research papers
Exploiting prior knowledge about biological macromolecules in cryoEM structure determination
^{a}MRC Laboratory of Molecular Biology, Cambridge, United Kingdom, ^{b}Department of Mathematics, Royal Institute of Technology (KTH), Sweden, ^{c}DeepMind, United Kingdom, and ^{d}Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, United Kingdom
^{*}Correspondence email: cbs31@cam.ac.uk, ozan@kth.se, scheres@mrclmb.cam.ac.uk
Threedimensional reconstruction of the electronscattering potential of biological macromolecules from electron cryomicroscopy (cryoEM) projection images is an illposed problem. The most popular cryoEM software solutions to date rely on a regularization approach that is based on the prior assumption that the scattering potential varies smoothly over threedimensional space. Although this approach has been hugely successful in recent years, the amount of prior knowledge that it exploits compares unfavorably with the knowledge about biological structures that has been accumulated over decades of research in structural biology. Here, a regularization framework for cryoEM
is presented that exploits prior knowledge about biological structures through a convolutional neural network that is trained on known macromolecular structures. This neural network is inserted into the iterative cryoEM structuredetermination process through an approach that is inspired by regularization by denoising. It is shown that the new regularization approach yields better reconstructions than the current state of the art for simulated data, and options to extend this work for application to experimental cryoEM data are discussed.Keywords: 3D reconstruction; image processing; singleparticle cryoEM; imaging; structure determination; cryoelectron microscopy.
1. Introduction
In cryoEM singleparticle analysis, the threedimensional structure of biological macromolecules is reconstructed from twodimensional projection images of multiple copies of these molecules in different relative orientations. The requirement to image under lowdose conditions, in order to limit damage to the radiationsensitive samples, leads to high levels of noise in cryoEM images and the need to average over many images. Moreover, individual macromolecules, or particles, adopt unknown orientations in the sample and may adopt multiple conformations. The resulting large numbers of unknown variables render the optimimization problem illposed, which when combined with the high noise level in the data leads to a challenging reconstruction problem.
The most widely used approach to cryoEM RELION program (Scheres, 2012a,b). Other software packages, such as CryoSPARC (Punjani et al., 2017) and THUNDER (Hu et al., 2018), have since built on the same approach. The MAP approach differs from common alternative approaches in two aspects. Firstly, it marginalizes over the unknown orientation (and conformational class) assignments of the particles, and secondly, it uses an explicit penalization term as a regularizer.
to date is based on the maximumaposteriori (MAP) estimator. This approach was introduced to the cryoEM field through its implementation in theWhereas previously popular software packages typically aimed to find the best orientation and class for each particle, the marginalization approach rather uses a weighted average of all possible orientations. Marginalization was introduced to the cryoEM field by Sigworth (1998). Using simulated data, he showed that marginalization over the orientations for 2D alignment against a single reference leads to reduced sensitivity to the choice of the starting reference and to the ability to align images with very low signaltonoise ratios. Marginalization over class assignments was later found to be particularly useful for 2D (Scheres et al., 2005) and 3D (Scheres et al., 2007) classification of experimental cryoEM images. Since its original conception in Xmipp (Scheres et al., 2005), marginalization over class assignments has been adopted in many software packages, including RELION (Scheres, 2012b), THUNDER (Hu et al., 2018), Frealign/CisTEM (Grant et al., 2018), Sphire (Moriya et al., 2017) and cryoSPARC (Punjani et al., 2017).
In general, regularization can be used to prevent overfitting when solving an illposed problem. Before the introduction of the MAP approach, cryoEM software tools would prevent overfitting mostly through heuristic lowpass approaches and Wiener filter approaches (Penczek, 2010; Frank, 2008). The MAP approach pioneered the optimization of an explicitly regularized target function by expressing the reconstruction problem in an empirical Bayesian framework. The prior information that underlies the regularization in this approach is the assumption that rapidly changing density values in realspace cryoEM reconstructions, or high powers at higher spatial frequencies in Fourier space, are unlikely. In other words, one assumes that cryoEM reconstructions are smooth. By expressing both the (marginal) likelihood function and the prior in Fourier space, a standard L_{2} Tikhonov regularization approach is formulated (Engl et al., 1996). Its MAP solution can be obtained through expectation–maximization (Dempster et al., 1977), where an analytical solution to the maximization step exists. The resulting algorithm yields a 3D Wiener filter that removes highresolution noise from the reconstruction, while estimating all necessary parameters from the data (Scheres, 2012b). The ability to obtain efficiently filtered reconstructions without the need for usertunable parameters probably played an important role in the rapid uptake of MAP optimization in the field (FernandezLeiro & Scheres, 2016).
Nevertheless, the smoothness prior seems an informationpoor choice when compared with the knowledge about biological macromolecules that has been acquired through decades of structural biology research. For example, we know that the density for the macromolecules in cryoEM reconstructions is surrounded by flat solvent density and that proteins are made up of aminoacid chains that fold into well defined secondarystructure elements such as αhelices and βstrands. Although a wealth of structural information exists, it has in practice been difficult to handcraft a formulation of a prior function that can be incorporated into the optimization algorithm that underlies cryoEM Despite this, recent work has shown that realspace filtering of maps performed iteratively throughout the reconstruction can help to mitigate the effect of noise on alignment and hence improve convergence and final map quality (Ramlaul et al., 2019, 2020).
Machine learning based on deep neural networks, or deep learning in brief, can capture prior information and has recently seen tremendous success in a wide range of computer vision tasks (Krizhevsky et al., 2012; Ren et al., 2015; Ronneberger et al., 2015). Convolutional neural networks have been shown to perform equally well or better than conventional stateoftheart methods in inverse problems for several imaging modalities (Arridge et al., 2019), including denoising (Zhang et al., 2017; Jifara et al., 2019; Gondara, 2016), computed tomography (Adler & Öktem, 2017, 2018b) and (Hammernik et al., 2018; Mardani et al., 2018). Generally, this performance is attributed to the ability of deep neural networks to efficiently learn statistical correlations about both the noise and the signal through the examples in the training data set.
Regularization by denoising (RED) is an imagerecovery framework that enables the plugin of various denoisers into a MAP optimization protocol (Romano et al., 2017). This includes many denoiser classes, including general noisereduction algorithms such as blockmatching and 3D filtering (BM3D), more specialized denoising algorithms (Jonić et al., 2016), and also deep learningbased denoisers such as denoising autoencoders and Unets.
The RED framework was originally centered around an explicit expression for a prior that incorporates the output of the denoiser. This framework requires that the denoiser function fulfills two conditions: local ).
and Jacobian symmetry. However, most stateoftheart denoisers, including denoising convolutional neural networks, fail to exhibit these properties, in particular Jacobian symmetry. Despite this, RED has been shown to also perform well with these denoiser functions. Hence, the explicit prior approach cannot explain the behavior of this framework. In response to these issues, Reehorst and Schniter proposed a new framework, scorematching by denoising (SMD), which showed that RED achieves its performance by approximating the `score' or the gradient of the prior distribution. This approach circumvents the requirement for an explicit prior expression and further does away with the abovementioned conditions on the denoiser function (Reehorst & Schniter, 2018Here, we propose a cryoEM structuredetermination procedure that is inspired by the RED protocol to integrate a convolutional neural network to express prior information. CryoEM reconstruction typically uses a vectorvalued regularization parameter to model variations in signaltonoise ratios across different spatial frequencies. The socalled `goldstandard' recommendations in the field are to estimate these through the Fourier shell correlation (FSC) between independently obtained reconstructions from halves of the data set (Henderson et al., 2012). This estimate typically varies greatly between data sets and throughout different steps. Therefore, we expand RED to account for a vectorvalued regularization parameter that varies throughout the reconstruction and present a theoretical framework, inspired by SMD, to show that this approach is valid in the case of more realistic Gaussian noise. We further present a simple heuristic to prevent any resolution overestimations that may arise owing to the new prior in the most difficult cases. We explore the measured performance of the denoiser in different resolution domains to determine the confidence in the new prior, which is subsequently used to weight between the new prior and the classical exponential prior. We call this approach confidenceweighted regularization by denoising (CWRED).
Because we train our convolutional neural network on tens of thousands of cryoEM reconstructions from simulated data and test it using data that are simulated in the same manner, this work does not yet represent a finalized solution to improve cryoEM reconstruction from experimental data. Rather, it serves as a general proof of principle that learned priors can improve cryoEM reconstruction. Future research directions, such as the exploration of alternative optimization algorithms and different strategies to design and train the neural networks, are discussed.
2. Theory
2.1. Mathematical formalization of structure recovery
We consider structure recovery in singleparticle cryoEM based on a Bayesian formalization of the problem. The starting point is after the particlepicking step where one extracts 2D projection images from the electron micrographs. Each of these shows a single macromolecular structure, or particle. Typically, following a classification step, images of the same macromolecular structure are selected for further
We may here assume that these 2D projection images are approximately centered (with inplane 2D translation) with respect to the corresponding 3D particle and that minor adjustments remain for an optimal translational alignment.To formalize the above, let denote the discrete 2D Fourier transforms of the aforementioned 2D projection images. Likewise, let denote the discrete 3D Fourier transforms of the corresponding particles. This Fourier transform acts on the electrostatic potential generated by the particle and its surrounding aqueous buffer. Since all particles represent the same macromolecule up to an orientation and a translation, we have that s_{i} = t_{i} ○ r_{i}(x), where is the 3D Fourier transform of the molecule structure, r_{i} ∈ SO(3) is its 3D rotation and t_{i} ∈ T(3) is its 3D isomorphic realspace translation. We define the composite transformation s_{i} := t_{i} ○ r_{i} and thus g ∈ G := SE(3) belongs to the special Euclidean group.
Structure recovery aims to reconstruct (the 3D Fourier transform of the structure of the molecule) from 2D data when the corresponding transformations g_{1}, … g_{m} ∈ G are unknown. It is natural to consider data as being generated by a random variable, i.e. there are valued random variables that generate the corresponding data. In the Bayesian setting one also adopts the viewpoint that the molecule structure and transformations are generated by random variables. Hence, we introduce a valued random variable that generates molecular structures (models) and the Gvalued random variable that generates particle transformations g_{1}, … g_{m}. Data generated by the single particle is then a single sample of the valued random variable , where
Here, is a valued random variable representing noise in data and is the digitized (linear) model for TEM image formation in frequency space. In particular, for some matrix for fixed g ∈ G. The systematic effects of optical aberrations, such as the contrast transfer function (CTF), can be precalculated and given during for each image. These effects can be incorporated into H, in which case H would be assigned a subscript i and considered unique for each particle image. However, to simplify the notation, this will be ignored in the following sections.
Assume next that the random variables are independent and marginalize over the transformation using a known (prior) probability density of orientations and translations in G. The joint probability density (joint data likelihood) for the entire data set conditioned on the model is then expressible as
Note here that for fixed g ∈ G and is given by (1), i.e. it is specified by the matrix H_{g} and the noise distribution .
The Bayesian approach to structure recovery aims to compute a suitable estimator that summarizes the (posterior) probability distribution of x given Y. The density of this posterior distribution is expressible by Bayes' theorem as
where P_{data} denotes the density for the joint distribution of .
2.2. MAP with a Gaussian prior
The MAP estimator aims to find the model that maximizes the posterior probability (or equivalently its logposterior probability), i.e. we seek a maximizer to . From (3) we obtain that a MAP estimator maximizes , where
To proceed, we need to specify (the logprior for 3D models) and (the joint log–data likelihood).
Assume that data in frequency space is corrupted with additive uncorrelated Gaussian noise. More precisely, assume in (1) is a complex circularly symmetric Gaussian with mean zero and a diagonal covariance matrix with diagonal vector that encodes the frequencydependent power of noise for each component of the ith image. Then,
Next, most implementations of MAPbased cryoEM structure recovery use an uncorrelated Gaussian prior for 3D models with zero mean. Stated more precisely, one assumes
Here, contains the frequencydependent estimate of the power of the signal in the model. Such a distribution arises from applying the central limit theorem to 3D models with a random distribution of atoms, and can thus be intuitively regarded as a minimal assumption criteria about the 3D model (Wilson, 1949). More intuitively, this prior biases the recovery to a smoother realspace representation by suppressing large Fourier components.
A MAP estimator maximizes , so in particular it solves . There is no closedform solution for this equation, so one has to resort to iterative schemes. One example is the expectation–maximization (EM) algorithm (Dempster et al., 1977).
For a data likelihood as in (5), the EM method generates iterates by computing from data Y and the previous iterate . Given (4), (2) and (43) in Appendix B it can be shown (Bendory et al., 2020) that this is performed by solving the equation
where H* is the adjoint of H with respect to the usual inner product on (see Appendix B) and the expectation is weighted by
To simplify the notation in the following sections, we define
In the above, is diagonal owing to the Fourier slice theorem and the offdiagonal components are discarded.
Henceforth, when multiplying or dividing two vectors of equal length by each other, it is to be interpreted as applying these operations componentwise. This allows us to rewrite (7) as
For a prior that is Gaussian as in (6), we obtain ∇logP(x) = −r^{−2}x, so one can solve (7) (perform the Mstep) analytically. This has a closedform solution, generating the scheme
Additionally, we set in (9) and the regularization parameter . One common approach in line with the Bayesian viewpoint is to assume these are generated by random variables and then either marginalize over them or use estimates that are updated at every iteration n. As an example, the estimate of the regularization parameter τ is commonly based on the radially averaged Fourier shell correlation (FSC) calculated by comparing 3D models reconstructed from two independent halves of the data set (Scheres, 2012b). More precisely, let be two 3D models with associated data sets Y_{1} and Y_{2}, which are arrays (of equal length m/2) with elements in . Next, define
with FSC: denoting the Fourier shell correlation (FSC). To use the above in (11), split the data into two sets Y_{1} and Y_{2}. Let denote the two iterative sequences of 3D models obtained by applying the MAP EM scheme in (11) to Y_{1} and Y_{2}, respectively. Instead of using a constant value for the regularization parameter τ, we instead use (12) to adaptively adjust its value based on these sequences, i.e. we replace the fixed in (11) with for k = 1, 2. Intuitively, this amounts to reducing the regularization applied in (11) when the FSC increases, which means that more signal is permitted to accumulate into each of the two reconstructions.
2.3. MAP with nonGaussian prior
Much of the usefulness of the EM method resides in the ability to perform the Mstep in (7) in a computationally feasible manner. This is possible for a Gaussian prior and results in the EM scheme given in (11). Here, we consider priors that are `close' to a Gaussian in the sense that the second term below varies slowly with x:
For this class of priors, (10) reads as
which has the following approximate solution:
Using this approximation, we can compute an approximate MAP estimator using the scheme
The above requires one to specify the `nonGaussian' part . One can adaptively set τ as in the Gaussian case by splitting the data into two parts and using (12). Next, one may consider a datadriven approach to learn the `nonGaussian' part instead of handcrafting it.
The regularization by denoising (RED) method allows us to approximate the gradient of the logprior using a neural network trained to denoise volumes. The RED prior formula has been shown to accurately estimate the gradient of the prior through score matching by denoising (SMD; Reehorst & Schniter, 2018). When dealing with independent additive Gaussian noise, RED can be used to integrate an external denoiser into an iterative imagerestoration protocol (Romano et al., 2017). We show in Appendix A that the MMSE estimator under Gaussian noise with covariance τ^{−2} approximates the gradient of the logprior according to ∇logP(x) ≃ f(x) − τ^{−2}(x). However, based on empirical observations, a more conservative approach is to also modulate f(x) with τ^{−2}, which would suppress the influence of the denoiser when certainty in the data is high. Hence, we propose the gradient logprior expression
Inserting (15) into (14) gives an approximate Mstep for priors using a learned denoiser:
The input to the denoiser in this update formula is x^{(n)}, which lacks the information extracted in the latest Estep. However, both K and B can be evaluated at this stage, hence through them it is computationally feasible to provide the denoiser with the most `uptodate' information. Additionally, based on empirical observations we introduce a heuristic parameter, λ, to tune the influence of the denoiser in the Mstep. Conclusively, we propose the following update scheme:
Note that (17) is equivalent to (11) when λ = 0 , so 0 ≤ λ ≤ 1 is an empirical parameter that balances the influence of denoiser versus conventional Gaussian prior. Furthermore, in (17) is equivalent to x^{(n+1)} in (11) if τ^{−2} = 0. Hence, the denoiser acts on the unregularized map that contains information from the most recent alignment of the experimental data, which alleviates the assumptions made for (14).
Further adaptation can be obtained by making use of the fact that RELION is used for the To do this, we consider a mimimum meansquare estimator (MMSE) that is trained to `denoise' RELION volumes. More precisely, we consider the following supervised statistical learning problem:
In the above, is the random variable that generates appropriately coarsened (lowpass filtered) 3D structures of macromolecular assemblies from the PDB and is the random variable that generates RELION structures.
Since we lack knowledge about the joint probability distribution of , the conditional expectation in the lefthand side of (18) is replaced by its empirical counterpart given by supervised training data for i = 1, …, n that are random draws from , i.e. is the true 3D structure derived from the PDB (after appropriate coarsening) and is a corresponding RELION structure. Furthermore, a deep neural network is used to parameterize valued mappings on , thereby replacing the infinitedimensional minimization over such functions with a finitedimensional minimization over the deep neural network parameters. The resulting counterpart of (18) can then be written as the following empirical riskminimization problem:
If the network has sufficient model capacity and there is sufficient supervised training data, then the above yields an approximation to the conditional mean estimator, i.e. .
3. Experimental design
3.1. Convolutional neural network
Synthetic training data were generated from 543 atomic structures that were downloaded from the Protein Data Bank (PDB; Berman et al., 2000). All downloaded structures were solved using Xray crystallography to at least 4 Å resolution, had molecular weights between 40 and 100 kDa, and consisted of a single protein chain; 110 structures also contained Atomic models were converted to volumetric density maps in the MRC format (Crowther et al., 1996) using the pdb2map tool in EMAN2 (Tang et al., 2007). In order to obtain simulated projection data sets at different signaltonoise ratios (SNRs), the intensity values of these maps were multiplied by three different constants: 0.014, 0.012 and 0.010.
Next, the density maps were projected onto images of 96 × 96 pixels, with a pixel size of 1.5 Å, and independent Gaussian noise of unity variance was added. The choice for a relatively large pixel size reduces computational costs, but was primarily motivated by the observation that SNRs in typical cryoEM images at spatial frequencies higher than 3 Å are so low that they no longer effectively contribute to particle alignment. The resulting projection data sets had SNRs in the range 0.0003–0.0100 and on average 0.0020, which was calculated as the average perpixel variance of the projection images divided by the variance of the added noise. Because RELION reconstructions are corrected for the effects of the CTF, no attempts were made to model the typical CTF of an electron microscope at this stage. Individual projection data sets of 10 000 projections each were generated for the different structures and SNRs using random projection directions that uniformly sampled the orientation space. Each of these projection data sets was then used for standard, unmasked `3D autorefinement' in RELION (Scheres, 2012b), which was started from a 30 Å lowpass filtered version of the groundtruth map. The new external reconstruct functionality in RELION3.1 (see below) was used to obtain unregularized halfmaps at each of the intermediate iterations of all refinements. In total, this process yielded over 25 000 unmasked 3D density maps at different stages of Of these maps, 23% had a resolution better than 4 Å and less than 2% had a resolution of 10 Å or worse. This data set was subsequently augmented through sizepreserving rotations.
To verify the validity of the underlying assumptions about Gaussian distribution of noise for the derivations in the theory section, we also trained a network using pairs of noisefree groundtruth and pure Gaussian noisedistorted maps. For this purpose, we used the estimated frequencyspace distribution of noise calculated from the RELION intermediate reconstructions. During training, random Gaussian noise was generated in frequency space matching this profile. The noise was then convoluted with a realspace mask that dampened the noise levels in the solvent region. However, unless specified, the presented results regard a denoiser trained on RELION intermediate reconstructions. All networks were trained and evaluated on realspace maps. In practice, each map update involves an inverse Fourier transform of and a subsequent Fourier transform of the denoiser, f, before (17) is evaluated. To improve training convergence and robustness during inference, the input to the network was standardized through subtraction of the volume average and division by the volume standard deviation. The effect of the standardization was subsequently reversed via addition of the average and multiplication by the standard deviation of the original volume.
The presented approach extends the reconstruction protocol with a denoiser, which is a common imagetoimage task with well established efficient network architectures. Hence, we limited the optimal architecture to hyperparameter optimization such as the depth and channel width of the network. For this work we chose a UNet (Ronneberger et al., 2015) for the denoiser, primarily motivated by the extensive research that has been performed on their application to denoising. Training and inference was performed using Python and Tensorflow 1.15 (Abadi et al., 2015). At the start of this project, Tensorflow had the largest community and support compared with other available deeplearning frameworks, which we determined to be important for the accessibility of the implemented code and method to the scientific community. The network was trained with residual learning (Zhang et al., 2017) and L^{2} regularization of network weights via Adam (Kingma & Ba, 2014), with a minibatch size of ten maps, for 27 epochs and with a constant learning rate of 10^{−4}. The network was trained on an NVIDIA 2080 Ti graphics card until convergence for 21 h. The Unet consisted of five downsampling and upsampling stages that were implemented through average pooling and transposed convolutional layers (Dumoulin & Visin, 2016), respectively. Each stage was made up of two consecutive blocks repeating BN+ReLU+Conv. The kernel sizes of all convolutional layers were set to 3, while the number of channels for the first hidden layer was set to four. The number of channels was increased by a factor of two for each lower stage in the Unet. Experiments suggested that wider networks perform better when assessed with the mean squared error (MSE) to the ground truth. However, owing to memory limitations the above number of channels was chosen for the first input layer as the best balance between performance and memory requirements.
The input–output pairs for training consisted of the unregularized and unmasked reconstructions from intermediate RELION iterations and their respective noisefree groundtruth maps. Here, the coarsening applied to the ground truth (see 19) was achieved by lowpass filtering it to match the resolution of the corresponding noisy input map. This was performed by multiplying the Fourier transform of the ground truth by the FSC of each noisy input map. The FSC was estimated for each map by RELION during the reconstruction of the training data set from the halfmaps.
3.2. Assessment of the RED approach
Four PDB structures were selected to test the performance of the RED approach (Table 1). The test structures were excluded from the training data set and had a minimum r.m.s.d. of 8 Å for at least 1000 aligned atom pairs with any other structure in the training data set. The four test structures represent different levels of difficulty for the reconstruction, mostly arising from differences in their molecular weight (MW) and their overall shapes (as projection images of nearspherical structures are harder to align than those with more anisotropic shapes). The test structures are listed in Table 1 in order of increasing difficulty, from the elongated, larger structure with PDB code 4ail to the smaller, nearspherically shaped structure with PDB code 4m82. Projection data sets for testing were made in the same way as the projection data sets that were used to generate the training set for the convolutional neural network. However, in contrast to the training data set, the test data set consists of images with a simulated CTF corresponding to a defocus range of 0.2μ to 2.0μ. We confirmed that using data with or without a CTF does not have any noticeable impact on the performance of the denoiser if changes in SNR are accounted for. This is expected, since the denoiser is only exposed to data where the effects of the CTF is mitigated through averaging over images with varying defocus. Test data sets were generated at four different SNRs by multiplying the original maps from the pdb2map tool by 0.022, 0.019, 0.016 and 0.013, which yields average SNRs of (i) 0.0038, (ii) 0.0028, (iii) 0.0019 and (iv) 0.0012, respectively.

The singlepass performance of the denoiser can be examined by applying it once to unregularized maps and evaluating the ratio between the L^{p} difference to ground truth from the denoised map and the unregularized map. We evaluated the average of this ratio (using both L^{1} and L^{2}) for the entire training data set as a function of the estimated nominal resolution of each map.
Standard, unmasked RELION 3D autorefinements, with updates based on (11), were compared with refinements with updates based on (17). Again, all refinements were started from initial models that were obtained by 30 Å lowpass filtering of the groundtruth map and used the same parameters. The results of both types of refinements were compared based on the reported halfmap FSC and the FSC against the ground truth, and the angular error relative to the true projection angles. All maps were first multiplied with a solvent mask with a smooth edge before the FSC calculation.
3.3. Implementation details
Instead of implementing the update formula in (17) directly in the C++ code of RELION, we created an external reconstruction functionality in its program. When using the external_reconstruct commandline option, the relion_refine program will write out MRC files containing B(x, y) and K(x, y) for both halfmaps at each iteration step, together with the current estimate for FSC and τ^{2}. The program then waits until reconstructions with a specified filename for both halfmaps exist on the file system, reads those maps back in and proceeds with the next iteration of This functionality was coupled to a script in Python to perform the update formula in (17) using the pretrained denoiser model. The Python script for external reconstruction using the proposed RED protocol, together with the architecture and the weights of the convolutional neural network that was used for the presented results, may be downloaded from github (https://github.com/3dem/externprior).
4. Results
4.1. Performance of the denoiser
We first tested the effects of applying the denoiser to individual, intermediate, unregularized reconstructions from a standard RELION [Figs. 1(a) and 1(b)]. After the first iteration of of the PDB entry 4ail projection data at SNR (i), the unregularized halfreconstruction contains only lower resolution components and exhibits notable rotational smearing. The latter is often observed in lowresolution intermediates and is caused by uncertainty in the angular assignments. Application of the denoiser reduces the rotational smearing and brings the map closer to the true signal, as confirmed by visual comparison and the FSC with the groundtruth map.
At iteration 16 of the same
the unregularized map contains higher resolution components, but noise components are clearly visible in the volume surrounding the protein, which should be devoid of features. The denoiser efficiently removes the latter while retaining the same level of sharpness in the protein region and, again, both visual comparison as well as FSC calculation with the groundtruth map confirm that the denoised map is closer to the groundtruth signal than the unregularized map.The average ratio between the difference to ground truth of the denoised map and the unregularized map was calculated for L_{1} and L_{2} at each nominal resolution [Fig. 1(c)]. When the resolution of the input map is worse than 10 Å the denoiser fails to produce a significant improvement on average. As the resolution improves beyond 10 Å the performance gradually improves and eventually plateaus beyond 4.5 Å.
4.2. Performance of the RED approach
Next, we tested the performance of the RED approach on projection data sets of the four test structures at four different SNRs. We first ran the RED approach with a denoiser trained on Gaussian noise and compared the results with those for a denoiser trained on RELION intermediate reconstructions (Fig. 2). The REDbased approach outperforms the standard approach in all refinements performed, as measured by the FSCs between the resulting reconstructions and the groundtruth maps (solid green and purple lines in Fig. 2). This suggests that Gaussian noise can partially explain the distortions. However, the denoiser trained on RELION intermediate reconstructions performs better in most cases, confirming that Gaussian noise is an incomplete model of the distortions observed in RELION intermediate reconstructions.
Both the RED approach and standard RELION autorefinement produced a reconstruction with a resolution close to the Nyquist frequency for the easiest test structure, PDB entry 4ail, at the highest SNR. Conversely, both approaches yielded a lowresolution solution that was devoid of recognizable protein features for the most difficult test structure, PDB entry 4m82, at the lowest SNR. Therefore, the range of structures and SNRs of our test data set represents a wide range of scenarios from easy to hard refinements. The gains in performance tend to be higher at higher SNRs; they are small for the most difficult refinements. It is noteworthy that in the most simple cases the RED approach achieves an FSC that is equal or slightly higher than that of the reconstruction with the true angles.
Measures of reconstruction quality that are reported by RELION without having access to the ground truth, i.e. the halfmap FSCs, also improve when using the RED approach (dotted green and purple lines in Fig. 2). However, for the of PDB entry 4m82 at the three lowest SNRs, we noticed a severe overestimation of resolution based on the halfmap FSCs using the RED approach. A visual examination of the maps [Fig. 3(b)] reveals highcontrast features that are similar in appearance to those of biological macromolecules but do not correspond to the ground truth. This points to the denoiser introducing false highresolution features into the reconstructions that are shared among the two halfmaps. This then leads to an overestimation of the resolution, which is based on the correlation between the halfmaps. Since the halfmap FSC is the only reliable estimate of resolution when the ground truth is missing, this issue poses a major problem for the validation of the reconstruction results.
4.3. Confidence weighting
The λ parameter in (17) can be used to fall back on the Gaussian prior in a scenario where confidence in the performance of the denoiser is lacking. Inspired by the results of the singlepass performance test of the denoiser [Fig. 1(c)], we tested a simple tuning scheme for λ to address the issue of resolution overestimation. We will refer to this modification as confidenceweighted RED (CWRED).
At nominal resolutions worse than 10 Å, as estimated by RELION's halfmap FSC, we assign a full fallback onto the Gaussian prior by setting λ = 0. This is where the singlepass performance indicates that the denoiser results in little improvement in the reconstruction [see Fig. 1(c)]. To avoid impairing the performance at high resolutions, however, we set λ = 1 at nominal resolutions better than 4.5 Å, which is approximately where the singlepass performance of the network begins to peak. For any nominal resolution between 4.5 and 10 Å we use a linear interpolation between 0 and 1.
Using this approach, we see that the overestimation of the halfmap FSC is significantly alleviated (dotted red lines in Fig. 4) without affecting the overall quality of the reconstructions as measured by the FSC with the ground truth (solid red lines in Fig. 4), the errors in the angular assignments (Fig. 5) and the visual appearance of the resulting structures (Figs. 6 and 7). A visual inspection of the map for PDB entry 4m82 resulting from the with the lowest SNR also suggests that the reconstruction no longer accumulates false structural features with poor correlation to the ground truth (Fig. 3).
5. Discussion
Deep learning faces four well known general challenges: domain shift, brittleness, explainability and fairness. Domain shift can prevent the deep neural network (DNN) from generalizing beyond the training domain owing to shifts in distribution between the training data set and the data encountered when the DNN is deployed; brittleness is the sensitivity to small changes in the data; explainability refers to transparency in the features learned and the degree that they can be made subject to human interpretability; and fairness reflects insensitivity to input variables that should be statistically independent from the output. The manner in which the DNN is incorporated into the reconstruction process, proposed here, intentionally retains most of the established statistical model that has been handcrafted from physical principles, based on the imageformation process. This approach alleviates many of the concerns related to brittleness and explainability by minimizing the role of the DNN. Furthermore, the data used to train the DNN will in principle introduce bias. However, this would in principle hold for any choice of priors. Generally, a handcrafted prior is more biased than a trained one (Adler & Öktem, 2018a). Therefore, fairness is of minor concern. However, domain shift is a considerable issue that will be further discussed in this section.
Convolutional neural networks for denoising are typically trained on pairs of noiseless and noisy representations of the signal. It is thus crucial to have access to an accurate noiseless groundtruth signal, which makes it challenging to apply these networks in areas where such a ground truth is impossible or expensive to acquire, such as medical imaging (Huo et al., 2018). To circumvent this problem, we used synthetic groundtruth cryoEM reconstructions and used a simplistic physical forward model to generate simulated projection images. In doing so, we explicitly did not aim to obtain a network to denoise reconstructions from experimental cryoEM data, and thereby avoided some of the challenges of domain shift. Rather, we used test data sets that were generated using the same forward model to provide a proof of principle that learned priors can improve the reconstruction process. Our results convincingly make this case: the RED approach outperformed standard autorefinement in RELION in all of the tests performed.
The results shown in Fig. 1 suggest that the denoiser has learned to differentiate protein from solvent regions and apply adequate amounts of filtering suited for each region, which is difficult to achieve using classical denoising algorithms. Additionally, in the easiest test cases the RED approach achieves an equal or higher FSC than the reconstruction using the true angles, which supports the idea that structural information can be injected into the reconstruction process through the proposed method.
The standard approach in RELION uses an L_{2} Tikhonov regularization on the Fourier components of the reconstruction (Scheres, 2012b). In practice, this purely Fourierbased regularization term is often complemented with an ad hoc regularization in real space by setting all densities to a constant value in the volume outside a userspecified 3D mask around the reconstruction. Often, such masks are generated after an initial has yielded a preliminary reconstruction. In the tests performed here, no solvent mask was provided. Thereby, the improvements observed using the RED approach reflect the difference between a purely Fourierbased L_{2} Thikonov regularization and a learned prior in real space. The observed differences with masked refinements would be smaller.
Although, aware of problems with domain shift, we used different macromolecular structures (i.e. with >8 Å r.m.s.d. on the atomic coordinates) in the test and training data sets, we still identified a risk of injecting similar, proteinlike features into both halfmaps, which then lead an overestimation of resolution based on the halfmap FSC. Without access to the ground truth, the halfmap FSC is the single most important measure of quality for the reconstruction, resulting in a major concern for the usefulness of the method. We observed that for our network this problem was most noticeable for reconstructions at resolutions lower than 10 Å, for which there were few examples in the training data set. More data in this domain will probably alleviate the problem. Still, we were able to address this issue with an empirical approach that we call confidenceweighted RED. By falling back onto the less informative Gaussian prior at nominal resolutions where the denoiser is known to produce little improvement in the reconstruction, we managed to sufficiently alleviate the problem while retaining equal overall performance. This approach is based on performance statistics gathered by applying the denoiser to the training data set, and required no significant additional effort compared with regular RED. To avoid the risk of overestimating the confidence in the denoiser, ideally another data set should be used instead of the training data set. It is also noteworthy that the performance of the denoiser is likely to vary over different resolution shells and that a local patchbased approach might also be required to better handle variations in local resolution. Therefore, a more detailed investigation into the confidenceweighting scheme could lead to further improvements.
We envision multiple other avenues for improving the approach set out in this paper. Firstly, different network architectures may be devised and networks may be trained through alternative loss functions to improve their performance. The macromolecular structures used for our training and test data sets were limited to contain only a single protein chain. Wider neural networks with more model capacity may be required to achieve comparable performance on experimental cryoEM data. A large variety of successful network architectures have been developed in other areas of image processing, for example FFDNet (Zhang, Zuo et al., 2018) and RCAN (Zhang, Li et al., 2018). Furthermore, to handle larger reconstructions the denoiser may be trained on patches, rather than on the entire map as performed here, and the patches may be combined using a sliding window, similar to previous work in tomography (Tegunov et al., 2020). This approach is appealing owing to the inherent flexibility in terms of particle/box size and memory requirements. Moreover, the use of patches may be intrinsically better suited to deal with the inherent variability in map quality and local resolution that is caused by the different extents of order that exist in many biological macromolecules. For instance, confidence weighting can easily be made to rely on the average local resolution in each patch rather than the entire volume. However, as networks trained on patches might no longer see the entire box, where a particle at the center is surrounded by solvent this approach may be less powerful in flattening the solvent. One solution could be the combination of multiple denoisers that are trained in different resolution domains. For instance, a patchbased denoiser dedicated to the highresolution domain could be combined with a denosier with a global view of the map at low resolution. The confidenceweighting scheme introduced in this paper is well suited to combine each of these denoisers according to the performance in each of the resolution domains.
Secondly, to limit problems with domain shift, the networks should be optimized for the denoising of reconstructions from experimental cryoEM data. Experimental cryoEM data have different characteristics in both the signal and the noise to the synthetic data used here. Experimental noise is not independent in the pixels, but falls off with higher spatial frequencies. Both the signal and part of the noise are affected by the CTF of the microscope, and in particular the lowest spatial frequencies in experimental cryoEM reconstructions are poorly modeled by the simplistic forward model of protein atoms in vacuum used in this paper. Several options exist to generate more realistic pairs of maps for training the denoiser. Refinements with experimental cryoEM data downloaded from the EMPIAR database (Iudin et al., 2016) may be used to generate training data. In such a scenario, generating image pairs with ground truth will be difficult, but one could generate intermediate reconstructions from relatively small subsets of the data and provide the highresolution maps from the complete data sets as substitutes for the ground truth. This is particularly feasible when the networks will only be trained on intermediateresolution reconstructions (also see below). Alternatively, one could train generative adversarial networks (GANs) to learn a datadriven forward model from pairs of atomic models and their corresponding noisefree cryoEM reconstructions in order to generate more realistic groundtruth maps of disordered regions, for example membrane patches. For this purpose, conditional GANs (cGANs) are a particularly suitable candidate, since pairs of atomic model (ground truth) and reconstruction density are available (Isola et al., 2017). Similarly, cycleconsistent adversarial networks (CycleGANs; Zhu et al., 2017) or styletransfer GANs (Gatys et al., 2016) may be used to relate the two data domains in cases where pairs are lacking, for example for lowresolution reconstructions without a matching ground truth or for generalizing the data set to new experimental conditions with few existing examples (Januszewski & Jain, 2019).
Thirdly, the approach described here may be adapted to use optimization algorithms other than expectation–maximization. A gradientdriven approach with minibatches would be based on and (15) to inject prior knowledge into the reconstruction more often, which has the potential to improve the convergence speed. For this class of algorithms, adversarial regularisers might be a more natural candidate compared with RED, since this method models the prior more directly and thus enables better control of the properties of the generated gradient (Lunz et al., 2018). Alternatively, algorithms based on ADMM and plugandplay denoisers are of potential interest for investigation (Venkatakrishnan et al., 2013; Bigdeli et al., 2019).
Finally, an important consideration when employing prior information in solving any inverse problem is that the prior information used in solving the problem can no longer be used for external validation of the results. This touches on the challenge of explainability and is relevant when injecting prior information about biological macromolecules into the cryoEM reconstruction process. In current cryoEM approaches such information is not used at all, and the appearance of protein and nucleic acidlike features in cryoEM reconstructions is often implicitly used to confer confidence in the correctness of the maps. However, one could imagine a more informative prior generating reconstructions in which such features are incorrectly `invented' from noise in the maps. Commonly used procedures to prevent overfitting, most notably splitting the data set into two independent halves, do not necessarily protect against the interpretation of such false features. Therefore, new validation tools may need to be devised to guard against the interpretation of false features. One option could be to only use information up to a given intermediate resolution when training the denoiser. In this way, the presence of protein or nucleic acidlike features beyond this resolution could still be used to provide confidence in the result. This approach is particularly appealing because the highresolution signal typically does not contribute much to the alignment of the individual noisy projection images anyway (Henderson et al., 2011).
We expect that with further improvements along the lines discussed above, the approach presented in this paper will result in a tool for
that will also outperform the current state of the art for experimental cryoEM data. This has the exciting potential to improve any reconstruction from existing data sets and to expand the applicability of the technique to samples that are currently beyond its scope.APPENDIX A
Dependent Gaussian noise
SMD describes the results of RED using a white Gaussian noise model with a fixed variance as the underlying assumption for the image distortion. This appendix describes details of how the RED approach was expanded to handle the type of distortions observed in cryoEM data. In particular, we derive a dynamically adjusted regularization parameter, as described in (15), and a modified training scheme by managing the significantly varying levels of distortion in the data. We show how this leads to a training protocol that minimizes the model mapping to a coarsened ground truth. The aim of this appendix is not to provide a full mathematical proof of (15), but rather to present an approach that highlights the assumptions and approximations that are required to derive it.
We define the random vector , , where P here is the distribution of undistorted groundtruth data. Here, we assume a Gaussian noise model formalized as , where and is a covariance matrix that depends on to describe the shape of the Gaussian. For the application in this paper, this dependency does not have to be explicitly defined; rather, an empirical approach is sufficient where the training data set is extended to also include observations of ν. In Section 3.1 we specify how this was performed in practice. One can device an empirical prior distribution that approximates P through kernel density estimation (KDE; Parzen, 1962), given a large representative corpus of training data, {x}^{T}_{t=1}, drawn from P, and the corresponding {ν}^{T}_{t=1}, as
For the continuum limit of KDE, we instead obtain
The gradient of the logKDEprior then becomes (see Appendix B for the precise definition of ∇)
We define an MMSE function as
In practice, owing to the higher relative error of highfrequency components, corresponds to a coarsened (lowpass filtered) version of . We also define the MMSE of as
It then holds that
Now, it follows that
In the above, (31) uses the orthogonality principle and (33) follows from (28). In the particular case of f, in the above, belonging to a class of universal approximator functions, e.g. CNN denoisers, the minimization (i.e. training) is carried out over the network parameters θ with pairs of . It thus follows that optimizing the weights θ of such a function, , through
equates matching the gradient of the logKDEprior through
In this paper, we assume that p_{KDE} is a good enough approximation of P given the trainingdata size and data dimensionality. Conclusively, f_{θ} is trained to map a noisy data point, , to a weighted combination of lowpass filtered ground truths. We are, however, still required to estimate ψ. Ignoring the offdiagonal components of ψ and estimating only the diagonal components through (12), i.e. ψ ≃ τ^{−2}, implies that (36) is equivalent to (15).
APPENDIX B
Wirtinger derivatives
To derive (7) one has to differentiate the expression for the posterior probability in (3). In this paper, it is expressed in the Fourier domain as a real function with complex input. Here, we define the gradient operator acting on such a function using the formalism of Wirtinger derivatives (see, for example, Appendix A of Fischer, 2005). More specifically, the Wirtinger derivatives of f with respect to the complex variables z_{i} = x_{i} + iy_{i} and z_{i}^{*} := x_{i} − iy_{i} are defined via
The corresponding gradients can then be defined as
We also use the shorthand notation . Note that is stationary if and only if or . It follows from Appendix A in Fischer (2005) that for and
where A* is the adjoint of A with respect to the usual inner product on . Hence, if we take and b = σ_{i}^{−1}y_{i} we find
Similar calculations give the gradient of the Gaussian logprior:
In this paper, to simplify notation outside this appendix, the subscript on ∇ is dropped when the differentiation is with respect to x or any variations thereof.
Acknowledgements
We thank Jake Grimmett and Toby Darling for help with highperformance computing.
Funding information
CBS acknowledges support from the Leverhulme Trust project on `Breaking the nonconvexity barrier', the Engineering and Physical Sciences Research Council (EPSRC; EP/M00483X/1, EP/S026045/1 and Centre EP/N014588/1), the RISE projects CHiPS and NoMADS, the Cantab Capital Institute for the Mathematics of Information and the Alan Turing Institute; OÖ acknowledges support from the Swedish Foundation for Strategic Research grants (AM130049 and ID140055); SHWS acknowledges support from the UK Medical Research Council (MC_UP_A025_1013) and Wave 1 of the UK Research and Innovation (UKRI) Strategic Priorities Fund under the EPSRC (EP/T001569/1), particularly the `AI for Science' theme within that grant and The Alan Turing Institute.
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