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Geometric and Kinematic Models of Proteins. Study of movement independent of the forces that cause them. What is Kinematics?. 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.

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protein
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
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
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
Geometric Models of Bio-Molecules
  • Hard-sphere model (van der Waals radii)
    • Van der Waals surface
van der waals potential
Van der Waals Potential

Van der Waals interactions between twoatoms result from induced polarization

effect (formation of electric dipoles). Theyare weak, except at close range.

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

12-6 Lennard-Jones potential

geometric models of bio molecules1
Geometric Models of Bio-Molecules
  • Hard-sphere model (van der Waals radii)
    • Van der Waals surface

Van der Waals radii in Å

geometric models of bio molecules2
Geometric Models of Bio-Molecules
  • Hard-sphere model (van der Waals radii)
    • Van der Waals surface
    • Solvent- accessible surface
    • Molecular surface
computed molecular surfaces
Computed Molecular Surfaces

Probe of 1.4Å

Probe of 5Å

computation of hard sphere surface grid method halperin and shelton 97
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 971
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 removalComputational Geometry: Theory and Applications 11 (2), 1998, 83-102.

trapezoidal decomposition1
Trapezoidal Decomposition

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

slide16

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
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
Visualization Models
  • Stick (bond) model
visualization models2
Visualization Models
  • Stick (bond) model
  • Small-sphere model
kinematic models of bio molecules

(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

peptide bonds make proteins into long kinematic chains
Peptide bonds make proteins into long kinematic chains

The atomistic model does not encode this kinematic structure( algorithms must maintain appropriate bond lengths)

kinematic models of bio molecules1
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 byinternalcoordinates (bond lengths and angles, and torsional angles around bonds)
issues with linkage model
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)
2 d case6

y

y

Rotation matrix:

cos q -sin qsin qcos q

j

i

q

ty

tx

x

x

2-D Case
2 d case7

y

y

Rotation matrix:

i1 j1i2j2

j

i

q

ty

tx

x

x

2-D Case
2 d case8

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?

homogeneous coordinate matrix

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
homogeneous coordinate matrix in 3 d

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
example

z

y

x

Example

cos q 0 sinq tx

0 1 0 ty

-sin q 0 cos q tz

0 0 0 1

q

rotation matrix

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
homogeneous coordinate matrix in 3 d1

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

questions
Questions?

What is the potential problem with homogeneous coordinate matrix?

building a serial linkage model
Building a Serial Linkage Model
  • Rigid bodies are:
  • atoms (spheres), or
  • groups of atoms
building a serial linkage model1
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
building a serial linkage model2
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
building a serial linkage model3
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
bond length1

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
bond angle1

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
torsional dihedral angle

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
transform t i 1

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

transform t i 11

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

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

Ti+1

z

x

t

i-1

d

i

b

i-2

relative position of two atoms
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

update
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)
tree shaped linkage

Why?

Tree-Shaped Linkage

Root group of 3 atoms

p atoms  3p -6 parameters

tree shaped linkage1

T0

world coordinate system

Tree-Shaped Linkage

Root group of 3 atoms

p atoms  3p -6 parameters

simplified linkage model
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

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

convention for f y angles

C

C

N

Ca

f=0

Convention for f-y Angles
  • fis 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 fory

C

Ca

N

C

f=p

ramachandran maps
Ramachandran Maps

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

ψ

φ

f y c linkage model of protein
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
kinematic models of bio molecules2
Kinematic Models of 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 byinternal 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
In linkage model a small local change may have big global effect

 Computational errors may accumulate

drawback of homogeneous coordinate matrix

x’ i1 j1 k1 tx x

y’ i2 j2 k2 ty y

z’ i3 j3 k3 tz z

1 0 0 0 1 1

=

Drawback of Homogeneous Coordinate Matrix
  • 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 unitquaternion
unit quaternion

R(r,q)

R(r,q+2p)

Unit Quaternion

R(r,q) = (cosq/2, r1sin q/2, r2sinq/2, r3sinq/2)

=cosq/2 + rsin q/2

Space of unit quaternions: Unit 3-sphere in 4-D space with antipodal points identified

operations on quaternions
Operations on Quaternions

P =p0+ p

Q =q0+ q

Product R = r0 + r = PQ

r0 = p0q0 – p.q(“.” denotes inner product)

r = p0q + q0p + pq (“” denotes outer product)

Conjugate of P:P* = p0-p

transformation of a point
Transformation 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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