Protein Structure-Structure Alignment With Discrete Fréchet Distance

Abstract

Matching two geometric objects in 2D and 3D spaces is a central problem in computer vision, pattern recognition and protein structure prediction. In particular, the problem of aligning two polygonal chains under translation and rotation to minimize their distance has been studied using various distance measures. It is well known that the Hausdorff distance is useful for matching two point sets, and that the Fréchet distance is a superior measure for matching two polygonal chains. The discrete Fréchet distance closely approximates the (continuous) Fréchet distance, and is a natural measure for the geometric similarity of the folded 3D structures of bio-molecules such as proteins. In this paper, we present new algorithms for matching two polygonal chains in 2D to minimize their discrete Fréchet distance under translation and rotation, and an effective heuristic for matching two polygonal chains in 3D. We also describe our empirical results on the application of the discrete Fréchet distance to the protein structure-structure alignment.

Protein Local Structure Alignment Under the Discrete Fréchet Distance

“…|A i | = |B i | = 1: both the person and the dog move (jump) forward. For 3D chains these bounds are O(k 4 l 4 log(k + l)) and O(k 7 l 7 log(k + l)) respectively [14]. They are significantly faster than the corresponding bounds for the continuous Fréchet distance (certainly due to a simpler distance structure), which are O((k + l) 11 log(k + l)) and O((k + l) 20 log(k + l))…”

“…Apparently, for any solution for PLSA we should allow translation and rotation. When m = 2 and when both translation and rotation are allowed, we can use a method similar to that in [14] to compute the optimal local alignment with fixed δ. The idea is as follows.…”

“…We can enumerate all possible configurations for A and τ (B) to realize a discrete Fréchet distance of δ. There are O((n 1 n 2 ) f ) = O(n 12 ) number of such configurations, following an argument similar to [22,14]. Then for each configuration, we can use the above Theorem 4.1 to obtain the optimal local alignment for each configuration and finally we simply return the overall optimal solution.…”

“…We comment that when m is larger, but still a constant, the above idea can be carried over so that we will still be able to solve PLSA in polynomial time. It follows from [22,14] that we have O(n mf ) = O(n 6m ) number of configurations between the m chains. Then we can again use…”

Radial Basis Function Process Neural Network Training Based on Generalized FRECHET Distance And GA-SA Hybrid Strategy

“…Fréchet distance [10] is a judgment to measure the similarity between polygonal curves. In applications, dynamic samples normally consist of a sequence of discrete sampling points, therefore, Eiter and Mannila [11] put forward discrete Fréchet distance on the basis of continuous Fréchet distance, and it achieved good application effect in protein structure prediction [12], online signature verification [13], etc. Time-varying process signal can be seen as a onedimensional curve about the time; therefore, discrete Fréchet distance can be extended to timevarying function space to measure nature difference between input samples of RBF-PNN.…”