A Mathematical Framework for Protein Structure Comparison

Comparison of protein structures is important for revealing the evolutionary relationship among proteins, predicting protein functions and predicting protein structures. Many methods have been developed in the past to align two or multiple protein structures. Despite the importance of this problem, rigorous mathematical or statistical frameworks have seldom been pursued for general protein structure comparison. One notable issue in this field is that with many different distances used to measure the similarity between protein structures, none of them are proper distances when protein structures of different sequences are compared. Statistical approaches based on those non-proper distances or similarity scores as random variables are thus not mathematically rigorous. In this work, we develop a mathematical framework for protein structure comparison by treating protein structures as three-dimensional curves. Using an elastic Riemannian metric on spaces of curves, geodesic distance, a proper distance on spaces of curves, can be computed for any two protein structures. In this framework, protein structures can be treated as random variables on the shape manifold, and means and covariance can be computed for populations of protein structures. Furthermore, these moments can be used to build Gaussian-type probability distributions of protein structures for use in hypothesis testing. The covariance of a population of protein structures can reveal the population-specific variations and be helpful in improving structure classification. With curves representing protein structures, the matching is performed using elastic shape analysis of curves, which can effectively model conformational changes and insertions/deletions. We show that our method performs comparably with commonly used methods in protein structure classification on a large manually annotated data set

Two-dimensional classification of interfacial and partitioning properties of amino acids

A classification of amino acid residues based on the interfacial and partitioning properties was introduced by Khokhlov et al. [1] [2] Amino acid residues are characterized by two parameters: the standard free energy of adsorption of an amino acid at an octanol/water interface and the standard free energy of the partition of an amino acid between octanol and water, both of them normalized by kT. As a result, five groups of amino acids having similar values of the parameters are identified. This classification for the amino acids is based in trace correlations between two one-dimensional parameters which are related with the interactions in the biological environment: hydrophilic / hydrophobic behaviour (partition) and activity at the interface (surface tension). This method is believed to be able to provide promising results in the search of correlation giving rise to protein sequences. A comparison of the parameters in question gives information on energetic preferences of the molecules to be located at the interface or in a bulk phase. This study is applied on serine, threonine, aspartic acid, glutamic acid and tyrosine.Popular Science Summary The protein folding is the physical process in which a protein withdraws into its characteristic three-dimensional structure and function. Each protein begins as a linear chain of amino acids, resulted from a sequence of our genetic material, and does not have three-dimensional structure. However, each amino acid chain has certain chemical characteristics that can influence to the folding like hydrophobia and hydrophilia. These amino acids interact with each other in their cellular environment to produce a well-defined three-dimensional shape, the folded protein, known as native state. The mechanism of protein folding is not completely understood. However, the three-dimensional protein structure is essential to perform its function. If the protein does not fold into the desired shape, typically produce inactive proteins with different properties including toxic. Some neurodegenerative diseases among others are considered the consequence of the accumulation of incorrectly folded proteins. Therefore it is important to know which factors affect the protein folding and how we can predict its final structure

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