Abstract
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 ES(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 ES(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 ES(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 ES(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.
Highlights
Proteins, the end products of the information encoded in the genome of any organism, play a central role in defining the life of this organism
SUMMARY AND CONCLUSIONS We have developed a new method for quantifying the compactness of a protein structure
In this new approach we compute the conformal map f between the surface of the protein and the 2-sphere that has minimal distortion, where distortion is defined as a symmetric elastic energy ES(f ) that measures the distance between f and an isometry
Summary
The end products of the information encoded in the genome of any organism, play a central role in defining the life of this organism. They catalyze most biochemical reactions within cells and are responsible, among other functions, for the transport of nutrients and for signal transmission within and between cells. The corresponding linear chain of amino acids becomes functional only when it adopts a three-dimensional shape, the so-called tertiary, or native structure of the protein. This is by no means different from the macroscopic world: most proteins serve as tools in the cell and as such either have a defined or adaptive shape to function, much as the shapes of the tools we use are defined according to the functions they need to perform
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