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Kinematic and Geometric Models of Proteins: Insights from a Multisite Course

This lecture delves into the geometric and kinematic models of proteins, emphasizing their atomistic structures in 3D space. It covers fundamental principles such as peptide bonding and kinematic chains, vital for understanding protein configurations. The atomistic model's limitations and the necessity for maintaining appropriate bond lengths in kinematic structures are discussed. Key features like molecular weight, amino acid frequency, and folding energy are also analyzed. This session, part of an ongoing course taught across various prestigious institutions, aims to deepen the understanding of protein dynamics in biochemistry.

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Kinematic and Geometric Models of Proteins: Insights from a Multisite Course

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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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