Geometric shape matching for recovering protein conformations from single-particle Cryo-EM data

TL;DR

This work addresses recovery of the three-dimensional backbone structure of single polypeptide proteins from single-particle cryo-electron microscopy (Cryo-SPA) data by using methods from shape analysis to recover the three-dimensional backbone structure.

Abstract

We address recovery of the three-dimensional backbone structure of single polypeptide proteins from single-particle cryo-electron microscopy (Cryo-SPA) data. Cryo-SPA produces noisy tomographic projections of electrostatic potentials of macromolecules. From these projections, we use methods from shape analysis to recover the three-dimensional backbone structure. Thus, we view the reconstruction problem as an indirect matching problem, where a point cloud representation of the protein backbone is deformed to match 2D tomography data. The deformations are obtained via the action of a matrix Lie group. By selecting a deformation energy, the optimality conditions are obtained, which lead to computational algorithms for optimal deformations. We showcase our approach on synthetic data, for which we recover the three-dimensional structure of the backbone.

Figures (10)

• Figure 1: The "anatomy" of the backbone of a protein.
• Figure 1: The initial and final conformation of the C_$\alpha$ atoms in the backbone of the closed-to-open adenylate kinase deformation. Note that this depicts the 3D arrangements, i.e., elements in $A$.
• Figure 2: The forward model is built up from three intermediate steps. First is to use $\operatorname{\mathcal{M}}$ to map $V$ (relative positions) to corresponding absolute atomic positions, i.e., elements in $A$. Next, we use $\operatorname{\mathcal{B}}$ to generate a (approximate) 3D map in $X$, which is then mapped by $\operatorname{\mathcal{P}}$ to a 2D TEM image in $Y$.
• Figure 2: The data (in $Y$) used to deform the template. The three images are obtained by applying the forward model to the protein conformation in \ref{['fig:target_bb']} and adding Gaussian noise with standard deviation $1.0$ to each pixel.
• Figure 3: The results of applying the indirect shape matching algorithm to the template (\ref{['fig:initial_bb']}) using projections along the $x$, $y$ and $z$ axes. Note that while not perfect, the deformation along these axes decently captures the changes between the initial and target conformation.

Theorems & Definitions (17)

• Remark 2.1
• Theorem 2.2
• Remark 2.3
• Remark 3.1
• Remark 3.2
• Remark 3.3
• Lemma A.1
• Proof 1
• Proof 2: Proof of \ref{['th:gradient_general']}
• Corollary B.1