Linear-time protein 3-D structure searching with insertions and deletions

Dynamic Programming Used to Align Protein Structures with a Spectrum Is Robust

Several efficient algorithms to conduct pairwise comparisons among large databases of protein structures have emerged in the recent literature. The central theme is the design of a measure between the C_α atoms of two protein chains, from which dynamic programming is used to compute an alignment. The efficiency and efficacy of these algorithms allows large-scale computational studies that would have been previously impractical. The computational study herein shows that the structural alignment algorithm eigen-decomposition alignment with the spectrum (EIGAs) is robust against both parametric and structural variation.

The difficulty of protein structure alignment under the RMSD

BACKGROUND

Protein structure alignment is often modeled as the largest common point set (LCP) problem based on the Root Mean Square Deviation (RMSD), a measure commonly used to evaluate structural similarity. In the problem, each residue is represented by the coordinate of the Cαatom, and a structure is modeled as a sequence of 3D points. Out of two such sequences, one is to find two equal-sized subsequences of the maximum length, and a bijection between the points of the subsequences which gives an RMSD within a given threshold. The problem is considered to be difficult in terms of time complexity, but the reasons for its difficulty is not well-understood. Improving this time complexity is considered important in protein structure prediction and structural comparison, where the task of comparing very numerous structures is commonly encountered.

RESULTS

To study why the LCP problem is difficult, we define a natural variant of the problem, called the minimum aligned distance (MAD). In the MAD problem, the length of the subsequences to obtain is specified in the input; and instead of fulfilling a threshold, the RMSD between the points of the two subsequences is to be minimized. Our results show that the difficulty of the two problems does not lie solely in the combinatorial complexity of finding the optimal subsequences, or in the task of superimposing the structures. By placing a limit on the distance between consecutive points, and assuming that the points are specified as integral values, we show that both problems are equally difficult, in the sense that they are reducible to each other. In this case, both problems can be exactly solved in polynomial time, although the time complexity remains high.

CONCLUSIONS

We showed insights and techniques which we hope will lead to practical algorithms for the LCP problem for protein structures. The study identified two important factors in the problem's complexity: (1) The lack of a limit in the distance between the consecutive points of a structure; (2) The arbitrariness of the precision allowed in the input values. Both issues are of little practical concern for the purpose of protein structure alignment. When these factors are removed, the LCP problem is as hard as that of minimizing the RMSD (MAD problem), and can be solved exactly in polynomial time.

TS-AMIR: a topology string alignment method for intensive rapid protein structure comparison

Background

In structural biology, similarity analysis of protein structure is a crucial step in studying the relationship between proteins. Despite the considerable number of techniques that have been explored within the past two decades, the development of new alternative methods is still an active research area due to the need for high performance tools.

Results

In this paper, we present TS-AMIR, a Topology String Alignment Method for Intensive Rapid comparison of protein structures. The proposed method works in two stages: In the first stage, the method generates a topology string based on the geometric details of secondary structure elements, and then, utilizes an n-gram modelling technique over entropy concept to capture similarities in these strings. This initial correspondence map between secondary structure elements is submitted to the second stage in order to obtain the alignment at the residue level. Applying the Kabsch method, a heuristic step-by-step algorithm is adopted in the second stage to align the residues, resulting in an optimal rotation matrix and minimized RMSD. The performance of the method was assessed in different information retrieval tests and the results were compared with those of CE and TM-align, representing two geometrical tools, and YAKUSA, 3D-BLAST and SARST as three representatives of linear encoding schemes. It is shown that the method obtains a high running speed similar to that of the linear encoding schemes. In addition, the method runs about 800 and 7200 times faster than TM-align and CE respectively, while maintaining a competitive accuracy with TM-align and CE.

Conclusions

The experimental results demonstrate that linear encoding techniques are capable of reaching the same high degree of accuracy as that achieved by geometrical methods, while generally running hundreds of times faster than conventional programs.

A spectral approach to protein structure alignment

A new intrinsic geometry based on a spectral analysis is used to motivate methods for aligning protein folds. The geometry is induced by the fact that a distance matrix can be scaled so that its eigenvalues are positive. We provide a mathematically rigorous development of the intrinsic geometry underlying our spectral approach and use it to motivate two alignment algorithms. The first uses eigenvalues alone and dynamic programming to quickly compute a fold alignment. Family identification results are reported for the Skolnick40 and Proteus300 data sets. The second algorithm extends our spectral method by iterating between our intrinsic geometry and the 3D geometry of a fold to make high-quality alignments. Results and comparisons are reported for several difficult fold alignments. The second algorithm's ability to correctly identify fold families in the Skolnick40 and Proteus300 data sets is also established.