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Geometric and Kinematic Models of Proteins

Geometric and Kinematic Models of Proteins. LECT_4 8 th Oct 2007. From a course taught firstly in Stanford by JC Latombe, then in Singapore by Sung Wing Kin, and now in Rome by AG…web solidarity. With excerpta from a course by D. Wishart. (x 4 ,y 4 ,z 4 ). (x 5 ,y 5 ,z 5 ).

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Geometric and Kinematic Models of Proteins

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  1. Geometric and Kinematic Models of Proteins LECT_4 8th Oct 2007 From a course taught firstly in Stanford by JC Latombe, then in Singapore by Sung Wing Kin, and now in Rome by AG…web solidarity. With excerpta from a course by D. Wishart.

  2. (x4,y4,z4) (x5,y5,z5) (x6,y6,z6) (x8,y8,z8) (x7,y7,z7) (x1,y1,z1) Kinematic Models of Bio-Molecules • Atomistic model: The position of each atom is defined by its coordinates in 3-D space (x3,y3,z3) (x2,y2,z2) p atoms  3p parameters Drawback: The bond structure is not taken into account

  3. Peptide bonds make proteins into long kinematic chains The atomistic model does not encode this kinematic structure ( algorithms must maintain appropriate bond lengths)

  4. Protein Features ACEDFHIKNMF SDQWWIPANMC ASDFDPQWERE LIQNMDKQERT QATRPQDS... Sequence View Structure View

  5. Where To Go** http://www.expasy.org/tools/

  6. Compositional Features • Molecular Weight • Amino Acid Frequency • Isoelectric Point • UV Absorptivity • Solubility, Size, Shape • Radius of Gyration • Free Energy of Folding

  7. Kinematic Models of Bio-Molecules • Atomistic model: The position of each atom is defined by its coordinates in 3-D space • Linkage model: The kinematics is defined by internal coordinates (bond lengths and angles, and torsional angles around bonds)

  8. T? T? Linkage Model

  9. Issues with Linkage Model • Update the position of each atom in world coordinate system • Determine which pairs of atoms are within some given distance(topological proximity along chain  spatial proximitybut the reverse is not true)

  10. z T(x) y T x x Rigid-Body Transform

  11. y x 2-D Case

  12. y y x x 2-D Case

  13. y y x x 2-D Case

  14. y y x x 2-D Case

  15. y y x x 2-D Case

  16. y y x x 2-D Case

  17. y y Rotation matrix: cos q -sin qsin qcos q j i q ty tx x x 2-D Case

  18. y y Rotation matrix: i1 j1i2j2 j i q ty tx x x 2-D Case

  19. y y Rotation matrix: a i1 j1i2j2 a b j i = b’ q ty a’ b’ b q a tx a a’ x x 2-D Case v Transform of a point?

  20. y y y’ q y ty x’cos q -sin qtxx tx + x cosq – y sin q y’ = sin q cos qtyy = ty + x sin q + y cos q 1 0 0 1 1 1 x x’ tx x x Homogeneous Coordinate Matrix i1 j1txi2 j2ty 0 0 1 • T = (t,R) • T(x) = t + Rx

  21. ? q2 q1 3-D Case

  22. R z y x y i z j k x Homogeneous Coordinate Matrix in 3-D i1 j1 k1txi2 j2 k2tyi3 j3 k3tz 0 0 0 1 with: • i12 + i22 + i32 = 1 • i1j1 + i2j2 + i3j3 = 0 • det(R) = +1 • R-1 = RT

  23. z y x Example cos q 0 sinq tx 0 1 0 ty -sin q 0 cos q tz 0 0 0 1 q

  24. k q Rotation Matrix R(k,q)= kxkxvq+ cqkxkyvq- kzsqkxkzvq+ kysq kxkyvq+ kzsqkykyvq+ cqkykzvq- kxsq kxkzvq- kysqkykzvq+ kxsqkzkzvq+ cq where: • k = (kx ky kz)T • sq = sinq • cq = cosq • vq = 1-cosq

  25. z y x y i z j k x x’ i1 j1 k1 txx y’ i2 j2 k2 tyy z’ i3 j3 k3 tzz 1 0 0 0 1 1 = Homogeneous Coordinate Matrix in 3-D (x,y,z) (x’,y’,z’) Composition of two transforms represented by matrices T1 and T2 : T2T1

  26. Building a Serial Linkage Model • Rigid bodies are: • atoms (spheres), or • groups of atoms

  27. Building a Serial Linkage Model • Build the assembly of the first 3 atoms: • Place 1st atom anywhere in space • Place 2nd atom anywhere at bond length

  28. Bond Length

  29. Building a Serial Linkage Model • Build the assembly of the first 3 atoms: • Place 1st atom anywhere in space • Place 2nd atom anywhere at bond length • Place 3rd atom anywhere at bond length with bond angle

  30. Bond angle

  31. z x y Coordinate Frame -1 0 Atom: -2

  32. Building a Serial Linkage Model • Build the assembly of the first 3 atoms: • Place 1st atom anywhere in space • Place 2nd atom anywhere at bond length • Place 3rd atom anywhere at bond length with bond angle • Introduce each additional atom in the sequence one at a time

  33. z x y 1 0 0 0cb -sb0 0 1 0 0 d 0 ct -st 0 sbcb 0 0 0 100 0 st ct 0 0 0 1 0 0 010 0 0 0 1 0 0 0 1 0 0 0 1 Ti+1 = Bond Length -1 1 0 -2

  34. z x y 1 0 0 0cb -sb0 0 1 0 0 d 0 ct -st 0 sbcb 0 0 0 100 0 st ct 0 0 0 1 0 0 010 0 0 0 1 0 0 0 1 0 0 0 1 Ti+1 = Bond angle

  35. z x y 1 0 0 0cb -sb0 0 1 0 0 d 0 ct -st 0 sbcb 0 0 0 100 0 st ct 0 0 0 1 0 0 010 0 0 0 1 0 0 0 1 0 0 0 1 Ti+1 = Torsional (Dihedral) angle

  36. z x y 1 0 0 0cb -sb0 0 1 0 0 d 0 ct -st 0 sbcb 0 0 0 100 0 st ct 0 0 0 1 0 0 010 0 0 0 1 0 0 0 1 0 0 0 1 Ti+1 = Transform Ti+1 y i+1 Ti+1 z x t i-1 d i b i-2

  37. z x y 1 0 0 0cb -sb0 0 1 0 0 d 0 ct -st 0 sbcb 0 0 0 100 0 st ct 0 0 0 1 0 0 010 0 0 0 1 0 0 0 1 0 0 0 1 Ti+1 = Transform Ti+1 y i+1 Ti+1 z x t i-1 d i b i-2

  38. Readings: J.J. Craig. Introduction to Robotics. Addison Wesley, reading, MA, 1989. Zhang, M. and Kavraki, L. E.. A New Method for Fast and Accurate Derivation of Molecular Conformations. Journal of Chemical Information and Computer Sciences, 42(1):64–70, 2002.http://www.cs.rice.edu/CS/Robotics/papers/zhang2002fast-comp-mole-conform.pdf

  39. Serial Linkage Model T1 0 1 T2 -1 -2

  40. Relative Position of Two Atoms Ti+2 k-1 Ti+1 i+1 Tk k i Tk(i) = Tk …Ti+2 Ti+1 position of atom k in frame of atom i

  41. Update • Tk(i) = Tk…Ti+2 Ti+1 • Atom j between i and k • Tk(i) = Tj(i)Tj+1Tk(j+1) • A parameter between j and j+1 is changed • Tj+1 Tj+1 • Tk(i)  Tk(i) = Tj(i)Tj+1 Tk(j+1)

  42. Why? Tree-Shaped Linkage Root group of 3 atoms p atoms  3p -6 parameters

  43. T0 world coordinate system Tree-Shaped Linkage Root group of 3 atoms p atoms  3p -6 parameters

  44. Simplified Linkage Model In physiological conditions: • Bond lengths are assumed constant [depend on “type” of bond, e.g., single: C-C or double C=C; vary from 1.0 Å (C-H) to 1.5 Å (C-C)] • Bond angles are assumed constant[~120dg] • Only some torsional (dihedral) angles may vary • Fewer parameters: 3p-6   p-3

  45. f C C N Ca 3.8Å Bond Lengths and Angles in a Protein w: Ca Ca f: C  C y: N  N w = p w

  46. peptide group side-chain group f-y Linkage Model

  47. C C N Ca f=0 Convention for f-y Angles • f is defined as the dihedral angle composed of atoms Ci-1–Ni–Cai–Ci • If all atoms are coplanar: • Sign of f: Use right-hand rule. With right thumb pointing along central bond (N-Ca), a rotation along curled fingers is positive • Same convention for y C Ca N C f=p

  48. Ramachandran Maps They assign probabilities to φ-ψ pairs based on frequencies in known folded structures ψ φ

  49. The sequence of N-Ca-C-… atoms is the backbone (or main chain) Rotatable bonds along the backbone define the f-y torsional degrees of freedom Small side-chains with c degree of freedom c c c c c Cb Ca f-y-c Linkage Model of Protein

  50. Side Chains with Multiple Torsional Degrees of Freedom (c angles) 0 to 4 c angles: c1, ..., c4

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