Abstract
Cryo-electron microscopy is a technique in structural biology for determining the 3D structure of macromolecules. A key step in this process is detecting common lines of intersection between unknown embedded image planes. We wish to characterize such common lines in terms of the unembedded geometric data detected in experiments. We use techniques from spherical geometry, real algebraic geometry, and linear algebra. We show that common lines are the solutions to a system of polynomial equalities and inequalities, i.e., they form a semi-algebraic set. These polynomials are low degree, and we explicitly derive them in this paper. The polynomials we derive provide the desired intrinsic characterization of common lines. We discuss possible applications of these polynomials to reconstruction algorithms that are robust to the high levels of noise present in cryo-electron images.
Highlights
Cryo-electron microscopy is a technique used to discover the structure of small molecules, usually proteins in the context of structural biology research [1]
Mathematical model We briefly describe the mathematical model for Cryo-electron microscopy (cryo-EM), following [3], Section 0
This is relevant to cryo-EM reconstruction, because the microscope orientations are unknown, it is possible to detect the common lines data the orientations realize from the images I1, . . . , IN [5]
Summary
Cryo-electron microscopy (cryo-EM) is a technique used to discover the structure of small molecules, usually proteins in the context of structural biology research [1]. A cryo-EM experiment produces images Ii and Ij from orientations Fi = (ai, bi) and Fj = (aj, bj) These frames define isometric embeddings ιi and ιj (Figure 3) of the unembedded image planes Pi and Pj into R3, given by ιi(x, y) = xai + ybi, ιj(x, y) = xaj + ybj. Algorithms have long been known (e.g., [4], Section 2.1) that recover a set of realizing frames from valid common lines data This is relevant to cryo-EM reconstruction, because the microscope orientations are unknown, it is possible to detect the common lines data the orientations realize. Common lines based approaches for cryo-EM reconstruction (Problem 2) assume that we can accurately detect the valid common lines realized by the unknown microscope orientations. The development of common lines reconstruction algorithms that are robust to this kind of error is an active area of research
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