Molecular Modeling lecture 4. Rotation Least-squares Superposition Structure-based alignment algorithms

Transcription

1 Molecular Modeling lecture 4 Rotation Least-squares Superposition Structure-based alignment algorithms

2 Rotation is addition in polar coordinates 2

3 What happens when you move the mouse to rotate a molecule? Mouse sends mouse coordinates (Δx,Δy) to the running program Rotation angles are calculated: θ x = Δx*scale, θ y = Δy*scale Rotation matrices are calculated: y x R x = [ ] cos θ x -sin θ x 0 sin θ x cos θ x

4 What happens when you move the mouse (cont'd): 4. New atom coordinates are calculated v = RyRxv 5. The scene is rendered using the new coordinates.

5 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)) Angles are measured counter-clockwise (right-handed).

6 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

7 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 the same for any r, any α.

8 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

9 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

10 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.

11 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 =

12 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

13 Exercise 3.1: rotate a point (x,y,z) = (1., 4., 7.) Rotate this point by 90 around the Z-axis Then... Rotate the new point by 90 around the Y-axis. What are the new coordinates?

14 supplementary slide: Special properties of rotation matrices They are square, 2x2 or 3x3(higher dimensions in principle) The product of any two rotation matrices is a rotation matrix. The inverse equals the transpose, R -1 = R T Every row/column is a unit vector. Any two rows/columns are orthogonal vectors. The cross-product of any two rows equals the third. x = Rx, where R is a rotation matrix. Read more about rotation matrices at: mathworld.wolfram.com/rotationmatrix.html

15 Least Squares Superposition minimizes the root mean squared deviation in aligned atoms 15

16 Exercise 3.2: Superimpose by hand Do these pairs: 1N9U vs 1N9V 1WFA (chain A vs chain B) File Open: RCSB PDB: code: xxxx in-class Delete parts not needed (e.g. waters). Select Chain label in SEQ. R-mouse/Delete. Hide All atoms Ribbon Style: oval, Color: chain or terminus Select synchronize (always check synchronize) SEQ (opens the sequence window) Double-click on chain label to select one molecule. In MOE window: Rotate selected : meta-middlemouse-drag. Translate selected : shift-meta-middlemouse-drag Rotate all: middlemouse-drag Translate all: shift-middlemouse-drag Until molecules are superposed. have the instructor check you off.

17 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.

18 Least squares superposition Problem: find the rotation matrix, M, and a vector, v, that minimize the following quantity: M x i + v y i 2 i Where x i and y i are the equivalent coordinates from molecules 1 and 2, respectively.

19 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.

20 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 translating the center of mass of both molecules to the origin. The equation simplifies to: We have one equation (i) for each atom, M has 9 unknowns. Mx i + v y i Mx' i y' i If there are more equations than unknowns, there is a unique solution.

21 Least squares Least squares solves a set of a linear equations : x 11 a 1 +x 12 a 2 + x 1N a N = y 1 x 21 a 1 +x 22 a 2 + x 2N a N = y 2 x M1 a 1 +x M2 a 2 + x MN a N = y M This is 'shorthand' notation for the linear equations. knowns unknowns a x = y knowns matrix vector vector

22 x a Least squares, continued = y Green elements are known. Orange are unknown. Fat Rectangles are matrices. Thin rectangles are vectors. x T a = x x T y Multiply both sides by transpose of x. "Squaring" x T x a = x T y "Squared" matrix can be inverted. (We can use the "LU decomposition".) (x T x) -1 x T x a = (xt x) -1 x T y Multiplying both sides by the inverse of "squared" matrix solves for a. a = (x T x) -1 x T y Summary: a = (x T x) -1 x T y

23 least-squares superimposed molecules

24 Structure-based alignment algorithms use structure to define the sequence alignment 24

25 Chicken/Egg Least squares superposition defines the alignment. The alignment defines the least squares superposition. 25

26 Structural alignment algorithm types Alignment algorithms create a one-to-one mapping of subset(s) of one sequence to subset(s) of another sequence. Structure-based alignment types: Geometric--intermolecular Algorithms may be do this by minimizing the intermolecular distances or rootmean-square deviation (rmsd) in superimposed alpha-carbon positions. Geometric--intramolecular Algorithms minimize the difference between aligned contact maps or distance matrices. Intramolecular distances are used Non-Geometric Algorithms align structural properties, such as %buried, or secondary structure type, usually using dynamics programming (DP)

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

28 Real (meaningful) structural alignment Aligned residues have same secondary (local) structure. Pairs of aligned residues have the same contacts (either both in contact or both not in contact) Erroneous structural alignment Aligned residues are simply superimposed in space. Are not the same secondary structure. Do not have the same contacts.

29 Structure-based alignment can identify remote homologs In general, the number of remote homologs exceeds the number of close homologs. But sequence similarity is useless below 25% identity likelihood the "twilight zone" percent identity for structural homologs

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

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

32 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

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

34 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.

35 DALI algorithm: Get a square from structure A Search alignment space 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

36 DALI algorithm DALI algorithm: select a square from the pairs list 1st pair 4dfr Axes = sequence position shade=distance, shorter distances are lighter shade. 1yac

37 DALI algorithm add a square, evaluate cross blocks 2nd pair 1st pair 4dfr cross blocks Calculate S. Better: Keep. Worse: Reject. 1yac

38 DALI alignment output

39 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.

40 Exercise 3.3: MOE Superpose Do these pairs: 2ptl.pdb 2gb1.pdb 3sdh.pdb 1h97.pdb 3sdh.pdb 1phn.pdb 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 Try different settings. Check RMSD. Take-home part!

41 Review questions 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? What does RMSD stand for? Why do we need an alignment to calculate RMSD? What 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 protein analogs? What does DALI maximize? What is the search space of DALI? 41

42 Supplementary slides 42

43 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. 43

44 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

45 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.

46 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

47 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

48 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.

49 HOW to use CE to find and align structural homologs Go to Find structural alignments by selecting from ALL or REPRESENTATIVES from the PDB. Submit your protein and chain. Or use 4dfr:A Select 2 structures. Then hit: Get alignment (or use 4DFR and 1YAC) Download as PDB file. Save it. For use in MOE, divide the file into 2, one for each protein.

50 CE alignment

51 Alignment of15 analogs helix sheet

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

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

54 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

55 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

56 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.

57 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.