Constrained Shape Analysis with Applications to RNA Structure

TL;DR

Constrained Shape Analysis with Applications to RNA Structure develops a unified framework for analyzing biomolecular shapes under length constraints by introducing polypolar coordinates and polysphere representations. It combines multicentring with polypolar coordinates to separate constrained radial lengths from unconstrained torsions, enabling tractable modeling and analysis on constrained shape spaces. The paper provides concrete RNA-centered examples, discusses concentration-based statistics and tangent-projection methods, and introduces distributions on polypolar spaces alongside an extended clustering approach (MINT-AGE) with parametric mode hunting for very small clusters. These ideas offer a principled route to analyzing RNA backbone conformations under partial information and pave the way for unsupervised learning in constrained biomolecular shape analysis.

Abstract

In many applications of shape analysis, lengths between some landmarks are constrained. For instance, biomolecules often have some bond lengths and some bond angles constrained, and variation occurs only along unconstrained bonds and constrained bonds' torsions where the latter are conveniently modelled by dihedral angles. Our work has been motivated by low resolution biomolecular chain RNA where only some prominent atomic bonds can be well identified. Here, we propose a new modelling strategy for such constrained shape analysis starting with a product of polar coordinates (polypolars), where, due to constraints, for example, some radial coordinates should be omitted, leaving products of spheres (polyspheres). We give insight into these coordinates for particular cases such as five landmarks which are motivated by a practical RNA application. We also discuss distributions for polypolar coordinates and give a specific methodology with illustration when the constrained size-and-shape variables are concentrated. There are applications of this in clustering and we give some insight into a modified version of the MINT-AGE algorithm.

Figures (9)

• Figure 1: The five landmarks (circled) of the low detail RNA structure with the backbone and bases starting at N1/N9.
• Figure 2: Coordinates for the five landmarks from low detail RNA backbone, see Figure \ref{['RNA']}. Left: Permuting landmarks. Middle: multicentring. Right: constrained MPP coordinates. See Examples \ref{['eg:multicentring']} and \ref{['eg:MUCEPOPES']}.
• Figure 3: Left: part of a molecular backbone chain with bond angles $\theta_j,\theta_{j+1}$ and dihedral angles $\phi_{j-1},\phi_j$. Right: the same part multicentred with $z_{j-1} = x_j-x_{j-1},z_{j} = x_{j+1}-x_{j},z_{j+1} = x_{j+2}-x_{j+1}$. Their directions give the $z$-axis of the local coordinate system, where the $y$-axis is determined by the preceding $z$-coordinate.
• Figure 4: Simplex MPP coordinates in case of $m=4$. For Type 1 coordinates, $u_4$ is orthogonal to the simplex spanned by $z_1,\ldots,z_4$ (not depicted) and for Type 2 coordinates, $u_4$ is orthogonal to each of $z_1,z_2,z_3$. For both types $u_3$ is orthogonal to the simplex (triangle) $\Delta^2$ spanned by $z_1,z_2,z_3$, and $u_2$ is orthogonal to the simplex (linear segment) $\Delta^1$ spanned by $z_1,z_2$, determining $u_1$ such that $(u_1,\ldots,u_4) \in SO(4)$ (Type 1), and such that $u^T_1z_1>0$ (Type 2).
• Figure 5: Left: four out of the five landmarks in 2D where $x_1$ and $x_5$ vary along circles of fixed radii about $x_2$ and $x_4$, respectively. Right: in multicentred polysphere coordinates. The two blue lines are of fixed constant lengths in both panels.

Theorems & Definitions (14)

• Definition 2.1
• Example 2.2
• Definition 2.3
• Example 2.4
• Example 2.5
• Lemma 2.6
• proof
• Remark 2.7
• Definition 2.8
• Remark 2.9