Molecular Modeling lecture 17, Tue, Mar. 19. Rotation Least-squares Superposition Structure-based alignment algorithms

Transcription

1 Molecular Modeling lecture 17, Tue, Mar. 19 Rotation Least-squares Superposition Structure-based alignment algorithms

2 Matrices and vectors Matrix algebra allows you to express multiple equations as one expression [ ] a b = c d means, 1a + 2b = c 3a + 4b = d The equations share the variables a, b.!2

3 Matrices and vectors Rows are multiplied by columns [ ] a b = c d 1a + 2b = c 3a + 4b = d!3

4 Matrices and vectors Identity matrix [ ] a b = a b!4

5 Matrices and vectors Inverse matrix If [ ][ ] = a c b d f h g j [ ] then, [ ] -1 [ ] f g = a c b d h j!5

6 Matrices and vectors Matrix and vector variables R contains 4 variables. R = [ ] a c b d v contains 2 variables. v = f g

7 Matrices and vectors Rv = u what is u? u = a f + b g c f + d g

8 In polar coordinates, Rotation is addition!8

9 Rotation is angular addition y atom starts at v=(x= rcosα, y= rsinα) axis of rotation = Cartesian origin r β (x,y ) α (x,y) x rotates around origin to v =(x =rcos(α+β), y'=rsin(α+β)) Rotations are always counter-clockwise (right-handed).

10 Sum of angles formuli cos (α+β) = cos α cos β sin α sin β sin (α+β) = sin α cos β + sin β cos α

11 A rotation matrix x = rcos α y (x,y ) y = rsin α β (x,y) r α x x' = r cos (α+β) = r (cos α cos β sin α sin β) = (r cos α) cos β (r sin α)sin β = x cos β y sin β y' = r sin (α+β) = r (sin α cos β + sin β cos α) = (r sin α) cos β + (r cos α) sin β = y cos β + x sin β! # " x' $! & = cosβ y' # % " sinβ sin β $! & rcosα $! # & = cos β cos β %" rsinα # % " sin β sin β $! cosβ % & # x$ " y & % rotation matrix is independent of r, α.

12 rotations around principal axes The Z coordinate stays the same. X and Y change. cos β -sin β 0 Rz = The Y coordinate stays the same. X and Z change. cos γ 0 sin γ Ry = sin β cos β sin γ 0 cos γ The X coordinate stays the same. Y and Z change R x = 0 cos α -sin α 0 sin α cos α

13 A series of rotations is the product of rotation matrices. cos β -sin β 0 cos γ 0 sin γ cos β cos γ -sin β cos β sin β cos β = sin β cos γ cos β -sin β sin γ sin γ 0 cos γ sin γ 0 cos γ Rotation around z Rotation around y 3D rotation This is a Y rotation followed by a Z rotation

14 multiplication order is right to left. X-rotation α degrees, followed by Y-rotation γ degrees cos γ 0 sin γ sin γ 0 cos γ cos α -sin α 0 sin α cos α R y R x = = cos γ sin α sin γ cos α sin γ 0 cos α -sin α -sin γ sin α cos γ cos α cos γ Y-rotation γ degrees, followed by X-rotation α degrees cos α -sin α 0 sin α cos α cos γ 0 sin γ sin γ 0 cos γ xr y = = cos γ 0 sin γ sin α sin γ cos α -cos γ sin α -cos α sin γ sin α cos α cos γ opposite order gives a different result.

15 Transposing the matrix reverses the rotation For the opposite rotation, flip the matrix. This is the transpose x y = A C B D cos β sin β -sin β cos β The inverse matrix = The transposed matrix, because = A B C D T x y cos β sin β -sin β cos β cos β -sin β sin β cos β =

16 Right-handed 90 principle axis rotations: 90 rotation around X 90 rotation around Y 90 rotation around Z Helpful hint: For a R-handed rotation, the -sine is up and to the right of the +sine. z y x y x z z x y

17 on paper...exercise 17.1: rotate a point v = (1., 4., 7.) 1. Write the matrix that rotates 90 around z 2. Rotate v -> v' 3. Write the matrix that rotates 90 around y 4. Rotate v' -> v'' What is v''? Show me your answer.

18 Least Squares Superposition minimizes the root mean squared deviation in aligned atoms!18

19 RMSD Root Mean Square Deviation in superimposed coordinates is the standard measure of structural difference. Σ(v 1 i - v 2 i) 2 i=1,n N Where v 1 i and v 2 i are the equivalent* coordinates from molecules 1 and 2, respectively. *Equivalent as defined by an alignment.

20 Least squares superposition Problem: find the rotation matrix, M, and vector, v, that minimize the following quantity: Σ(Ma i + v - b i ) 2 Where a i and b i are the equivalent coordinates from molecules a and b, respectively.

21 Alignment defines structurally equivalent positions. Only aligned positions are included in the RMSD calculation. 4DFR:A ISLIAALAVDRVIGMENAMPWNLPADLAWFKRNTLDKPVIMGRHTWESIG-RPLPGRKNI 1DFR:_ TAFLWAQNRNGLIGKDGHLPWHLPDDLHYFRAQTVGKIMVVGRRTYESFPKRPLPERTNV 4DFR:A ILSSQ-PGTDDRVTWVKSVDEAIAAC--GDVPEIMVIGGGRVYEQFLPKAQKLYLTHIDA 1DFR:_ VLTHQEDYQAQGAVVVHDVAAVFAYAKQHLDQELVIAGGAQIFTAFKDDVDTLLVTRLAG 4DFR:A EVEGDTHFPDYEPDDWESVFSEFHDADAQNS--HSYCFKILERR 1DFR:_ SFEGDTKMIPLNWDDFTKVSSRTVEDT---NPALTHTYEVWQKK Unaligned positions are not.

22 Finding structural equivalences by linear Least squares (1) At the position of best superposition, we have an approximate equality: (2) We can eliminate v by superposing the center of mass of both molecules. Simplifies to: Ma i + v = b i Ma' i = b i

23 Least squares superposition Set of a 3N linear equations, 3 equations for each atom : M 11 a(i) x +M 12 a(i) y + M 13 a(i) z = b(i) x M 21 a(i) 1 +M 22 a(i) 2 + M 23 a(i) z = b(i) y M 31 a(i) 1 +M 32 a(i) 2 + M 33 a(i) z = b(i) z for all i=1,n known unknown When we set these equations to be equal although they are not, the solution is a minimum of the squared differences or least-squares. known Shorthand: M a = b For the mathematically inclined, here's the real algorithm:

24 a Least squares (shorthand) M = b Green are known. Orange are unknown. a T M= a a T b Square of a is a T a a T a M = a T b Invert squared matrix. (Cholesky decomposition) (a T a) -1 a T a M = (at a) -1 a T b Multiplying both sides by the inverse of squared matrix solves for the rotation matrix M. M = (a T a) -1 a T b Summary: M = (a T a) -1 a T b

25 Square, 2x2 or 3x3 (2D or 3D) Determinant = 1 Non-scaling. x = Rx. Orthogonal Special properties of rotation matrices The inverse equals the transpose, R -1 = R T Every row/column is a unit vector ( x =1) Any two rows/columns are orthogonal (dot product = 0). The cross-product of any two rows equals the remaining row. The product of any two rotation matrices is a rotation matrix. Read more about rotation matrices at: mathworld.wolfram.com/rotationmatrix.html

26 least-squares superimposed molecules

27 Structure-based alignment algorithms use structure to define the sequence alignment!27

28 Chicken/Egg problem Least squares superposition defines the alignment. The alignment defines the least squares superposition.!28

29 Structural alignment algorithm types Geometric--intermolecular Algorithms may be do this by minimizing the intermolecular distances or rootmean-square deviation (rmsd) of superimposed alpha-carbon positions. Geometric--intramolecular Algorithms minimize the difference between aligned distance matrices. DALI. Non-Geometric Algorithms align structural properties, such as %buried, secondary structure type, or structural environment usually by dynamic programming. SSAP.

30 Structure-based alignment tools DALI Programs VAST CE FATCAT MAMMOTH PRiSM *SCALI *SARF *FlexSNAP Databases FSSP HOMSTRAD COMPASS PALI SSAP SuperPose *these also do non-sequential alignment.

31 Meaningful structural alignment Aligned residues have same secondary (local) structure. Aligned residues have the same local contacts Erroneous structural alignment Alpha helix aligned to beta strand. Paired beta aligned to un-paired beta.

32 What we are saying when we assert that two positions align. 1. The two positions have a common ancestor position. 2.The two positions have the same environment, perhaps function.!32

33 Twilight-zone homologs far exceed close homologs database frequency of true structural homologs "twilight zone" % identity Q: Why are there more distant homologs than close homologs?

34 Example of structural homologs (sometimes called analogs) 4DFR: Dihydrofolate reductase 1YAC: Octameric Hydrolase Of Unknown Specificity 5.9% sequence identity (best alignment!) 1YAC structure was solved without knowing its function. Alignment to 4DFR and others implies it is a hydrolase of some sort, probably uses NAD cofactors.

35 Viewing structural homologs by viewing the secondary structure Sometimes you can see the structural similarity better in structural layers. DHFR in yellow and orange. YAC in green and purple sheets only helices only

36 Let's study an algorithm DALI: a intramolecular geometric structural alignment algorithm DALI: (Distance matrix-based ALIgnment) Liisa Holm & Chris Sander (1) Generate a distance matrix for each protein The distance matrix contains all pairwise distances.(symmetrical) j i D ij = distance between alpha carbon i and alpha carbon j Geometric--intramolecular

37 Aligning two distance matrices Cut-andpaste alignment of distance matrices Resulting sequence alignment

38 DALI algorithm S = L i =1 L j =1 DALI maximizes S. φ R ( i, j) where, φ R ( i, j) = θ R d A dij A - d B kl ij θ R is just a constant. dij A is the distance from residue i to residue j, and dkl B is the distance from residue k to residue l. i is aligned to k and j is aligned to l. S is maximal when d ij A = d kl B everywhere. The alignment is the independent variable, i.e. the search space.

39 DALI algorithm: Get a square from structure A Aligns structures using distance matrices. Randomly align with structure B Keep the best S scores vs send high scores to pairs list Each pair of 6x6 s corresponds to a gapped alignment

40 DALI algorithm : Aligns structures using distance matrices. (1) select a pair of segments from the pairs list 1st pair 4dfr Axes = sequence position shade=distance, shorter distances are lighter shade. 1yac

41 DALI algorithm (2) Add second pair from pairs list. Calculate S. Better: Keep. Worse: Reject. 2nd pair 1st pair 4dfr cross blocks 1yac

42 DALI alignment output

43 Uses of structural alignment in modeling A structure-based alignment is the Gold Standard for a sequence alignment. Aligned structures tell you where structurally conserved regions are, versus where insertions/deletions are allowed. Structural analogs provide a source of plausible loop structures. Multiple aligned structures show evolutionary plasticity.

44 Exercise 17.2: MOE Superpose Do these pairs: 2ptl 2gb1 3sdh 1h97 3sdh 1phn File Open: RCSB PDB: code: xxxx Delete extra copies if there are multiple chains. Hide All atoms Ribbon Style: oval, Color: terminus Select synchronize (check) Protein Superpose Check RMSD, %identity. Try again with options "gaussian distance weighting" and "accent secondary structure". Find residues that are aligned but should not be, according to the requirement for the same secondary/local structure.

45 Review How do you multiply a matrix by a vector? Rotate the vector (1,2,3) by 90 around Z. How is the Z axis related to the X and Y axes? Can I apply rotations in any order and get the same result? Write the RMSD equation. Why do we need an alignment to calculate RMSD? What value is minimized in least-squares superposition? Is least squares optimal or heuristic? What is the output of a structure-based alignment? Is structure-based alignment optimal or heuristic? What kind of structure-based alignment algorithm is DALI? What two properties should be conserved between structurally aligned positions? What are analogs? What does DALI maximize? What is the search space of DALI?!45

46 Supplementary slides!46

47 supplementary slides: Principal axes unit vectors are matrix columns. ( a b c d e f g h i )x = a d g rotated x axis = b e h rotated y axis = c f i rotated z axis You can create a rotation matrix by defining three mutually orthogonal unit vectors, then lining them up side-by-side in a 3x3 matrix.!47

48 supplementary slides: two 3D rotation conventions: Euler angles, α β γ axis of z x z rotation: # cosγ sinγ 0& # &# cosα sinα 0& % (% (% ( % sinγ cosγ 0( % 0 cos β sinβ( % sinα cosα 0( % (% (% ( Order of $ 0 0 1' $ 0 sinβ cos β ' $ 0 0 1' rotations: Each rotation is around a principle axis. z Polar angles, φψκ y z -y -z # cosφ sinφ 0& # cosϕ 0 sinϕ& # cosκ sinκ 0& # cosϕ 0 sinϕ &# cosφ sinφ 0& % (% (% (% (% ( % sinφ cosφ 0( % (% sinκ cosκ 0( % (% sinφ cosφ 0( % (% (% (% (% ( $ 0 0 1' $ sinϕ 0 cosϕ ' $ 0 0 1' $ sinϕ 0 cosϕ' $ 0 0 1' Net rotation = κ, around an axis axis defined by φ and ψ

49 supplementary slides: Polar angle convention: # cosφ sinφ 0& # cosϕ 0 sinϕ& # cosκ sinκ 0& # cosϕ 0 sinϕ &# cosφ sinφ 0& % (% (% (% (% ( % sinφ cosφ 0( % (% sinκ cosκ 0( % (% sinφ cosφ 0( % (% (% (% (% ( $ 0 0 1' $ sinϕ 0 cosϕ ' $ 0 0 1' $ sinϕ 0 cosϕ' $ 0 0 1' z = north pole ψ κ y φ x = prime equator Rotation of κ degrees around an axis axis located at φ degrees longitude and ψ degrees latitude. Rotation order is Z(-φ), Y(-ψ), Z(κ), Y(ψ), Z(φ) Notice the nested rotations.

50 SSAP alignment A View is the set of all vectors from one residue. Each residue has its own "View", which is a set of vectors to nearest neighbor residues. Algorithm type: Geometric--intramolecular

51 SSAP alignment: views i and j must have similar backbone angles, otherwise the score is zero. View for Template residue i View for Target residue j j The difference between the two views is a measure of how similar the structures are, when viewed from i and j. i residue level score matrix

52 SSAP algorithm: Double Dynamic Programming residue level scoring matrix DP1 summary matrix DP2 For each ij pair, we find the best DP alignment that includes ij. Keep the DP score at position (i,j) in teh summary matrix. The summary matrix is subjected to a second round of DP, to give the optimal alignment.

53 Alignment of15 analogs helix sheet

54 Two other servers for structurebased alignment FATCAT VAST: Algorithm type: Geometric--intermolecular

55 Non-sequential alignment! 1alk vpt 4 3 SCALI non-sequential alignment Yuan & Bystroff, 2005

56 SCALI Non-sequential structure-based alignments can be used to identify similar motifs in the packing geometry of SSEs. For example, it can find two proteins that have 2/4/2 α/β/α 3-layer sandwich architecture, regardless of how the SSEs are connected. (1) Exhaustive gapless alignments (2) FASTA-style assembly of alignment from fragment pairs. (3) Alignment score based on sequence similarity, local structure similarity and contact map similarity. Algorithm type: Geometric--intramolecular

57 Exhaustive alignment, no gaps. protein structure B Each dot represents two positions that have the same local structure. Darker means similar in sequence. The SCALI alignment is constructed from fragment pairs (diagonal rows). protein structure A

58 FASTA-style search in alignment space alignments generation... SCALI uses a near-greedy algorithm. Each candidate parent alignment generates up to 50 children alignments. Children are selected based on similarity (sequence, local structure, contacts). Selected children become parents in the next generation. The program is done when there is nothing left to align.

59 Non-sequential alignment space alignment space vs β α β β β α Expressing a non-sequential alignment like a standard sequence alignment loses information. Using an alignment matrix is better, as in this example of two different ββα units.