Zhijun Wu Department of Mathematics Program on Bio-Informatics and Computational Biology

1. A Novel Geometric Build-Up Algorithm for Solving the Distance Geometry Problem and Its Application to Multidimensional Scaling Zhijun Wu Department of Mathematics Program on Bio-Informatics and Computational Biology Iowa State University Joint Work with Tauqir Bibi, Feng Cui, Qunfeng Dong, Peter Vedell, Di Wu

2. Distance Geometry mapping from semi-metric to metric spaces Euclidean and non-Euclidean T S Multidimensional Scaling data classification geometric mapping of data fundamental problem: find the coordinates for a set of points, given the distances for all pairs of points B Cayley-Menger determinant necessary & sufficient conditions of embedding singular-value decomposition method strain/stress minimization Molecular Conformation embedding in 3D Euclidean space protein structure prediction and determination sparse, inexact distances, bounds on the distances, probability distributions

3. Proteins are building blocks of life and key ingredients of biological processes. A biological system may have up to hundreds of thousands of different proteins, each with a specific role in the system. A protein is formed by a polypeptide chain with typically several hundreds of amino acids and tens of thousands of atoms. A protein has a unique 3D structure, which determines in many ways the function of the protein. HIV Retrotranscriptase an example: 4200 atoms 554 amino acids

4. Molecular Distance Geometry Problem Given n atoms a1, …, an and a set of distances di,j between ai and aj, (i,j) in S

5. Problems and Complexity problems with all distances: solvable in O (n3) using SVD problems with sparse sets of distances: NP-complete (Saxe 1979) problems with distance ranges (NMR results): NP-complete (More and Wu 1997), if the ranges are small problems with probability distributions of distances: stochastic multidimensional scaling, structure prediction

6. Current Approaches • Embed Algorithm by Crippen and Havel • CNS Partial Metrization by Brünger et al • Graph Reduction by Hendrickson • Alternating Projection by Glunt and Hayden • Global Optimization by Moré and Wu • Multidimensional Scaling by Trosset, et al

7. Embed Algorithm time consuming in O(n3~n4) • bound smooth; keep distances consistent • distance metrization; estimate the missing distances • repeat (say 1000 times): • randomly generate D in between L and U • find X using SVD with D • if X is found, stop • select the best approximation X • refine X with simulated annealing • final optimization costly in O(n2~n3) Crippen and Havel 1988 (DGII, DGEOM) Brünger et al 1992, 1998 (XPLOR, CNS)

8. Geometric Build-Up Independent Points: A set of k+1 points in Rk is called independent if it is not a set of points in Rk-1. Metric Basis: A set of points B in a space S is a metric basis of S provided each point of S is uniquely determined by its distances from the points in B. Fundamental Theorem: Any k+1 independent points in Rk form a metric basis for Rk. Blumenthal 1953: Theory and Applications of Distance Geometry

9. Geometric Build-Up in two dimension

10. Geometric Build-Up in three dimension

11. Geometric Build-Up in three dimension

12. Geometric Build-Up 1 x1 = (u1, v1, w1) x2 = (u2, v2, w2) x3 = (u3, v3, w3) x4 = (u4, v4, w4) ? xi = (ui, vi, wi) i ||xi - x1|| = di,1 ||xi - x2|| = di,2 ||xi - x3|| = di,3 ||xi - x4|| = di,4 2 4 j ||xj - x1|| = dj,1 ||xj - x2|| = dj,2 ||xj - x3|| = dj,3 ||xj - x4|| = dj,4 ? xj = (uj, vj, wj) 3

13. The geometric build-up algorithm solves a molecular distance geometry problem in O(n) when distances between all pairs of atoms are given, while the singular value decomposition algorithm requires O(n2~n3) computing time!

14. The X-ray crystallography structure (left) of the HIV-1 RT p66 protein (4200 atoms) and the structure (right) determined by the geometric build-up algorithm using the distances for all pairs of atoms in the protein. The algorithm took only 188,859 floating-point operations to obtain the structure, while a conventional singular-value decomposition algorithm required 1,268,200,000 floating-point operations. The RMSD of the two structures is ~10-4 Å.

15. Problems with Sparse Sets of Distances

16. Control of Rounding Errors

17. Control of Rounding Errors

18. Tolerate Distance Errors

19. Tolerate Distance Errors i (i,j) in S j xj are determined.

20. The objective function is convex and the problem can be solved using a standard Newton method. Each function evaluation requires order of n floating point operations, where n is the number of atoms. (i,j) in S xj are determined. In the ideal case when every atom can be determined, n atoms require O(n2) floating point operations.

21. NMR Structure Determination The distances are given with their possible ranges. i j

22. (i, j) in S

23. Computational Results Computational Results The structure of 4MBA (red lines) determined by using a geometric build-up algorithm with a subset of all pairs of inter-atomic distances. The X-ray crystallography structure is shown in blue lines.

24. Computational Results Computational Results The total distance errors (red) for the partial structures of a polypeptide chain obtained by using a geometric build-up are all smaller than 1 Å, while those (blue) by using CNS (Brünger et al) grow quickly with increasing numbers of atoms in the chain.

25. Extension to Statistical Distance Data the distributions of the distances in structure database i j structure prediction