4,239 research outputs found
How round is a protein? Exploring protein structures for globularity using conformal mapping.
We present a new algorithm that automatically computes a measure of the geometric difference between the surface of a protein and a round sphere. The algorithm takes as input two triangulated genus zero surfaces representing the protein and the round sphere, respectively, and constructs a discrete conformal map f between these surfaces. The conformal map is chosen to minimize a symmetric elastic energy E S (f) that measures the distance of f from an isometry. We illustrate our approach on a set of basic sample problems and then on a dataset of diverse protein structures. We show first that E S (f) is able to quantify the roundness of the Platonic solids and that for these surfaces it replicates well traditional measures of roundness such as the sphericity. We then demonstrate that the symmetric elastic energy E S (f) captures both global and local differences between two surfaces, showing that our method identifies the presence of protruding regions in protein structures and quantifies how these regions make the shape of a protein deviate from globularity. Based on these results, we show that E S (f) serves as a probe of the limits of the application of conformal mapping to parametrize protein shapes. We identify limitations of the method and discuss its extension to achieving automatic registration of protein structures based on their surface geometry
A Metric for genus-zero surfaces
We present a new method to compare the shapes of genus-zero surfaces. We
introduce a measure of mutual stretching, the symmetric distortion energy, and
establish the existence of a conformal diffeomorphism between any two
genus-zero surfaces that minimizes this energy. We then prove that the energies
of the minimizing diffeomorphisms give a metric on the space of genus-zero
Riemannian surfaces. This metric and the corresponding optimal diffeomorphisms
are shown to have properties that are highly desirable for applications.Comment: 33 pages, 8 figure
Neural Semantic Surface Maps
We present an automated technique for computing a map between two genus-zero
shapes, which matches semantically corresponding regions to one another. Lack
of annotated data prohibits direct inference of 3D semantic priors; instead,
current State-of-the-art methods predominantly optimize geometric properties or
require varying amounts of manual annotation. To overcome the lack of annotated
training data, we distill semantic matches from pre-trained vision models: our
method renders the pair of 3D shapes from multiple viewpoints; the resulting
renders are then fed into an off-the-shelf image-matching method which
leverages a pretrained visual model to produce feature points. This yields
semantic correspondences, which can be projected back to the 3D shapes,
producing a raw matching that is inaccurate and inconsistent between different
viewpoints. These correspondences are refined and distilled into an
inter-surface map by a dedicated optimization scheme, which promotes
bijectivity and continuity of the output map. We illustrate that our approach
can generate semantic surface-to-surface maps, eliminating manual annotations
or any 3D training data requirement. Furthermore, it proves effective in
scenarios with high semantic complexity, where objects are non-isometrically
related, as well as in situations where they are nearly isometric
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