Distance Geometry-Matrix Completion

Transcription

1 Christos Konaxis Algs in Struct BioInfo 2010

2 Outline Outline Tertiary structure Incomplete data

3 Tertiary structure Measure diffrence of matched sets Def. Root Mean Square Deviation RMSD = 1 n x i y i 2, n i=1 where x i, y i R 3 are (C α ) atom coordinates in SAME coordinate frame. X = [x 1,..., x n ], Y = [y 1,..., y n ], then RMSD(X, Y ) = 1 n X Y F, where M 2 F = i,j M 2 ij = tr(m T M), is the Frobenious metric, and tr(a) = i A ii is the trace of matrix A = [A ij ].

4 Tertiary structure Optimal alignment of matched sets Translate to common origin by subtracting centroid x c = 1 x i. n Rotate to optimal alignment by rotation matrix Q : Q T = I. Also should have detq = 1. Exists deterministic linear algebra algorithm [Kabsch]. Overall cost= O(n 3 ), but least-squares approximation in O(n). i

5 Tertiary structure Optimal rotation Assume common centroid: RMSD(X, Y ) = min Q Y XQ F, Q T Q = I 3. Y XQ F = tr(y T Y ) + tr(x T X ) 2tr(Q T X T Y ), so must maximize tr(q T X T Y ). Consider SVD: X T Y = UΣV T, U T U = V T V = I, Σ = diag[s i ] : s 1 s 3. Then tr(q T X T Y ) = tr(q T UΣV T ) = tr(v T Q T UΣ) tr(σ), because T = V T Q T U orthonormal T ij 1. Hence maximum at T = I Q = UV T. If dett = 1 then just negate T 33.

6 Introduction Nuclear Magnetic Resonance (NMR) and Nuclear Overhauser Effect (NOE) spectroscopy provide approximate inter-atomic distances for molecular structures as large as atoms. The distances measured by NMR and NOESY experiments (usually a small subset of all possible pairs) must be converted into a 3D structure consistent with the measurements. In general the distances are imprecisely measured: for each distance d ij we have l ij d ij u ij.

7 The Method is based on the foundational work of Cayley (1841) and Menger (1928) who showed how convexity and other basic geometric properties could be defined in terms of distances between pairs of points. The problem can be reduced to the completion of a partial matrix M satisfying certain prorperties.

8 Distance matrices Definition (and structure): A distance matrix D is square, D ii = 0, D ij = D ji 0. Definition. A distance matrix D is euclidean and embeddable in R k iff points p i R k : D ij = 1 2 dist(p i p j ) 2. Embeddable matrices in R 3 correspond to 3D conformations.

9 Embedding matrices in R k Thm [Schoenberg 35,Blumenthal 53] Take border (Cayley-Menger) matrix B =. D. 1 Then, D embeds in R k iff rank(b) k + 2. Cor. A distance matrix D expresses a 3D conformation iff rank(b) = 5

10 Cyclohexane s distance matrix p 1 p 2 p 3 p 4 p 5 p 6 p 1 p 2 p 3 p 4 p 5 p u c x 14 c u 1 u 0 u c x 25 c 1 c u 0 u c x 36 1 x 14 c u 0 u c 1 c x 25 c u 0 u 1 u c x 36 c u 0 Known: u (adjacent), φ o c (triangle). Rank condition (= 5) equivalent to the vanishing of all 6 6 minors. This yields a 3 3 system of quadratic polynomials in the x 14, x 25, x 36. If all c, u same, then 2 isolated conformations, one 1-dim set. If the c, u perturbed, then 16 solutions R [Emiris-Mourrain].

11 Points from distances Given distance matrix D, compute coordinate matrix X. D = 0 v 1 2 v v n 2 v v 1 v v 1 v n 2 v 2 2 v 2 v v 2 v n 2. Find Gram matrix G = v n 2 v n v 1 2 v n v v 1 v 1 v 1 v 2... v 1 v n v 2 v 1 v 2 v 2... v 2 v n v nv 1 v nv 2... v nv n Elements of G computed using: 2v i v j = v i 2 + v j 2 v i v j 2, or 1. Subtract the first row of D from each row. 2. Subtract the first column from each column. 3. Delete the first row and column. Result: 2G.

12 Points from distances Given G, find n 3 matrix X s.t. G = X T X using eigenvectors matrix V (s.t. V T V = I ), and eigenvalues diagonal matrix E. Then, GV = EV. Since G is symmetric and comes from a 3D distance matrix, it has 3 non-zero real eigenvalues with real eigenvectors. Construct a diagonal matrix E whose entries on the diagonal are the square roots of the entries of E. Then, G = V E EV T = X T X, where X := EV T.

13 Positive (semi)definite matrices Def. An n n real matrix M is positive (semi)definite if Denoted M 0, M 0. [ ] Examples: x T Mx > 0 (x T Mx 0) x Lem. [Sylvester]. M 0 det M i > 0, i i upper-left minor M i, i = 1,..., n. Cor. M 0 det M > 0. If M 0 (M 0), then all its eigenvalues are positive (non-negative).

14 Embeding via rank Thm. {p i } embed in R 3 (and not R 2 ) iff D 0 & rkd = 3. [ ] y R n, y T PP T y = (y T P)(y T P) T 0: positive semidefinite. rk(ab) = min{rka,rkb} rkd =rkp. embed p a, p b, p c linearly independent rkp = 3. [ ] Singular Value Decomposition (SVD): symmetric real D = USU T, where U T U = UU T = I, S =diag[s 1,..., s n ], s i = eigenvalue 2 0. rkd = 3 S = [s 1, s 2, s 3, 0, ], S = [ s i ], D = U S SU T. P := U S is n 3: defines n points p i R 3.

15 Bound smoothing For any three points i, j, k in R 3 the triangle inequality holds: d ik d jk d ij d ik + d jk. Meausered distances: l ij d ij u ij l ik d ik u ik l jk d jk u jk Improved upper bound: ū ij = min{u ij, u ik + u jk } Improved lower bound: l ij = max{l ij, l ik u jk, l jk u ik }

16 The tightened upper bounds can be computed independently of the lower. The tightened upper bounds further improve the tightened lower bounds: ū ik u ik, ū jk u jk, l ij = max{l ij, l ik u jk, l jk u ik }, so lij = max{l ij, l ik ū jk, l jk ū ik } If l ij > ū ij (e.g. when an upper bound is too low) we have a triangle inequality violation.

17 Incomplete data Matrix completion problems Problem: Given a partial matrix M, can M be completed to a positive semidefinite matrix (PSD), positive definite matrix (PD), Euclidean distance matrix (EDM)? EDM is reduced to PSD: If D = (d ij ), d ii = 0, is a symmetric n n matrix, define (n 1) (n 1) symmetric matrix X = (x ij ), where x ij := 1 2 (d in + d jn d ij ), i, j = 1,..., n 1. D is a distance matrix iff X is positive semidefinite. Moreover v i R k iff rank(x ) k.

18 Incomplete data It is not known if PSD is in NP. PSD can be solved with an arbitrary precision in polynomial time (interior point, elipsoid method). We will examine polynomial instances of PSD. If a matrix contains a fully determined line or column then its completion problem reduces to the completion of a smaller matrix.

19 Incomplete data Assumptions: M = Hermitian (M = M), all diagonal entries of M are specified (for positive semidefinite matrices), moreover they are equal to zero (for distance matrices). If m ij is specified, then m ji = m ij is also specified. For every n n matrix M we define the graph (pattern of M) G = ([1, n], E) with vertices [1,..., n]. There is an edge ij between vertices i, j if entry m ij is specified.

20 Incomplete data Partial PSD matrices Def. A matrix M is partial positive semidefinite (partial-psd) if every principal specified submatrix of M is positive semidefinite. Lem. If the incomplete matrix M has a PSD (PD, distance matrix) completion, then M is partial-psd (partial-pd, partial-distance matrix). Hence we have necessary (but not sufficient) conditions for the existance of a completion of M. Counter-example: Partial-PSD but PSD-completion: 1 1? 0 1 1? 1 1 1

21 Incomplete data Chordal graphs A chord of a circuit C of a graph G is an edge of G which is not in C but which joins two vertices of C. A graph is chordal if every circuit of length 4 has at least one chord. Th. Every partial positive semidefinite matrix M with pattern G has a positive semidefinite completion iff G is chordal. Constructive proof; can be turned into a polynomial time algorithm. Th. PSD can be solved in polynomial time if the matrix has a chordal pattern. (bit model).