Mathematical model for statistical manifold learning

The goal of GTM is to find a representation for the distribution p(t) of a dataset in a J-dimensional data space t = (t₁, …, t_J) in terms of L-dimensional latent variables s = (s₁, …, s_L). One may consider a non-linear, parametric function t = A(s; W), which maps points s in the latent space into corresponding points A(s; W) in the data space [33] (Fig 1A). The mapping is controlled by a set of parameters W, which represent the weights and biases in the case of a feed-forward neural network as the mapping. In the situation in which the dimensionality L of the latent space is less than the dimensionality J of the data space, the nonlinear transformation A(s; W) maps the latent space onto an L-dimensional non-Euclidean manifold embedded within the data space. Previous studies have established GTM as an alternative to self-organizing maps (SOM) [45] and that the GTM framework overcomes most limitations of SOM while introducing no significant disadvantages [33,34].

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Fig 1. Strategy for unsupervised single-particle clustering via statistical manifold learning.

(A) The fundamental principle of GTM is to establish a numerical relationship between variables in the latent space and a non-Euclidean manifold composed of the Fourier transformed image data in the data space. The manifold embedding can be determined by a set of nonlinear basis functions and a weighted parametric matrix. The likelihood function for the nonlinear mapping is solved by the expectation-maximization algorithm. (B) The workflow of implementing the unsupervised clustering strategies in ROME is as follows: (I) All images are aligned using MAP2D in a reference-free manner, and are subsequently classified into many groups by unsupervised GTM. (II) The unsupervised classes obtained in step (I) are further classified into many sub-classes by unsupervised GTM in a hierarchical fashion.

https://doi.org/10.1371/journal.pone.0182130.g001

To adapt the general framework of GTM to the single-particle cryo-EM data clustering problem, we built the contrast transfer function (CTF) into the non-linear mapping function in Fourier or reciprocal space, which allows for an efficient correction of the aberration effect of the objective lenses in electron microscopy [1]. A vector in data space x_i represents the Fourier transform of a particle image. A latent variables s reflects the inherent degrees of freedom controlling the 2D structural differences observed in single-particle images, which arise either from distinct conformational states of the imaged biomolecules or from different viewing angles. For the sake of simplicity, we used t_i to represent the translated and rotated version of x_i, so that t_i = T_-τ(x_i), where T_-τ denotes the operator of in-plane image rotation and translation; τ = (θ, r_x, r_y) is the in-plane rotation angle and translation, which can be determined through a 2D image alignment procedure, such as the regularized ML method [25–27]. Thus, an image t_i in the frequency domain can be modelled as: (1) where t_ij is the j-th component of the 2D Fourier transform of the i-th experimental image; CTF_ij is the j-th component of the contrast transfer function for the i-th image; n_ij is Gaussian noise; and A_j(s_k; W) is the j-th component of the 2D Fourier transform of the underlying k-th molecular projection.

The nonlinear function A(s; W) is expanded by a set of basis functions {ϕ₁, ϕ₂, …, ϕ_M} through the weight matrix W: (2) where A(s; W) is a K × J matrix with elements A_j(s_k; W); Ф(s) is a K × M matrix with elements ϕ_km = ϕ_m(s_k); and W is an M × J matrix containing the weight and bias parameters. In our algorithmic design, we used a combination of one fixed basis function and many Gaussians basis functions in the form: (3) where M_NL is the number of Gaussian basis functions; μ_m is the mean of the Gaussian distribution; and σ is the variance of the Gaussian distribution.

Although the noise in cryo-EM data may take multiple forms, potentially including both white and non-white noises, previous studies have established that cryo-EM noise can be largely approximated with the normal distribution without obvious detrimental effects on data analysis [1,25–27]. Thus, in our algorithm we chose the Gaussian distribution for the probability density function in the data space: (4) where 1/β_ij is the variance of noise for the j-th pixel of the i-th image.

Since we manage to classify a dataset consisting of N images {t₁, t₂, …, t_N} into K classes, this can be translated into the problem of finding K points {s₁, s₂, …, s_K} in the latent space that are mapped onto {t₁, t₂, …, t_N} in the data space through Eq (1). To make the problem tractable, we considered a specific form for the probability distribution p(s) given by a sum of delta functions centered on K nodes of a regular grid in the latent space: (5)

By integrating over the s-distribution on the manifold, the distribution p(t_i|W, β) in the data space for a given value of W is: (6)

The joint probability density of observing the collection of N images t = {t₁, t₂, …, t_N} is . The ML estimate of the model parameters Θ = {W, β} can be found by maximizing the logarithmic form of the joint probability, namely, . As we would like to consider the prior probability of the weight matrix, a regularized ML estimator, also called the maximum a posteriori (MAP) estimate, with respect to Θ can be used to describe the nonlinear mapping problem. Thus, the model parameters are sought to maximize the MAP: (7)

Here we assume a Gaussian prior distribution over W: (8) Where 1/α is the variance of the Gaussian prior distribution over the weight matrix W, and W is the total number of elements in the matrix W.

Expectation-maximization algorithm

The unsupervised clustering problem is ill-posed due to a high level of noise, potentially missing orientations and discontinuity between different conformational states in single-particle cryo-EM data. To enable effective computation, the expectation-maximization (E-M) algorithm can be used to estimate the model parameters Θ corresponding to the optimal clustering. The E-M algorithm alternates between two steps: the expectation step (E-step) and the maximization step (M-step). In the E-step, the old model parameters W⁽ⁿ⁾ and are used to evaluate the posterior probabilities, or responsibilities, R_ki(Θ⁽ⁿ⁾) of each latent variable s_k for every image t_i using Bayes’ theorem: (9)

In the M-step, Q(Θ, Θ⁽ⁿ⁾) is maximized with respect to the model parameters Θ: (10)

To this end, the partial derivatives of Eq (10) with respect to each model parameter should equal zero when Q(Θ, Θ⁽ⁿ⁾) is maximized. This allows us to estimate the model parameters iteratively. Maximizing Q(Θ, Θ⁽ⁿ⁾) with respect to W thus gives rise to the following equation, which uses to update the weight matrix W⁽ⁿ⁺¹⁾ in the (n+1)-th iteration: (11)

Similarly, by maximizing Eq (10) with respect to β, we obtained the following re-estimation formula that uses R_ki(Θ⁽ⁿ⁾) and W⁽ⁿ⁺¹⁾ to update : (12)

During the iteration of the E-M algorithm, no class averages need to be computed. Nonetheless, upon convergence of the E-M algorithm, one can calculate the class averages using the posterior probability R_ki: (13)

Note that this equation is derived from Eqs (11) and (12) and resembles the Wiener filter.

Note that Eq (11) is a system of linear equations. Solving this system of equations is equivalent to separately solving J systems of linear equations C_jw_j = b_j, j = 1, …, J, where , , and . This decomposition allows us to distribute the numerical computation of Eq (11) to many computer nodes in parallel. To further reduce the computational cost, we used a mean to replace in the numerical solution of Eq (11), as follows: (14) where N is the total number of the images and J is the total number of pixels in each image.

The convergence of the E-M algorithm can be monitored by the weighted loss function (15)

Thus, a convergence criterion can be devised based on the weighted loss function , i.e., the program is terminated when does not change more than 0.1% for six consecutive iterations.

Implementation of the GTM algorithm

Based on the above mathematical framework, we have implemented the following algorithm in the ROME software package.

Input: Particle images t_i, and their corresponding CTF_i, i = 1, 2, 3, …, N.

Output: Class assignment and class averages A_k

1. Set the fixed number of latent points {s_k}, k = 1, 2, 3… K. The space between two adjacent points is 1.
2. Set the values of basis function centers{μ_m}, m = 1, 2, 3… M.
3. Select the value of basis function width .
4. Calculate the basis function ϕ from Eq (3).
5. Initialize the weighted matrix W from a Gaussian circle and β; and set α to a fixed value of 0.01.
6. Iterate until the convergence criterion, (i = 0, 1, …, 5), is satisfied:
   1. Set iteration number n = n + 1;
   2. E-step:
      1. Compute the δ_ij distance between the latent space and data space
      2. Compute the probability matrix P from Eq (4) using ϕ, δ_ij and W;
      3. Compute the responsibility matrix R from Eq (9) using P, δ_ij and W.
   3. M-step:
      1. Use Eq (11) and R to update W;
      2. Use R and δ_ij to update β_ij and calculate its mean from Eq (14).
7. Calculate class averages from Eq (13).

Hierarchical strategy for unsupervised clustering

Although our GTM-based algorithm may be advantageous for unsupervised classification, it is not computationally efficient for image alignment, a procedure that determines three geometrical parameters for each image: x-y translation and in-plan rotation. Thus, we employed the adaptive MAP-based 2D alignment method (hereafter referred to as MAP2D) [19,46] to align single-particle images prior to the GTM-based clustering procedure (Fig 1B). Averages from random subsets of unaligned images were used to initialize the MAP2D-based image alignment. Furthermore, a Gaussian model was used to initialize the parameters W of the E-M algorithm in GTM solution. Thus, in both steps of the MAP2D-based alignment and GTM-based clustering, no external initial model or reference was needed, ensuring the unsupervised nature of our approach.

The unsupervised single-particle clustering may be conducted in a hierarchical fashion. First, all particles are aligned based on the translations and rotations determined by MAP2D. Then GTM is applied to partition these shifted and rotated particles into different classes. However, some classes might be mixtures of non-identical conformational states. For these classes, the aligned particles in each class can be further classified into tens or hundreds of reference-free sub-classes by GTM clustering. This hierarchical clustering strategy can be further iterated and used in different scenarios. For example, if large numbers of classes are needed to assess sample heterogeneity, the strategy can be used to obtain relatively more unsupervised classes without any human intervention. If one needs to verify the structural homogeneity within individual classes, the hierarchical clustering strategy can help examine structural variability within specific classes at a deeper level.

Comparative benchmarking with simulated data

Each simulated dataset, including 50,000 projections, was generated by projecting the density map of the Escherichia coli 70S ribosome along random orientations [20]. We uniformly chose 500 orientations covering half a sphere. Each orientation was regarded as a Gaussian center, around which 100 projections were generated with a Gaussian distribution. To simulate realistic situations, we added random shifts in the range of -2.0 to 2.0 pixels relative to the image center. We further generated defocus values in the range of -1.0 to -3.5 μm and modified the projection images with the CTF in the Fourier domain. The projections were then additively contaminated with Gaussian noise at SNRs of 1/50, 1/100, 1/150, and 1/200, which allowed us to benchmark the proposed algorithm at different noise levels.

Using the simulated data, we compared the results from our approach with those obtained using RELION [19], EMAN2 [47] and SPIDER [20]. All procedures were set to classify these 50,000 particles into 500 classes. In all cases described below, the alignment or classification was run for 30 rounds. Both in our approach and in the unsupervised 2D classification in RELION, we set up an in-plane rotational search with an angular increment of 2.5°, a translational search in the range of -3 to 3 pixels, and 100 classes for MAP2D alignment. We also conducted a classification based on an alternative method in RELION, which was implemented using the ‘skip_align’ option to classify pre-aligned particles without doing further alignment during classification. To enable a fair comparison, in the pre-alignment step, we set up the same control parameters for MAP2D alignment as the ones without using the ‘skip_align’ option in RELION.

EMAN2 uses multivariate statistical analysis (MSA) [1,22–24] with multi-reference alignment (MRA) method [24,47]. PCA was used in the MSA to reduce the dimensionality of images [47]. The particles were first CTF-corrected through phase flipping. Then, translational and rotational invariants were calculated and used for initial clustering. Next, PCA in the MSA step was used to capture the principal features, which were then used as multiple references for image alignment and K-means clustering in the MRA step. We set up 20 MSA basis vectors, 12 alignment references and 8 iterations for the EMAN2 procedure.

SPIDER combines reference-free alignment (RFA) with PCA and K-means clustering [48]. It involves three steps. First, all CTF-corrected noisy images were filtered through a low-pass Butterworth filter, with the pass band and stop band frequencies at 0.08 and 0.12, respectively. These particle images were then aligned globally in a reference-free manner, in order to find rotations and translations for all images that minimize the sum of squared deviations from their mean using the AP SR command in SPIDER. Second, PCA was used to solve the eigenvalue and eigenvector of all rotated and shifted images in the MSA step. Third, all particles were classified by K-means clustering [20].

Experimental cryo-EM datasets

We used three experimental cryo-EM datasets to examine if our clustering approach could yield useful classes and detect subtle structural difference among classes. The experimental cryo-EM datasets were collected using an FEI Tecnai Arctica microscope, operated under an acceleration voltage of 200 kV and equipped with Gatan K2 Summit direct electron detector. The inflammasome [8] and 26S proteasome [7] were imaged at a nominal magnification of 21,000 times. The proteasomal 19S regulatory particle (RP) [49] was imaged at a nominal magnification of 54,000 times. Cryo-EM data were collected semi-automatically using Leginon [50]. Movies comprising 36 frames per exposure were collected for the inflammasome and the 19S RP with an accumulated dose of 50 electrons/Ų. For the 26S proteasome, movies of 30 frames per exposure and an accumulated dose of 30 electrons/Ų were obtained. The physical pixel size and the super-resolution pixel size of the inflammasome and 26S proteasome datasets were 1.72 Å and 0.86 Å, respectively. The calibrated physical pixel size of 19S RP was 0.98 Å. The defocus in data collection were set in the range of -1.0 to -3.0 μm. See S1 Fig for typical micrographs and class averages.

Ab initio 3D reconstruction

To examine the effect of GTM-based clustering on ab initio 3D reconstruction, 117,471 RP complex particles were classified, by unsupervised GTM clustering implemented in ROME, into 1,000 classes (S2 Fig). We set the in-plane rotational search to a 2.5° increment, the translational search in the range of -4 to 4 pixels and 100 classes for MAP2D alignment. We used 234 class averages of the highest quality from the 1,000 GTM-produced classes to generate an initial model of the RP complex using e2initialmodel.py in EMAN2. In this step, 8 refinement iterations were performed with outputs of 10 initial models in search of a good one. As a control, we used the traditional method to obtain an initial model of the RP complex in EMAN2 from 10,000 raw RP complex images. These particles were first classified into 100 classes using e2refine2d.py. 12 MSA basis vectors, 5 alignment references and 8 iterations were set up for this EMAN2 procedure. 72 good classes were used to generate an initial reconstruction of RP complex by e2initialmodel.py in EMAN2. In this step, 8 refinement iterations were performed with outputs of 10 initial reconstructions. The best reconstruction was chosen for further analysis.

Parallel implementation and code optimization

Both the MAP2D and GTM implementations in ROME are optimized specifically for Intel^® Xeon^® processors and Intel^® Xeon Phi^™ coprocessors. In analogy to MAP2D in RELION, algorithm-level optimizations such as an adaptive E-M approach were adopted in ROME. In addition, the particle image stack was divided based on the unit of individual images across multiple computer nodes in the parallelization of the data flow in the MAP2D module. However, the image stack was divided based on the unit of individual image pixels in the GTM implementation. This allowed the problem to be distributed across multiple cluster nodes and attached coprocessors, while minimizing data transfer and memory requirements. Next, complex data structures were converted from arrays of structures to structures of arrays, making it easier to implement the inner loops of compute-intensive kernels with processor instructions that can be executed on multiple data elements in parallel (vectorization). Loops were also restructured so that independent threads of execution work on data elements that are near one another in memory. This ‘cache blocking’ attempts to lower the pressure that the algorithms put on the memory subsystem as the number of threads climbs, and tries to make the most effective use of the processor memory caches. To further improve the use of memory and decrease serialization overhead, memory allocation and deallocation were moved from the middle of the algorithm to the initialization phase. Finally, a single-precision data type instead of a double-precision one was used to store the image data in parts of the algorithm where the extra precision is not necessary. This treatment can improve performance by about a factor of two.

Classification accuracy with simulated data

To quantify the classification accuracy of our approach, we generated a series of synthetic datasets with various SNRs, each of which was composed of 50,000 simulated images as described above. Since all the original angles of the simulated data were known, the angular difference between any pair of images classified into each class (hereafter named ‘angular distances’) could be computed. We adopted the statistical behavior of the angular distances as a measure of the quality of the corresponding classes [13]. Supposing that n images are classified into one class, we have pairs of angular distances in this class. The histogram distribution of the angular distances from all the classes is a good way to compare the results from different algorithms [13]. A sharper, higher peak at lower angular distances in the distribution curve is expected from a better and more accurate clustering. We also used a second plot that ranks the number of particles over all classes to show the effective classes in the results. With the dataset at a SNR of 0.02, the classification accuracy of our approach is almost the same as that of SPIDER, and was better than that of both EMAN2 and RELION (S3C Fig). For the data at a SNR of 0.01, our approach yielded the sharpest and highest peak at lower angular distances in the distribution plot (Fig 2D), with a total of 495 effective classes (Fig 2E). This feature remained when the SNR was further reduced to 0.005. By contrast, the MAP2D classification implemented in RELION yielded a lower peak (Fig 2D), but only 162 effective classes (Fig 2E). The RELION results obtained with the ‘skip_align’ option were much worse than those from the standard MAP2D classification procedure (Fig 2D and 2E). Conventional PCA/K-means approaches in EMAN2 and SPIDER resulted in a much lower, wider peak at greater angular distances (Fig 2D and 2E). When the SNR was decreased to 0.005, the classification accuracies of these approaches were further reduced (Fig 2F and 2G and S3E and S3F Fig). These results suggest that a high level of noise significantly compromises the performance of data clustering by conventional PCA/K-means clustering approaches. By contrast, our GTM approach significantly improved the classification accuracy under conditions of reduced SNRs.

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Fig 2. Benchmarking the performance of unsupervised clustering using simulated data.

(A) A projection of the 70S ribosome model. (B and C) Examples of the simulated images of the 70S ribosome with SNRs of 1/100 (B) and 1/200 (C). The right panel in (B) and (C) shows the low-pass filtered version of each simulated image. (D and F) The normalized histogram exhibits the distributions of angular distances resulting from the five classification methods that were applied to the simulated images with SNRs of 1/100 (panel D) and 1/200 (panel F). (E and G) The sizes of classes were ranked for the five classification methods with SNRs of 1/100 (panel E) and 1/200 (panel G).

https://doi.org/10.1371/journal.pone.0182130.g002

We further examined the impact of the dimensionality of the latent space on the classification accuracy in the GTM algorithm. Using the same datasets of simulated images, we compared the measurements of the classification accuracies for the latent spaces assumed in one, two and three dimensions. The experiment with 2D latent space yielded slightly better classification accuracies than those with 1D and 3D latent space in our GTM algorithm for SNRs of 0.01, 0.0067 or 0.005 (Fig 3). This result suggests that the increased dimensionality in manifold learning does not necessarily improve classification accuracy.

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Fig 3. Classification accuracy with one-, two- and three-dimensional latent space in our GTM algorithm.

(A) Normalized histograms exhibit the angular distances for the one- and two-dimensional latent space under different SNRs. (B) The sizes of classes are for different latent space dimensions with varying SNRs. The label ‘GTM_D’ in (A) and (B) represents the number of dimensions. GTM_1D denotes that 500 points in one dimensional latent space were sampled in the GTM algorithm. GTM_2D denotes that 100 points in one dimension and 5 points in the other dimension, a total of 500 points, were sampled by the GTM algorithm. GTM_3D denotes that 20 points in the first dimension and 5 points in each of the other two dimensions, giving a total 500 points, were sampled in the GTM algorithm.

https://doi.org/10.1371/journal.pone.0182130.g003

We also examined the impact of several control parameters in the GTM algorithm on classification accuracy. First, there are two independent hyper-parameters in GTM: M_NL and α. Using the simulated datasets with a SNR of 1/100, we compared the measurements of the classification accuracy for different M_NL and α values. The experiment with M_NL = 0.8×K, where K is the class number, yielded better classification accuracy than those with smaller M_NL values (S4 Fig). This result suggests that an increased number of nonlinear basis functions can strengthen the classification capability of GTM. For the hyper-parameter α, the experiment with α = 0.01 yielded a better classification accuracy than those with smaller or larger α values (S5 Fig). This result suggests that an appropriate α value can penalize the ‘peaky’ elements in the weight matrix W to prevent overfitting and to improve the classification ability of GTM. Next, we compared the classification accuracies for different numbers of classes, namely, K = 100, 300 and 500. We found that the classification accuracy increases with increasing class number (S6 Fig). However, this was not the case for the other classification methods implemented in RELION and SPIDER (S6 Fig). Finally, we investigated the robustness of our GTM clustering algorithm by varying the initialization. We used different characteristic radii of the Gaussian circles and different values of β, as well as the outputs of PCA, for GTM initialization (S7 Fig). The results indicate that the performance of our GTM clustering is insensitive to the parameters of Gaussian circle inputs, which generally gave rise to better performance than those obtained by the PCA-based initialization (S7 Fig).

Applications to experimental cryo-EM datasets

To demonstrate the applicability of our clustering approach, we applied our method to several experimental cryo-EM datasets (S1 Fig). First, ROME was used for an unsupervised clustering of a 17,103-particle dataset of the inflammasome complex [8]. The resulting class averages directly presented three different conformations of the complexes corresponding to 10-, 11- and 12-fold symmetry (Fig 4A and S8A Fig). For comparison, 300 classes were produced by reference-free MAP2D classification implemented in RELION [19] (S8B Fig). Second, unsupervised GTM clustering of 96,488 particles of the proteasomal regulatory particle associated with the core particle (RP-CP) sub-complex directly revealed different local features among class averages (Fig 4B) [2,7]. These local differences were verified as corresponding to distinct conformational states through further 3D reconstruction [2,7]. Our results thus indicate that unsupervised GTM clustering can sort out not only the projections of underlying structures into relatively homogenous 2D classes but can also unearth incomplete structures or junk particles, allowing the efficient in silico purification of single-particle datasets.

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Fig 4. Unsupervised clustering by GTM.

(A) Typical class averages of inflammasome particles generated by unsupervised GTM clustering in ROME. Red, yellow and green boxes indicate the top views (first row) and the side views (second row) of 10-, 11-, and 12-fold inflammasome complex, respectively. The side views of the complex structure differ by length. Besides, the purple box denotes the class average of an incomplete inflammasome complex. (B) Typical class averages of RP-CP sub-complexes generated by unsupervised GTM in ROME. The red or yellow boxes indicate a pair of class averages showing differences in local features corresponding to the local movement of the Rpn5 subunit of the RP-CP subcomplex [7]. The green box indicates a pair of class averages showing the movement of the Rpn1 subunit of RP-CP subcomplex [7]. The purple box labels the class average of the incomplete RP-CP subcomplex. (C) Typical side-view class averages of the inflammasome were initially classified using the MAP2D classifier in a reference-free manner. Two classes among 50 classes visually resemble the 11-fold inflammasome complex particles. (D) The class average highlighted by red box in panel (C) was further classified by GTM. The red boxes indicate the 11-fold inflammasome particles. The green boxes indicate the 10-fold inflammasome particles that were misclassified by MAP2D into the same class as the rest 11-fold structures. The yellow boxes indicate the 12-fold inflammasome particles that were misclassified by MAP2D into the same class as the rest of the 11-fold structures. (E) A 57,001-particle dataset of free RP was initially classified using the MAP2D classifier in a reference-free manner. (F) The class marked by the red box in panel (E) was further classified by GTM in ROME. Several classes of RP-CP sub-complex particles (red boxes) were found to be misclassified into this free RP class.

https://doi.org/10.1371/journal.pone.0182130.g004

Unsupervised GTM clustering with improved accuracy can help identify hidden heterogeneity in the reference-free classes generated by other methods. For instance, after initial reference-free clustering, particle images of specific classes could be selected for deeper unsupervised clustering by GTM in a hierarchical fashion. Delicate differences between GTM-generated class averages may reveal features corresponding to distinct conformational states. To test this idea, a class of 281 particles resulting from MAP2D-based classification (Fig 4C), whose average visually resembles the side view of the 11-fold inflammasome complex, was further classified into 30 sub-classes by GTM (Fig 5A). Based on the length of the intrinsic structure, the GTM-based reference-free clustering identified the side views of 10-fold and 12-fold inflammasome complexes among the 30 sub-classes (the red boxes in Fig 4D), suggesting that the MAP2D-generated initial class is a mixture of the complexes with 10-, 11- and 12-fold symmetry. Another MAP2D-classified image group with 3961 particles, whose average visually represents a tilted view of the free 19S regulatory particle (RP) that is not associated with a 20S core particle (CP), was further classified by GTM (Fig 4E). This deeper clustering identified 448 single-particle images (11.3%) of RP-CP subcomplexes that were misclassified into the dataset of free RP (Fig 4F).

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Fig 5. Comparison of hierarchical unsupervised clustering using ROME, RELION, SPIDER and EMAN2.

(A) Unsupervised classification of a MAP2D-generated class into 30 sub-classes using the GTM algorithm in ROME. The red box marks the side view projection of the 11-fold inflammasome complex. The green box marks the side view projection of the 10-fold inflammasome complex, whose length is smaller than that of the others. The yellow box marks the side view projection of the 12-fold inflammasome complex, whose length is larger than that of the others. (B) Unsupervised classification of the same MAP2D-generated class into 30 sub-classes in RELION. The major class exhibits the side view projections of 11-fold inflammasome complex, whereas all the other classes present ‘junk’ features. (C) Unsupervised K-means clustering of the same MAP2D-generated class into 30 sub-classes in SPIDER. The green boxes highlight the side views of the 10-fold inflammasome complex, whose length is smaller than others. The yellow boxes label the side view projections of the 12-fold inflammasome complex, whose length is larger than that of the others. (D) Unsupervised K-means clustering of the same MAP2D-generated class into 30 sub-classes in EMAN2. The yellow boxes label the side view projections of the 12-fold inflammasome complex, whose length is longer than that of the others. The green boxes label the side view projections of 10-fold inflammasome complex, whose length is shorter than that of the others.

https://doi.org/10.1371/journal.pone.0182130.g005

Fig 5 presents a further comparison with other algorithms including MAP2D and K-means clustering, which suggests that our GTM outperforms those algorithms in unambiguously identifying hidden heterogeneity with the hierarchical clustering strategy. In this comparison, we selected a MAP2D-classified image group comprising 281 side views of the inflammasome complexes (red boxes in Fig 4C), which was further classified into 30 classes by unsupervised GTM clustering in ROME, by reference-free MAP2D classification in RELION, and by K-means clustering in SPIDER and EMAN2. The sub-classes resulting from the GTM algorithm all showed clear 2D structural features, indicating a mixture of the side views of 10-, 11- and 12-fold complexes in this dataset, as manifested by their length variation (green box and yellow box in Fig 5A). By contrast, only two meaningful classes were obtained by RELION, while the other sub-classes were either featureless or empty (Fig 5B). The results from SPIDER and EMAN2 appear to be improved over RELION. However, there were still more than 10 sub-classes in both cases that were either featureless or presented aberrant features likely resulting from mis-classification. Our comparative studies suggest that, unlike our GTM-based clustering, other unsupervised clustering approaches, such as MAP2D, PCA and K-means, are either less efficient or ineffective for directly detecting subtle structural variations using the hierarchical clustering strategy.

GTM-based clustering improves ab initio reconstruction

To demonstrate the clustering capacity of our approach, 117,471 particles of the free RP complex were classified into 1,000 classes by GTM implemented in ROME in an unsupervised manner. The computation was completed in 228 minutes with 30 rounds of E-M optimization on a computing cluster comprising 512 Intel Xeon E5 CPU cores (S1 Table) and 858 effective classes were obtained (S2 Fig). We used 234 class averages of the highest quality from the 1,000 unsupervised classes to generate an initial 3D reconstruction of the RP complex in EMAN2 (Fig 6A, S2 Fig). As a control, we used the conventional multivariate data analysis (MDA) and K-means clustering to obtain an initial reconstruction of RP complex in EMAN2 from 10,000 raw images of the RP complex (Fig 6B). The FSC curves between the atomic model and the two initial reconstructions using a 0.5-cutoff suggest that the resolution of the initial model using GTM-based class averages was 20.6 Å, which was significantly higher than the resolution, 29.1 Å, of the other initial reconstruction obtained by the conventional approach (Fig 6C). Next, we performed rigid-body fitting of the atomic model of the free RP into the two initial reconstructions. The atomic model fits into the GTM-based initial reconstruction with an excellent agreement (Fig 6A). By contrast, considerable elements in the atomic model were out of the density of the initial reconstruction obtained by the conventional approach (Fig 6B). The contrasting results show that the initial reconstruction built upon our GTM clustering method is of significantly improved quality compared with the conventional one. Taken together, these results suggest that the increased class number and improved classification accuracy of our unsupervised GTM clustering gives rise to a marked improvement of ab initio 3D reconstruction with an affordable computational cost.

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Fig 6. Initial 3D reconstruction from the reference-free class averages of ROME and EMAN2.

(A) The initial reconstruction calculated by the ROME-generated class averages is superimposed with the atomic model of free RP shown in a ribbon representation, suggesting that they are highly compatible with each other. (B) The initial reconstruction calculated by the EMAN2-generated class averages is superimposed over the atomic model of free RP shown in a ribbon representation. A substantial part of the atomic model is outside of the density of the initial reconstruction, suggesting poor map quality and a large reconstruction error. (C) FSC curves between the RP atomic model and the initial reconstructions generated by ROME- and EMAN2-based class averages.

https://doi.org/10.1371/journal.pone.0182130.g006

Optimization of computational performance and efficiency

Our implementation of the GTM-based clustering algorithm in the ROME software has been optimized for modern HPC hardware, such as Intel^® MIC architecture used in the Intel Xeon Phi^™ Knights Corner and Knight Landing processors. Our modernized code in ROME outperforms the existing software in both speed and magnitude. First, we compared the computational performance between ROME and RELION using several datasets comprising different numbers of particles. It took 143 minutes for ROME to classify 57,001 particles of RP into 300 unsupervised classes on a computing cluster comprising 512 Intel Xeon E5 CPU cores (Fig 7A). By contrast, it took ~40 hours for RELION to classify the same dataset into 300 unsupervised classes on the same cluster. In all cases, the E-M algorithm was run for 30 iterations. Thus, under the same circumstances, ROME was about 10–20 times faster than RELION in unsupervised 2D classification. Second, a 96,488-particle dataset of the RP complex was used to benchmark the performance of ROME in unsupervised clustering with different class numbers on the 512-core computing cluster (Fig 7B). With increasing class numbers from 100 to 1,000, the running time of GTM clustering was increased in a polynomial trend from 14 to 214 minutes. Specifically, ROME completed the unsupervised clustering of 1,000 classes in 306 minutes with 30 iterations of expectation-maximization (S1 Table), whereas RELION did not complete the same number of iterations of optimization in a week on the same hardware condition. Furthermore, we also used this dataset to test reference-free classification at the 1,000-class level by MSA and K-means clustering in SPIDER [1,20]. Unfortunately, the program kernel crashed in the reference-free alignment step.

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Fig 7. Performance evaluation of unsupervised clustering with ROME.

(A) Performance of unsupervised single-particle clustering in ROME versus RELION using different datasets. Unsupervised 2D classification into 300 classes using both software programs were performed on four experimental datasets: Dataset1 refers to the 16,306-particle dataset of the inflammasome with 250×250 box size; dataset2 refers to the 35,407-particle dataset of the free RP complex with 160×160 box size; dataset3 refers to the 96,488-particle dataset of the RP-CP complex with 160×160 box size; dataset4 refers to the 57,001-particle dataset of the free RP complex with 180×180 box size. MAP2D alignment in ROME and GTM clustering for 300 classes wes also performed. The blue, green, and red histograms represent the running time of RELION, MAP2D in ROME, and GTM in ROME, respectively. For more comparison, see S9 Fig and S1 Table. (B) The 96,488-particle dataset of the RP-CP subcomplex was used to test the performance of GTM in ROME (blue dots). The green dots represent the total running time including both the MAP2D alignment and GTM clustering in ROME. The running time was polynomially related to the number of classes.

https://doi.org/10.1371/journal.pone.0182130.g007