{"id":"86f41b99-e7eb-43cd-a720-f2a1164624a3","arxiv_id":"2509.05541","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Cryo-EM reconstruction can be cast as a stochastic inverse problem over molecular-structure distributions and solved by particle-based Wasserstein gradient flow.","lead":"This paper reformulates cryo-EM reconstruction as a problem of recovering a probability distribution over molecular shapes from noisy 2D images, not just a single structure. It solves this with a particle-based Wasserstein gradient flow and shows on simulated data, including a flexible protein, that continuous shape distributions can be recovered.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section 4.3 contradicts itself on viewing directions: data and Algorithm 4.1 use random rotations, yet the text says directions were fixed 'throughout the experiment'; if the HSP90 run used fixed orientations, the central claim about unknown rotations is untested.","rationale":"The reader's weakest assumption concerns the generative model: i.i.d. observations, known noise, uniform rotations, and identifiability. My concern is more specific and is located inside the paper's own text: Section 4.3 first says random rotations are used, then says viewing directions were fixed throughout the experiment. This is not merely a missing robustness test; it is an internal inconsistency about which experiment was actually run. If the rotation-fixing sentence describes the main experiment, the central claim about recovering conformations under unknown random orientations is not supported by the one realistic experiment. If it describes only the PCA analysis, then the main experiment is probably as claimed and the contradiction is a presentation error. A single rerun or code inspection settles which reading is correct. I do not move the verdict to REJECT because the paper's framework is coherent, the energy-distance experiments are plausible, and the ambiguity is resolvable; the appropriate disposition remains CONDITIONAL, with the condition being a clear statement and verification of the rotation protocol in Section 4.3. My agreement with the reader is partial because the reader correctly flags untested models and orientation bias, but my concern is a narrower, text-located contradiction rather than a general lack of validation.","tokens_in":22501,"tokens_out":22302,"duration_ms":210972,"concrete_test":"Rerun the HSP90 experiment exactly as Algorithm 4.1 is written: for every iteration, draw independent uniform random orientations for each particle and for the forward evaluation, with the same 3000 observed images generated under R∼U(SO(3)) and the same hyperparameters. Then compare the final mode-marginal energy distance and the Fig. 6 marginals to the reported values. If the energy distance degrades materially (e.g., from 19.44 toward the initial 89.21) or the optimization fails to converge, the current results likely relied on fixed viewing directions rather than on the random-rotation model the paper claims to handle.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The main realistic validation is the HSP90 experiment in Section 4.3. The text first says 'random rotations are drawn from R∼U(SO(3))' for the observed images, and Algorithm 4.1 explicitly instructs the user to 'Generate random orientations from uniform quaternion distribution' at every iteration. Section 5 then advertises as a differentiator that the method, unlike common cryo-EM pipelines, does not estimate viewing directions but draws them randomly. However, the same section later states, in the PCA-comparison paragraph: 'To simplify the analysis, we fix the viewing directions throughout the experiment. By removing ambiguity due to random rotations, this setup ensures that variations in the images are primarily due to differences in molecular conformation, rather than orientation.' As written, this is a direct contradiction. If the reported mode-marginal recovery in Fig. 6 was produced with fixed viewing directions, then the experiment never tests the random-rotation model that is central to the claimed contribution; the method's success could be an artifact of aligned projections, and the Section 5 claim about not estimating orientations would be unsupported. If the sentence refers only to a separate PCA diagnostic, the wording is still misleading and must be corrected. The paper never reconciles these statements, and because the random-orientation capability is the key differentiator claimed in the abstract and conclusion, this ambiguity is load-bearing for the central claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript formulates single-particle cryo-EM reconstruction with structural heterogeneity as a stochastic inverse problem over the space of probability measures. The authors model the observed images as an empirical distribution obtained by pushing an unknown distribution over molecular conformations through a random forward operator that includes random rotations, projection, convolution, and Gaussian noise. Reconstruction is posed as a variational problem (Eq. 2.6) minimizing a discrepancy D (KL divergence, MMD, or energy distance) between predicted and observed image distributions, and is solved by a Wasserstein gradient flow (Eq. 2.10), discretized as an interacting particle system (Eq. 2.11). The paper derives a connection between this optimize-then-discretize approach and MAP-based discretize-then-optimize methods (Section 3), including a Gamma-convergence result for the large-data limit and a consistency result for particle discretization. Numerical experiments on a 1D toy problem, a 2D nanocluster model, and a synthetic HSP90 protein with low-dimensional normal-mode dynamics aim to show recovery of continuous conformational distributions from simulated cryo-EM images.","tokens_in":22768,"tokens_out":10415,"duration_ms":84128,"significance":"The core idea of treating the conformational distribution as the object of inference and using gradient flows in Wasserstein space is well motivated and, to my knowledge, a novel combination with the cryo-EM push-forward model and MMD/energy discrepancies. The theoretical contribution includes a clean derivation of how the MAP/RELION-style objective emerges as a DTO discretization of a KL-based variational problem (Section 3.1), and the Gamma-convergence and consistency results (Propositions 3.1-3.3) provide a useful framework, albeit under assumptions that are not verified for the specific objectives used. The particle method is natural and leverages JAX reverse-mode adjoints. However, the numerical validation is entirely synthetic and lacks baselines; more importantly, the viewing-direction contradiction in Section 4.3 undermines the main experimental claim. If these issues are resolved, the paper could be a valuable contribution to continuous-heterogeneity analysis in cryo-EM and to stochastic inverse problems generally.","major_comments":[{"comment":"There is a direct contradiction about viewing directions in the HSP90 experiment. Section 4.3 states that 'random rotations are drawn from R∼U(SO(3))' for generating the observed images, and Algorithm 4.1 instructs the user to 'Generate random orientations from uniform quaternion distribution' at every iteration; Section 5 advertises as a differentiator that the method 'does not explicitly assume that the viewing direction for each image has been estimated a priori but instead draw it randomly.' Yet the PCA-comparison paragraph in Section 4.3 says: 'To simplify the analysis, we fix the viewing directions throughout the experiment.' If the reported recovery (Fig. 6, energy distances 89.21 to 19.44) was obtained with fixed viewing directions, then the method's advertised ability to handle unknown random rotations is not tested by the main realistic experiment, and the Section 5 claim is unsupported. If the sentence refers only to the PCA diagnostic in Fig. 8, the wording is still misleading and must be corrected. The authors must state unambiguously, for the run that produced Fig. 6, whether the random orientations were drawn per Algorithm 4.1 or fixed, and if fixed, provide an experiment with random rotations to support the central claim.","section":"§4.3 (PCA paragraph), §5"},{"comment":"The direction of the KL divergence is inconsistent between the gradient formula and the MAP derivation. Eq. (2.7) defines D_KL(µ||ν); if the variational objective in (2.6) uses D = D_KL(F(ρθ) || ρ^δ_y), then the Fréchet derivative in Section 2.5.1, δD/δρ_y = log(ρ_y/ρ^δ_y) + 1, is correct. However, Section 3.1, Eq. (3.1), defines the objective with D_KL(ρ^δ_y || ρ_y), the reversed direction. The Fréchet derivative of the reversed KL with respect to ρ_y is -ρ^δ_y/ρ_y, not the formula given. This matters for the sign of the particle update (2.11). The numerical experiments in Section 4 state that they 'use the KL divergence (2.7)' but do not specify which argument order is used. The authors must specify the exact functional form used for D in all numerical examples, correct the formula in Section 2.5.1 if needed, and check that the resulting gradient direction is consistent with the reported convergence of the algorithm.","section":"§2.5.1, Eq. (2.7), §3.1 Eq. (3.1)"},{"comment":"The paper does not provide convergence analysis for the particle gradient flow (2.11), which is the algorithm used in all experiments. Section 2.4 only notes that the empirical measure approximation is subject to 'numerical error, random error and potential (density) estimation error,' and Section 5 explicitly says that 'a comprehensive convergence analysis for the proposed particle method' is future work. For a paper whose main methodological contribution is the OTD approach, the absence of any theoretical guarantee for the discretized gradient flow (in number of particles N, step size Δt, and Monte Carlo samples K for the random rotation) leaves the core numerical scheme unsupported. At minimum, the paper should state this limitation in Section 2 where the method is introduced, and ideally add a convergence result for the linear 1D case (the setting of Section 4.1) to establish consistency of the scheme with the continuous Wasserstein gradient flow.","section":"§2.4, Eq. (2.11), §5"},{"comment":"The numerical validation is entirely synthetic and lacks baseline comparisons, error bars, or repeated-seed statistics. The 1D and nanocluster examples are illustrative, but the HSP90 experiment, described as 'realistic,' is the main evidence for the abstract's claim that the method 'demonstrates its ability to recover continuous distributions over structural states.' This claim is weakened by (i) the viewing-direction ambiguity raised above, (ii) the absence of any comparison with existing continuous-heterogeneity methods such as RECOVAR, 3DVA, or cryoDRGN, and (iii) no assessment of sensitivity to the KDE bandwidth, learning rate, or random initial conditions. I recommend adding at least one baseline comparison on the same synthetic data and reporting statistics over multiple random seeds, or tempering the abstract's validation claim.","section":"§4, abstract"}],"minor_comments":[{"comment":"The superscript formatting in '104' and '9 × 104' is lost; these should clearly read '10^4' and '9 × 10^4', otherwise the stated sample sizes and iteration counts are implausible.","section":"§4.1, §4.3"},{"comment":"In the sentence 'In contrast, four our initial guess distribution ρθ', 'four' should be 'for'.","section":"§4.3"},{"comment":"The energy distance definition has a formatting error: the expression inside the square root appears to end with a stray vertical bar, and the definition should be checked against standard references to ensure the signs are correct.","section":"Eq. (2.9)"},{"comment":"The statement that Eq. (3.5) is 'precisely [29, Eqn. (2)]' is stronger than what is shown; the derivation involves dropping constants and setting λ = K/N, and RELION's practical objective also contains terms for per-image normalization and the noise model. The equivalence should be qualified as holding after the stated simplifications.","section":"§3.1, after Eq. (3.5)"},{"comment":"Lemma 3.2 assumes that E is continuous in the weak topology, but the paper does not state whether this continuity holds for the KL or MMD objectives used in the examples; the applicability of the lemma to the numerical settings should be addressed.","section":"§3.2, Lemma 3.2"}],"recommendation":"major_revision","confidential_remarks":"The viewing-direction contradiction and the KL-direction inconsistency are load-bearing and must be resolved before the paper can be accepted. The manuscript also builds heavily on the authors' own earlier SIP papers [31-33]; the editor should weigh the incremental novelty for this venue. The synthetic-only validation without baselines is below the standard expected for a methods paper in cryo-EM, though the theoretical framework is coherent."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core idea is genuinely worth your time: formulate cryo-EM reconstruction as a stochastic inverse problem over probability measures, using a random push-forward operator, and solve the variational problem with a Wasserstein gradient flow. Earlier SIP work used deterministic push-forwards, and cryo-EM heterogeneity methods did not take this route, so the random-push-forward formulation is new. The particle update is derived cleanly, the MMD/energy distance sidesteps density estimation, and the connection to MAP/DTO methods is a useful perspective even if the Gamma-convergence arguments are routine.\n\nThe two simple numerical experiments are fine as sanity checks, and the HSP90 experiment with 3000 images is suggestive. But there is a direct self-contradiction that you must flag if you referee this. Section 4.3 says the observed images use random rotations drawn from U(SO(3)), and Algorithm 4.1 instructs the reader to generate random orientations at every iteration. Then, in the PCA-comparison paragraph, the text says 'we fix the viewing directions throughout the experiment.' If viewing directions were actually fixed, the central claim about not needing to estimate orientations is untested in the most realistic experiment. The authors need to clarify, and if they meant only the PCA diagnostic, the wording is still misleading.\n\nOther soft spots are addressable but real. There is no convergence analysis for the particle discretization; the conclusion explicitly defers it. All data are generated under the same forward model used by the method, with no CTF, orientation bias, or model-error tests. There are no baselines or error bars, and no code or data are released. Also, the KL orientation in the gradient formula of Section 2.5.1 (KL(rho_y || rho_y^delta)) does not match the orientation in the MAP derivation of Section 3.1 (KL(rho_y^delta || rho_y)). That matters for consistency and should be fixed.\n\nThe math largely holds together. The self-citation to the authors' earlier Wasserstein-gradient-flow papers is not a problem here because the random push-forward setting is a real extension.\n\nThis paper deserves a serious referee. The framework is coherent, the idea is useful, and the flaws are fixable. But the HSP90 contradiction is load-bearing, and without resolving it the experimental support for the main claim is compromised. I would send it to peer review with a request for major revision, not desk-reject.","headline":"A coherent variational framework for cryo-EM as a stochastic inverse problem, with a clean particle gradient flow and a useful MAP connection—but the experiments are synthetic-only and the HSP90 section directly contradicts itself on whether viewing directions were random or fixed.","tokens_in":23338,"tokens_out":2095,"would_cite":false,"duration_ms":20394,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65M32","49Q22","65M75","65K10"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper argues that cryo-EM reconstruction is a stochastic inverse problem: the unknown is a probability distribution over molecular conformations, recovered by a Wasserstein gradient flow, and demonstrated on simulated HSP90 data with…","keywords":["cryo-EM","structural heterogeneity","stochastic inverse problem","Wasserstein gradient flow","particle method","push-forward measure","maximum mean discrepancy"],"falsifier":"Take the simulated HSP90 setup and repeat with images corrupted by an unmodeled contrast transfer function or with orientations drawn from a strongly biased, non-uniform distribution; if the recovered conformational distribution shifts systematically away from the known ground truth, the claim that the flow recovers the true structural distribution fails.","tokens_in":22230,"feed_emoji":"🔬","tokens_out":6953,"duration_ms":56637,"temperature":0.7,"pith_summary":"The paper seeks to establish that cryo-EM reconstruction is a stochastic inverse problem over probability measures: the unknown is not a single 3D structure but an entire distribution $\\rho_\\theta$ over molecular conformations. It formulates reconstruction as the variational problem of minimizing a statistical discrepancy (KL, MMD, or energy distance) between the observed image distribution $\\rho_y^\\delta$ and the distribution $F(\\rho_\\theta)$ predicted by pushing a candidate structural distribution through the random imaging operator. The paper shows this variational problem can be solved by a Wasserstein gradient flow, implemented with a particle system, and demonstrates in simulations—including a realistic HSP90 protein with only 3,000 images—that continuous conformational distributions are recoverable. It also shows that established MAP-based cryo-EM pipelines fit within the same framework as discretize-then-optimize limits, and it supplies consistency conditions under which those limits converge to the infinite-dimensional solution. If these claims hold, cryo-EM would gain a way to study continuous protein dynamics rather than discrete classes.","feed_headline":"Cryo-EM recast as stochastic inverse problem over shapes","feed_subtitle":"Treating the unknown as a distribution over conformations lets 3,000 images recover continuous protein motion.","key_machinery":"The central object is the random push-forward operator $F(\\rho_\\theta) = \\int T_{\\hat\\omega\\#}\\rho_\\theta \\, d\\mu_\\omega(\\hat\\omega)$, which sends a distribution over structures to the predicted distribution of images by composing each structure with a random rotation, projection, and convolution, then mixing over the law of the noise and geometry. The paper solves the variational problem (2.6) by evolving the structural distribution along the Wasserstein gradient flow, equation (2.10), whose velocity field is the composition of the adjoint Jacobian $\\nabla T_\\omega^\\top$ with the Fréchet derivative of the chosen discrepancy $\\delta D/\\delta\\rho_y$; this flow is simulated by the particle ODE (2.11), with the adjoint-vector product evaluated by reverse-mode automatic differentiation.","core_discovery":"Cryo-EM reconstruction is posed as a variational problem over probability measures: minimize $E(\\rho_\\theta) = D(F(\\rho_\\theta), \\rho_y^\\delta)$, where $F$ is the random push-forward of the unknown structural distribution through rotations, projection, and additive noise. The authors show that this objective admits a Wasserstein gradient flow, whose particle discretization gives a deterministic ensemble update for conformational samples. Numerically, the flow recovers continuous conformational distributions—including a simulated HSP90 protein from only 3,000 images, without pre-estimated viewing directions. The paper further shows that common MAP-based pipelines are discretize-then-optimize instances of the same variational problem, and it gives $\\Gamma$-convergence conditions under which such schemes converge to the infinite-dimensional optimum as data and particle counts grow.","pith_inferences":["If the variational objective is well-behaved, the number of images needed may scale with the intrinsic dimension of the conformational landscape rather than the number of discrete states, which would change how cryo-EM experiments are designed for dynamic proteins.","The OTD/DTO distinction suggests a testable benchmark: on the same dataset, a DTO reconstruction and an OTD flow should agree in the large-data limit; discrepancies would localize where the discretization prematurely constrains the solution space.","The method's reliance on kernel choices (energy distance versus other MMD kernels) could be probed by varying the kernel and measuring whether the recovered distribution shifts in Wasserstein distance, a sensitivity the paper leaves unexplored.","Applying the flow to experimental data will require coupling it to per-image orientation estimation; whether the uniform-rotation assumption can be relaxed to learned pose distributions is a natural untested extension."],"forward_implications":["Continuous conformational variability, such as hinge bending or domain opening, can in principle be recovered without discrete classification or pre-estimated viewing directions.","The particle flow automatically produces samples from the inferred conformational distribution, so downstream quantities such as ensemble averages or approximate free-energy differences follow directly.","Because the framework only needs a differentiable simulator of the image-formation process, it transfers to other stochastic inverse problems with random forward operators.","MAP-based cryo-EM pipelines are unified with this framework as a discretize-then-optimize limit, and the consistency conditions specify when their output converges to the continuous solution."],"supporting_citations":[{"why":"Supplies the measure-theoretic formulation of stochastic inverse problems that the paper lifts to a random forward operator.","marker":"[6]"},{"why":"Establishes the variational/stochastic optimization viewpoint over probability measures that the paper adapts to cryo-EM.","marker":"[33]"},{"why":"Provides the Wasserstein gradient flow approach and the existence/stability conditions the variational problem relies on.","marker":"[31]"},{"why":"Supplies the Bayesian/MAP objective that the paper re-derives as a discretize-then-optimize instance.","marker":"[39]"},{"why":"The concrete MAP objective whose equation (2) the paper matches exactly when the structural distribution is approximated by particles.","marker":"[29]"},{"why":"Establishes the distribution-over-structures statistical perspective that this paper builds on.","marker":"[46]"},{"why":"Provides the empirical-measure convergence theorem used in the Γ-convergence consistency proof.","marker":"[17]"}],"fun_headline_variants":["Cryo-EM solved as continuous shape distribution from 3,000 images","Turns cryo-EM into a probability flow over protein shapes","Wasserstein flow recovers continuous conformations without view angles","Cryo-EM without class averaging: shape distribution from raw projections"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on the assumption that the observed images really are independent draws from the push-forward of some structural distribution under the known forward model with uniform random rotations and known noise—if that generative model is wrong, the recovered distribution is not guaranteed to match reality.","fun_headline_variants_meta":{"raw":{"variants":["Cryo-EM solved as continuous shape distribution from 3,000 images","Turns cryo-EM into a probability flow over protein shapes","Wasserstein flow recovers continuous conformations without view angles","Cryo-EM without class averaging: shape distribution from raw projections"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000444,"raw_usage":{"total_tokens":2255,"prompt_tokens":959,"completion_tokens":1296,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":575,"completion_tokens_details":{"reasoning_tokens":1218}},"tokens_in":575,"tokens_out":1296,"duration_ms":9168,"temperature":1.0,"reasoning_tokens":1218,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T16:24:46.965063+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the simulated HSP90 setup and repeat with images corrupted by an unmodeled contrast transfer function or with orientations drawn from a strongly biased, non-uniform distribution; if the recovered conformational distribution shifts systematically away from the known ground truth, the claim that the flow recovers the true structural distribution fails.","supporting_citations":[{"cited_title":"Breidt, T","cited_arxiv_id":null,"evidence_quote":"Supplies the measure-theoretic formulation of stochastic inverse problems that the paper lifts to a random forward operator."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the variational/stochastic optimization viewpoint over probability measures that the paper adapts to cryo-EM."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Bayesian/MAP objective that the paper re-derives as a discretize-then-optimize instance."},{"cited_title":"Kimanius, L","cited_arxiv_id":null,"evidence_quote":"The concrete MAP objective whose equation (2) the paper matches exactly when the structural distribution is approximated by particles."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the distribution-over-structures statistical perspective that this paper builds on."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the empirical-measure convergence theorem used in the Γ-convergence consistency proof."}],"review_version":1}