# Reconstructing Heterogeneous Biomolecules via
Hierarchical Gaussian Mixtures and Part Discovery

Synthetic Data Results
Experimental Data Results

## Abstract

Cryo-EM is a transformational imaging technique in structural biology where computational methods are used to infer 3D molecular structure at atomic resolution from extremely noisy 2D electron microscope images. At the forefront of research is how to model the structure when the imaged particles exhibit non-rigid conformational flexibility and compositional variation where parts are sometimes missing. We introduce a novel 3D reconstruction framework with a hierarchical Gaussian mixture model, inspired in part by Gaussian Splatting for 4D scene reconstruction. In particular, the structure of the model is grounded in an initial process that infers a part-based segmentation of the particle, providing essential inductive bias in order to handle both conformational and compositional variability. The framework, called CryoSPIRE, is shown to reveal biologically meaningful structures on complex experimental datasets, and establishes a new state-of-the-art on CryoBench, a benchmark for cryo-EM heterogeneity methods.

## CryoSPIRE

We introduce a novel two-stage GMM-based framework to tackle both conformational and compositional heterogeneity. It begins with a part discovery stage where we optimize a coarse-grained GMM, with each Gaussian component augmented with a learnable feature vector, and learn latent-conditioned MLPs to modulate Gaussian locations and amplitudes. We observe that the learned features encode meaningful information about structural regularities. In particular, Gaussian components that coherently deform or consistently appear or disappear receive similar features. This enables inference of a part-based segmentation of the particle, from which we define a Scaffold Part-aware Gaussian Mixture (CryoSPIRE) model in terms of a set of anchors, one per part, each with a corresponding set of Gaussians. Optimizing this representation recovers a high-resolution representation of 3D density maps with compositional and conformational variability.

## Results on IgG-1D Synthetic Dataset

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## Results on Pre-Catalytic Spliceosome

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