessed to be at 13.5 Å (Fig. 4D). However, for the dataset with SNR of 0.06, our conventional SPA procedure could not yield a good reconstruction due to the extremely low SNR. As a result, in the following LO-refinement procedure, only the datasets with SNR of 0.25 and 0.11 were tested.

Assessed resolutions at FSC = 0.5 of reconstructions from three simulated datasets by conventional SPA procedures and LO-refinement procedures

Dataset | Subunit | Conventional SPA reconstruction before LO-refinement | LO-refinement by searching shifts and angles simultaneously | LO-refinement by searching shifts and angles separately |
---|---|---|---|---|

With Gaussian noise at SNR = 0.25 | Small | 13.4 Å | 11.2 Å | 11.1 Å |

Large | 11.1 Å | 10.6 Å | 10.4 Å | |

With Gaussian noise at SNR = 0.11 | Small | 15.5 Å | 12.8 Å | 12.2 Å |

Large | 13.1 Å | 11.7 Å | 11.6 Å | |

Generated from InSilicoTEM | Small | 9.7 Å | 9.1 Å | 8.9 Å |

Large | 9.1 Å | 8.9 Å | 8.7 Å |

For the large subunit in the dataset of SNR 0.25,the LO-refinement method also yielded a less-noisy density map with a better fit to the ground-truth structure (Fig. 5D) and improved the resolution (FSC = 0.5) from 11.1 Å to 10.6 Å for the exhaustive searching strategy and from 11.1 Å to 10.4 Å for the separate searching strategy (Fig. 5E and Table 1), which is further proved by local resolution analysis using ResMap (Kucukelbir et al., 2014).

We observed that, after LO-refinement, the quality of the density map and the assessed resolution at FSC = 0.5 for the target subunit were significantly improved while those for the non-target subunit became worse (Figs. 5C, 5F, 6C and 6F). The reason for this observation is that the LO-refinement procedure increases the accuracy of the parameters for the target module and thereby at the same time the accuracy of those for the non-target module is decreased due to the varied relative position and orientation between the target (refined) and non-target modules (not refined).

We also observed that, LO-refinement improves the reconstruction more significantly for the small subunit than for the large subunit (Table 1). This is likely due to that the large subunit has more weight to contribute to the final projection, leading to a smaller error of orientation determination. As a consequence, the room for improvement in the accuracy of orientation determination is smaller for the large subunit than for the small subunit.

Furthermore, we found that the two different optimization strategies yield different reconstruction resolutions for the same target module (Table 1). The separate searching strategy (Fig. 3B) had a slightly better result than the exhaustive simultaneous searching strategy (Fig. 3A). One reason for this is that for the separate searching strategy the final shift parameters are the average of ten optimized values from ten randomly selected trial angles, which overcomes the limitation of sampling only in integer steps, thereby increasing the accuracy of shift parameter determination. It is worth noting that, the separate searching strategy is also more efficient, requiring less intensive computation than that of the exhaustive simultaneous searching strategy.

### Reconstruction and optimization for the datasets generated by InSilicoTEM

Computation consumptions during optimizing parameters of the InSilicoTEM generated dataset for the two parameter-searching strategies

Simultaneous search strategy | Separate search strategy | |
---|---|---|

Node configuration (CPU type, MHZ, cache size, memory) | Intel Xeon X5650, 2.67 GHz, 12 MB, 36 GB | Intel Xeon X5650, 2.67 GHz, 12 MB, 36 GB |

Number of processors per node | 12 | 12 |

Number of nodes | 7 | 7 |

Network (I/O) | 1GB Ethernet | 1GB Ethernet |

Storage | NFS Disk array (SATA II, 7200 rpm, raid5) | NFS Disk array (SATA II, 7200 rpm, raid5) |

Number of particles | 58542 | 58542 |

Size of particle | 128 × 128 | 128 × 128 |

Number of projections | 5000 | 5000 |

Computation time of LO-refinement for the small subunit | 36.7 h | 7.8 h |

Computation time of LO-refinement for the large subunit | 37.6 h | 8.1 h |

All of above, for the dataset close to experimental electron microscopic conditions with CTF modulation and camera effect, the LO-refinement algorithm can still work effectively to improve the map quality and the reconstruction resolution of the target module.

## DISCUSSION

The conventional method in single particle analysis of heterogeneous sample with multiple conformations is to perform 2D or 3D classifications to try to separate different conformations into independent classes. The success of the conventional method is based on the assumption that the target macromolecular complex could only exhibit a small number of conformations. However, this assumption is challenged by the fact that many macromolecular complexes behave in a dynamic equilibrium with continuous conformational changes.

The work presented herein describes an image-processing algorithm, named as local optimization refinement (LO-refinement), to improve the reconstruction quality and resolution in single particle analysis of macromolecular complexes with infinite conformations. The assumption of LO-refinement is that the macromolecular complex can be treated as a combination of multiple modules, with each module exhibiting a relatively rigid conformation within our interested resolution. And the multiple conformations of the complex can be regarded as slightly varied relative positions and orientations among different rigid modules. Although the assumption is demanded, we realize that it reflects the nature of many macromolecular machines and is applicable to a large number of cases.

The main idea of the LO-refinement procedure is to focus on each rigid module and optimize its orientation and position individually. By maximizing the cross correlation coefficient (CCC) between the experimental projection and a series of simulated projections that are comparable to the experimental projection with varied orientation and position of the target module, we could obtain optimized parameters to improve the reconstruction of the target module. During the calculation of CCC, we apply a mask around the target module to reduce the contribution from the non-target modules, thereby increasing the accuracy of parameter determination for the target module. For parameters searching, we used two strategies, the exhaustive search for both angles and shifts and the separate search for angles and shifts, which is similar to the previously reported image alignment algorithms (Joyeux and Penczek, 2002). For the reconstruction step, we move the focused part to the center of the volume where the resolution is always higher than the surrounding since a better tolerance for the errors of angular parameters.

For a proof of principle of our LO-refinement procedure, we generated two types of datasets using the structures of ribosomes with two relative rigid modules (large and small subunits). One dataset incorporated Gaussian noises of different levels into the projections. The other dataset was generated using the program InSillicoTEM (Vulovic et al., 2013) to incorporate near experimental microscopic effects including contrast transfer function and detective quantum efficiency of camera. Testing the LO-refinement procedure against both types of datasets showed significant improvements on both the map quality and the assessed resolution.

Besides the ribosome molecules with two assumed rigid modules, this LO-refinement procedure could in principle be applicable to the complexes with multiple modules. In these cases, the non-target modules can be treated as one integral part with their parameters of orientations and positions unchanged. With this procedure, all the modules of the complex can be reconstructed into a better resolution. Furthermore, it is clear that this LO-refinement procedure can be iterated to further optimize the parameters of all the target modules and improve the resolutions of their reconstructions.

It should be pointed out that our LO-refinement procedure requires the modules of a complex in rigid conformations within a specific resolution. In reality, various degrees of conformational variations likely exist for any given module, which can be further affected by the changes between modules, especially at higher resolutions. This may limit the improvement by this LO-refinement procedure to achieve atomic resolution in some cases. Nonetheless, in many cases the rigid module assumption is valid to atomic resolutions, from extensive experience learned from the practice of multi-domain/module non-crystallography symmetry (NCS) averaging in X-ray crystallographic studies. The challenge may be the appropriate identification of the rigid modules, which could be facilitated by the recently reported normal mode analysis method (Jin et al., 2014) that can analyze the internal conformational flexibility of a target module and a new analytical approach for determining the free-energy landscape and the continuous trajectories of molecular machines (Dashti et al., 2014). In addition, another challenge is to clearly define the interaction interfaces among different modules, which may not be resolved by this LO-refinement procedure.

This LO-refinement procedure could be further improved in the following aspects. First, the assumption of that both the terms (1) and (5) can reach maxima simultaneously at the same orientation of target module may not be valid in many cases. The inference of other non-target modules, term (8), during calculating CCC should be avoided in the next improvement. One solution to this problem is to remove the information of non-target modules from the experimental particle image. Besides, the real experimental particle image involves CTF (contrast transfer function) modulation of the particle projection while the CTF effect in the simulated model projection image has been corrected. One could not compare the experimental image and simulated image directly without considering the CTF effect. Thus, the theory of LO-refinement from term (1) to term (8) can be adapted and improved as follows.

*k*.

As the result, the target CCC defined in term (21) can reach the maxima only if the cross-correlation of projections between experimental and simulated module A defined in term (25) reaches the maxima. This improved theory of LO-refinement described from term (17) to term (25) can fully avoid the inference of non-target modules and account the effect of CTF modulation, and thereby would yield further improved reconstruction of the target modules, especially when dealing with the real experimental data.

In addition, besides back projection, other reconstruction algorithms, i.e. SIRT (Bangliang et al., 2000) and NUFFT (Chen and Förster, 2014), can be applied. Furthermore, maximum likelihood probability (Dempster et al., 1977) and Bayesian analysis (Scheres, 2012a) could also be implemented in this LO-refinement procedure.

During the revision of the present paper, we noticed that the recent publication by Liu and Cheng (2015), where they developed an image processing method to reconstruct the high-resolution map of viral internal structure within the capsid, described the detailed math of how to subtract the information of viral internal structure from the raw experimental whole virus particle. The idea of their information subtraction is similar to our proposed adjusted LO-refinement theory in this discussion from term (17) to term (25).

Furthermore, we also noticed that Nguyen et al. recently reported the cryoEM structure of pre-assembled spliceosomal complex and they developed an image processing approach called “multi-body refinement” to improve the density for the flexible arm domain (Nguyen et al., 2015). The idea of their “multi-body refinement” is similar to our LO-refinement but implemented differently in Fourier space and combined together with Bayesian approach. The success of their “multi-body refinement” approach has become another proof of the idea described in this paper. By implementing the adapted LO-refinement theory with improved codes for efficient computation, our LO-refinement algorithm will provide an alternative solution in real space to deal with the conformational flexibility of macromolecular complexes for single particle analysis.

## MATERIALS AND METHODS

### Test the datasets with Gaussian noise

A 3D map of the 80S ribosome (Taylor et al., 2009) was generated from PDB file (PDB entry 4V7H) by using the command e2pdb2mrc.py in EMAN2 (Ludtke et al., 1999) with a pixel size of 4 Å. Subsequently, a rotation around an axis through the subunit center and a shift in 3D space were applied to each subunit independently using the commands CG, ROT L and SH in SPIDER (Frank et al., 1996).

For simulating the scenario of slight and random flexibility between the large and small subunits, the direction of the rotational axis was selected randomly, and the rotational angle was assigned randomly in a normal distribution with an average value of 0° and a standard deviation of 1.67°. The shifts x, y, z were also assigned randomly in a normal distribution with an average value of 0 pixel and a standard deviation of 1 pixel. The two randomly moved subunits were then combined together to generate a whole 80S molecule. In total, 50,000 density maps were generated in this way and each map was projected once with the projection direction randomly selected. As a result, we simulated a dataset of a molecular complex with multiple conformations that are fixed in ice with random orientations.

In the final step of generating the simulation data, we added Gaussian noise into each projection by using the commands FS, MO and ADD in SPIDER (Frank et al., 1996). Three different (0.25, 0.11 and 0.06) SNR of noises were used according to previous studies (Baxter et al., 2009), yielding three datasets with different levels of noises (Fig. 4A).

To carry out 3D reconstruction, we first applied a conventional SPA routine using a customized SPIDER script in the Liu lab (Huang et al., 2012) to perform reconstruction refinement against the above simulated datasets. The density map of the whole ribosome that was generated from the corresponding PDB file was low-pass filtered to 20-Å resolution as an initial model. For each cycle of the refinement, the FSC (Fourier shell correlation) curve between the refined map and the PDB-generated density map was calculated by using the command FSC in SPIDER.

After the conventional SPA refinement became converged (Fig. 4C and 4D), we applied the LO-refinement to both the small and large subunits respectively. Both procedures of the LO-refinement described above (Fig. 3) were tested. The subunits were segmented using Chimera (Pettersen et al., 2004). A round mask with a diameter of 30 pixels for the small subunit, or 40 pixels for the large subunit was used during CCC computation to select the target module while excluding the background noise and the signal from the non-target module. The projection direction of the target module was searched within a cone of 10° and the in-plane shift was searched within a range from −4 to 4 pixels. For the second optimization strategy (Fig. 3B), the number of randomly selected angles was set to 10. To ensure that the improvement by using this LO-refinement was not due to increased sampling rate, both the angular and the shift sampling steps were kept the same (2° and 1 pixel respectively) as the last cycle of refinement in the conventional SPA procedure. Similarly, the 3D reconstruction methods were also kept the same as the weighted back projection (WBP) that was carried out using the command BP 32F in SPIDER.

After one iteration of LO-refinement, the density of the target module was segmented using a soft mask in the shape of the module, and the FSC curve was calculated against the density map generated from PDB file for comparison and further analysis (Scheres and Chen, 2012).

### Test the dataset generated from InSilicoTEM

We further validated our LO-refinement by using a simulated dataset with experimental conditions considered. We used the software package InSilicoTEM (Vulovic et al., 2013) to generate a new dataset of ribosome projections. This procedure takes into account the most relevant physical parameters of cryo-electron microscopy including both contrast transfer function and camera factors.

In this test, the coordinates of the 70S ribosome subunits (PDB entry 4V7C) (Brilot et al., 2013) were rotated and shifted respectively in the same way described above for the Gaussian type dataset on the 80S ribosome. The randomly moved subunits were then combined into a whole structure of the 70S ribosome.

All the physical parameters used in InSilicoTEM to generate a simulated dataset

Specimen | |||||

Motion blur | 0 Å | Thickness of the specimen | 38 × 10 | Amplitude contrast | Plasmon of ice and protein |

Electron-specimen interaction | |||||

Interaction type | Weak phase | ||||

Microscope | |||||

Spherical aberration | 2.7 × 10 | Illumination aperture | 0.030 × 10 | Chromatic aberration | 2.7 × 10 |

Energy spread of the source | 0.7 eV | ||||

Aperture | |||||

Diameter of objective aperture | 100 × 10 | Focal distance | 4.7 × 10 | ||

Acquisition settings | |||||

Pixel size | 0.2 × 10 | Defocus | 2.0, 2.3, 2.6, 2.8, 3.0, 3.2, 3.4, 3.7, 4.0 μm | Astigmatism | 0 m |

Acceleration voltage | 200 kV | Dose rate | 20 e-/Å | ||

Detector-camera | |||||

Camera type | Gatan US4000 | DQE | With DQE |

CTF correction (phase flipping) of the dataset generated from InSilicoTEM was performed using the command e2ctf.py in EMAN2. Then a conventional SPA procedure (Fig. 4E) and subsequent LO-refinement for each subunit were performed in the same way as described above for the Gaussian type dataset. Slightly differently, a binning factor of 2 was used for two-dimensional image alignment to reduce the computation time, while the reconstruction was calculated without binning. During LO-refinement, the angle and shift sampling steps (2° and 1 pixel respectively) and the reconstruction method (WBP using BP 32F in SPIDER) were kept the same as those in the last cycle of refinement in the conventional SPA procedure.

After one iteration of LO-refinement, the density of the target module was segmented using a soft mask in the shape of the module, and the FSC curve was calculated against the density map from PDB file for comparison and further analysis (Scheres and Chen, 2012). All the reconstructed maps were analyzed by ResMap (Kucukelbir et al., 2014).

## DATA ACCESSION

All the simulated datasets together with the final reconstructed maps are available at http://feilab.ibp.ac.cn/shared in case that those interested readers want to test their algorithms for comparison.

## Notes

### ACKNOWLEDGEMENTS

This work was supported by grants from the Strategic Priority Research Program of Chinese Academy of Sciences (XDB08030202) to F.S. and F.Z., the National Basic Research Program (973 Program) (Nos. 2014CB910700 to F.S. and 2012CB917200, 2010CB912400 to C.C.Y.), and the National Natural Science Foundation of China (Grant Nos. 61232001 and 61202210) to F.Z.. All the intensive computations were performed by using the high performance computers at Center for Biological Imaging (CBI, http://cbi.ibp.ac.cn), Institute of Biophysics, Chinese Academy of Sciences.

### COMPLIANCE WITH ETHICS GUIDELINES

Hong Shan, Zihao Wang, Fa Zhang, Yong Xiong, Chang-Cheng Yin and Fei Sun declare that they have no conflict of interest. And this article does not contain any studies with human or animal subjects performed by any of the authors.

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