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)