cryoSPHERE: Single-Particle HEterogeneous REconstruction from cryo EM
Authors: Gabriel Ducrocq
Lukas Grunewald
Sebastian Westenhoff
Fredrik Lindsten
Venue: ICLR 2025 Poster (2025)
Keywords: cryoEM; protein structure; Deep Learning; Machine Learning; generative modelling
Primary area: generative models

Abstract: The three-dimensional structure of proteins plays a crucial role in determining their function. Protein structure prediction methods, like AlphaFold, offer rapid access to a protein’s structure. However, large protein complexes cannot be reliably predicted, and proteins are dynamic, making it important to resolve their full conformational distribution. Single-particle cryo-electron microscopy (cryo-EM) is a powerful tool for determining the structures of large protein complexes. Importantly, the numerous images of a given protein contain underutilized information about conformational heterogeneity. These images are very noisy projections of the protein, and traditional methods for cryo-EM reconstruction are limited to recovering only one or a few consensus conformations.

In this paper, we introduce cryoSPHERE, which is a deep learning method that uses a nominal protein structure (e.g., from AlphaFold) as input, learns how to divide it into segments, and moves these segments as approximately rigid bodies to fit the different conformations present in the cryo-EM dataset. This approach provides enough constraints to enable meaningful reconstructions of single protein structural ensembles. We demonstrate this with two synthetic datasets featuring varying levels of noise, as well as two real dataset. We show that cryoSPHERE is very resilient to the high levels of noise typically encountered in experiments, where we see consistent improvements over the current state-of-the-art for heterogeneous reconstruction.

Published as a conference paper at ICLR 2025
CRYOSPHERE: SINGLE-PARTICLE HÉTEROGENEOUS RECONSTRUCTION FROM CRYO EM
Gabriel Ducrocq
Lukas Grunewald
Division Statistics and Machine Learning Linköping University, Linköping, Sweden gabriel wwtrrie.ducrooco ducroocee si
Department of Chemistry
Uppsala University, Uppsala, Sweden
lukas. grunewalddkent .u.
Sebastian Westenhoff
Fredrik Lindsten
Department of Chemistry
Division of Statistics and Machine Learning Linköping University, Linköping, Sweden fredrik. lindsten@ winiiiiie... ii..s
Uppsala University, Uppsala, Sweden sebast ina, westenhof f@kemi... uu. se
ABSTRACT
The prree-iimensional structure of proteins plays a crucial role in determining their function. Protein structure prediction methods, like AlphaFold, offer rapid access to protein's structure. However, large protein complexes cannot be reli- ably predicted, and proteins are dynamic, making important resolve their full conformational distribution. Single-particle cryo-electron microscopy (cryo-EM) is a powerful tool for determining the structures of large protein complexes. Im- portantly, the numerous images of a given protein contain underutilized informa- tion about conformational heterogeneity. These images are very noisy projections of the protein, and traditional methods for cryo-EM reconstruction are limited to recovering only one a few consensus conformations. this paper, we introduce cryoSPHERE, which is deep learning method that uses nominal protein struc- ture (c.g., from AlphaFold) as input, learns how to divide it into segments, and these segments approximately rigid bodies fit the different conforma- tions present in the cryo-EM dataset. This approach provides enough constraints enable meaningful reconstructions of single protein structural ensembles. We demonstrate this with two synthetic datasets featuring varying levels of noise, as well two real dataset. We show that cryoSPHERE is very resilient to the high levels of noise typically encountered in experiments, where we see consistent im- provements over the current of-te-of-the-art for heterogeneous reconstruction.
INTRODUCTION
Single-particle cryo-electron microscopy (cryo-EM) is powerful technique for determining the three-dimensional structure of biological macromolecules, including proteins. In a cryo-EM exper- iment, millions of copies of the same protein are first frozen in thin layer of vitreous ice and then imaged using an electron microscope. This yields micrograph: noisy image containing 2D pro- jections of individual proteins. The protein projections are then located on this micrograph and out so that experiment typically yields 10* to 10? images of size Npx x Npx of individual pro- teins, referred as particles. Our goal is reconstruct the possible structures of the proteins given these images. Frequently, proteins are conformationally heterogeneous and each copy represents a different structure. Conventionally, this information has been discarded, and all of the sampled structures assumed to be in only Here, we would like to recover of the structures in heterogeneous reconstruction.
few conformations (homogeneous reconstruction).
Structure reconstruction from cryo-EM presents a number of challenges. First, each image shows particle in a different, unknown orientation. Second, because of the way the electrons interact with the protein, the spectrum of the images is flipped and reduced. Mathematically, this corresponds to convolution of each individual image with the Point Spread Function (PSF). Third, the images typi- cally have very low signal-to-noise ratio (SNR). For these reasons, is very challenging to perform

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Experimental crydEM Image, 4
Learnable
Encoder
Decoder
Fixed mapping
Transformation per sogesont
Sample
fo
GMM
Base structure, Sa
Conrelation loss
Segmentation
Predicted image, (RR, S,.Saow)
Transformed structure, Snow
Map residues
Project PSF
Transform structure
Transtormation por residue
Figure 1: Flow chart our network, The learnable parts of the model are the encoder, the decoder and the Gaussian mixture. Note that even though the transformations predicted by the decoder on a per image basis, that is not the case of the Gaussian mixture, which is shared across all particles.
de novo cryo-EM reconstruction. Standard methods, produce electron densities averaged over many, if not all conformations (Scheres 2012 Punjani et al. (017), performing discrete heterogeneous re- construction. More recent methods attempt to extract continuous conformational heterogeneity, e.g.. by imposing constraints on the problem through
underlying structure deformed to fit the differ- ent conformations present in the dataset, see e.g. Rosenbaum et al. 20:1;: Zhong et al. (2021b); provide
Li al. 2023). AlphaFold (Jumper et al., 2021) and RosettaFold (Back al.] (2021) such a structure based on the primary sequence of the protein only. In spite of this strong prior, it is still difficult recover meaningful conformations. The amount of noise and the fact that observe only 2D projections creates local minima that are difficult to escape (Zhong et al. 2021b Rosenbaum al. 202T), leading to unrealistic conformations.
To remedy this, we root our method in the observation that different conformations can often be explained by large scale movements of domains of the protein (Mardt et al., (2022). Specifically, we develop a variational auto-encoder (VAE) (Kingma & Welling 2014) that, from nominal structure and set of cryo-EM images:
Learns how divide the amino-acid chain into segments, given defined maximum number of segments; see Figure 2 The nominal structure can for instance be obtained by
AlphaFold (Jumper et 2021
For each image, learns approximately rigid transformations of the identified segments of the nominal structure, which effectively allows us to recover different conformations on an image-by-image (single particle) basis.
These two steps happen concurrently, and the model is end-to-end differentiable. The model is illustrated in Figure ! The implementation of the model is available on github
Note that what call segment is conceptually different from a domain in the structural biology sense. The domains of protein play pivotal role in diverse functions, engaging in interactions with other proteins, DNA/RNA, or ligand, while also serving catalytic sites that contribute significantly to the overall functionality of the protein, see e.g. Schulz & Schirmer (1979); Nelson et al. (2017). By comparison, the segments not strictly necessary for the function of the method, experiments in Section 5 show that our VAE often recovered the actual domains corresponding different conformations.
learn do not necessarily have a biological function. However, while
NOTATIONS AND PROBLEM FORMULATION
In what follows, we consider only the C. atoms of the protein. protein made of number Ries N+ of residues r, is denoted = !ires where the coordinates of residue are the coordinates of its Ca atom. The electron density map of structure S, also called a volume, is function Vss R + R. where V15(2)) proportional the probability density function of an electron of S being present in an infinitesimal region around 2: E R3, That is, the expected number of electrons in B C R is proportional to (s(x))..
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Assume have set of 2D images ( of size Npix Npxx representing 2D projections of different copies of the same protein in different conformations. Traditionally, the goal cryo-EM heterogeneous reconstruction has been to recover, for each image the electron density map V, corresponding to the underlying conformation present in image i; see Section for a review of these methods. However, following recent works, e.g., sg.,|Rosenbaum al. (202T); [Zhong et al. (2021b), aim at recovering, for each image i, the underlying structure S; explaining the image. That is, we try to recover the precise position in I of each residue.
METHOD CRYOSPHERE
In this section, we present our method for single-particle heterogeneous reconstruction, denoted cryoSPHERE. The method focuses on structure instead of volume reconstruction. It differs from the previous (Rosenbaum et al., (021) and concurrent (Li et al. 2023) works along this line in the way the movements of the residues on a residue level and then imposing a loss on the reconstructed structure, our method learns to decompose the amino-aeid chain of the protein into segments and, for each image 4, rigidly move the learnt segments of a base structure Sp to match the conformation present in that image. This is motivated by the fact that different conformations of large proteins often can be explained by large scale movements of its domains (Mardt et al. 2022).
constrained: instead of deforming the base structure
The base structure So can be obtained using methods like AlphaFold (Jumper al.) 2021 and RosettaFold (Back et al. 2021), based on the amino-acid sequence of the protein. In Section S we further fit the AlphaFold predicted structure into a volume recovered by a custom backprojection algorithm provided by [Zhong et al. (2020).
We use a type of VAE architecture, see Figure [ We map each image to a latent variable by stochastic encoder, which is then decoded to rigid body transformation per segment. Based on these transformations and the segment decomposition, the underlying structure Sp is deformed, posed and turned into volume that is used to create a projected image. This image is then compared to the input image. After that, the backward pass updates the parameters of the encoder, decoder and Gaussian mixture. We now describe the details of our model.
3.1 IMAGE FORMATION MODEL
To compute the 2D projection of the protein structure S, we first estimate its 3D electron density map V:
(- ( 20
(-Ilr-al a
Vs(r) = Σ An exp
(1)
aES
where Aa is the average number of electrons per atom in residue a, r E RR3 and by default. Hence, the protein's 's electron density is approximated as the sum of Gaussian kernels centered on its C. atoms. From these density maps, we then compute an image projection Je. X as:
I (Tr = g / VRs+e(r)drs.
where (Ta, Ty) E R² are the coordinates a pixel. T: E is the coordinate along the 2 axis, R E (3)) a rotation matrix and E 3R3 is a translation vector. The abuse of notation RS + means that every atom of S is rotated according to R and then translated according to t. The image finally convolved with the point spread function (PSF) g, which in Fourier space is the contrast transfer function (CTF), see Vulovié et al. (2013). Note that the integral can be computed exactly for our choice of approximating the density map as a sum of Gaussian kernels, which significantly reduces the computing time.
3.2 MAXIMUM LIKELIHOOD WITH VARIATIONAL INFERENCE
To learn distribution of the different conformations, we hypothesize that the conformation seen in image I, depends on a latent variable 2; E R with prior p(z;). Let fo(S,, 2) a function which, for given base structure Sn and latent variable 2, outputs new transformed structure S. This
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function depends set of learnable parameters 9. Then, the conditional likelihood of an image I E R Nan x Na with a pose given by a rotation matrix R and a translation vector t modeled as Po(I*|R, So. = ([ (o 2)), o noise). where moise is the variance of the observation noise. The marginal likelihood is thus given by
Po [ R. So) = f S (o S).2))(2)d..
In practice, the pose (R, t) of given image is unknown. However, following similar works (Zhong et al. 2021b et al. p. we suppose that we can estimate and to sufficient accuracy using
off thellll- methods (Scheres 2012 Punjani al. (017).
Directly maximizing the likelihood (3) is infeasible because one needs to marginalize over the la- tent variable. For this reason, adopt the VAE framework, conducting variational inference p=(z[?*) xx po(7"| (z))),) and simultaneously performing maximum likelihood estimation on the parameters
Let go (z) denote an approximate posterior distribution over the latent variables. We can then maximize the evidence lower-bound (ELBO):
C(O. v) llog po ([*[=]] (KL(<(<//""))l(>
(4)
which lower bounds the log-likelihood logpo( Here DKL. denotes the Kullback-Leibler (KL) divergence. In this framework fo is called the decoder and (=(=//") the encoder.
3.3 SEGMENT DECOMPOSITION
To handle the often very low SNR encountered in cryo-EM data, we regularize the transformation of the structure produced by the decoder by restricting it to transforming whole segments of the protein. We fix maximum number of segments Nogmm E {l..... ] and we represent the decomposition of the protein by stochastic matrix G E Ree X.egn, The rows of represent "how much of each residue belongs to each segment", and our objective is to ensure that each residue primarily belongs to one segment, that is:
Vi {1.... fL....,Rrees Pres), 3m* E =f1....,
s.t Σ Gim <1
(5)
m
We also aim for the segments to respect the sequential structure of the amino acid chain, and the model to be end-to-end differentiable. Without end-to-end differentiability. could not apply the reparamcterization trick and we would have to resort to Monte Carlo estimation of the gradient of the segments, which has higher variance, e.g. Mohamed al. (219)).
To meet these criteria, we fit a Gaussian mixture model (GMM) with Nsegm components on the real line supporting the residue indices. Each component has mean Pmn- standard deviation o. and a logit weight an- The {am} } are passed into softmax to obtain the weights { } of the GMM, ensuring they are positive and summing one. We further anneal the Gaussian components by temperature > O, and define the probability that a residue belongs to segment m as:
fo(illm mm
Gim
(6)
Σ Naaen (o(ilue, of m
ZK=1
where d(rlu, o²) is the unidimensional Gaussian probability density function with mean and variance o² and is a fixed hyperparameter. If sufficiently large, we can expect condition (5) to be verified. See Figure for an example of segment decomposition using Gaussian mixture.
In this "soft" decomposition of the protein, each residue can belong to more than one segment, allowing for smooth deformations. In addition, the differentiable architecture is amenable to gradient descent methods, and a well chosen 7 can approximate a "hard" decomposition of the protein. We 20 in the experiment section. In experience, this segmentation procedure is very robust to different initialization and converges in only a few epochs.
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3.4
DECODER ARCHITECTURE
The decoder describes the distribution of the images given the latent variables, which include:
0.0025
0.0020
0.0015
00.0010
1. One latent variable E IR per image, parame- terizing the conformation.
0,0000
Residue
2. The global parameters (J4m; Fw: Omm I m= of the GMM describing the segment decomposition.
Mag
Figure 2: Example segments recov- ered with Gaussian mixture of 6 com- ponents.
Given these latent variables and a base structure So, we parameterize the decoder fe in three steps. First, neural network with parameters maps 2; E RR to a set of rigid body transformations, one for each segment m 1, ................................ Nesgm- The transformation of segment m represented by a translation vector im and unit quaternion 4n (Vicci 2001). which can further be decomposed into an axis of rotation 6n, and rotation angle 5m. Second, given the parameters of the GMM, we compute the matrix G. Finally, for each residue i of So, we update the coordinates of all its atoms {ass 14,
First, agk is successively rotated around the axis om with an angle Gsmóm for m E {1...., N.ggm obtain updated coordinates d'g.
2. Second, it is translated according to: ax = d/a Ej=m Gim
This way, the transformation for residue incorporate contributions from all segments, proportion- ally on how much they belong to the segments. If condition (5) met, a roughly rigid motion for each segment can be expected.
3.5 ENCODER AND PRIORS
We follow the classical VAE frame- work. The distribution (v((((* is given by a normal distribution N(u(T*), diag(o?(T*))) where μ RL and a E T2 are generated by a neural network with parameters vi taking an image 1 as input. Additionally, the ap- proximate posterior distribution on the parameters of the GMM is chosen to be Gaussian and independent of the input image:
0.10
The di disterran
60
Fredicedd distar ces
0.08
30
0.06
0.04
40
0.02
x-y
0.00
30
35
40
45
50
55
80
30
30
Figure 3: MD dataset SNR 0.001. Left: Histograms of the distances of the two upper domains. The true dis- tances are in green. The recovered distances are in blue. Right: Predicted against black line represent x The correlation between the predicted and true distances 0.73. For the same plot for cryoStar, see Appendix B.2 of the supplementary file.
Mm NV 3
distances in Angström. The
mm NVo Po.n)
Om N(vom B2m)
where [V/mm= 9pm Vams Boms Vam. Bom Mer are parameters that are directly optimized. In practice we use ELU+1 layers for o.n to avoid negative or null standard deviation.
Finally, we assign standard Gaussian priors both the local latent variable 2, ~ /o Iz), and the
Nugn global GMM parameters {pm, Tm ams Fyme1 This reparameterization (Kingma & Welling 2014) straightforward for a Gaussian distribution. Calculating the Kl.-divergence between two Gaussian distributions as in equation 4 is also straightforward.
3.6 LOSS
Since the images may be preprocessed in unknown ways before running cryoSPHERE, we use a correlation loss between predicted and ground truth image instead of mean squared error loss,
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present in the dataset and perform maximum Published as a conference paper at ICLR 2025
similar to (Li et al. 2023):
-I I(R ts. (Su, 2))
Loore
||4711 x ([, 44 (So z))1
where denotes the dot product. The total loss to minimize writes:
(8) In experience, it is unnecessary to add any regularization term to the correlation and KL di- vergence losses, except for datasets featuring a very high degree of heterogeneity. In that case, we offer the option of adding continuity loss to avoid breaking the protein and clashing loss to avoid
((,,1*) Leorr + DKL (xL((9(2(( (x({1))))))))
clashing residues, as it is done in (Rosenbaum et al. 2021; Li et al. 2023 Jumper et al. 2021) We describe these losses in Appendix A. of the supplementary file.
RELATED WORKS
Two of the most popular methods for cryo-EM reconstruction, which are not based on deep learning. are RELION [Scheres] 2012) and cryoSPARC (Punjani et al., (017). Both methods perform volume reconstruction, hypothesize that & conformations a posteriori estimation over the k density maps, thus performing discrete heterogeneous reconstruc- tion. Both of these algorithms operate in Fourier space using an popeecatioo-maximizzae Dempster et al. (197)) and are -mootiooti the poses are refined for each image. Other ap- proaches perform continuous heterogeneous reconstruction. For example, 3DVA (Punjani & Fleet, 202 uses probabilistic principal component analysis model to learn latent space.
algorithm
Another class of methods involve deep learning and typically performs continuous heterogeneous reconstruction using VAE architecture. Of those that attempt to reconstruct density map, cryo- DRGN (hhng et al. 2020) (020)2021a) and CryoAI (Levy et al. 2022) use a VAE acting on Fourier space to learn latent space and mapping that associates a 3D density map with each latent variable. They perform non-amortized and amortized inference over the poses, respectively. Other methods are de- fined in the image space, e.g. 3DFlex (Punjani & Fleet 202 and cryoPoseNet (Nashed et al,, 202I]. They both perform non-amortized inference over the poses. These methods either learn, for a given image T,, (V+()))) the values set of N3, fixed 3DD coordinates {ax), representing
the volume a grid (explicit parameterization), or they learn an actual function Vi : R - IR in the form of neural network that can be queried chosen coordinates (implicit parameterization) These calmmebbsed methods cannot use external structural restraints or force fields as additional information. This limits their applicability to low SNR data sets, which are frequent in protein cryo EM.
Other deep learning methods attempt to directly reconstruct structures instead volumes and share common process: starting from plausible base structure, obtained with e.g. AlphaFold (Jumper jct al. 2021). ). for each image, they move each residue of the base structure to fit the conformation present in that specific image. These methods differ how they parameterize the structure and in the prior they impose on the deformed structure or the motion of the residues. For example Atom VAE (Rosenbaum al. 2021) considers only residues and penalizes the distances between two subsequent residues that deviate too much from an expected value. CryoFold (Zhong et al.) 20216) considers the residue centers and their side-chain and also imposes loss on the distances between subsequent residues and the distances between the residue centers and their side-chain. Unfortunately, due to the high level of noise and the fact that we observe only projections of the structures, these "per-residue transformation" methods tend to be stuck in local minima, yielding unrealistic conformations unless the base structure taken from the distribution of conformations present in the images (Zhong al. 2021b), limiting their applicability real datasets. Even though Atom VAE (Rosenbaum et al. 2021 1) could roughly approximate the distribution of states of the protein, it was not able to recover the conformation given specific image.
To reduce the bias that the base structure brings, DynaMight (Schwab et al. (023) fits pseudo-atone in consensus map with neural network directly. Similar to work, several other methods constrain the atomic model to rigid body motions. For example e2gmm (Chen & Ludtke, 202T] Chen al. 2023) deform nominal structure So based on how much its residues close

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learnt representation Ssmall of So- This is similar to GMM, except that their takes place in R and is not used to perform rigid body motion. Instead, they ask the user to define the segmentation in later step. This is in contrast to cryoSPHERE, which learns the motion and the segmentation concurrently. Using DynaMight (Schwab et al., 2024), Chen et al. (2024) developed a focused refinement patches of the GMM representation of the protein. These patches are learnt using k- means on the location of residues and do not depend on the different conformations of the data set. This in contrast to cryoSPHERE where the learning of the segments of the protein is tightly linked to the change of conformation. Concurrently to our work, et al. (2023) developed cryoStar which learns translate each residue independently using a variational auto-encoder. They enforce the local rigidity of the motion of the protein by imposing a similarity loss between the base structure and the deformed structure as well as clash loss. The interested reader can see Donnat et al. (2022) for an in-depth review of deep learning methods for cryo-EM reconstruction.
The reconstruction methods relying on an atomic model, such as cryoStar, DynaMight or cryoSPHERE offer the possibility to the user to provide prior information via this atomic model. They also offer the possibility of deforming the protein according to chemical force fields, This is not the case of the methods performing volume reconstruction without such an atomic model.
EXPERIMENTS
In this section, we test cryoSPHERE on a set of synthetid]] and real datasets with vary- ing level of noise and compare the results to cryoDRGN (Zhong et al.) 2020 and cryoStar (Li cet al. 2023). CryoDRGN state-of-the-art method for continuous heterogeneous recon- struction, in which the refinement occurs at the level of electron densities, while cryoStar is structural method similar to ours. To our knowledge, the code for AtomVAE and CryoFold not available and non-trivial to reimplement. For this reason we focus our comparison on the aforementioned methods, which have furthermore reported state-of-the-are performance. In Appendix B. we demonstrate that cryoSPHERE able to recover the exact ground truth when it exists. We also discuss its performances with varying SNR and Nsegm and show how to debias cryoSPHERE results using DRGN-AI or cryoStar volume method in Appendix B.2 Finally, Appendix cryoStar.
(Tycans
Caree on
Care
0,500
0.145
0.000
DC
10.0
5.0
3.5
2.5
7.0
DC
10.0
5.0
3.5
2.5
7.0
Recolution
Recolution
Figure 4: MD dataset. Left: cryoSPHERE Recovered segments. The colors denotes different contiguous domains. Middle and right: mean FSC comparison +/- one standard deviation, for cryoSphere and cry- oDRGN and cryoStar. For a comparison between cryoStar and cryo- DRGN, Appendix B.2 in the supplementary file.
B.5 compares the computational costs of cryoSPHERE and
5.1 MOLECULAR DYNAMICS DATASET: BACTERIAL PHYTOCHROME.
As a more difficult test case we simulate a continuous motion of a bacterial phytochrome, with PDB entry 4Q0J (Burgie al. 2014). The trajectory starts at the closed conformation of Figure IT and ends at the most open conformation on the same figure. It corresponds to a dissociation of the two top parts of the protein. This dataset has very low SNR of 0.001. Our base structure is obtained by AlphaFold and is subsequently fitted into a homogeneous reconstruction given by the backprojection algorithm. We train cryoSPHERE with Nagmm 25, cryoStar, and cryoDRGN for 24 hours each, us- ing the same single GPU. We get one predicted structure per image for crYOSPHERE and cryoStar, that we turn into volumes using (T). and one predicted volume per image for cryoDRGN. See Ap- pendix B.2 in the supplementary file for details and comparison with different values of N Note that since both cryoSPHERE and cryoStar use a nominal structure, we fit the structure we obtained through AlphaFold in the consensus reconstruction obtained by backprojection and use that exact same structure as the nominal one for both methods.
"ee Appendix B of the supplementary file for details on how we created the synthetic datasets.

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SB3
Helicase
Helicawe
Hellicase
Foot
Left and middle left: different views of the structures corresponding to the red dots of Figure (48 The motion goes from red (left in the first principal component) to white blue (right of the principal component). Only the Ca atoms are shown. Right and middle right: different views of two volumes recovered by training DRGN-AI on the latent space of cryoSPHERE. The U2 domain disappears on the volume because compositional heterogeneity.
Figure 5: EMPIARI0180.
Figure 3 shows the predicted distance between the two upper parts of the protein being dissociated, against the ground truth distance for each image. In spite of the very low SNR, cryoSPHERE roughly recovers the right distribution of distances. More importantly, the correlation between the predicted distance and ground truth distance 0.74, showing that eryoSPHERE is able to recover the correct conformation given an image. This is in stark contrast with Rosenbaum al. (2021) who could not recover the conformation conditionally on an image. In addition, our model has learnt to separate the two mobile top domains from the fix bottom one, as shown by the segment decomposition in Figure Appendix B.2 in the supplementary file shows the same figures for cryoStar.
We plot the mean of the FSC curves between the predicted volumes and the corresponding ground truth volumes in Figure 4 for cryoSPHERE, cryoDRGN and cryoStar. CryoSPHERE performs better than both cryoDRGN and cryoStar at both the 0.5 and 0. 0.143 cutoffs. We attribute this to three key properties. Firstly. we fit our base structure into consensus reconstruction. This step corrects the position of the medium-scaal boosting the FSC of cryoSPHERE at the 0.5 cutoff.. Secondly, acting directly on the structure level offers finer resolution than cryoDRGN given the level of noise. Figure 32 shows that cryoDRGN underestimated the opening of the protein and sometimes gives very noisy volumes. That explain why we outperform cryoDRGN at the 0.143 cutoff. Finally, cryoSPHERE is rigidly moving larger segments of the protein. This provide better resistance to high levels of noise and overfitting compared to moving each residue individually like cryoStar does, providing possible explanation to the improvement compared to cryoStar at the 0.143 cutoff.
elements of the base structure that could have been misplaced,
5.2 EMPIAR 10180
We now demonstrate that cryoSPHERE is applicable to real data as well as large proteins. We run cryoSPHERE on EMPIAR- -0180 Plaschka et al. (2017), comprising 327 490 images of pre- catalytic spliceosome with 13 941 residues, making it computationally heavy dataset to tackle. We use the atomic model by Plaschka et al. (2017) (PDB: 5NRL).
Figure 5 shows set of ten structures taken evenly along the first principal component of the latent space. To interrogate if these structures contain bias from the structural constraints, we perform a volume reconstruction step similar to cryoStar Phase II, see Figure 5
Traversing the first principal component shows that the Sf3b domain gets incurvated down while the helicase move closer to the foot of the protein. This is in line with the literature (Li et al. 2023 Plaschka et al. 2017). The motion of the protein also brings the alpha helix of the Spp381 domain closer to the foot, as corroborated by Li al. (2023). Comparison between the recovered structures and volumes (Figure 50) shows similar movements, indicating small amount of bias from the structural constraints. In addition, the absence of density corresponding to the U2 domain in the volume indicates that it there is compositional heterogeneity that cryoSPHERE could not detect, see Figure 5 We provide movie of the motion and more structures and volumes in appendix B.3 in the supplementary file.
5.3 EMPIAR- 12093
We now tackle the recently published EMPIAR- 12093 (Bódizs et al. 2024). This dataset comprises two sets of images: non-activated (Pfr) and one activated (Pr). These dataset are very challeng-
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CryoSTAR
CryoSPHERE
Pr state
90
Clashes & detormed a helllcos
Scissoring moiion
Pfr state
90
Figure 6: EMPIARI Left of the black line: Ten structures sampled along PCI, for cryoSPHERE and cryoStar. Right of the black line: examples of volumes reconstructed by training the cryoStar volume method on the latent space cryoSPHERE for debiasing. The blue and red volumes correspond to the first and last volumes along PC1. Top row corresponds to Pr; bottom row to Pfr.
ing because of the high level of noise and heterogeneity of the protein. especially in the Pfr dataset. Traditional methods like cryoSpare (Punjani et al. 2017) or cryoDRGN (Zhong et al. 2020: 202Ia) fail at reconstructing the upper part of the protein, see Bódizs al. (024) and Appendix [B.4
Figure shows principal component traversal for cryoStar and cryoS For Pr, both methods in strong agreement and reveal rotation of the upper domain around its axis, while the lower part remains stationary. This aligns with previous studies (Wahlgren et al.) Malla et al. 2024)).
The Pfr dataset showcases an lower SNR and more dynamical protein: the protein opens up completely. From consensus reconstructions alone, one could suspect that the upper domains are cut off in the sample prepara- tion procedure. However, the protein is complete in Pr (light-activated) structure and the photocycle is re- versible, (Takala al.) 2014) suggesting that this is not the case and that strong conformational heterogeneity that is at play.
Number clashing residues
250
200
150
100
50
cryoSPHERE
cryoStar
For Pfr cryoStar is unable to produce physically plausi- ble results: the top part of the protein appears disordered and shows random motion. In addition, cryoStar does not recover the "scissoring" motion of protein, which is thought to be active (Bódizs et al.222024) On the contrary, cryoSPHERE gives a high level of motion in a structured manner and recovers the "scissors" opening of the protein.
Figure 7: EMPIAR 12093. Distribu- tion of the number of clashes for 2000 randomly chosen structures Pfr dataset, for cryoSPHERE and cryoStar. Two non contiguous residues are said be clashing if their distance is less than 4 A.
Without any clashes (Fig. 7 [Bódizs et al. 2024).
Analysis of the dataset on phytochromes illustrates the scope and limitations of the different meth- ods. Pure image-based methods (i.e. cryo DRGN) already fail on the Pr state with its intermediate disorder, while cryoSTAR and cryoSPHERE succeed in obtaining reasonable reconstructions (Fig- ure 6). For the Pfr state it becomes evident that cryoSTAR struggles with the high noise and large motions encoded in the dataset. Its deformation- based approach results in unphysical motions along the first principal component, often leading to structural clashes. In contrast, cryoSPHERE handles the noise effectively, producing physically plausible large-scale motions both the upper and lower domains, see Figure 7 the supplementary movies and B.4 We assign this superior performance to the higher degree of structural constraints that are used in cryoSPHERE compared to cryoSTAR.
We also performed debiasing of cryoSPHERE with volume method and show examples of re- constructed volumes in Figure 6 For Pr, recovered densities are visible for the entire protein and confirm the dynamics of the upper domains, confirming the absence of compositional heterogeneity and a minimum bias due to structural constraints. However, for Pfr, meaningful density of the upper (dynamic) part of the protein cannot be recovered, because the signal level in the averaged

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density is too low. Thus, for this most dynamic protein case, volume-based debiasing is not possible, despite the fact that the structure based cryoSPHERE finds solutions that fit the data set.
DISCUSSION
CryoSPHERE presents several advantages compared to other methods for volume and structure reconstruction.
Deforming a base structure into a density map avoids the compu- tationally expensive NN2xx evaluation required by a decoder neural network in methods implicitly parameterising the grid, such as Zhong al. (2021a); Levy al. (2022). Furthermore, direct defor- mation of structure directly avoids the need for subsequent fitting into the recovered density map.
Efficiency in Deformation:
Learning one rigid transformation per segment, where the number of segments is much smaller than the number of residues, reduces the dimension- ality of the problem. This results in a smaller neural network size compared to approaches acting on each residues, such Rosenbaum al. (202T). Rigidly moving large portions of the protein corresponds to low-frequency movements, less prone to noise pollution than the high-frequency movements associated with moving each residue independently. In addition, since our goal is to learn one rotation and one translation per segment, latent variable of dimension 6 x NNsgm is, in principle, sufficiently flexible choice to model any transformation of the base structure. Choos- ing the latent dimension is more difficult for volume reconstruction methods such as (Zhong et al.) 2021a).
Reduced Dimensionality and Noise Resilience:
Interpretability: CryoSPHERE outputs segments along with one rotation and one translation per segment, providing valuable and interpretable information. Practitioners can easily interpret how different parts are moving based on the transformations the network outputs. This interpretability is often challenging for deep learning models such as Zhong et al. (2021a); Rosenbaum et al. (202T).
Section 5 and Appendix B.2 demonstrate cryoSPHERE's capability to recover conformational het- erogeneity while performing structure reconstruction. The division into Nesmm is learned from the data and only marginally impacts the FSC to the ground truth. Moreover, cryoSPHERE recovers the correct motion for the entire range of Neegm values and is able to keep the minimum necessary number of domains when the user sets too high (Appendix B.I.
Structural restraints allow interpretation of low SNR datasets: Is evident that structural re- straints as implemented in cryoSPHERE (this work) and cryoSTAR provide additional restraints that pure volume methods (i.e. cryoDRGN) lack, thus giving better reconstructions for high noise data sets. The additional restraints may introduce bias, which needs to be alleviated using a backprojec- tion algorithm. This, combined with cryoSPHERE's latent space, achieves better 0.5 cutoffs than cryoDRGN, indicating its effectiveness in resolving conformational heterogeneity and debiasing the results. If such volume is unavailable, simply increasing Nugm can reduce the bias. As a note of caution find that for most dynamic protein studies here (the Pfr state of the phytochrome), we find that volume- debiasing fails because of the very low electron density levels in the reconstructions. Here, other metrics should be developed in the future.
Summary: Our study opens up for significant advancements in predicting protein ensembles and
dynamics, critically important for unraveling the complexity of biological systems. By predicting all-atom structures from cryo-EM datasets through more realistic deformations. our work lays the foundation for extracting direct insights into thermodynamic and kinetic properties. This work an important milestone in showing that one can learn a segmentation of the protein that is intimately linked to the change of conformation of the underlying protein, in an end-to-end fashion. In the future, we anticipate the ability to predict rare and high-energy intermediate states, along with their kinetics, a feat beyond the reach of conventional methods such as molecular dynamics simulations.
It would be interesting to assess how much our segmentation correlates with bottom-up segmen- tation into domains conducted on the "omics" scale, see e.g. Lau et al.] (2023). To achieve this quantitatively, would need many examples of moving segments from cryo-EM investigations to match the millions of segments from the "omics" studies. Therefore, we leave this investigation to later work.
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ACKNOWLEDGEMENT
This work was financially supported by the Wallenberg AI, Autonomous Systems and Software Program (WASP) and the Data-Driven Life Science Program (DDLS) funded by the Knut and Al- ice Wallenberg Foundation through the WASP-DDLS collaboration, the Swedish Research Council (project 2020-04122, 2024-05011) 1). and the Excellence Center Linköping Lund in Informa- tion Technology (ELLIIT). Our computations were enabled by the Berzelius resource at the National Supercomputer Centre, provided by the Knut and Alice Wallenberg Foundation.
We thank Claudio Mirahello at the National Bioinformatics Infrastructure Sweden at Scil Lifel.ab for providing access his AlphaFold installation on Berzelius and Nancy Pomarici for providing input files and explanation of the metadynamics simulation.
7 REPRODUCIBILITY STATEMENT
As part of the current paper, we provide a github link the source code in Section T We also describe in detail how we generate the synthetic datasets in Appendix B.2 and the hyperparameters chosen to run cryoStar, cryoSPHERE and cryoDRGN in Appendix for each of the experiments.
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14


Reviews:
[{"review_id": "syr25MFaXZ", "reviewer": "Reviewer_RJxo", "summary": "The authors introduced a new method, cryoSPHERE, that resolves the continuous heterogeneity problem in cryo-EM reconstruction. Similar to some other methods, cryoSPHERE uses an atomic model as the input, makes segmentations, and uses the results as the regularization in finding the flexible movement in the cryo-EM dataset.", "questions": "Can the authors elaborate how the density maps (volumes) are reconstructed in cryoSPHERE (like in Fig. 6)? I understand how backprojection in cryo-EM works, but still find it difficult to understand how this is performed exactly.", "limitations": "", "rating": 6, "confidence": 5, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "Instead of finding the deformation for each residue, cryoSPHERE learns a segmentation of the amino acid or nucleotide chains and deforms the segment to fit the heterogenous cryo-EM data. This is indeed a valid assumption in many cases especially for large complexes  like the spliceosome (EMPIAR-10180).", "weaknesses": "Major:\n1. CryoSPHERE uses an atomic model as the reference and formulate the conformational heterogeneity in cryo-EM dataset as the deformation of the segments of the reference model. In finding the optimal segments, the approach is similar to what used in e2gmm, where a GMM with N_{segm} components is fitted. This segmentation result is then used in a cryoSTAR/DynaMight-like setting, replacing the regularization losses and compute predicted projections to compare with the particle images. The author should also compare the result with other similar methods, including DynaMight, e2gmm, and maybe 3DFlex, and discuss the outcomes.\n\n2. In cryo-EM, density maps are extremely important in evaluating the validity of the results. However, the authors only presented the density maps for one of the synthetic dataset. Especially in the case of under the hypothesis that the heterogeneity is modeled as the deformations of a canonical density, it is very important that the reconstructed canonical density is free from any obvious artifact.\n\n3. The FSC comparison in Sec. 5.2 between cryoSPHERE and cryoDRGN and cryoSTAR may not be very fair, since cryoSPHERE's density is reconstructed under the assumption that there exist a canonical density, while cryoDRGN does not hold the same hypothesis, and cryoSTAR explicitly avoids this to reduce the unwanted bias from the reference model. A fair comparison would be made against DynaMight and 3DFlex.\n\n4. Lack of ablation study. The authors should discuss how N_{segm}, which is a key hyperparameter that is related to the degree of freedom in the segmentation, affects the results in both synthetic and real data.\n\n5. The authors compare cryoSPHERE to cryoDRGN and cryoSTAR, but from the experiments, it is unclear what the advantage that cryoSPHERE brings to the field. The only advantage seems to be the increase FSC resolution, but it is only shown in one synthetic dataset.\n\nMinor:\n1. The sequence of Fig.6 and Fig.7 in the paper is reversed.\n2. In the related works, e2gmm should also be considered as a method using deep learning.\n3. In the FSC comparison, if not comparing two half maps, the cutoff of 0.5 should be used and 0.143 is not meaningful. See [1] appendix for the reason (and also why 0.143 is the number widely used for half maps comprison). Therefore the discussion at the end of Sec. 5.2 about 0.5 and 0.143 cutoff improvement is not very correct.\n4. The writing could be better polished.\n5. I do not think the \"Ethics Statement\" content is proper as it lacks evidence.\n\n\n[1] Rosenthal, Peter B., and Richard Henderson. \"Optimal determination of particle orientation, absolute hand, and contrast loss in single-particle electron cryomicroscopy.\" Journal of molecular biology 333.4 (2003): 721-745.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "X9CPbccse6", "reviewer": "Reviewer_NciV", "summary": "- The authors propose a VAE and GMM-based approach to atomistic heterogeneous Cryo-EM reconstruction\n- The GMM part learns how to divide the amino acid chan in segments\n- The VAE learns how to encode images into latent variables and decode these variables along with the GMM output into segment-wise conformational changes\n- The authors claim that their method overcomes current challenges that atomistic models face, while also achieving improved interpretability by learning these segments (that ideally correspond to domains in the biological sense)", "questions": "Questions for clarification \n- In section 3.6 the authors say “Similar to Li et al. (2023), instead of using a mean squared error loss between the predicted and ground truth image, we use a correlation loss between the true and predicted image”. However, they do not motivate this choice. Would you please care to elaborate?\n- In section 5.1 the authors write “Testing the segment decomposition, we then run cryoSPHERE by requesting division into Nsegm = 4. The program learnt a first and third segment with 0 residues, a second segment with 1353 residues and a fourth segment with 157 residues (Figure 5). Thus, cryoSPHERE learnt segments according to the ground truth.” Is there a reason why we should expect that the model prefers to learn two segments?\n\nAdditional feedback\n- It reads as if equation 7 is the only loss that is being used. I assume that the VAE latent distribution is also trainable? It would benefit the readability if the authors clarify whether the loss is just the decoder loss or the whole loss.“\n- In section 5.3 the authors write “The first two principal components explain more than 96 percent of the variance.” In general, I would be very reluctant to make any claims about PCA in the latent space as you could come up with a transformation that scrambles the latent space and still gives the same decoding, but covers much less variance. I can imagine that Figure 7 is a nice illustration, but I would recommend to leave it with that.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The work is of high quality and overall very well-written. I’d like to highlight the following strong points of the paper:\n- The authors seem very well aware of the difficulties atomistic models have in terms of the optimisation landscape and address the problem adequately by regularising while keeping sequential information intact\n- The authors give useful biological remarks such as that their methods seems to recover domains, which is useful to know as a practitioner\n- The authors show that their method works on real data, which is actually the first time I’ve read this for atomistic models", "weaknesses": "There is one main weak point in the paper in my regard:\n- The authors do not address limitations of their method very thoroughly:\n    - whether or not their method is able to overcome poor initialisation of the initial structure S_0. This is one of the major risks of working with (pseudo)-atomistic models as shown in works such as DynaMight (Schwab et al. 2024) \n    - Whether or not local physical information such as bond lengths between atoms at the boundaries of the segments is preserved. This might be challenging", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "g07Q8fOIDb", "reviewer": "Reviewer_JGA5", "summary": "This paper introduces cryoSPHERE, a deep learning model that uses nominal protein structures (e.g., from AlphaFold) to break proteins into segments and fit these segments to various conformations in cryo-EM data.", "questions": "Could you include additional comparisons with methods like CryoSTAR, or Dynamight (e.g., computational cost, novelty, etc)", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "1. The approach of learning to decompose the protein’s amino acid chain into segments to represent various conformations is novel.\n2. The interpretability of moving part is highly beneficial for practitioners, offering a clearer understanding of conformational dynamics.", "weaknesses": "1. No meaningful improvements were observed over CryoStar\n    - Figures 3, 12, 13: It’s unclear what meaningful differences exist compared to CryoStar.\n    - Figures 14, 21: There doesn’t appear to be a significant difference between CryoStar and CryoSphere (CryoStar may even appear better).\n    - Figure 29: It would be great to compare CryoStar with CryoSphere rather than with CryoDRGN.\n2. Should change Figure 6:\n    - Instead of showing only CryoDRGN results and G.T, a comparison with CryoSTAR or Dynamight would be more appropriate (perhaps include Figure 32 and the CryoSphere result, as there are no CryoSphere results with SNR 0.001).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)