Digital twin of biomacromolecular thermodynamics in cryo-EM data

Abstract

The biological functions of biomacromolecules are linked to their thermodynamics. Thus, a quantitative understanding of the full scope of these dynamics, ranging from local fluctuations to large-scale conformational change on the free-energy landscape, is essential for elucidating the biomacromolecular mechanisms. Recently, cryo-electron microscopy (cryo-EM) has garnered increasing attention for determining novel static structures with large molecular sizes and complex structural assemblies. However, identifying their comprehensive thermodynamic behaviors remains challenging. In this study, we present a deep learning framework that reconstructs a provable “digital twin” of the thermodynamics recorded in cryo-EM data with a theoretically guaranteed isometric autoencoder. This framework enables a thermodynamic analysis of the heterogeneous conformational states and their transition pathways in a semi-automatic manner. We demonstrated the application of the framework to the cryo-EM data obtained from a 50S-Ribosome and two SARS-CoV-2 spike proteins, which yielded novel biological insights without supplementary assumptions. This work establishes a provable approach to achieve the quantitative analysis of biomacromolecular thermodynamics using only cryo-EM data, whereas current similar methods do not guarantee the exact construction of the free-energy landscape. Overall, our approach has the potential to advance biological foundations by uncovering the thermodynamics of complex biological systems.

Understanding the thermodynamics of biomacromolecules is crucial for elucidating their mechanisms yet remains challenging with current cryo-electron microscopy (cryo-EM) methodology. Here, the authors introduce a deep learning framework that reconstructs a “digital twin" of cryo-EM data, enabling precise thermodynamic analysis using only cryo-EM data and revealing insights into biomolecular dynamics without supplementary assumption.

Introduction

Biological phenomena are governed by the functions of biomacromolecules, which are intrinsically linked to their conformational dynamics on the free-energy landscape, i.e., their thermodynamic characteristics^1–3. The dynamic behaviors of biomacromolecules are represented as conformational transitions among heterogeneous structural states, and they span a wide range of timescale and magnitudes, from rapid local fluctuations to large-scale structural changes. Solving all their relevant conformations, including transient states and capturing the transition pathways among them in a quantitative manner is paramount in the field of life sciences.

Analyses of static three-dimensional (3D) structures provide a basis for understanding their associated dynamic behaviors. In recent years, single particle analysis of cryo-electron microscopy (cryo-EM)⁴ has gained popularity because of its capability to solve the high-resolution 3D structures of large and complex biomacromolecules, of which structural determination has been difficult when using other techniques, such as X-ray crystallography. A particularly notable advantage of cryo-EM is its capability to record all structural states present in solution, allowing for solving heterogeneous conformations, such as heterogeneous states during ribosome assembly⁵, from data acquired under a single experimental condition. Consequently, cryo-EM has substantially advanced our understanding of the static structures of biomacromolecules.

Despite these advances, understanding the thermodynamics of biomacromolecules targeted using cryo-EM remains challenging. Molecular dynamics at an atomic resolution has been investigated via experimental and computational approaches, such as nuclear magnetic resonance (NMR) and molecular dynamics (MD) simulations^6,7. Nonetheless, applying these methods to large and complex macromolecules is often limited because of several obstacles, such as signal overlapping, signal broadening, and requirements of high computational costs. Therefore, while cryo-EM has dramatically advanced structural biology by elucidating static snapshots, a critical gap remains in the comprehensive understanding of their complex systems.

Considering these limitations, several machine learning (ML) methods have been reported to capture the molecular dynamics from a collection of biomacromolecular cryo-EM particle images, which record all the dynamic behaviors^8–16. For example, deep learning (DL) methods, such as cryoDRGN⁹ and 3DFlex¹², compute the continuous conformational changes of biomacromolecules, employing the flexible regression capabilities of deep neural networks. However, currently existing ML methods are not guaranteed to generate an exact duplication of the entire free-energy landscape that is recorded in cryo-EM data, hindering identification of the reliable behaviors through quantitative analyses. Furthermore, the characterization of complex structural transition pathways, such as parallel assembly pathways in 50S-Ribosome, often necessitates reliance on laborious manual estimation that is guided by supplementary experimental information and prior knowledge. Davis et al. reconstructed heterogeneous static structures of multiple assembly intermediates of the 50S-Ribosome large subunit with an ML software cryoSPARC¹⁷ and inferred their assembly pathways by manually ordering them in a consistent manner based on experimental observations, such as “increasing molecular weight”, and the functional context from prior knowledge⁵. Consequently, an automated methodology for the comprehensive and unbiased extraction of the full thermodynamic ensemble of biomacromolecules directly from cryo-EM data is currently lacking.

Herein, we propose PaStEL with cryoTWIN (Fig. 1), a DL method that enables an unbiased extraction of the full thermodynamic behavior solely from a collection of biomacromolecular cryo-EM images, which is supported by rigorous theoretical guarantees. In the method, an isometric autoencoder, cryoTWIN, is first trained with the images. Then, the trained cryoTWIN exactly duplicates the whole free-energy landscape recorded in the images into a latent space via a theoretically guaranteed isometric embedding, while allowing for the accurate reconstruction of 3D structures through its decoder. Subsequently, PaStEL generates energetically favorable conformational pathways as minimum free-energy paths¹⁸ along with their occurrence probabilities using the latent space and the trained decoder. Thus, our method provides a provable digital twin of the thermodynamics recorded in biomacromolecular cryo-EM data, enabling the quantitative understanding of the comprehensive biomacromolecular thermodynamics.

Fig. 1. PaStEL with cryoTWIN: a framework for the quantitative analysis of biomacromolecular thermodynamics.

The proposed method, PaStEL with cryoTWIN, provides a digital twin of the thermodynamics recorded in biomacromolecular cryo-electron microscopy (cryo-EM) data, thereby enabling quantitative analysis of the thermodynamics. The method is performed through the following three steps. (Step 1) Train an isometric autoencoder, cryoTWIN, with cryo-EM particle images that record the free-energy landscape. (Step 2) These particle images are input into the trained encoder, and an exact duplication of the free-energy landscape is constructed in a low-dimensional latent space. This duplication is expressed as a simple mathematical formula. Using this formula and a pathway searching algorithm PaStEL, optimal paths between various latent variables representing conformational states are calculated. (Step 3) The calculated optimal paths are passed through the decoder to construct the minimum free-energy paths as sequences of 3D density maps. Each pathway is assigned its probability of occurrence. These sequences, occurrence probabilities, and the obtained simple mathematical formula are then used to perform a quantitative analysis.

Our main results are threefold. First, we mathematically proved that our method creates a digital twin of the thermodynamics recorded in the cryo-EM data. Second, using simulated cryo-EM images for Chignolin and Trp-cage, we verified accurate reconstructions of their diverse 3D structures and extensive free-energy landscapes. Finally, we demonstrated the efficiency and potential of our method in the field of structural biology research using real-world cryo-EM images of the 50S-ribosome, which undergoes complicated structural assembly of the subunits, and SARS-CoV-2 spike proteins, which undergo conformational changes to interact with a host cell receptor. A series of their analyses was performed using modest GPU resources, indicating that researchers in the cryo-EM community can readily apply this method to their targets. The derived thermodynamic behaviors of complex biological systems will provide unprecedented insights into their molecular mechanisms, advancing the field of life science research.

Results

Theory behind PaStEL with cryoTWIN

PaStEL with cryoTWIN generates optimal free-energy paths only from the particle images. This approach is guaranteed by two original theoretical findings: (i) optimally trained cryoTWIN using the particle images builds an isometric latent space for a conformational space, i.e., the free-energy landscape is expressed by the trained latent distributional model of cryoTWIN, and (ii) a generated path by PaStEL based on the trained cryoTWIN is equivalent to the MaxFlux path: optimal solution of the MaxFlux objective [ref. ¹⁸, Eq. (17)]. Intuitively, isometricity in (i) means that we can create a copy of the conformational distribution, which is equivalent info to the free-energy landscape for generating MaxFlux paths, in the low-dimensional latent space. Isometricity facilitates accessing the free-energy info via the latent space of cryoTWIN. As for (ii), it is a consequence of the isometricity and mathematical formula of the MaxFlux objective. Because the MaxFlux path is known to be an optimal conformational path on the free-energy landscape¹⁸, finding (ii) implies the generation of optimal free-energy paths. In the following, we present the findings (i) and (ii) in Theorems 1 and 2, respectively.

For the theoretical finding (i), let us first introduce two assumptions related to the particle image dataset: (A1) for the target biomolecule, a large number of sufficiently denoised 2D projection particle images with their pose orientations are collected without bias, and (A2) the target is under the equilibrium condition.

Theorem 1

(Isometric property and free-energy expression by cryoTWIN) Suppose that cryoTWIN is optimally trained with the particle image dataset, which satisfies (A1). Then, trained cryoTWIN builds an isometric latent space for a conformational space, as follows:

$${\underbrace{⟨\left(z+{\delta }_1\right)-z,\left(z+{\delta }_2\right)-z⟩}}_{=:\mathrm{d}{z}_1\cdot \mathrm{d}{z}_2}\propto {\underbrace{⟨V_{z+{\delta }_1}-V_z,V_{z+{\delta }_2}-V_z⟩}}_{=:\mathrm{d}{V}_1\cdot \mathrm{d}{V}_2},$$ 1

where $z\in {\mathbb{R}}^d$ is a latent variable, δ is an infinitesimal vector with an arbitrary direction, V_z is a conformation in the form of a 3D density map corresponding to the variable z, and d is the intrinsic dimension of the conformational manifold. For a latent variable z and an arbitrary infinitesimal vector δ, let $z^{\prime }:=z+\delta$. Then, as an immediate consequence of Eq. (1), $\parallel z-z^{\prime }{\parallel }_2\propto {\parallel V_z-V_{z^{\prime }}\parallel }_2$ holds. Moreover, $\forall z;p_{{\psi }^{*}}\left(z\right)\propto p\left(V_z\right)$ holds, where $p_{{\psi }^{*}}\left(z\right)$ is the trained latent distributional model in cryoTWIN, and $p\left(V_z\right)$ is the conformational distribution. Under the equilibrium condition (A2), the free-energy of V_z, denoted by E(V_z), can be expressed as follows:

$$E\left(V_z\right)=-k_{B}T\log p_{{\psi }^{*}}\left(z\right)+\mathrm{c},$$ 2

where k_B is the Boltzmann constant, T is the temperature, and c is a constant related to the intrinsic dimension and partition function of the conformational distribution. From Eq. (2), the free-energy difference between two conformational states V_z and $V_{z^{\prime }}$ is expressed by $\mathrm{\Delta }E:=E\left(V_z\right)-E\left(V_{z^{\prime }}\right)=-k_{B}T\log \frac{p_{{\psi }^{*}}\left(z\right)}{p_{{\psi }^{*}}\left(z^{\prime }\right)}$.

Proof

The proof can be found in Supplementary Information Section SI-1.1.

The training objective of cryoTWIN in Theorem 1, namely the expected training objective, is described in Definition 1; the empirical version of the expected training objective is given in Eq. (5). In Fig. 2a, we illustrate how to compute the empirical training objective, using the image dataset. Moreover, the conformation V_z is reconstructed from z using the decoder of trained cryoTWIN (Fig. 2b). For the latent distributional model p_ψ of cryoTWIN, we employ the Gaussian mixture model (GMM).

Fig. 2. Architecture of cryoTWIN.

a Diagram of how to define the empirical training objective of cryoTWIN. First, in the preliminary stage, for each particle image, the pose orientation $\widehat{R}$ is estimated. Then, the loss of each image is defined as the summation of two terms inside red rectangles: rate and weighted reconstruction error; the weight induces the isometric latent space. b Diagram illustrating how to reconstruct a 3D density map from a latent variable z after training cryoTWIN. First, the corresponding 3D Fourier volume to z is computed using all coordinates and the trained decoder. Then, the corresponding 3D density map is reconstructed by applying inverse FT to the volume.

We next present the theoretical finding (ii) as follows:

Theorem 2

(Generation of optimal paths on the free-energy landscape by PaStEL) Consider optimally trained cryoTWIN under (A1) and (A2). First, based on trained cryoTWIN, let us define PaStEL in two steps: (Step 1) obtain a minimizer of the following function with respect to the sequence of latent variables z^(0: m+1) ≔ (z⁽⁰⁾, z⁽¹⁾, z⁽²⁾, …, z^(m), z^(m + 1)):

$${\underbrace{{\sum }_{j=1}^{m+1}{\left.∣\right)\left.∣\right)z^{\left(j-1\right)}-z^{\left(j\right)}\left.∣\right)\left.∣\right)}_2/p_{{\psi }^{*}}\left(z^{\left(j\right)}\right)}}_{\mathrm{Ridgeline}\mathrm{length}\mathrm{from}{z}^{\left(0\right)}\mathrm{to}{z}^{\left(m+1\right)}},$$ 3

where z^(0:m+1) satisfies ∥z^(j−1) − z^(j)∥₂ ≈ 0 for j = 1, …, m + 1 and z⁽⁰⁾ (reps. z^(m + 1)) is a fixed start (resp. end) point. (Step 2) Return a sequence $\left(V_{z^{\left(0\right)}},V_{z^{{\left(1\right)}^{*}}},\dots ,V_{z^{{\left(j\right)}^{*}}},\dots ,V_{z^{{\left(m\right)}^{*}}},V_{z^{\left(m+1\right)}}\right)$, where $V_{z^{*}}$ is the reconstructed 3D density map from an optimized variable z^* using the trained decoder of cryoTWIN. Next, based on the isometric property of the cryoTWIN, we can rewrite the original MaxFlux objective [ref. ¹⁸, Eq. (17)] using the above-mentioned constrained sequence z^{(0:m + 1)}, as follows: $\arg {\min }_{z^{\left(0:m+1\right)}}{\sum }_{j=1}^{m+1}{\mathrm{e}}^{\frac{1}{k_{B}T}\left(E\left(V_{z^{\left(j\right)}}\right)-E_{\mathrm{ref}}\right)}\parallel V_{z^{\left(j-1\right)}}-V_{z^{\left(j\right)}}{\parallel }_2$, where E_ref is the reference energy. Then, the minimizer obtained at (Step 1) of PaStEL can optimize the rewritten MaxFlux objective.

Proof

The proof is given in the Supplementary Information Section SI-1.2.

The minimizer obtained in (Step 1) of PaStEL corresponds to black solid/dash paths on the latent energy landscape in Fig. 1, and the returned sequence in (Step 2) is indicated by black solid/dash paths on the output free-energy landscape in Fig. 1. Moreover, the last sentence in Theorem 2 implies that PaStEL generates the MaxFlux paths, i.e., optimal paths on the free-energy landscape, based on optimally trained cryoTWIN by the particle images.

Remark 1

The isometry guaranteed by Theorem 1 is local, not global. A local isometry preserves only the local distance structure of the original space, whereas a global isometry preserves the distance information between any two points. Consequently, even when a highly accurate local isometry is achieved, distances between distant points on the energy landscape estimated by cryoTWIN may be distorted and may not accurately reflect the distances between the corresponding conformations in conformational space. In contrast, Theorem 2 ensures that the thermodynamics computed by PaStEL can be distortion-free, even between distant start and end conformations. This is because the MaxFlux path is defined by integrating local thermodynamic segments. Each local segment is distortion-free, as it benefits from the underlying local isometry in Theorem 1. By integrating these accurate local segments, the resulting global path avoids accumulated distortion. In practice, as the local scale approaches zero, i.e., as the number of partitions m in Theorem 2 approaches infinity, the distortion in the thermodynamics computed by PaStEL converges to zero.

CryoTWIN builds an isometric latent space for a synthetic protein conformation space from the particle images

PaStEL is theoretically guaranteed to be able to generate optimal free-energy paths once cryoTWIN is optimally trained by the expected training objective. According to our theoretical analysis related to Theorem 2, PaStEL and the subsequent structural analysis require three properties of the trained cryoTWIN: (i) the isometric property of dz₁ ⋅ dz₂ ∝ dV₁ ⋅ dV₂ (Eq. (1)), (ii) free-energy equality by the trained latent distributional model p_ψ (Eq. (2)), and (iii) accurate reconstruction of the 3D density maps by the trained decoder. In practice, as we do not obtain the ideal training objective (Eq. (4)) in using a real-world dataset, we must employ the empirical version of the expected training objective (Eq. (5)). Hence, for practical use, it is important to hold the properties of (i)–(iii) for trained cryoTWIN based on the empirical objective.

To evaluate the aforementioned points, we analyzed three essential properties using the synthetic particle images for two proteins: Chignolin and Trp-cage. The diverse conformations on almost the entire free-energy landscape were prepared using multi-canonical MD (mcMD) simulations^19,20, and the 2D projection images were generated from these simulations according to their free-energy values (see Methods). The cryoTWIN trained with the synthetic images and their pose orientations demonstrated that the Pearson correlation coefficient between dz₁ ⋅ dz₂ and dV₁ ⋅ dV₂ for Chignolin and Trp-cage were 0.80 and 0.87, respectively (Fig. 3a, b). In addition, the pseudo free-energy landscapes depicted by $-\log p_{{\psi }^{*}}$, were in agreement well with the reference free-energy landscapes obtained from the generalized ensemble MD simulations (Fig. 3c, d; the correlation coefficients for Chignolin and Trp-cage were 0.89 and 0.93, respectively). These results demonstrated that (i) the isometric property and (ii) the free-energy equality were held in cryoTWIN trained via the empirical objective, allowing for the generation of ridgelines on the built latent space corresponding to the free-energy landscapes. Furthermore, the reconstructed 3D density maps from the latent variables were fitted well by the reference all-atom models at the corresponding position on the reference free-energy landscapes (Fig. 3e, f), demonstrating (iii) the accurate reconstruction of the 3D density maps by the trained decoder. Collectively, it was confirmed that PaStEL, combined with a trained cryoTWIN based on the empirical objective, can generate optimal free-energy paths for practical use. These results prompted us to apply our method to experimental cryo-EM data for yielding biological findings from the dynamic behaviors.

Fig. 3. Accurate reproduction of free-energy landscapes and 3D structures by cryoTWIN in synthetic proteins.

a, b Scatter plot related to the isometric property of cryoTWIN: dz₁ ⋅ dz₂ ∝ dV₁ ⋅ dV₂. Left and right plots were based on Chignolin and Trp-cage datasets, respectively. Quantity ρ expresses the correlation coefficient between dz₁ ⋅ dz₂ and dV₁ ⋅ dV₂. The orange line is a linear regression line for the blue points. c, d PCA visualizations of MD reference free-energy and estimated pseudo free-energy by cryoTWIN. Paired visualizations in (c) (resp. d) were based on the Chignolin (resp. Trp-cage) dataset. Color-bar expresses the standardized free-energy values. e, f Reconstructed 3D density maps by cryoTWIN and the corresponding reference all-atom model. Four representative density maps I to IV in (e) (resp. f) were reconstructed using four latent variables corresponding to coordinates I–IV in the cryoTWIN energy map of (c) (resp. d), while the four all-atom models correspond to four coordinates I–IV in the MD energy map of (c) (resp. d).

We remark that the results of the qualitative and quantitative comparisons between the proposed method and the state-of-the-art methods^8–16 are presented in Table SI-1 and Table SI-2 of Supplementary Information Section SI-3, respectively. Representative simulated cryo-EM particle images used in the quantitative comparison are shown in Supplementary Information Fig. SI-4. According to Table SI-1, among existing methods, only our method is theoretically capable of generating a digital twin of the comprehensive thermodynamics recorded in cryo-EM data. Furthermore, Table SI-2 shows that, in practice, our method generated a highly accurate duplication of the free-energy landscape, outperforming these existing methods with a large margin.

In Supplementary Information Section SI-4.1, we quantitatively evaluated the uncertainty introduced into the energy landscape estimated by cryoTWIN by variations in the training data and the random seeds used for learning. For the accuracy of the estimated values (reconstructed maps and their corresponding estimated energies) against the ground truth (all-atom models and their associated free-energy values) for each energy level, please refer to Supplementary Information Section SI-5.

PaStEL with cryoTWIN depicts the whole assembly pathways of 50S-Ribosome on the free-energy landscape from the experimental particle images

Ribosome is a machinery for protein synthesis, and the ribosomal assembly is known to be a multi-step process. To further demonstrate the efficiency of our method in real-world datasets, we analyzed cryo-EM 50S-ribosomal particle images (EMPIAR-10076).

CryoTWIN with latent GMM p_ψ was trained with the particle images and their pose orientations estimated by cryoSPARC¹⁷. After training, we reconstructed a set of density maps from 30 significant Gaussian mean vectors, ${\left\{{\mu }_i\right\}}_i$, representing a high density value $p_{{\psi }^{*}}\left({\mu }_i\right)$ that corresponds to the energetically stable conformation on the landscape. The previously defined labels to represent the heterogeneous structural states⁵ were assigned to the obtained maps according to FSC calculations with the experimentally solved ones (Fig. 4a). The Fourier shell correlation (FSC) curves and the map visualization showed a good agreement between the currently reconstructed and the experimental cryo-EM maps (Fig. 4b). These results demonstrated that our method can reconstruct diverse 50S-Ribosomal conformations equivalent to those solved by the experimental procedure and they covered all ribosomal structural states with the distinct assembly, such as the immature and fully matured states, which are labeled as B and E5, respectively. Furthermore, our approach provided an important insight that the structural states found in the previous studies are energetically stable in the conformational space during the ribosome assembly process. In addition to the experimental maps, the rRNA helix 68-bound states (labeled as C4), newly found using cryoDRGN⁹, were reconstructed as one of the energetically stable states.

Fig. 4. Generated parallel assembly pathways for 50S-Ribosome on the free-energy landscape.

a Class assignments of density maps reconstructed by cryoTWIN. The accumulated FSC values between the maps are displayed as a heatmap. Horizontal and vertical axes indicate the 14 structural labels in the reference maps and the 30 significant mean vectors' indexes in our trained GMM, respectively. Except for cryoDRGN-derived C4, the experimental maps are used as the references. b Representative FSC curves between cryoTWIN-derived and experimentally derived density maps and their visualization. FSC curves for all the cryoTWIN-derived maps are shown in Fig. SI-1. The density maps used in the FSC calculations (top panel) are visualized. The reference experimental maps are colored in gray. c Parallel assembly pathways generated by PaStEL with cryoTWIN. Nodes and edges represent heterogeneous structural states as the 3D density map and the MaxFlux path between two states, respectively. Nodes and the representative 3D density map are colored according to the structural super-classes assigned by Davis et al.; B, C, D, and E are colored by orange, green, yellow, and cyan, respectively. The edge distance was defined by the ridgeline length of Eq. (3). Colored thick arrows p₁ to p₅ represent the known 50S-Ribosomal pathways (p1–p4) and a novel pathway (p₅) found in this study.

The reconstructed maps were set as graph nodes, and then the pathway graph$\mathcal{G}$ (see Methods) were computed using the trained cryoTWIN and PaStEL, showing that multiple plausible paths were available from the immature state B to the mature state E5 (Fig. SI-2). To explore the energetically favorable pathways during ribosome maturation, we generated pathways passing through multiple nodes by setting B and E5 as the start and end nodes, respectively, and ranked them based on total edge distance (the higher the rank, the shorter the total edge distance). In this calculation, we set the number of transit nodes to four or five according to the previous study on 50S-ribosomal assembly pathways⁵. The high-ranked generated pathways were identical to or quite similar to the parallel pathways proposed by Davis et al. (p₁–p₄ in Fig. 4c, a complete list of the generated pathways is presented in Extended data). These results demonstrated that we successfully generated the energy-based reliable assembly pathways only from the cryo-EM images without prior biological knowledge. In addition to their previously investigated pathways, an energetically favorable pathway, which involves the binding of rRNA helix 68 in a conformation lacking the central protuberance, was newly found: a pathway passing through C4 state (p₅ in Fig. 4c). These findings would advance our understanding of the ribosome assembly process, highlighting the power of our method to investigate conformational transitions of biological macromolecules in the structural biology research field. In addition to their high-ranked representative pathways, other pathways were found in this analysis. They could be minor pathways during the assembly process.

PaStEL with cryoTWIN uncovers a thermodynamic basis for increased infectivity in the SARS-CoV-2 spike protein D614G mutant

Pathogenic amino acid substitutions in macromolecules often perturb the dynamic behaviors, leading to their functional alterations or dysfunctions. Therefore, elucidating the molecular mechanism from a view of free-energy governing molecular dynamics will be of great importance for drug development targeting a pathogenic mutant. In our approach, when the wildtype and mutant cryo-EM data are acquired under the same temperature, the individually obtained free-energy differences can be quantitatively analyzed on the same scale. This property prompted us to investigate the thermodynamic mechanism of functional alteration induced by pathogenic amino acid substitution using cryoTWIN with PaStEL.

SARS-CoV-2 infection is triggered by a structural change in the spike protein receptor-binding domain (RBD) from closed to open conformations required for angiotensin-converting enzyme 2-binding of host cells. Although the wildtype predominantly forms the Closed conformation, the D614G mutant with enhanced infectivity primarily forms the Open conformation (particularly the Open 1up state)²¹. However, the thermodynamic basis for these conformational differences under the equilibrium condition remains unclear, hampering the comprehension of the molecular mechanism of the enhanced infectivity of the mutant.

We prepared particle images of both the wildtype and D614G mutant spike proteins from the micrographs acquired under the same experimental conditions (EMPIAR-10469 and EMPIAR-10725, respectively) and individually trained cryoTWIN with latent GMM p_ψ. The density maps reconstructed from the significant Gaussian mean vectors ${\left\{{\mu }_i\right\}}_i$ for the wildtype were categorized into the Closed or Open 1up states, as observed in experiments (Fig. 5a). Meanwhile, the density maps for the mutant adopting the Closed, Open 1up, or Open 2up states, agreed well with the experimental observations (Fig. 5b). These reconstructed maps were well fitted by the experimentally determined structures, indicating that the known structures were energetically stable states in the conformational space, as expected.

Fig. 5. Generated transition pathways for spike protein (wildtype/D614G mutant) with free-energy curves.

a, b Reconstructed 3D density maps (colored in white) by cryoTWIN and their corresponding all-atom models (purple or yellow). These all-atom models were from the Protein Data Bank. Five density maps were used as either the start or end conformation for generating pathways by PaStEL. c Visualization of three generated conformational transition pathways by PaStEL on free-energy surfaces. Path is expressed by either a black triangle, red triangle, or orange circle. Surfaces were obtained using the trained latent distribution of cryoTWIN. d, e Generated conformational transition pathways corresponding to three trajectories in (c), with their free-energy curves. Vertical axis denotes the free-energy difference, ΔE, measured from the lowest free-energy value on the pathway. For the start and end density maps in each generated pathway, their corresponding all-atom models were fitted. f Zoom-in comparison for RBD and NTD during transition. Left panel is the most unstable intermediate state (blue) vs. Open 1up state (white) in the energy curve of (d). Middle one is the most unstable intermediate state (red) vs. Open 1up state (white) in the red energy curve of (e). Right panel is the most unstable intermediate state (yellow) vs. Open 2up state (white) in the yellow energy curve of (e). Conformational changes in RBD and NTD are indicated by orange and black arrows.

We further generated the conformational transition pathways between the representative conformations of the wildtype and their free-energy curves using PaStEL based on the trained cryoTWIN (left panel of Fig. 5c, d). The reconstruction of the intermediate maps illustrated gradual structural changes in the RBD between the Closed and Open states. The free-energy profile showed that the Closed state was more stable than the Open 1up state, and there existed a high energy barrier between the Closed and Open 1up states. In contrast, the D614G mutant was shown to favor the Open 1up state compared with the other states (right panel of Fig. 5c, e). The Open 2up state, uniquely found in the mutant, was more unstable than the Closed state, as estimated in the previous experiment²¹. Notably, the free-energy barrier between the Closed and Open 1up states in the D614G mutant was lower than that in the wildtype. This observation indicated that the conformations between the inactive Closed and active Open 1up states were readily exchangeable in the mutant compared with the wildtype. This mutation-induced perturbation in the dynamic property could offer enhancement of infectivity. The previous study reported that, in wildtype, the outward movement of the N-terminal domain (NTD) was accompanied by the structural change in the RBD to the Up state, while the NTD of the D614G mutant adopted a nearly identical outward conformation in the Closed and Up states (Fig. 5f)²¹. The reconstructed intermediate states in wildtype showed that the RBD and NTD domains were located at the intermediate position between the Up and Closed states, and their movements were concerted during the conformational exchange (Fig. 5f, left panel). On the other hand, in the D614G mutant, the NTD conformation was not changed during the exchange (Fig. 5f, middle and right panels). These results indicated that ignoring the concerted NTD movement in the D614G mutant could be attributed to the lowering of the energy barrier between the two states. These energy-based insights cannot be obtained from static structures and existing methods, highlighting the advantages of our method in understanding molecular mechanisms only from cryo-EM data.

For a quantitative evaluation of the uncertainty in the energy profile shown in Fig. 5, please see Supplementary Information Section SI-4.2.

Discussion

Herein, based on the theoretical findings for PaStEL with cryoTWIN, we developed a method to generate the free-energy landscape and compute optimal free-energy transition pathways of multiple conformations using only the cryo-EM particle images. Several ML methods have been reported for reconstructing heterogeneous conformations from the particle images, but they do not generate the free-energy landscape and free-energy-based pathways, hindering the analysis of the dynamic behavior. In this regard, MD simulation could be an approach to complement existing ML methods. However, considering that large structural changes occur on millisecond or longer timescales in general, the application to macromolecules targeted by cryo-EM is often unrealistic. Meanwhile, our approach can automatically reconstruct the energetically stable conformations and uncover the dynamic properties from cryo-EM particle images only, thereby reducing experimental costs (see e.g., Fig. SI-3), highlighting its potential to accelerate the analysis of complex biological systems.

Our framework, like other cryo-EM analyses that interpret state occupancies in thermodynamic terms^15,22,23, assumes that the analyzed vitrified particles approximate an equilibrium ensemble, such that relative occupancies can be related to free-energy differences through Boltzmann statistics. Accordingly, the recovered free-energy landscape should be interpreted as an estimate for the analyzed particle set under these assumptions, rather than as an exact description of the full solution-phase conformational ensemble. Rapid cooling can alter the structural ensemble during vitrification²⁴, and discarding low-quality particles during processing can further bias state populations. To accurately reconstruct the 3D density maps and estimate their free-energy values with our method, a sufficient number of high-quality particle images and reliable pose estimates is required. Practical limitations in real cryo-EM pipelines–namely, finite data, high noise levels, and pose estimation errors–can directly bias the learned latent density, which forms the basis of our free-energy estimation. Finite data particularly hampers the analysis of low-populated conformations, because the probability density for these rare states is poorly estimated, whereas image noise and pose errors can bias the learned landscape more broadly across the full conformational distribution. Because the use of these reconstructed maps currently relies on visual inspection, a method for estimating map quality should be developed and integrated with our method in future work. These limitations are not unique to cryoTWIN but are shared by current cryo-EM-based conformational heterogeneity analyses^9,15, and addressing them will require future advances in vitrification, particle-processing pipelines, and statistical methods that account for sampling bias and uncertainty. Advancements in cryo-EM equipment, such as cold field emission gun²⁵, and image processing methods, will expand the application scope of the ML-based methods, including our method.

In conclusion, this article is the first to propose a DL method for generating plausible conformational paths as a 3D density map on the free-energy landscape from only the particle images. Our method was designed based on novel theoretical guarantees—cryoTWIN and PaStEL theories. Through intensive experiments, PaStEL based on cryoTWIN successfully analyzed the dynamics of several biomacromolecules. Thus, our method will advance the biological foundation by elucidating the dynamics of biomacromolecules with large sizes and complex structural assemblies.

Methods

Implementation of cryoTWIN and PaStEL

We describe the formal definition for the expected training objective of cryoTWIN in Definition 1. For this, let us first recall the assumption (A1) introduced at “Theory behind PaStEL with cryoTWIN".

Definition 1

(Expected training objective of cryoTWIN) Assume (A1) for a particle image dataset. Let $X_R\in {\mathbb{R}}^{s\times s}$ (s is pixel size) denote the Fourier transformation of the particle image, whose pose orientation is R in the corresponding conformation V as the 3D density map. Let f_θ, g_ϕ, and p_ψ denote the encoder, decoder, and trainable latent distributional model in cryoTWIN, respectively, where θ, ϕ, ψ are a set of trainable parameters. The size of the autoencoder in cryoTWIN is supposed to be sufficiently large. The decoder g_ϕ is designed to output the reconstruction to X_R as inputs of $z\in {\mathbb{R}}^d$ and R, i.e., ${\widehat{X}}_R:=g_{\varphi }\left(z,R\right)$, where z is a latent variable corresponding to V and is obtained by the encoder f_θ. The quantity d is the intrinsic dimension of the conformational distribution p(V). Let ${\breve{X}}_R:=g_{\varphi }\left(z+\epsilon ,R\right)$ denote the noisy reconstruction to X_R, where ε is a random noise vector $\epsilon :=\left({ϵ}_1,{ϵ}_2,\dots ,{ϵ}_d\right)\in {\mathbb{R}}^d$ satisfying $\forall i;\mathbb{E}\left[{ϵ}_i\right]=0$ and $\forall \left(i,j\right);\mathbb{E}\left[{ϵ}_i{ϵ}_j\right]={\sigma }^2{\mathbf{1}}_{\left[i=j\right]}$. The function 1 is the indicator function: 1_[i = j] = 1 if i = j, otherwise zero. The expected training objective of cryoTWIN is given by

$${\theta }^{*},{\varphi }^{*},{\psi }^{*}=\arg {\min }_{\theta ,\varphi ,\psi }{\mathbb{E}}_{V,\epsilon }\left[L\right],$$ 4

where $L:=-\log p_{\psi }\left(z\right)+{\lambda }_1{\mathbb{E}}_R\left[\parallel W\odot \left(X_R-{\widehat{X}}_R\right)\parallel \genfrac{}{}{0ex}{}{2}{2}\mid V\right]+{\lambda }_2{\mathbb{E}}_R\left[\parallel W\odot \left({\widehat{X}}_R-{\breve{X}}_R\right)\parallel \genfrac{}{}{0ex}{}{2}{2}\mid V\right]$, W is s × s weight matrix whose (i, j)-th ($-s/2\le i,j\le s/2\&i,j\in \mathbb{Z}$) entry is $\sqrt[4]{i^2+j^2}$,  ⊙ is Hadamard product, and the symbol * means the optimality, and both λ₁ and λ₂ are the positive hyper-parameters.

In this study, we defined the empirical training objective of cryoTWIN via applying the Monte Carlo method to Eq. (4), as follows:

$$\arg {\min }_{\theta ,\varphi ,\psi }\frac{1}{N}{\sum }_{i=1}^N-\log p_{\psi }\left(z_i\right)+{\lambda }_1\parallel W\odot \left(X_i-{\widehat{X}}_{R_i}\right){\parallel }_2^2+{\lambda }_2\parallel W\odot \left({\widehat{X}}_{R_i}-{\breve{X}}_{R_i}\right){\parallel }_2^2,$$ 5

where z_i ≔ f_θ(X_i), each element in the noise vector ε was sampled from an identical and independent zero-mean Gaussian distribution, and the latent distribution model was defined by GMM $p_{\psi },\psi :={\left\{\left({\pi }_c,{\mu }_c,{\mathrm{\Sigma }}_c\right)\right\}}_{c=1}^C$. The symbol N denotes the number of particle images, and π_c, μ_c, Σ_c are the weight, mean vector, and variance matrix of the c-th Gaussian distribution in the GMM, respectively. Both the encoder and decoder were defined by a multi-layer perceptron (MLP). The GMM parameters are optimized jointly with the network. Updates are driven by encoder outputs produced through backpropagation, and we compute responsibility-based sufficient statistics in a DAGMM-inspired formulation²⁶. For the details of the GMM parameter updates and notation used in the pseudo-code, please refer to Supplementary Information Section SI-6 and Table SI-3.

Next, we describe the details of our solver for the minimization problem at (Step 1) of PaStEL in Theorem 2, as algorithm ${\mathcal{A}}_1$. This algorithm requires four inputs: (i) trained latent GMM p_ψ with $\psi ={\left\{\left({\pi }_c,{\mu }_c,{\mathrm{\Sigma }}_c\right)\right\}}_{c=1}^C$, (ii) trained decoder, (iii) index i of significant mean vector μ_i in p_ψ as start point & index j of significant mean vector μ_j in p_ψ as end point, where i, j ∈ {1, …, C}, and (iv) hyper-parameters $m\in \mathbb{N}\&\omega >1$. The significant μ is defined via a large density value p_ψ(μ). The algorithm ${\mathcal{A}}_1$ outputs z^{(0:m + 1)} via three steps as follows: (Step 1) z⁽⁰⁾ ← μ_i & z^(m+1) ← μ_j, (Step 2) for all t ∈ {1, …, m}, firstly, generate a sufficient amount of C-dimensional random weight vectors ${\left({\alpha }_1,\dots ,{\alpha }_C\right)}^{\top }$ satisfying ${\sum }_{c=1}^C{\alpha }_c=1$, where ∀ c; α_c ≥ 0 & (α_i, α_j) = ((1−t/m)^ω, t/m), secondly, define a set of candidate latent variables {z}, where $\mathrm{z}={\left({\sum }_{c=1}^C{\alpha }_c{\mathrm{\Sigma }}_c^{-1}\right)}^{-1}\left({\sum }_{c=1}^C{\alpha }_c{\mathrm{\Sigma }}_c^{-1}{\mu }_c\right)$, and thirdly, compute z^(t) by solving $z^{\left(t\right)}=\arg {\min }_{\left\{\mathrm{z}\right\}}\frac{1}{p_{\psi }\left(\mathrm{z}\right)}{\parallel z^{\left(t-1\right)}-\mathrm{z}\parallel }_2$. (Step 3) $z^{\left(0:m+1\right)}\leftarrow \left(z^{\left(0\right)},z^{\left(1\right)},z^{\left(2\right)},\dots ,z^{\left(m\right)},z^{\left(m+1\right)}\right)$.

Hyper-parameter selection

Through this study, the MLP architecture was 1024 × 3 (1024: number of nodes per layer; 3: number of layers) for both the encoder and decoder, and the dimension of the latent space was fixed to 8. For Eq. (5), λ₁ = λ₂ = 8 and C = 100 were used. Regarding the variance σ in the random noise ε, it was fixed at 0.9 for the simulation data. For the real-world data, it was scheduled to increase gradually from 0.9 at the start of training to 1.0 at the end. The trainable parameters in the autoencoder were initialized by the He initialization method and were optimized by the RAdam²⁷ with the learning rate 0.0001, where the number of epochs was 100, and the mini-batch size was 128 (only the 50S-ribosomal experiment employed a size of 64). For the solver algorithm ${\mathcal{A}}_1$ in PaStEL, we used (m, ω) = (11, 1.2).

Computational environment

For the experiments with the synthetic proteins and 50S-Ribosome (resp. spike proteins), 4 NVIDIA V100 SXM2 GPU and 2 Intel Xeon Gold 6148 processors (resp. 8 NVIDIA A100 SXM4 GPU and 2 Intel Xeon Platinum 8360Y processors) were used.

Preparation of Chignolin/Trp-cage synthetic particle images

Synthetic particle images for Chignolin (resp. Trp-cage) were prepared using the diverse conformations generated by mcMD simulations^19,28. We prepared a dataset of 1.5 million simulated particle images consisting of various molecular orientations with 128 × 128 (Å/pixel: 0.2) (resp. (Å/pixel: 0.32)) size by following steps 1–6 as described below.

1. Calculate the Boltzmann factor, exp(-E/RT); T = 300 K, for each sampled structure from the free-energy value calculated by mcMD.

2. Perform N (target total number of images: 1.5 million) weighted restoration extractions from the population of sampled structures, using the Boltzmann factors as weights. This determines the number of 2D projection images, n, to be computed from that structure for each sampled structure.

3. To determine the orientation of the projection image for each of the n assigned to each sampled structure, a uniform restoration extraction from a uniform hemispheric orientation list (10,240 orientations) is used to create a projection orientation list for each sample structure.

4. Each structure is best fitted based on all atoms, excluding hydrogen atoms, and a 3D density map was calculated using electron density calculation. Based on the projection orientation list created in step 3, 1.5 million images were obtained from the 3D density map for each structure to create synthetic cryo-EM particle images.

5. Only for the Trp-cage dataset, each particle image was enlarged by 25% via applying bicubic interpolation. After this, the Å/pixel is 0.32, while the original one was 0.40.

6. The images of Chignolin (resp. Trp-cage) were blurred by applying a Gaussian filter. The standard deviation of the filter was 0.050 (resp. 0.075).

Details of numerical experiments with Chignolin/Trp-cage

We trained cryoTWIN using the empirical objective in Eq. (5). For obtaining each scatter plot in Fig. 3a, b, we at first randomly selected the 15,000 latent variables {z} using the trained encoder, and then we computed dz₁ ⋅ dz₂ and dV₁ ⋅ dV₂ in Eq. (1) for each z using the trained decoder. To define “cryoTWIN” (resp. “MD”) energy map in Fig. 3c, d, PCA in scikit-learn OSS library was performed for a set of all the latent variables obtained from the trained encoder, and then the pseudo free-energy value $-\log p_{{\psi }^{*}}$ (resp. the MD free-energy value) was assigned to each projected 2D point. The all-atom models in Fig. 3e, f were employed via the following procedure: first, for each mean vector related to the 3D density map in e and f, we computed the nearest empirical latent variable, and then we picked the all-atom model of a particle image corresponding to the nearest variable. Model fitting into the maps is performed using UCSF ChimeraX²⁹. The reconstructed density maps and the corresponding all-atom models are publicly available, see Data availability.

Details of experiments using cryo-EM data of the 50S ribosome

We trained cryoTWIN by the empirical training objective of Eq. (5) with publicly available 131,899 50S-ribosomal particle images (https://github.com/zhonge/cryodrgn_empiar/tree/main/empiar10076 [Last accessed on 6th Sep., 2024]) of size 128 × 128 (Å/pixel: 3.275) and their corresponding pose orientations estimated by cryoSPARC. The accumulated FSC score s in Fig. 4a was calculated based on the FSC curve between a previously reported reference map and a reconstructed map: $s={\sum }_i{\mathrm{e}}^{-{\left(i\sigma \right)}^2}{f}_i$, where i is the frequency ranging from 0 to 0.5, σ = 0.1, and f_i is the FSC value at i. The 30 density maps reconstructed by cryoTWIN are publicly available, see Data availability. The structural labels for the reconstructed maps were assigned according to the maximum s values. The reference maps, except for C4, were obtained from EMD: 8440, 8442–8444, 8446–8449, 8451–8456, while the C4 map was from the URL of https://zenodo.org/records/4355284#.YCq_dI9Kj0o [Last accessed on 17th Feb., 2025]. The pathway graph $\mathcal{G}$ was constructed in the latent space by the following steps. We first defined the node set by the top 30 significant mean vectors, where the significance of a mean vector μ is defined by the density value $p_{{\psi }^{*}}\left(\mu \right)$; the higher the density value, the more significant the vector. Then, the ridgeline lengths (Eq. (3)) were calculated for all edges between the 30 nodes. At last, we removed the edges with the largest top 66% lengths from the graph. Note that the node corresponding to the Gaussian mean vector index No. 33 (see the horizontal axis of Fig. 4a) was removed by the edge removal. For the visualized pathway graph in Fig. 4c, we applied NetworkX³⁰ to $\mathcal{G}$.

Details of experiments using cryo-EM data of spike proteins

The particle images for the wildtype and the mutant D614G were derived from the micrographs, EMPIAR-10469 and EMPIAR-10725, respectively, using RELION software³¹. After conducting motion correction and CTF estimations, particle images were picked up by a template-based particle picking algorithm. All particle images were classified into five classes by a 3D classification algorithm, and the images associated with clear structural density maps were selected. Finally, 84,945 (256 × 256, 1.2229 Å/pixel) and 40,954 (256 × 256, 0.8566 Å/pixel) images together with the corresponding pose orientations were obtained for the wildtype and the mutant D614G, respectively. We trained cryoTWIN by the empirical objective of Eq. (5). The training datasets are publicly available, see Data availability. In Fig. 5a, b, the 3D density maps were obtained via the following procedure. First, we reconstructed 30 density maps from the GMM mean vectors with the 30 lowest pseudo free-energy values. Then, based on the existing structural labels, we classified the 3D maps via visual inspections using UCSF ChimeraX²⁹. Finally, among the 3D density maps with the same label, we picked one with the lowest energy. The all- atom models shown in a and b were derived from the protein data bank: 6zge (wildtype Closed), 6zgg (wildtype Open 1up), 7bnm (D614G Closed), 7bnn (D614G Open 1up), and 7bno (D614G Open 2up). Regarding Fig. 5c, we first applied PCA to a set of latent variables on the conformational pathway. Let A denote the PCA projector from the latent space to the PCA plane. Then, we generated random samples {y} on the plane and provided the pseudo free-energy value to each y by $-\log p_{{\psi }^{*}}\left(A^{-1}y\right)$. Model fitting into the maps shown in Fig. 5a, b, and f is performed using UCSF ChimeraX.

Author contributions

Y.O., S.M., T.K., and A.T. conceived the idea of writing a paper on this study. K.Y., Y.W., and A.N. created the method by finding its theoretical properties. K.Y. developed the implementation codes. K.O., K.Y., and M.W. equally contributed to the analyses of simulation and experimental cryo-EM image data. A.T., Y.A., and Y.S. prepared the simulation and spike protein datasets. K.Y, M.W., Y.W., K.O., and S.M. wrote the manuscript. All authors discussed the research, edited the manuscript, and approved its final version.

Peer review

Peer review information

Communications Chemistry thanks Nandan Haloi and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. A peer review file is available.

Funding

This work was supported by the FOCUS Establishing Supercomputing Center of Excellence project, and MEXT as “Program for Promoting Research on the Supercomputer Fugaku” (Simulation- and AI-driven next-generation medicine and drug discovery based on “Fugaku”, JPMXP1020230120). We used the supercomputer Fugaku provided by the RIKEN Center for Computational Science through HPCI System Research Projects (Project IDs: hp220078, hp230102, hp230216, hp240109, hp240211, ra000018) and the supercomputer system at the Information Initiative Center, Hokkaido University, Sapporo, Japan. Camphor3 in Kyoto University through HPCI System Research Projects (Project IDs: hp220078, hp230102, hp240109). The computational resources of AI Bridging Cloud Infrastructure (ABCI) provided by the National Institute of Advanced Industrial Science and Technology (AIST) were used. This research was also partially supported by the Fujitsu Small Research Lab “Large Scale Medical AI”, a joint research center between Fujitsu Limited and Kyoto University.

Data availability

All datasets and materials generated and/or analysed during the current study are publicly available in a Zenodo repository at https://zenodo.org/records/19813080. The repository contains: (i) all cryoTWIN-derived 3D density maps presented in this study, (ii) atomic models used for generating synthetic particle images of Chignolin and Trp-cage, (iii) the training datasets described in the spike protein section, (iv) the trained cryoTWIN models, and (v) all source data underlying the figures in the main text and Supplementary Information. Specifically, the repository includes numerical data for the PCA plots in Fig. 3a, b, as well as the free-energy landscapes in Fig. 3c, d. For Fig. 4a, c, a CSV file is provided that contains the numerical data used to generate the FSC heatmap and the pathway rankings based on MaxFlux values. This CSV file includes multiple tabs: “pathways_inter4” and “pathways_inter5”, where “interX” denotes the number of intermediate nodes used in generating the 50S-ribosomal pathways (excluding the start and end nodes). Each row corresponds to a generated pathway, including the sequence of Gaussian mean vector indices and the corresponding total ridgeline length (i.e., pathway distance). Structural labels assigned to each Gaussian index are determined based on the maximum accumulated FSC values. The tab “heatmap” provides the 14 × 30 matrix corresponding to the FSC heatmap in Fig. 4a, and the tab “labels” provides the mapping between Gaussian indices and their assigned structural labels. Source numerical data used to generate the plots in Fig. 5c, d, as well as Fig. SI-5, are also included in the repository. In addition, source data for the plots shown in Fig. SI-6 are provided. The FSC curves presented in Fig. 4b and Fig. SI-1 can be reproduced from the deposited 3D density maps available in the repository. The full training datasets for Chignolin and Trp-cage (approximately 400 GB) are not publicly hosted due to their large size but are available from the corresponding authors upon reasonable request.

Code availability

The custom code and algorithms central to this study are described in detail in the Methods section and Supplementary Information to allow for independent implementation. The source code contains proprietary components developed as part of collaborative research with Fujitsu Ltd. and is therefore not publicly hosted. However, the code is available from the corresponding author(s) upon reasonable request, and we are committed to providing access for the purpose of academic review and verification of our results.

Competing interests

The authors declare the following competing financial interest(s): This research was conducted as part of a collaborative research effort with Fujitsu Limited. Fujitsu Limited provided funding and technical support for this study. The authors ensured that the research design, data analysis, and interpretation remained unbiased and independent.

Supplementary information

The online version contains supplementary material available at https://doi.org/10.1038/s42004-026-02077-5.