Geometric and Kinematic Models of Proteins 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. LECT_4 8 th Oct 2007

Kinematic Models of Bio-Molecules  Atomistic model: The position of each atom is defined by its coordinates in 3-D space (x 4,y 4,z 4 ) (x 2,y 2,z 2 ) (x 3,y 3,z 3 ) (x 5,y 5,z 5 ) (x 6,y 6,z 6 ) (x 8,y 8,z 8 ) (x 7,y 7,z 7 ) (x 1,y 1,z 1 ) p atoms  3p parameters Drawback: The bond structure is not taken into account

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

Protein Features ACEDFHIKNMFSDQWWIPANMCASDFDPQWERELIQNMDKQERTQATRPQDS... Sequence ViewStructure View

Where To Go**

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

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)

Linkage Model T?

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 proximity but the reverse is not true)

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

2-D Case x y

x y x y

x y x y

x y x y

x y x y

x y x y

x y x y txtx tyty  cos  -sin  sin  cos  Rotation matrix: ij

x y 2-D Case x y txtx tyty  i 1 j 1 i 2 j 2 Rotation matrix: ij

x y 2-D Case x y txtx tyty  a b abab v a’ b’ =    a’ b’ i 1 j 1 i 2 j 2 Rotation matrix: ij Transform of a point?

Homogeneous Coordinate Matrix i 1 j 1 t x i 2 j 2 t y 001 x’ cos  -sin  t x x t x + x cos  – y sin  y’ = sin  cos  t y y = t y + x sin  + y cos  x y x y txtx tyty  x’ y’ y x  T = (t,R)  T(x) = t + Rx

3-D Case 11 22 ?

Homogeneous Coordinate Matrix in 3-D i 1 j 1 k 1 t x i 2 j 2 k 2 t y i 3 j 3 k 3 t z 0001 with: –i i i 3 2 = 1 –i 1 j 1 + i 2 j 2 + i 3 j 3 = 0 –det(R) = +1 –R -1 = R T x z y x y z j i k R

Example x z y  cos  0sin  t x 010t y -sin  0cos  t z 0001

Rotation Matrix R(k,  ) = k x k x v  + c  k x k y v  - k z s  k x k z v  + k y s  k x k y v  + k z s  k y k y v  + c  k y k z v  - k x s  k x k z v  - k y s  k y k z v  + k x s  k z k z v  + c  where: k = ( k x k y k z ) T s  = sin  c  = cos  v  = 1-cos  k 

Homogeneous Coordinate Matrix in 3-D x z y x y z j i k x’i 1 j 1 k 1 t x x y’i 2 j 2 k 2 t y y z’i 3 j 3 k 3 t z z = (x,y,z) (x’,y’,z’) Composition of two transforms represented by matrices T 1 and T 2 :T 2  T 1

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

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

Bond Length

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

Bond angle

Coordinate Frame z x y Atom: -2 0

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

1000c-s00100d0c-s0sc sc c-s00100d0c-s0sc sc T i+1 = Bond Length z x y

1000c-s00100d0c-s0sc sc c-s00100d0c-s0sc sc T i+1 = Bond angle z x y

Torsional (Dihedral) angle z x y 1000c-s00100d0c-s0sc sc c-s00100d0c-s0sc sc T i+1 =

Transform T i+1   i-2 i-1 i i+1 T i+1 d 1000c-s00100d0c-s0sc sc c-s00100d0c-s0sc sc T i+1 = z x y x y z

Transform T i+1 Transform T i+1   i-2 i-1 i i+1 T i+1 d z x y x y z 1000c-s00100d0c-s0sc sc c-s00100d0c-s0sc sc T i+1 =

Readings: J.J. Craig. Introduction to Robotics. Addison Wesley, reading, MA, 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, comp-mole-conform.pdf comp-mole-conform.pdf

Serial Linkage Model T1T1 T2T2

Relative Position of Two Atoms i k T k (i) = T k … T i+2 T i+1  position of atom k in frame of atom i T i+1 TkTk i+1 k-1 T i+2

Update  T k (i) = T k … T i+2 T i+1  Atom j between i and k  T k (i) = T j (i) T j+1 T k (j+1)  A parameter between j and j+1 is changed  T j+1  T j+1  T k (i)  T k (i) = T j (i) T j+1 T k (j+1)

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

Tree-Shaped Linkage Root group of 3 atoms p atoms  3p  6 parameters world coordinate system T0T0

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

Bond Lengths and Angles in a Protein  : C   C   : C  C  : N  N  =  3.8Å  C CC N C 

 Linkage Model peptide group side-chain group

Convention for f-y Angles  f is defined as the dihedral angle composed of atoms C i-1 –N i –Ca i –C i  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 CC N C  C CC N C 

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

 The sequence of N-C  -C-… atoms is the backbone (or main chain)  Rotatable bonds along the backbone define the  -  torsional degrees of freedom  Small side-chains with  degree of freedom     CC CC   -  -  Linkage Model of Protein

Side Chains with Multiple Torsional Degrees of Freedom (  angles) 0 to 4  angles:  1,...,  4

Kinematic Models of Bio-Molecules  Atomistic model: The position of each atom is defined by its coordinates in 3-D space Drawback: Fixed bond lengths/angles are encoded as additional constraints. More parameters  Linkage model: The kinematics is defined by internal parameters (bond lengths and angles, and torsional angles around bonds) Drawback: Small local changes may have big global effects. Errors accumulate. Forces are more difficult to express  Simplified (f-y-c) linkage model: Fixed bond lengths, bond angles and torsional angles are directly embedded in the representation. Drawback: Fine tuning is difficult

In linkage model a small local change may have big global effect  Computational errors may accumulate

Drawback of Homogeneous Coordinate Matrix x’i1j1k1txx y’i2j2k2tyy z’i3j3k3tzz = Too many rotation parameters  Accumulation of computing errors along a protein backbone and repeated computation  Non-redundant 3-parameter representations of rotations have many problems: singularities, no simple algebra  A useful, less redundant representation of rotation is the unit quaternion

Unit Quaternion R(r,  ) = ( cos  /2, r 1 sin  /2, r 2 sin  /2, r 3 sin  /2 ) = cos  /2 + r sin  /2 R(r,  ) R(r,  +2  ) Space of unit quaternions: Unit 3-sphere in 4-D space with antipodal points identified

Operations on Quaternions P = p 0 + p Q = q 0 + q Product R = r 0 + r = PQ r 0 = p 0 q 0 – p.q(“.” denotes inner product) r = p 0 q + q 0 p + p  q(“  ” denotes outer product) Conjugate of P: P * = p 0 - p

Transformation of a Point Point x = (x,y,z)  quaternion 0 + x Transform of translation t = (t x,t y,t z ) and rotation (n,q) Transform of x is x’ 0 + x’ = R(n,q) (0 + x) R * (n,q) + (0 + t)