  We consider the problem of recovering the three-dimensional atomic structure
of a flexible macromolecule from a heterogeneous cryo-EM dataset. The dataset
contains noisy tomographic projections of the electrostatic potential of the
macromolecule, taken from different viewing directions, and in the
heterogeneous case, each image corresponds to a different conformation of the
macromolecule. Under the assumption that the macromolecule can be modelled as a
chain, or discrete curve (as it is for instance the case for a protein backbone
with a single chain of amino-acids), we introduce a method to estimate the
deformation of the atomic model with respect to a given conformation, which is
assumed to be known a priori. Our method consists on estimating the torsion and
bond angles of the atomic model in each conformation as a linear combination of
the eigenfunctions of the Laplace operator in the manifold of conformations.
These eigenfunctions can be approximated by means of a well-known technique in
manifold learning, based on the construction of a graph Laplacian using the
cryo-EM dataset. Finally, we test our approach with synthetic datasets, for
which we recover the atomic model of two-dimensional and three-dimensional
flexible structures from noisy tomographic projections.
