nk university of athens - Εθνικόν και …...generic / minimal rigidity i graph g...
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Distance Geometry for computing comformations
Ioannis Z. Emiris
NK University of Athens
Algs in Struct BioInfo, April 6, 2020
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Outline
Motivation
Rigidity theory
Distance geometry
MoleculesSmall moleculesDistances to coordinatesNoisy dataMatrix perturbations
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Outline
Motivation
Rigidity theory
Distance geometry
MoleculesSmall moleculesDistances to coordinatesNoisy dataMatrix perturbations
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Structure ab initio
Structure from Distances: Treat, e.g., 5,000 atoms with:
I NMR spectroscopy yields approximate distances (exact if< 5A), hence 3d structure, in solution [K.Wuthrich (ETHZ),
Chemistry Nobel’02] ”for his development of NMR spectroscopy for
determining the 3-dimensional structure of biological
macromolecules in solution”
I X-ray crystallography: more accurate distances (error ≤ 1A)but in crystal state, which takes ∼ 1 year.
I Electron microscopy, etc
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NMR
I Software: Dyana [Guntert,Mumenthaler,Wuthrich’97],Embed [Crippen,Havel’88], Disgeo [Havel,Wuthrich’98],Dgsol [More,Wu], Abbie [Hendrickson], etc
I Physics: Specific isotopes have spin (±1/2) e.g.: H, C13.Each isotope absorbs / radiates back energy fromelectromagnetic (EM) pulse at specific “resonance” frequency.
I Steps:1. Constant magnetic field applied, spins aligned (polarized).2. EM pulse applied, specific nuclei stimulated/radiate energy3. Distance of nuclei-pairs depends on EM frequency:measured, and assigned to nuclei (semi-automatic).4. Nuclei coordinates in some frame (embedding) computedfrom (noisy) distances: our focus.
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Mechanisms / Robots
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Engineering
I Architecture, tensegrity
I Topography (Surveyors)
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Outline
Motivation
Rigidity theory
Distance geometry
MoleculesSmall moleculesDistances to coordinatesNoisy dataMatrix perturbations
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Euclidean embedding
A graph (network) is described by its vertices (nodes) V and itsedges E .
Problem Embed-Rd (find coordinates):Given a weighted (distance) undirected graph (V ,E ), find anembedding (coordinate vectors)
f : V → Rd , d ≥ 1,
which also maps the given weights to (Euclidean) distances, i.e.,
dist2(f (v), f (v ′)) = weight(v , v ′), ∀(v , v ′) ∈ E .
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Generic / Minimal Rigidity
I Graph G is generically rigid in Rd iff for generic edge lengthsit has a finite number of embeddings in Rd , modulo (ignoring)rigid motions.
I Graph G is minimally rigid iff it becomes non-rigid (flexible)once an edge is removed.
We call generically minimally rigid graphs simply rigid.
A rigid vs a non-rigid (flexible) graph in the plane R2.
Quad becomes rigid with one extra distance: 2 configurations possible
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Planar rigidity
Theorem (Maxwell:1864,Laman:1970)
Graph G = (V ,E ) is rigid in R2 iff:
I |E | = 2|V | − 3, and
I |E ′| ≤ 2|V ′| − 3, ∀ vertex-induced subgraph (V ′,E ′).
Intuition: |E | constraints/equations, 2|V | − 3 coordinates (x , y)per node except for 2 for point at origin, one for point on x-axis.
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Rigidity in R3
Generalized Laman: |E | = 3|V | − 6,|E ′| ≤ 3|V ′| − 6, ∀(V ′,E ′) ⊂ (V ,E ).
Counterexample: Double Banana:
Theorem (Cayley)
The 1-dimensional skeleta of triangulated/simplicial convexpolyhedra are rigid in R3.
For triangulated/simplicial polyhedra:|E | = 3|V | − 6, |E ′| ≤ 3|V ′| − 6, (V ′E ′) ⊂ (V ,E )
Thm applies when there exists an embedding s.t. V and E lie on aconvex polytope (or on the sphere): Double banana does not.
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Algebraic system
#embeddings = #real solutions of a polynomial system expressingweighted edges E , and
(d+12
)+ 1 constraints for “pin-down” and
removing scaling.
in R2 :
x1 = y1 = 0,x2 = d12, y2 = 0,(xi − xj)
2 + (yi − yj)2 = d2
ij , (i , j) ∈ E .
in R3 :
x1 = y1 = z1 = 0,x2 = d12, y2 = z2 = 0,z3 = 0,(xi − xj)
2 + (yi − yj)2 + (zi − zj)
2 = d2ij , (i , j) ∈ E .
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n = 7: 56 conformations [E-Moroz’11]
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The case n = 6
I The cyclohexane has 16 real embeddings [E-Mourrain’99].
I The “jigsaw” parallel robot has 16 real configurations.
2 chairs, 2 twisted-boats/crowns given 6 equal distances, 6 equalangles, 10% perturbation ⇒ 12 distances (Laman).These are precisely the conformations mostly observed in nature.
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Outline
Motivation
Rigidity theory
Distance geometry
MoleculesSmall moleculesDistances to coordinatesNoisy dataMatrix perturbations
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Matrix algebra
DefinitionFor any (rectangular) matrix A,
I rank(A) = r if r = #positive singular values.Recall: singular values ≥ 0.
A submatrix determinant is called minor.
LemmaFor any matrix A, rank(A) = max dimension of nonzero minor.Formally, rank(A) = r iff ∃r × r nonzero minor, and all k × k,minors vanish, k > r .
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Distance matrix
DefinitionA distance matrix M is square with real entries Mii = 0,Mij = Mji ≥ 0.
DefinitionA distance matrix M is embeddable in Euclidean space Rd iff
∃ points pi ∈ Rd : Mij =1
2dist(pi , pj)
2.
Embeddable matrices in R3 correspond to 3D conformations sinceone can assign one or more 3d coordinate vector to each atom.
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Cayley-Menger (or border) matrix
DefinitionDefine a Cayley-Menger (or border) matrix by appending a 0th rowand a 0th column to distance matrix M:
0 1 · · · 11...1
M
.
Again symmetric, 0-diagonal, non-negative entries.
Notice rank(CM matrix) = rank(M) + 2.
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Distance geometry
Theorem (Cayley:1841,Menger’28)
M embeds in Rd iff Cayley-Menger (border) matrix has
rank
0 1 · · · 11...1
M
= d + 2,
and, for any (k + 1)× (k + 1) “border” minor D(i1, . . . , ik)indexed by rows/columns 0, i1, . . . , ik :
(−1)k D(i1, . . . , ik) ≥ 0, k = 2, . . . , d + 1.
∃ strict inequality D(i1, . . . , id+1) 6= 0 iff cannot embed in Rd−1.Trivially M embeds in Rδ: δ > d .
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3D
Corollary
A distance matrix expresses 3D conformation iff border matrix hasrank= 5, and satisfies the triangle and tetrangular inequalities:
I For k = 2, D(i , j) = det
0 1 11 0 Mij
1 Mij 0
= 2Mij ≥ 0,
I for k = 3, by the triangular inequalities: −D(1, 2, 3) =
(d12+d13+d23)(d12+d13−d23)(d12+d23−d13)(d13+d23−d12)
I for k = 4, by the tetrangular inequalities.
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Outline
Motivation
Rigidity theory
Distance geometry
MoleculesSmall moleculesDistances to coordinatesNoisy dataMatrix perturbations
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Outline
Motivation
Rigidity theory
Distance geometry
MoleculesSmall moleculesDistances to coordinatesNoisy dataMatrix perturbations
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Robotics
Given some fixed geometric characteristics (angles, lengths) andthe position of the end-effector (here a ring), compute all possibleconfigurations defined by 6 consequent rotational DOFs.Inverse kinematics of a 6R robot with consecutive axes intersecting.
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Cyclohexane
p1 p2 p3 p4 p5 p6
p1p2p3p4p5p6
0 1 1 1 1 1 11 0 u c x14 c u1 u 0 u c x25 c1 c u 0 u c x361 x14 c u 0 u c1 c x25 c u 0 u1 u c x36 c u 0
Known u, c from bond distance du = 1.526A (adjacent), bondangle φ ' 109.5o ⇒ dc ' 2.49A (law of cosines in rigid triangle)Rank = 5⇔ all 6× 6 minors = 0, some 5× 5 minor 6= 0.For unknowns x14, x25, x36, use 3 such (quadratic) equations.
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Cycloheptane
v1 v2 v3 v4 v5 v6 v7
v1v2v3v4v5v6v7
0 1 1 1 1 1 1 11 0 c12 c13 x14 x15 c16 c171 c12 0 c23 c24 x25 x26 c271 c13 c23 0 c34 c35 x36 x371 x14 c24 c34 0 c45 c46 x471 x15 x25 c35 c45 0 c56 c571 c16 x26 x36 c46 c56 0 c671 c17 c27 x37 x47 c57 c67 0
14 known entries cij = d2
ij/2, 7 unknown xij ’s.2 · 7− 6 = 8 unknown coordinates ⇒ generically infite number(curve) of conformations.
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Outline
Motivation
Rigidity theory
Distance geometry
MoleculesSmall moleculesDistances to coordinatesNoisy dataMatrix perturbations
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Distances and inner products
Consider n + 1 (unknown) points/vectors p0, . . . , pn ∈ R3, and allpossible distances among them:
d2ij = |pi − pj |2, d2
i0 = |pi |2, by setting p0 = 0 ∈ R3.
Vector length can be written in terms of inner/dot product:
(x , y , z) · (x , y , z) = x2 + y2 + z2 = |(x , y , z)|2.
Inner/dot product can be written using the transpose vector:
(x , y , z) · (x , y , z) = (x , y , z)T (x , y , z).
Then, d2ij = |pi−pj |2 = (pi−pj)·(pi−pj) = |pi |2−2pTi pj+|pj |2.
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Matrix of inner products
Define Gram matrix G = [p1, . . . , pn]T [p1, . . . , pn] =: PTP, as then × n matrix of inner products:
G =
pT1 p1 pT1 p2 . . . pT1 pn
pT2 p1 pT2 p2 . . . pT2 pn
......
. . ....
pTn p1 pTn p2 . . . pTn pn
G is determined from distances, once a point is the origin:
d2ij = |pi |2 − 2pTi pj + |pj |2 ⇔ pTi pj =
d2i0 − d2
ij + d2j0
2=: Gij .
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Point coordinates from inner products (I)
Input: Gram matrix G of inner products pTi pj , p0 = 0.
[Gij ] = [pTi pj ] = PT · P, where P = [p1, . . . , pn] is 3× n.
G real symmetric ⇒ U = V . Singular Value Decomposition yields
G = UΣUT , UTU = I , Σ diagonal, entries σi ≥ 0.
rank[d2ij ] = 3⇒ rank(G ) = 3 ⇒ σ1, . . . , σ3 > 0 = σ4 = · · · = σn.
So all info is contained in 3× 3 up-left (principal) submatrix of Σ:
UΣUT =
[VU2
] [Σ′ 00 0
][V T UT
2 ].
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Point coordinates from inner products (II)
Define 3× 3 diagonal Σ′ and n × 3 V s.t. G = VΣ′V T :
G = V
σ1σ2
σ3
V T , σi > 0.
Now let 3× 3 diagonal√
Σ′ = diag(√σ1,√σ2,√σ3).
Then, G = V√
Σ′√
Σ′V T = PTP ⇒ P :=√
Σ′V T .
Output: point coordinates P (up to rigid transforms) in R3.
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Embeddability Theorem
Corollary (of Cayley-Menger)
Points pi embed in Rd , for min d , iff corresponding Gram matrixPTP has rank d .
In R3 : {pi} embed in R3 (not R2) iff G = PTP has rkG = 3.
TheoremFor matrix A = UΣBT (SVD), UΣ′V T is A’s best approximant ofrank ρ ≤ rank(A), where σ′k = σk , k = 1, . . . , ρ, σ′i = 0, i > ρ.
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Embedding via SVD
Input: full distance graph (clique) on n + 1 points, distances maybe inaccurate (noisy).
Embedding Algorithm
0. Pick point as origin (indexed 0).1. Compute all distances dij .2. Determine G , and run SVD: G = VΣV T .
Goal: embedding P =√
ΣV T (size n × 3).3. Force rank(G ) = 3 by defining diagonal matrix Σ′ s.t.
σ′k = σk , k = 1, 2, 3, σ′i = 0, i = 4, . . . , n.4. Output coordinates P =
√Σ′V T (size n× 3), and p0 = (0, 0, 0)
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Outline
Motivation
Rigidity theory
Distance geometry
MoleculesSmall moleculesDistances to coordinatesNoisy dataMatrix perturbations
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Minors
First use the inequalities:
det Border(2 points) = det
0 1 11 0 d2
ij/2
1 d2ij/2 0
≥ 0⇔ d2ij ≥ 0.
Triangular inequality: det Border(3 points) =(d12+d13+d23)(d12+d13−d23)(d12+d23−d13)(d13+d23−d12) ≥ 0
Tetrangular inequality: det Border(4 points) ≥ 0.
Eventually, use the rank condition:
All 6× 6 minors vanish ⇔ det Border(5 points) = 0, for all5-tuples.
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Smoothing triangular inequalities
Triangle inequalities (equality iff coliner): For any 3 points inEuclidean space of any dimension (including R3) the triangleinequality holds:
|dik − dkj | ≤ dij ≤ dik + dkj .
Left-hand side inequality follows from right-hand side inequality.
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Bound smoothing
Inaccurate distances dij given as intervals [lij , uij ], s.t. lij ≤ dij ≤ uij .
Improve upper bound uij by forcing triangular inequality:
uij ≤ uik + ukj .
All-min-paths in single pass, any order [Havel]: O(V 3).
Improve lower bound lij by forcing:
lij ≥ max{lik − ukj , lkj − uik},
where only one difference is positive e.g. uik > lik > ukj > lkj .This implies
lij ≥ lkm − uik − umj ,
where indices i , j , k,m are not necessarily all distinct.Independently of upper bounds by single-pass all-minpaths [Havel]
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Example
ubd ≤ ubc + ucd = 12uac ≤ uad + ucd = 13uab ≤ uad + ubd = 20lbd ≥ lab − uad = 2lac ≥ lab − ubc = 3
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Four atoms
For nonplanar atoms 1, 2, 3, 4, the Cayley-Menger determinant is:
0 1 1 1 11 0 d2
12 d213 d2
14
1 d221 0 d2
23 d224
1 d231 d2
32 0 d234
1 d241 d2
42 d243 0
> 0
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Heron’s formula
Triangular: CM(a, b, c) = 16(Area of T)2 =
= (a + b + c)(−a + b + c)(a− b + c)(a + b − c).
Tetrangular: CM(a, b, c , d , e, f ) = 288(Volume of T)2.
′Hρων o Aλεξανδρευs (c. 10-70 AD) was an ancient Greek
mathematician and engineer who was active in his native city of
Alexandria [wikipedia]
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Independence
If the given bounds satisfy triangle inequality, only 7 inequalitiesare non-redundant, derived from the tetrangular inequalities:
Consider the (3,4) distance: For upper limit u34:CM(l12, u13, u14, u23, u24, u34) ≥ 0,CM(u12, l13, l14, u23, u24, u34) > 0,CM(u12, u13, u14, l23, l24, u34) > 0,
For the lower limit l34 we have:CM(u12, u13, l14, l23, u24, l34) > 0,CM(u12, l13, u14, u23, l24, l34) > 0,CM(l12, l13, u14, l23, u24, l34) > 0,CM(l12, u13, l14, u23, l24, l34) > 0.
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Bound smoothing
Input: intervals [lij , uij ], s.t. lij ≤ dij ≤ uij , for unknown distancedij , where lij ≤ uij ; notice lij = uij iff dij = lij = uij .
Algorithm:0. Tighten intervals using the triangle inequality (linear pass ofgraph).1. Fix a Tolerance value > 0.2. Check all
(n4
)quadruples of nodes, applying 7 inequalities.
3. Repeat (2) until max change in any bound is < Tolerance.
Order of quadruples/inequalities does not affect output.BUT: step (2) may progress very slowly to final result dependingon order.Tetrangle inequalities much tighter than triangular, but slow.
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Slow progress
Regular pentagon:5 edges: u = l = 1,3 shown diagonals: l = 1.617,true diagonal = 1.618 = 2 cos 36o
Triangle-Bound-Smoothing yields upper bound = 2 for 5 diagonals;u24 by quadruple (2,3,4,5), then used in (2,4,5,1) for u25.
After 30 passes, tolerance = 10−14, we haveu24 = 1.6207323507579925, u25 = 1.6207323507579441.
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Outline
Motivation
Rigidity theory
Distance geometry
MoleculesSmall moleculesDistances to coordinatesNoisy dataMatrix perturbations
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Structure-preserving matrix perturbations
Let σi (A) ≥ 0 be the i-th singular value of matrix A.
Theorem. [Wicks-Decarlo’95] Given matrix B, there exists t ∈ R,P ∈ {0, 1}n×n (perturbation) s.t. f (t) = σn(B − tP) is continuousand f ′(t) = −uTPv , where u, v are the n-th singular vectors.So a Newton-like iteration finds P, t: σn(B − tP) ' 0.
Heuristic. [Nikitopoulos-E’02] If, moreover, border matrix B issufficiently close to an embeddable matrix (local minimum), thealgorithm applies for any σk , 6 ≤ k ≤ n.
Iterative algorithm.– Minimize σ6(B − tP) thus minimizing σk , 6 < k ≤ n, too.– Suitable t > 0, P (symmetric, 0-diagonal, 0’s on 1st row /column) found in O(n2), preserves B’s structure, reduces σ6.– Repeat until σ6 reduces by less than some threshold ε > 0.
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Performance on ring molecules
Matlab code perturbes matrix B to minimize 6-th singular value,preserves B’s structure (symmetric, diagonal 0, entries > 0);precision = 16 digits [Nikitopoulos-E:J.Math.Chem’02].
#atoms Init. σ6 Final σ6 Iterations Time [sec.] KFlops
10 2.38e-02 2.95e-13 3 0.11 10911 3.16e-02 2.60e-12 3 0.16 16512 8.13e-02 1.20e-07 3 0.22 28213 8.09e-02 8.49e-08 3 0.30 45014 3.72e-02 6.04e-13 3 0.49 60615 3.53e-02 2.02e-14 3 0.77 94016 3.78e-02 1.72e-12 3 1.15 140417 3.83e-02 1.70e-13 3 1.54 208218 3.53e-02 3.93e-13 3 2.14 303919 3.80e-02 4.59e-14 3 2.91 434420 4.00e-02 7.09e-13 3 3.79 6136
Flop=Floating-point operation.