Assign 4
Protein Structure Alignment
Due Date: April 5th, Before Midnight
In this assignment you will align two protein structures using the iterative superposition and dynamic programming algorithm (R20, Algorithm 9.1, and R19, sec 8.3.1.2 -- Structal).
Parsing: First, you must read the PDB files and extract the (x,y,z) coordinates for the CA atom (the alpha-carbon) from chain 'A' (model '0') from both the proteins. Let A be the 3 \times n matrix of protein A's coordinates, and let B be the 3 \times m matrix of protein B's coordinates. You may use the biopython's PDB module to read the structures (see Attach:biopdb_faq.pdf).
Initialization:Given two protein structures/coordinates A and B, you begin with an initial random equivalence E_0 of length k (where k is a parameter), i.e., E_0 = \{(A_{i_1}, B_{j_1}), \cdots, (A_{i_k}, B_{j_k})\}. Set t=0, which denotes the iteration number.
Superposition: Next, given E_t you extract only those amino acids that are part of the equivalence for both A and B, and optimally superimpose them using the singular value decomposition method (see R19, sec 8.3.1.1). Let A' = (A_{i_1}, A_{i_2}, ..., A_{i_k}) be the set of k 3D residues for A that are part of the equivalence. Likewise let, B' = (B_{j_1}, b_{j_2}, ..., B_{j_k}). For the superpositioning, you have to first compute the mean of A' and B' and subtract it from all coordinates so that A' and B' are centered. Denote the means as \mu_{A'}, \mu_{B'}. Next, compute the matrix C = B' A'^T. Then, you can use the numpy svd function to decompose C into U \Delta V^T matrices. Next compute the determinant (use numpy det) of VU^T, and note its sign, i.e., it is positive or negative. Define the matrix W as follows
W = \pmatrix{1 & 0 & 0\\0 & 1& 0\\ 0 & 0 &sign}
where sign is either +1 or -1. Finally the rotation matrix is given as R = V W U^T
Dynamic Programming: Given the rotation matrix R, you have to first compute the scoring matrix, which is a n \times m matrix S, that gives the score of aligning residue A_i with B_j. To find S, you have to first subtract \mu_{A'} from all points in A, and \mu_{B'} from all points in B, and then you should rotate B, by computing Br = RB, so that the entire structures A and B are superposed, based on the optimal residue of the residue pairs in the equivalence set E_t. Now define the scoring matrix S as follows
S(i,j) = \frac{M}{1 + \left( \frac{dist(A_i, B_j)}{d_0} \right)^2}
where M is the maximum score and d_0 is a distance threshold. Finally, compute the optimal global path for aligning A with B, using global dynamic programming, using the scoring matrix S, and a gap penalty of -M/2. You may use your dynamic programming implementation from assign1 for this, but keep in mind that this is global alignment with linear gap penalty.
Let P be the optimal path. For each pair of residues (a_i, b_j) that are aligned (corresponding to a diagonal move) in P, choose at least k pairs that are closest, and furthermore, add those pairs that are within a distance of \delta^{\max} to the new alignment E_{t+1}.
Iterate: Now that you have a new E_{t+1} you can use the superposition step to obtain a new rotation matrix, which can in turn be used to find another equivalence set E_{t+2} and so on. Repeat these two steps until the maximum number of iterations is reached, say 20, or the equivalence set does not change from one iteration to the next.
What to turn in
Write a script called 'RCSID-assign4.xx' (xx is py or r) that implements this algorithm. Your program should accept two file names as command line parameters, so I will run it as RCSID-assign4.py A.pdb B.pdb. Make sure you do not hard-code any paths.
Use the following parameters: M=20, d_0 = \sqrt{5}, k=20, maxiter=20, \delta^{\max} = 5, gap penalty = -M/2.
Run your code on the two proteins A=Attach:2A5G.pdb and B=Attach:1KGD.pdb. Since the program uses a random initialization, run it several times and report the best finding in terms of the rmsd and length of the final superposition.
A sample output on A=Attach:2IGD.pdb and B=Attach:2PTL.pdb is as follows.
Output from different iterations in the format:
ITER RMSD LEN E
where ITER is the iteration, RMSD is the RMSD of the residue pairs in the E set, LEN is the number of pairs in E, and finally E gives the actual pairs.
0 13.3247903031 20 [(0, 3), (1, 5), (5, 13), (6, 16), (7, 26), (13, 29), (16, 30), (17, 32), (21, 33), (23, 41), (24, 42), (30, 43), (39, 45), (40, 47), (46, 52), (54, 54), (56, 57), (57, 65), (59, 67), (60, 70)]
1 2.52174194727 42 [(1, 9), (2, 13), (3, 14), (4, 15), (7, 19), (8, 20), (9, 21), (10, 22), (11, 23), (12, 24), (18, 30), (19, 31), (20, 32), (21, 33), (22, 34), (24, 36), (27, 39), (28, 40), (30, 42), (31, 43), (33, 45), (34, 46), (35, 47), (37, 49), (38, 50), (39, 51), (40, 52), (41, 53), (42, 54), (45, 60), (46, 61), (47, 62), (48, 63), (49, 64), (50, 65), (51, 66), (55, 70), (56, 71), (57, 72), (58, 73), (59, 74), (60, 75)]
2 2.47808434193 52 [(1, 12), (2, 13), (3, 14), (4, 15), (6, 18), (7, 19), (8, 20), (9, 21), (10, 22), (11, 23), (12, 24), (13, 25), (17, 29), (18, 30), (19, 31), (20, 32), (21, 33), (22, 34), (23, 35), (24, 36), (25, 37), (26, 38), (27, 39), (28, 40), (29, 41), (30, 42), (31, 43), (32, 44), (33, 45), (34, 46), (35, 47), (36, 48), (37, 49), (38, 50), (39, 51), (40, 52), (41, 53), (42, 54), (44, 59), (45, 60), (46, 61), (47, 62), (48, 63), (49, 64), (50, 65), (51, 66), (55, 70), (56, 71), (57, 72), (58, 73), (59, 74), (60, 76)]
3 2.18804120087 52 [(3, 15), (4, 16), (5, 17), (6, 18), (7, 19), (8, 20), (9, 21), (10, 22), (11, 23), (12, 24), (13, 25), (14, 26), (17, 29), (18, 30), (19, 31), (20, 32), (21, 33), (22, 34), (23, 35), (24, 36), (25, 37), (26, 38), (27, 39), (28, 40), (29, 41), (30, 42), (31, 43), (32, 44), (33, 45), (34, 46), (35, 47), (36, 48), (37, 49), (38, 50), (39, 51), (40, 52), (41, 53), (42, 54), (44, 59), (45, 60), (46, 61), (47, 62), (48, 63), (49, 64), (50, 65), (51, 66), (55, 70), (56, 71), (57, 72), (58, 73), (59, 74), (60, 76)]
4 1.91720521889 54 [(3, 15), (4, 16), (5, 17), (6, 18), (7, 19), (8, 20), (9, 21), (10, 22), (11, 23), (12, 24), (13, 25), (14, 26), (15, 27), (17, 30), (18, 31), (19, 32), (20, 33), (21, 34), (22, 35), (23, 36), (24, 37), (25, 38), (26, 39), (28, 40), (29, 41), (30, 42), (31, 43), (32, 44), (33, 45), (34, 46), (35, 47), (36, 48), (37, 49), (38, 50), (39, 51), (40, 52), (41, 53), (42, 54), (44, 59), (45, 60), (46, 61), (47, 62), (48, 63), (49, 64), (50, 65), (51, 66), (52, 67), (54, 69), (55, 70), (56, 71), (57, 72), (58, 73), (59, 74), (60, 76)]
5 2.00373283905 55 [(3, 15), (4, 16), (5, 17), (6, 18), (7, 19), (8, 20), (9, 21), (10, 22), (11, 23), (12, 24), (13, 25), (14, 26), (15, 27), (16, 29), (17, 30), (18, 31), (19, 32), (20, 33), (21, 34), (22, 35), (23, 36), (24, 37), (25, 38), (26, 39), (28, 40), (29, 41), (30, 42), (31, 43), (32, 44), (33, 45), (34, 46), (35, 47), (36, 48), (37, 49), (38, 50), (39, 51), (40, 52), (41, 53), (42, 54), (44, 59), (45, 60), (46, 61), (47, 62), (48, 63), (49, 64), (50, 65), (51, 66), (52, 67), (54, 69), (55, 70), (56, 71), (57, 72), (58, 73), (59, 74), (60, 76)]
6 1.99949320528 55 [(3, 15), (4, 16), (5, 17), (6, 18), (7, 19), (8, 20), (9, 21), (10, 22), (11, 23), (12, 24), (13, 25), (14, 26), (15, 27), (16, 29), (17, 30), (18, 31), (19, 32), (20, 33), (21, 34), (22, 35), (23, 36), (24, 37), (25, 38), (26, 39), (27, 40), (29, 41), (30, 42), (31, 43), (32, 44), (33, 45), (34, 46), (35, 47), (36, 48), (37, 49), (38, 50), (39, 51), (40, 52), (41, 53), (42, 54), (44, 59), (45, 60), (46, 61), (47, 62), (48, 63), (49, 64), (50, 65), (51, 66), (52, 67), (54, 69), (55, 70), (56, 71), (57, 72), (58, 73), (59, 74), (60, 76)]
7 1.88581355948 54 [(4, 16), (5, 17), (6, 18), (7, 19), (8, 20), (9, 21), (10, 22), (11, 23), (12, 24), (13, 25), (14, 26), (15, 27), (16, 29), (17, 30), (18, 31), (19, 32), (20, 33), (21, 34), (22, 35), (23, 36), (24, 37), (25, 38), (26, 39), (27, 40), (29, 41), (30, 42), (31, 43), (32, 44), (33, 45), (34, 46), (35, 47), (36, 48), (37, 49), (38, 50), (39, 51), (40, 52), (41, 53), (42, 54), (44, 59), (45, 60), (46, 61), (47, 62), (48, 63), (49, 64), (50, 65), (51, 66), (52, 67), (54, 69), (55, 70), (56, 71), (57, 72), (58, 73), (59, 74), (60, 76)]
final: 8 1.88581355948 54 [(4, 16), (5, 17), (6, 18), (7, 19), (8, 20), (9, 21), (10, 22), (11, 23), (12, 24), (13, 25), (14, 26), (15, 27), (16, 29), (17, 30), (18, 31), (19, 32), (20, 33), (21, 34), (22, 35), (23, 36), (24, 37), (25, 38), (26, 39), (27, 40), (29, 41), (30, 42), (31, 43), (32, 44), (33, 45), (34, 46), (35, 47), (36, 48), (37, 49), (38, 50), (39, 51), (40, 52), (41, 53), (42, 54), (44, 59), (45, 60), (46, 61), (47, 62), (48, 63), (49, 64), (50, 65), (51, 66), (52, 67), (54, 69), (55, 70), (56, 71), (57, 72), (58, 73), (59, 74), (60, 76)]
This means that these two proteins align very well, the final RMSD is only 1.885 Armstrongs, which is excellent, and the length of E is 54. It took 8 iterations to converge.
To verify the initial steps, here is the rotation matrix for the equivalence shown for iter 0 (i.e., the initial equivalence):
[[-0.26270193 0.95855713 -0.11025375]
[ 0.87425673 0.28481713 0.39313394]
[ 0.40824348 0.00688694 -0.91284704]]
Also, here is the distance matrix {$dist(A_i, B_j) $} for the full A and B proteins using the R matrix above (and after subtracting the mean of A' and B'):
[[ 25.1905632 21.49979401 19.19536209 15.43400478 12.66535664
9.23564529 7.99627686 8.24728489 5.50331879 5.5723424
8.220994 9.56705189 8.30479145 6.77224159 10.43939972
11.67946529 13.58983612 16.70627213 18.28353691 22.05890465
24.78569221 28.48748016 31.24429893 34.45605087 36.97476578
40.60082245 43.95594406 45.20505142 42.96960068 40.13074493
37.11952209 33.7072525 30.2229805 27.49041176 23.90198898
22.08301926 18.68135262 17.31081963 18.95576859 21.09442902
24.88700104 24.5606842 24.37025261 27.28245354 29.62311935
28.1989193 29.0222187 32.49860382 33.23902893 31.77645111
34.64390564 36.94028473 37.13645554 38.12525177 41.13920212
43.7687149 46.37696075 45.28638077 45.88804626 42.35799026
40.18754196 39.0904274 35.4871788 34.3348465 31.9474926
29.8416729 26.3657856 26.85677147 28.36254501 25.46079254
26.94579124 30.28253365 32.68797302 34.75585556 38.03684998
40.70904922 44.27345276 46.9213562 ]
[ 26.95505142 23.27863693 21.36565018 17.62249565 15.34382629
11.96963024 9.66260147 9.24251366 6.43422508 4.13778019
5.91330147 6.10856247 4.54999018 3.97565126 7.63130569
8.60414886 11.19109917 14.04884052 15.16766548 18.93677521
21.50052834 25.13386726 27.84525299 30.95096779 33.55895615
37.13139343 40.54790878 41.70214844 39.53320312 36.60364914
33.66374207 30.18024635 26.71854019 23.90129852 20.34877205
18.51791573 15.2625618 14.12201214 16.23814774 18.53061295
22.24834442 21.74761581 21.22352219 24.25079536 26.50723076
24.85173607 25.62771606 29.18831062 29.79864883 28.20090675
31.10331345 33.42543411 33.50121307 34.4658165 37.53960037
40.10217667 42.77712631 41.7338829 42.44293213 38.94993973
36.87746048 35.97213745 32.44078445 31.45022774 29.30546761
27.3631134 24.07466888 24.81881714 25.9737606 22.81719017
24.07498741 27.26457214 29.55447197 31.46740723 34.73976517
37.29336929 40.90842819 43.45294952]
[ 30.34098244 26.73019218 24.64323807 20.84477997 18.57393837
14.97834206 12.83662891 12.92505646 10.2173748 7.71899605
8.77955246 7.50327873 4.05335045 1.55119443 4.91832924
5.31976032 7.47109652 10.4297142 12.00563717 15.73970795
18.60967064 22.29222298 25.22283936 28.47800636 31.093853
34.72015381 38.09857941 39.51159286 37.47782516 34.66146469
31.53812408 28.12725449 24.50375938 21.65031433 17.94051552
16.00900269 12.45275879 11.18008709 12.92653275 14.93155956
18.69495392 18.48163986 18.09147453 20.92922592 23.36515617
22.04716301 22.7418766 26.17758369 27.07646179 25.69075012
28.47666359 30.77002907 31.18558884 32.15954208 35.04554367
37.80696487 40.27445602 39.16392136 39.63007355 36.06509781
33.88725662 32.77286148 29.15945816 28.13989067 25.8404007
23.97968292 20.68579102 21.20061874 22.33961487 19.27456665
20.80101204 24.03675652 26.55628586 28.52174759 31.8878479
34.54548264 38.1037178 40.80623627]
[ 31.00114632 27.58112335 25.38829803 21.60493469 19.76748466
16.02895737 13.71173382 14.55034351 12.60955429 9.74145603
11.33586407 9.2539854 5.98233461 2.7573061 2.25823975
4.98306513 7.02983236 8.88901711 10.16380215 13.92272854
16.60038185 20.39876366 23.13523102 26.57988167 28.88146019
32.60386276 35.82228088 37.3739357 35.17208481 32.58275223
29.30894089 26.09918785 22.48565865 20.02163506 16.40707397
15.15310383 11.61688328 11.32389164 13.03824234 14.16435909
17.83637047 18.30172157 17.48394394 19.8428688 22.72008705
21.62576103 21.70299721 25.1057148 26.42177773 24.83496475
27.24162865 29.84331894 30.48590851 31.06358147 33.91952515
36.66289139 39.00373459 37.57262802 38.01370239 34.51615524
32.05435562 30.74850655 27.16361237 25.75102425 23.37034035
21.14603806 17.71247482 18.54765129 20.02093124 17.07792282
18.33623886 21.8029232 24.17665863 26.50340652 29.70804214
32.60554123 36.04206085 38.88313675]
[ 33.12666702 29.76586723 27.8965168 24.19423866 22.66514778
19.0477066 16.27955437 16.68074417 14.8600359 11.43966961
12.13600731 9.04712677 6.15707207 5.19177771 1.58353317
3.52776766 6.36952972 6.83053923 6.89113188 10.68425465
13.06340504 16.83343315 19.49308777 22.8676796 25.23820686
28.92878914 32.20788574 33.68028641 31.53490067 28.87028885
25.63507652 22.36154366 18.74105835 16.23140907 12.64197636
11.60720444 8.2992878 8.93317127 11.26587391 12.26039028
15.66145039 16.05812263 14.62359905 16.97698975 19.83965492
18.50621986 18.32502556 21.8336525 23.09124374 21.28024673
23.63236237 26.34661293 26.90476036 27.35826683 30.27957726
32.94779968 35.34169388 33.90187836 34.46783829 31.02118301
28.62097359 27.52656746 24.03876114 22.77299309 20.7332859
18.69807434 15.57806683 16.86616898 17.82915688 14.54236889
15.35454369 18.65483665 20.84194183 23.03968811 26.21753883
29.0202179 32.50988007 35.25742722]
[ 35.58585358 32.3938179 30.52281761 26.89382935 25.64430428
22.0167942 19.2256012 19.926548 18.42396736 14.94118881
15.66284084 12.32766151 9.81251621 8.84264469 5.0265274
6.27309465 7.72069597 6.14905596 5.03374386 8.31692696
10.28317261 14.08049965 16.5457058 20.09838676 22.17467117
25.9373455 29.0644474 30.72319794 28.51906204 26.09180069
22.68669319 19.6493969 16.0268364 13.96666718 10.61112976
10.51520634 7.73981524 9.61333466 11.59152794 11.48659992
14.39071655 15.46064186 13.53654766 15.18964195 18.4107666
17.37734604 16.48070526 19.83588028 21.52762985 19.5864315
21.45720482 24.42717552 25.26893044 25.32757378 28.13097
30.79837036 33.00415802 31.27618599 31.77194405 28.40317535
25.75413132 24.49805069 21.07680893 19.49364281 17.49106979
15.23022842 12.17247581 13.95410728 14.95188618 11.74357605
12.08215523 15.52705288 17.54970169 20.05517387 23.07925034
26.08779144 29.44678497 32.32622147]
[ 37.15762329 34.09597397 32.53568268 29.04720688 28.14312172
24.67304039 21.57821274 22.03246117 20.81339836 17.11674309
17.40901184 13.75095177 12.03876877 11.95987988 8.27559566
9.05821896 10.66898537 8.28077316 5.31997824 7.32658482
7.86191654 11.42613983 13.38090897 16.81889915 18.78867912
22.52498627 25.67073441 27.15405273 24.86278915 22.38652992
19.04189873 15.98468685 12.46028137 10.7064333 7.80451965
8.9034214 7.48658037 10.58695889 12.94528103 12.61324215
14.77848911 15.76386547 13.04862499 14.39495754 17.47725487
16.12234879 14.62904644 18.02816391 19.63275719 17.2254467
18.81307411 22.05824471 22.70810127 22.38460541 25.32761192
27.7610817 30.06513786 28.21494865 28.9870472 25.78686523
23.16809082 22.24535179 19.10250282 17.58455467 16.22177887
14.11998653 11.74429417 14.29021835 14.6760149 11.32525921
10.49549007 13.55142403 14.93937778 17.38369751 20.17251587
23.09179688 26.47181129 29.19706726]
[ 40.91591644 37.82372284 36.21535873 32.68210602 31.62857056
28.08392334 25.09719086 25.48663521 23.98748398 20.32725716
20.31369972 16.61704826 14.66299725 14.63918209 10.97243786
10.73449039 11.26016521 7.92374468 4.54468203 4.32681942
4.33997393 7.99128151 10.33571815 13.97109985 16.04834366
19.80817604 22.95560074 24.75083351 22.78564644 20.43000031
16.91059113 13.99263287 10.27863407 8.53483582 5.78360271
7.35623312 6.82149887 10.23035622 11.88363647 10.74352837
12.10647488 13.46261597 10.50549793 11.1476841 14.36887741
13.43053532 11.48072815 14.62890339 16.61420631 14.43506241
15.70993614 18.93254852 20.00408936 19.64879417 22.308815
24.96136093 27.00381088 25.11607742 25.61612129 22.33346367
19.62680054 18.5590229 15.34434605 13.93390942 12.685709
11.01811409 9.22981548 11.77901745 11.47541523 8.05551147
6.86557484 9.80384159 11.45326996 13.87148571 16.8470993
19.84089851 23.20005798 26.07851791]
[ 43.46969604 40.50943375 39.07800293 35.6540184 34.84004211
31.38196754 28.26245117 28.59373474 27.29643059 23.55918503
23.38152695 19.59955406 18.05034065 18.28991699 14.6284647
14.42139912 14.94215488 11.47107983 8.06503677 6.31059456
3.50970578 5.60836554 6.90538168 10.56032848 12.32751846
16.11365509 19.20277405 20.9973774 19.0044899 16.7692337
13.17312813 10.4720993 6.900774 6.25800896 5.35067797
8.46204567 9.4938097 13.12702084 14.66502285 13.22336578
13.61917782 15.01879692 11.64078999 11.44104958 14.32804394
13.40051651 10.57876492 13.31741238 15.39714432 12.85501671
13.35938931 16.9015255 17.9838047 17.02293587 19.64200401
22.06879997 24.03699875 21.88472366 22.56794357 19.51316071
16.71668625 15.91973591 13.14759445 11.72468567 11.41519833
10.12348461 9.60474873 12.65886211 11.71638584 8.83942127
6.03364897 7.87077808 8.40571976 10.88809776 13.48261452
16.50124168 19.79580688 22.61240959]
[ 46.89094162 43.88819122 42.54684067 39.11042786 38.20999908
34.75667191 31.61109543 31.70435333 30.21383095 26.47812843
25.90806198 22.12085533 20.65947342 21.22354889 17.71023941
16.88826942 17.04752922 13.48694324 10.24519825 7.32576609
3.57974505 2.2335391 3.46799445 7.19058561 9.43898392
13.10474968 16.36336517 18.27907753 16.81248856 14.52860928
10.99975395 8.40891457 4.97370577 4.6490922 5.52357435
8.6184206 10.78718472 14.20633602 15.35002804 13.74212837
13.12437057 14.30724525 10.77833271 9.84762287 12.12443256
11.33172989 7.92291641 10.13881397 12.27574825 9.77131844
9.71960258 13.30804157 14.61409855 13.52436352 15.93776989
18.48340797 20.31942749 18.23576355 18.84346199 15.74311543
13.07702732 12.63219833 10.10695076 9.47635078 10.19223309
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