A Surface of Minimum Area Metric for the Structural Comparison of Proteins

Falicov, Alexis; Cohen, Fred E.

doi:10.1006/jmbi.1996.0294

Cited by 56 publications

(49 citation statements)

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“…[4][5][6][7][8]. Such a metric is of interest for comparison between folded structures, as well as to quantify how close an unfolded or partly folded structure is to the native.…”

Section: Discussionmentioning

confidence: 99%

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Generalization of distance to higher dimensional objects

Plotkin

2007

Proc. Natl. Acad. Sci. U.S.A.

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The measurement of distance between two objects is generalized to the case where the objects are no longer points but are one-dimensional. Additional concepts such as nonextensibility, curvature constraints, and noncrossing become central to the notion of distance. Analytical and numerical results are given for some specific examples, and applications to biopolymers are discussed.T he distance, as conventionally defined between two zerodimensional objects (points) A and B at positions r A and r B , is the minimal arclength travelled in the transformation from A to B. A transformation r(t) between A and B is a vector function that may be parametrized by a scalar variable t: 0 Յ t Յ T, r(0) ϭ r A , r(T) ϭ r B , and the distance travelled is a functional of r(t). The (minimal) transformation r*(t) is an object of dimension one higher than A or B; i.e., it yields a distance that is one-dimensional. The distance D* is found through the variation of the functional (1):Here, ẋ ϭ dx/dt, ṙ ϭ dr/dt, and we have let g v ϭ ␦ v (Euclidean metric). The boundary conditions mentioned above are present at the end points of the integral. The Einstein summation convention will be used where convenient (e.g., eq. 1b); however, all the analysis here deals with spatial coordinates, ϭ 1, 2, 3, on a Euclidean metric. Generalizations to dimension higher than 3, as well as non-Euclidean metrics, are straightforward to incorporate into the formalism. On a Euclidean metric, the minimal distance becomes the diagonal of a hyper-cube. However, formulated as above, the solutions minimizing D are infinitely degenerate, because particles moving at various speeds but tracing the same trajectory over the total time T all give the same distance. To circumvent this problem, what is typically done is to let one of the space variables (e.g., x) become the independent variable. However, for higher dimensional objects or zero dimensional objects on a manifold with nontrivial topology, there is no guarantee that the dependent variables (y, z) constitute single valued functions of x. Alternatively, one can study the ''time'' trajectory of the parametric curve defined above, but under a gauge that fixes the speed to a constant, v o , for example. One can either fix the gauge from the outset with Lagrange multipliers, or choose a gauge that may simplify the problem after finding the extremum equations. The latter is often simpler in practice.To be specific, the effective Lagrangian appearing in the above problem is ͌ ṙ 2 , and the Euler-Lagrange (EL) equations arewith v the unit vector in the direction of the velocity. The boundary conditions are r*͑0͒ ϭ r A and r*͑T͒ ϭ r B.[3]Because the derivative of a unit vector is always orthogonal to that vector, Eq. 2 says that the direction of the velocity cannot change, and therefore straight line motion results. Applying the boundary conditions gives v ϭ (r B Ϫ r A )/͉r B Ϫ r A ͉. However, any function v(t) ϭ ͉v o (t)͉v satisfying the boundary conditions is a solution, so long as ͐ 0 In what follows, we generalize...

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“…[4][5][6][7][8]. Such a metric is of interest for comparison between folded structures, as well as to quantify how close an unfolded or partly folded structure is to the native.…”

Section: Discussionmentioning

confidence: 99%

“…In fact to lowest order F(kL) Ϸ Ϫ( 2 /2160) (kL) 6 . The extremum corresponding to pure rotation of curve r A into r B is a maximum!…”

Section: [6]mentioning

confidence: 99%

Generalization of distance to higher dimensional objects

Plotkin

2007

Proc. Natl. Acad. Sci. U.S.A.

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“…Structures of all proteins were determined by X-ray crystallography to at least 2.5 Å resolution. Structural matches were determined by the structural superposition program MINAREA 43 , using the same cutoff as in the previous study (a ratio score of 0.2 or lower) for assigning structural similarity. In order to mimic blind structure prediction challenges, the true structures of the 58 test sequences and their homologs (more than 25% sequence identity) were not considered.…”

Section: Data Setsmentioning

confidence: 99%

New Local Potential Useful for Genome Annotation and 3D Modeling

Chandonia

Cohen

2003

Journal of Molecular Biology

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“…Refs. 1,3,4,7,8,9,11,13,14,15,16,17,18,19,21,22,23,24,25), among tein structure alignment based on flexible transformation) are commonly used. We have developed a feedback algorithm for pairwise protein structure alignment and our web alignment tool is available for public access.…”

Section: Introductionmentioning

confidence: 99%

Feedback Algorithm and Web-Server for Protein Structure Alignment

Zhao

Fu²,

Alanis³

et al. 2008

Journal of Computational Biology

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We have developed a feedback algorithm for protein structure alignment between two protein backbones. A web portal implementing this method has been constructed and is freely available for use at http://fpsa.cs.uno.edu/ with a mirror site at http://fpsa.cs.panam.edu/FPSA/. We compare our algorithm with three other, commonly used methods: CE, DaliLite and SSM. The results show that in most cases our algorithm outputs a larger number of aligned positions when the (Cα) RM SD is comparable. Also, in many cases where the number of aligned positions is larger or comparable, our learning method is able to achieve a smaller (Cα) RM SD than the other methods tested. This trend of larger number of aligned positions and smaller (Cα) RM SD is observed more frequently in cases where the similarity between protein structures is weak.

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A Surface of Minimum Area Metric for the Structural Comparison of Proteins

Cited by 56 publications

References 0 publications

Generalization of distance to higher dimensional objects

Generalization of distance to higher dimensional objects

New Local Potential Useful for Genome Annotation and 3D Modeling

Feedback Algorithm and Web-Server for Protein Structure Alignment

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