Local Geometric Distortion

Overview

When comparing two proteins with GW, the local geometric distortion (LGD) of a residue quantifies its contribution to the overall GW distance. This provides a residue-level measure of structural conservation.

Definition

Formally, when comparing proteins \(X\) and \(Y\), the local geometric distortion of residue \(x_i\) is defined as:

\[LGD(x_i) = \sum_{j, k, l} |d_X(x_i, x_j) - d_Y(y_k, y_l)|^2 T_{i, k} T_{j, l}\]

where \(T\) is the optimal correspondence between residues of \(X\) and \(Y\).

Relationship to GW Distance

The sum of local geometric distortions relates directly to the GW distance:

\[\frac{1}{2} \sqrt{\sum_i LGD(x_i)} = GW(X, Y)\]

and similarly for residues in \(Y\).

Interpretation and Applications

Low LGD: Indicates that a residue is structurally well-conserved relative to the other protein.
High LGD: Indicates lower conservation, which may correspond to flexible regions (e.g. switch regions).

LGD can be used to:

Identify structurally conserved regions in evolutionarily related proteins.
Detect flexible or variable regions by comparing multiple conformations of the same or similar proteins.

Note

For robust results, it is recommended to average LGD values across multiple protein comparisons rather than relying on a single pairwise comparison. Avoid using LGD values from downsampled proteins.

Fused Local Geometric Distortion

When using fused GW, the fused local geometric distortion extends the concept to include biochemical data. It is defined as:

\[LGD(x_i) = \sum_{j, k, l} \left[ \alpha \cdot |d_X(x_i, x_j) - d_Y(y_k, y_l)|^2 + (1 - \alpha) \cdot \delta(x_i, y_j) \right] T_{i, k} T_{j, l}\]

where \(\delta(x_i, y_j)\) measures the biochemical difference between residues, and \(\alpha\) controls the balance between geometric and biochemical contributions.