Geometric and kinematic models of proteins
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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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Geometric and Kinematic Models of Proteins

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Geometric and kinematic models of proteins

Geometric and Kinematic Models of Proteins


What is kinematics

Study of movement independent of the forces that cause them

What is Kinematics?


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


Is it art

Is it art?


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 decomposition

Trapezoidal Decomposition


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.


Geometric and kinematic models of proteins

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 models1

Visualization Models


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)


Linkage model

T?

T?

Linkage Model


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)


Rigid body transform

z

T(x)

y

T

x

x

Rigid-Body Transform


2 d case

y

x

2-D Case


2 d case1

y

y

x

x

2-D Case


2 d case2

y

y

x

x

2-D Case


2 d case3

y

y

x

x

2-D Case


2 d case4

y

y

x

x

2-D Case


2 d case5

y

y

x

x

2-D Case


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 001

  • T = (t,R)

  • T(x) = t + Rx


3 d case

?

q2

q1

3-D Case


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 k3tz0001

with:

  • i12 + i22 + i32 = 1

  • i1j1 + i2j2 + i3j3 = 0

  • det(R) = +1

  • R-1 = RT


Example

z

y

x

Example

cos q0sinqtx

010ty

-sin q0cos qtz

0001

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’i1j1k1txx

y’i2j2k2tyy

z’i3j3k3tzz

100011

=

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


Bond length

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


Bond angle

Bond angle


Coordinate frame

z

x

y

Coordinate Frame


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

1000cb-sb00100d

0ct-st0sbcb000100

0stct000100010

000100010001

Ti+1 =

Bond Length


Bond angle1

z

x

y

1000cb-sb00100d

0ct-st0sbcb000100

0stct000100010

000100010001

Ti+1 =

Bond angle


Torsional dihedral angle

z

x

y

1000cb-sb00100d

0ct-st0sbcb000100

0stct000100010

000100010001

Ti+1 =

Torsional (Dihedral) angle


Transform t i 1

z

x

y

1000cb-sb00100d

0ct-st0sbcb000100

0stct000100010

000100010001

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

1000cb-sb00100d

0ct-st0sbcb000100

0stct000100010

000100010001

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


Serial linkage model

Serial Linkage Model

T1

0

1

T2

-1

-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


F y linkage model

peptide group

side-chain group

f-y Linkage Model


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


Side chains with multiple torsional degrees of freedom c angles

Side Chains with Multiple Torsional Degrees of Freedom (c angles)

0 to 4 c angles: c1, ..., c4


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’i1j1k1txx

y’i2j2k2tyy

z’i3j3k3tzz

100011

=

Drawback of Homogeneous Coordinate Matrix

  • Too many rotation parameters

  • Accumulation of computing errors along a protein backbone and repeated computation

  • Non-redundant 3-parameter representationsof 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 spacewith 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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