Protein Conformation Prediction (Part I)

1. Doug Raiford Lesson 17 Protein Conformation Prediction (Part I)

2. Two folding models • Framework model • Secondary structure first • Assemble secondary structure segments • Hydrophobic collapse • Molten: compact but denatured • Formation of secondary structure after: settles in • van der Waals forces and hydrogen bonds require close proximity Protein Conformation Prediction (Part I)

3. Experimentally determining • Isolate protein and crystalize • Time consuming process • Slowly evaporate • Many experiments in parallel • Different conditions • X-ray crystallography • Get XYZ spatial coordinates Protein Conformation Prediction (Part I)

4. PDBs • Store these XYZ coordinates in text files • PDB website X Y Z Occu Temp Element ATOM 1 N THR A 5 23.200 72.500 13.648 1.00 51.07 N ATOM 2 CA THR A 5 23.930 72.550 12.350 1.00 51.27 C ATOM 3 C THR A 5 23.034 72.048 11.220 1.00 50.34 C ATOM 4 O THR A 5 22.819 72.747 10.228 1.00 51.19 O ATOM 5 CB THR A 5 25.221 71.703 12.416 1.00 51.94 C ATOM 6 OG1 THR A 5 26.159 72.326 13.305 1.00 53.51 O ATOM 7 CG2 THR A 5 25.849 71.583 11.046 1.00 53.33 C Protein Conformation Prediction (Part I)

5. Modeling • To fully model the folding action of a polypeptide chain • Must know all the forces acting on each aa • Must be able to predict the motion of the aa’s given the forces Protein Conformation Prediction (Part I)

6. How to aa’s move? • Recall that proteins are able to fold because of the torsional rotation of the aa bonds R  almost always 180    R Protein Conformation Prediction (Part I)

7. In order to model folding… • Must be able to take phi and psi angles and transform into xyz coordinates of various atoms • Don’t forget about R groups • What places in space are occupied? • Bump checking Protein Conformation Prediction (Part I)

8. Anatomy of a carbon atom • Tetrahedron Protein Conformation Prediction (Part I)

9. Remember • Know distances • Each angle is 109.5 R  almost always 180    R Protein Conformation Prediction (Part I)

10. The angles • 4 atoms on same plane • , , and ω all relative to R group (O in case of ω) Protein Conformation Prediction (Part I)

11. Torsion angles to xyz • One approach • Given xyz of last three, and next torsion angle… • Transform so that C is at origin, BC on new X, AB on plane of new Y • Then apply torsion • Start D on X • Swing out 70.5 (180-109.5; in the plane of Y) • Rotate by torsion angle Protein Conformation Prediction (Part I)

12. New vector space • To transform a vector space… X C Z Y B A Protein Conformation Prediction (Part I)

13. New vector space • To transform a vector space… X C Z Y B New X axis New Z axis New Y axis A Protein Conformation Prediction (Part I)

14. New vector space • It’s all about projections • If target vector is a unit vector then simple dot product A B Protein Conformation Prediction (Part I)

15. New vector space • Dot product of a row with vector yields the projection of the vector onto the vector represented by the row • All three dot products yields all three components X C Z B Y A New X New Z New Y Protein Conformation Prediction (Part I)

16. What is the new X? • The new X is BC (as a unit vector) X’ C Z’ Y’ B A Protein Conformation Prediction (Part I)

17. But what is BC? • Remember, all we have is the last xyz coordinates • All vectors are assumed to originate at the origin • So BC is actually [XC,YC,ZC]-[XB,YB,ZB] C B Origin Protein Conformation Prediction (Part I)

18. And what is ||BC||? • Magnitude of BC X’ C Z’ Y’ B A Protein Conformation Prediction (Part I)

19. New vector space • First row of transformation matrix X C Z B Y A New X Protein Conformation Prediction (Part I)

20. Second Row • AB in plane of new Y • so Z component is zero X C Z B Y Important piece: Y component A Protein Conformation Prediction (Part I)

21. New vector space • Second row of transformation matrix X C Z B Y A New Y Protein Conformation Prediction (Part I)

22. New vector space • Third row of transformation matrix easy once have first two: Cross Product X C Z B Y A New Y Protein Conformation Prediction (Part I)

23. The next point: it’s all trig • Know distance to next atom • Know angle is 70.5° (180-109.5) • X component = ||CD|| cos(70.5°) • Y component starts out at ||CD|| sin(70.5°) • This is the distance from X to the new D X D C Z Y B A Protein Conformation Prediction (Part I)

24. Final torsional rotation Y • Z component is that distance times sinθ(torsion angle) • Y = ||CD|| sin(70.5°)*cos θ • Z = ||CD|| sin(70.5°)*sin θ Dnewin plane of xy 70.5° Z C X Dfinal Θ (torsional angle) C Dnewin plane of xy Y Protein Conformation Prediction (Part I)

25. Going from xyz to angles • Transform next xyz into new vector space coordinates (same as before • Determine ||CD|| X D C Z Y B A Protein Conformation Prediction (Part I)

26. An example • XYZ coordinates for an amino acid • Build the linear transform matrix used to transform the original vector space into the space defined by the three atoms above. Protein Conformation Prediction (Part I)

27. Example • BC? X Calculator makes life easier: [2.863,-15.219,-0.703] sto A [3.920,-14.209,-0.705] sto B [5.265,-14.836,-1.065] sto C unitV (C-B) unitV under “VECTR / MATH” [XC,YC,ZC]-[XB,YB,ZB] [5.265 -14.836 -1.065]-[3.920 -14.209 -0.705] [1.345 -0.627 -0.36] Magnitude of BC? C Z distance B to C: 1.527 B New X axis: [0.880 -0.410 -0.236] Y A Protein Conformation Prediction (Part I)

28. Example Calculator A-C sto A B-C sto B C-C sto C B-Asto AB C-Bsto BC unitV BC (same answer) unitV under “VECTR / MATH” • Actually forgot a step • Need to translate all three points • Move in direction of negative C • Will place C and origin and keep A and B relative to C X C Z B Y A No change to X Protein Conformation Prediction (Part I)

29. Example • New Y? X Calculator unitV(AB-(dot(AB,BC)/(norm BC)2 * BC)) Norm under “VECTR / MATH” C Z B Y A New Y axis: [0.440 0.894 0.088] Protein Conformation Prediction (Part I)

30. Example Calculator unitV BC entersto X unitV(AB-(dot(AB,BC)/(norm BC)2 * BC))entersto Y cross(X,Y) Cross under “VECTR / MATH” • New Z? X C Z B Y A New Z axis: [0.174 -0.181 0.968] Protein Conformation Prediction (Part I)

31. Two approaches • De novo • From first principles • Comparative/Homology Based • Sequence similarity Protein Conformation Prediction (Part I)

32. Protein Conformation Prediction (Part I)