Geometric and Kinematic Geometric and Kinematic Models of ProteinsModels 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 8th Oct 2007
Kinematic Models of Bio-Molecules
Atomistic model: The position of each atom is defined by its coordinates in 3-D space
(x4,y4,z4)
(x2,y2,z2)(x3,y3,z3)
(x5,y5,z5)
(x6,y6,z6)
(x8,y8,z8)(x7,y7,z7)
(x1,y1,z1)
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)
NN
NN
C’
C’
C’
C’
O
O O
O
Cα
Cα
Cα
Cα
Cβ
Cβ Cβ
Cβ
φ
ψ φ
ψ φ
ψ φ
ψ
Resi Resi+1 Resi+2 Resi+3
Protein Features
ACEDFHIKNMFACEDFHIKNMFSDQWWIPANMCSDQWWIPANMCASDFDPQWEREASDFDPQWERELIQNMDKQERTLIQNMDKQERTQATRPQDS...QATRPQDS...
Sequence View Structure View
Where To Go**
http://www.expasy.org/tools/
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?
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 proximitybut the reverse is not true)
Rigid-Body Transform
x
z
y
x
T
T(x)
2-D Case
x
y
2-D Case2-D Case
x
y
x
y
2-D Case2-D Case
x
y
x
y
2-D Case2-D Case
x
y
x
y
2-D Case2-D Case
x
y
x
y
2-D Case2-D Case
x
y
x
y
x
y
2-D Case2-D Case
x
y
tx
ty
cos -sin sin cos
Rotation matrix:
ij
x
y
2-D Case2-D Case
x
y
tx
ty
i1 j1i2 j2
Rotation matrix:
ij
x
y
2-D Case2-D Case
x
y
tx
ty
a
b
ab
v
a’b’ =
α
α
a’
b’
i1 j1i2 j2
Rotation matrix:
ij
Transform of a point?
Homogeneous Coordinate MatrixHomogeneous Coordinate Matrix
i1 j1 tx
i2 j2 ty
0 0 1
x’ cos -sin tx x tx + x cos – y sin y’ = sin cos ty y = ty + x sin + y cos 1 0 0 1 1 1
x
y
x
y
tx
ty
x’
y’
y
x
T = (t,R) T(x) = t + Rx
3-D Case3-D Case
1
2
?
Homogeneous Coordinate Homogeneous Coordinate Matrix in 3-DMatrix in 3-D
i1 j1 k1 tx
i2 j2 k2 ty
i3 j3 k3 tz
0 0 0 1
with: – i12 + i22 + i32 = 1– i1j1 + i2j2 + i3j3 = 0– det(R) = +1– R-1 = RT
x
z
y xy
z ji
k
R
ExampleExample
x
z
y
cos 0 sin tx
0 1 0 ty
-sin 0 cos tz
0 0 0 1
Rotation Matrix
R(k,) =
kxkxv+ c kxkyv- kzs kxkzv+ kys
kxkyv+ kzs kykyv+ c kykzv- kxs
kxkzv- kys kykzv+ kxs kzkzv+ c
where:
• k = (kx ky kz)T
• s = sin• c = cos• v = 1-cos
k
Homogeneous Coordinate Matrix in 3-D
x
z
y xy
z ji
k
x’ i1 j1 k1 tx xy’ i2 j2 k2 ty yz’ i3 j3 k3 tz z1 0 0 0 1 1
=(x,y,z)
(x’,y’,z’)
Composition of two transforms represented by matrices T1 and T2 : T2 T1
Building a Serial Linkage ModelBuilding a Serial Linkage Model
Rigid bodies are:• atoms (spheres), or• groups of atoms
Building a Serial Linkage ModelBuilding a Serial Linkage Model
1. Build the assembly of the first 3 atoms:a. Place 1st atom anywhere in spaceb. Place 2nd atom anywhere at bond length
Bond LengthBond Length
Building a Serial Linkage ModelBuilding a Serial Linkage Model
1. Build the assembly of the first 3 atoms:a. Place 1st atom anywhere in spaceb. Place 2nd atom anywhere at bond lengthc. Place 3rd atom anywhere at bond length with bond angle
Bond angleBond angle
Coordinate FrameCoordinate Frame
z
x
y
Atom: -2
-1
0
Building a Serial Linkage ModelBuilding a Serial Linkage Model
1. Build the assembly of the first 3 atoms:a. Place 1st atom anywhere in spaceb. Place 2nd atom anywhere at bond lengthc. Place 3rd atom anywhere at bond length with
bond angle2. Introduce each additional atom in the sequence
one at a time
1 0 0 0 cβ -sβ 0 0 1 0 0 d
0 c -s 0 sβ cβ 0 0 0 1 000 s c 0 0 0 1 0 0 0 100 0 0 1 0 0 0 1 0 0 0 1
Ti+1 =
Bond LengthBond Length
z
x
y
-2
-1 10
1 0 0 0 cβ -sβ 0 0 1 0 0 d
0 c -s 0 sβ cβ 0 0 0 1 000 s c 0 0 0 1 0 0 0 100 0 0 1 0 0 0 1 0 0 0 1
Ti+1 =
Bond angleBond angle
z
x
y
Torsional (Dihedral) angle
z
x
y
1 0 0 0 cβ -sβ 0 0 1 0 0 d
0 c -s 0 sβ cβ 0 0 0 1 000 s c 0 0 0 1 0 0 0 100 0 0 1 0 0 0 1 0 0 0 1
Ti+1 =
Transform Ti+1
β
i-2
i-1
i
i+1Ti+1
d
1 0 0 0 cβ -sβ 0 0 1 0 0 d
0 c -s 0 sβ cβ 0 0 0 1 000 s c 0 0 0 1 0 0 0 100 0 0 1 0 0 0 1 0 0 0 1
Ti+1 =
z
x
y
x
y
z
Transform TTransform Ti+1i+1
β
i-2
i-1
i
i+1Ti+1
d
z
x
y
x
y
z
1 0 0 0 cβ -sβ 0 0 1 0 0 d
0 c -s 0 sβ cβ 0 0 0 1 000 s c 0 0 0 1 0 0 0 100 0 0 1 0 0 0 1 0 0 0 1
Ti+1 =
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
Serial Linkage ModelSerial Linkage Model
-1
1
-2
0
T1
T2
Relative Position of Two AtomsRelative Position of Two Atoms
i
k
Tk(i) = Tk … Ti+2 Ti+1 position of atom k
in frame of atom i
Ti+1 Tki+1
k-1Ti+2
UpdateUpdate
Tk(i) = Tk … Ti+2 Ti+1
Atom j between i and k Tk
(i) = Tj(i) Tj+1 Tk
(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)
Tree-Shaped LinkageTree-Shaped Linkage
Root group of 3 atoms
p atoms 3p 6 parameters
Why?
Tree-Shaped LinkageTree-Shaped Linkage
Root group of 3 atoms
p atoms 3p 6 parameters
world coordinate system
T0
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 Bond Lengths and Angles in a Proteinin a Protein
: Cα Cα: C Cψ: N N
=
3.8Å
C
CαN
C
ψ Linkage Model
peptide group
side-chain group
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
CαN
C
C
CαN
C
Ramachandran Maps
They assign probabilities to φ-ψ pairs based on frequencies in known folded structures
φ
ψ
The sequence of N-Cα-C-… atoms is the backbone (or main chain)
Rotatable bonds along the backbone define the -ψ torsional degrees of freedom
Small side-chains with degree of freedom
Cα
Cβ
--ψψ-- Linkage Model of Linkage Model of ProteinProtein
Side Chains with Multiple Torsional Degrees of Freedom
( angles)
0 to 4 angles: 1, ..., 4
Kinematic Models Kinematic Models of Bio-Moleculesof Bio-Molecules
Atomistic model: The position of each atom is defined by its coordinates in 3-D spaceDrawback: 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 In linkage model a small local change may have big global effecthave big global effect
Computational errors may accumulate
Drawback of Homogeneous Coordinate Matrix
x’ i1 j1 k1 tx xy’ i2 j2 k2 ty yz’ i3 j3 k3 tz z1 0 0 0 1 1
=
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 QuaternionUnit Quaternion
R(r,) = (cos /2, r1 sin /2, r2 sin /2, r3 sin /2)
= cos /2 + r sin /2
R(r,)
R(r,+2)
Space of unit quaternions:Unit 3-sphere in 4-D spacewith antipodal points identified
Operations on QuaternionsOperations on Quaternions
P = p0 + p
Q = q0 + q
Product R = r0 + r = PQ
r0 = p0q0 – p.q (“.” denotes inner product)
r = p0q + q0p + pq (“” denotes outer product)
Conjugate of P:P* = p0 - p
Transformation of a PointTransformation of a Point
Point x = (x,y,z) quaternion 0 + x
Transform of translation t = (tx,ty,tz) and rotation (n,q)
Transform of x is x’
0 + x’ = R(n,q) (0 + x) R*(n,q) + (0 + t)
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