A statistical pipeline for identifying physical features that differentiate classes of 3D shapes

Abstract

The recent curation of large-scale databases with 3D surface scans of shapes has motivated the development of tools that better detect global patterns in morphological variation. Studies, which focus on identifying differences between shapes, have been limited to simple pairwise comparisons and rely on prespecified landmarks (that are often known). We present SINA-TRA, the first statistical pipeline for analyzing collections of shapes without requiring any correspondences. Our novel algorithm takes in two classes of shapes and highlights the physical features that best describe the variation between them. We use a rigorous simulation framework to assess our approach. Lastly, as a case study we use SINATRA to analyze mandibular molars from four different suborders of primates and demonstrate its ability recover known morphometric variation across phylogenies.

A Topological Data Analytic Approach for Discovering Biophysical Signatures in Protein Dynamics

“…2(a)-(c)). This observation is similar to previous works which show coordinate-based regularization to be most effective when variation between 3D structures occurs on a global scale and in the same direction on the unit sphere [13]. In the cases of random spherical perturbations, the variance of the distribution of atoms in the ROI widens; hence, the Elastic Net and Neural Network have a more difficult time identifying features that differentiate two protein classes, unless those variations happen on a global scale (again see Figs.…”

“…Adopted from its predecessor [13], the second step of the SINATRA Pro pipeline uses a tool from integral geometry and differential topology called the Euler characteristic (EC) transform [14–17]. As a brief overview of this approach, given the mesh representation ℳ of a protein structure, the Euler characteristic is an accessible topological invariant defined as where the collection # V ( ℳ ), # E ( ℳ ), # F ( ℳ ) denotes the number of vertices (atoms), edges (connections between atoms), and faces (triangles enclosed by edges) of the mesh, respectively.…”

“…In previous studies, it has been observed that the Euler characteristic can be a less-than-optimal shape summary statistic when inter-class variation between 3D objects is high and driven by local fluctuations in morphology [13, 16, 17, 52, 66]. Given that this situation can be quite common in molecular dynamics, we introduce a new topological invariant which we refer to as the differential Euler characteristic (DEC) (see Fig.…”

“…For simplicity, we assume that the implied posterior distribution of β (deterministically given in Eq. (4)) is approximately multivariate normal with an empirical mean vector µ and positive semi-definite covariance/precision matrix Σ = Λ − 1 [13, 79]. Given these values, we iteratively partition such that, for each j -th topological feature: Under normality assumptions, Eq.…”

“…From a classical hypothesis testing point-of-view, the null hypothesis for Eq. ( 6) assumes that every DEC feature equally contribute to the total variance between proteins, while the alternative hypothesis proposes that some DEC features are better associated with biophysical changes in protein structures than others [13,79].…”