Geometric and Kinematic Geometric and Kinematic Models of ProteinsModels of Proteins

What is Kinematics?

Study of movement independent of the forces that cause them

Protein Long sequence of amino-acids (dozens to thousands), also

called residues from a dictionary of 20 amino-acids

Role of Geometric and Kinematic Models

Represent the possible shapes of a protein (compare/classify shapes, find motifs)

Answer proximity queries: Which atoms are close to a given atom? (computation of energy)

Compute surface area (interaction with solvent)

Find shape features, e.g., cavities (ligand-protein interaction)

What are the issues? Large number of atoms

Combinatorial problems Large number of degrees of freedom

Large-dimensional conformation space Need to efficiently update information during

simulation (surface area, proximity among atoms):• What is the position of every atom in some given

coordinate system?• Which atoms intersect a given atom?• What atoms are within some distance range from another

one?

Complex metric in conformational space Many shape matching issues

Geometric Models of Bio-Molecules

Hard-sphere model (van der Waals radii) Van der Waals surface

Van der Waals Potential

12-6 Lennard-Jones potential

The van der Waals force is the force to which the gecko's unique ability to cling to smooth surfaces is attributed!

Van der Waals interactions between twoatoms result from induced polarization effect (formation of electric dipoles). Theyare weak, except at close range.

Geometric Models of Bio-Molecules

Hard-sphere model (van der Waals radii) Van der Waals surface

H C N O F P S Cl

1.2

1.7

1.5

1.4

1.35

1.9

1.85

1.8

Van der Waals radii in Å

Geometric Models of Bio-Molecules

Hard-sphere model (van der Waals radii) Van der Waals surface Solvent- accessible surface Molecular surface

Computed Molecular Surfaces

Probe of 1.4Å Probe of 5Å

Is it art?

Computation of Hard-Sphere Surface

(Grid method [Halperin and Shelton, 97])

Each sphere intersects O(1) spheres

Computing each atom’s contribution to molecular surface takes O(1) time

Computation of molecular surface takes Θ(n) time

Why?

Computation of Hard-Sphere Surface

(Grid method [Halperin and Shelton, 97])

Each sphere intersects O(1) spheres

Computing each atom’s contribution to molecular surface takes O(1) time

Computation of molecular surface takes Θ(n) time

Why?

D. Halperin and M.H. Overmars Spheres, molecules, and hidden surface removal Computational Geometry: Theory and Applications 11 (2), 1998, 83-102.

Trapezoidal Trapezoidal DecompositionDecomposition

Trapezoidal Trapezoidal DecompositionDecomposition

D. Halperin and C.R. Shelton A perturbation scheme for spherical arrangements with application to molecular modeling Computational Geometry: Theory and Applications 10 (4), 1998, 273-288.

Possible project: Design software to update surface area during molecule motion

Other approach: Alpha shapes http://biogeometry.duke.edu/software/alphashapes/pubs.html

Simplified Geometric Models

United-atom model: non-polar H atoms are incorporated into the heavy atoms to which they are bonded

Lollipop model: the side-chains are approximated as single spheres with varying radii

Bead model: Each residue is modeled as a single sphere

Visualization Models

Stick (bond) model

Visualization Models

Visualization Models

Stick (bond) model

Small-sphere model

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

C

C

C

C

C

C C

C

Resi Resi+1 Resi+2 Resi+3

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’ =

a’

b’

i1 j1i2 j2

Rotation matrix:

ij

Transform of a point?

Homogeneous Coordinate Homogeneous Coordinate MatrixMatrix

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

Questions?

What is the potential problem with homogeneous coordinate matrix?

Building a Serial Linkage Building a Serial Linkage ModelModel

Rigid bodies are:• atoms (spheres), or• groups of atoms

Building a Serial Linkage Building a Serial Linkage ModelModel

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 Building a Serial Linkage ModelModel

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

Building a Serial Linkage Building a Serial Linkage ModelModel

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

2. Introduce each additional atom in the sequence one at a time

1 0 0 0 c -s 0 0 1 0 0 d

0 c -s 0 s c 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

1 0 0 0 c -s 0 0 1 0 0 d

0 c -s 0 s c 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 c -s 0 0 1 0 0 d

0 c -s 0 s c 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

i-2

i-1

i

i+1Ti+1

d

1 0 0 0 c -s 0 0 1 0 0 d

0 c -s 0 s c 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

i-2

i-1

i

i+1Ti+1

d

z

x

y

x

y

z

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

1 0 0 0 c -s 0 0 1 0 0 d

0 c -s 0 s c 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 =

Serial Linkage ModelSerial Linkage Model

-1

1

-2

0

T1

T2

Relative Position of Two Relative Position of Two AtomsAtoms

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 ModelIn 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: 3p6 p3

Bond Lengths and Angles Bond Lengths and Angles in a Proteinin a Protein

: C C: C C: N N

=

3.8Å

C

CN

C

Linkage Model

peptide group

side-chain group

Convention for Angles

is defined as the dihedral angle composed of atoms Ci-1–Ni–Ci–Ci

If all atoms are coplanar:

Sign of : Use right-hand rule. With right thumb pointing along central bond (N-C), a rotation along curled fingers is positive

Same convention for

C

CN

C

C

CN

C

Ramachandran MapsThey 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

---- 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 (--) 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 In linkage model a small local change may have big global change may have big global

effecteffect

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 Operations on QuaternionsQuaternions

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 PointPoint x = (x,y,z) quaternion 0 + x

Transform of translation t = (tx,ty,tz) and rotation (n,)

Transform of x is x’

0 + x’ = R(n,) (0 + x) R*(n,) + (0 + t)