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Geometric and Kinematic Models of Proteins

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Study of movement independent of the forces that cause them

- Long sequence of amino-acids (dozens to thousands), also called residues from a dictionary of 20 amino-acids

- 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)

- 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

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

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

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

Van der Waals radii in Å

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

Probe of 1.4Å

Probe of 5Å

- 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?

- 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.

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.

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

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

- 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

- Stick (bond) model

- Stick (bond) model
- Small-sphere model

(x4,y4,z4)

(x5,y5,z5)

(x6,y6,z6)

(x8,y8,z8)

(x7,y7,z7)

(x1,y1,z1)

- 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

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

- 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)

T?

T?

- 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)

z

T(x)

y

T

x

x

y

x

y

y

x

x

y

y

x

x

y

y

x

x

y

y

x

x

y

y

x

x

y

y

Rotation matrix:

cos q -sin qsin qcos q

j

i

q

ty

tx

x

x

y

y

Rotation matrix:

i1 j1i2j2

j

i

q

ty

tx

x

x

y

y

Rotation matrix:

a

i1 j1i2j2

a

b

j

i

=

b’

q

ty

a’

b’

b

q

a

tx

a

a’

x

x

v

Transform of a point?

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

i1 j1txi2 j2ty 001

- T = (t,R)
- T(x) = t + Rx

?

q2

q1

R

z

y

x

y

i

z

j

k

x

i1 j1 k1txi2 j2 k2tyi3 j3 k3tz0001

with:

- i12 + i22 + i32 = 1
- i1j1 + i2j2 + i3j3 = 0
- det(R) = +1
- R-1 = RT

z

y

x

cos q0sinqtx

010ty

-sin q0cos qtz

0001

q

k

q

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

z

y

x

y

i

z

j

k

x

x’i1j1k1txx

y’i2j2k2tyy

z’i3j3k3tzz

100011

=

(x,y,z)

(x’,y’,z’)

Composition of two transforms represented by matrices T1 and T2 :T2T1

What is the potential problem with homogeneous coordinate matrix?

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

- Build the assembly of the first 3 atoms:
- Place 1st atom anywhere in space
- Place 2nd atom anywhere at bond length

- 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

z

x

y

- 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

z

x

y

1000cb-sb00100d

0ct-st0sbcb000100

0stct000100010

000100010001

Ti+1 =

z

x

y

1000cb-sb00100d

0ct-st0sbcb000100

0stct000100010

000100010001

Ti+1 =

z

x

y

1000cb-sb00100d

0ct-st0sbcb000100

0stct000100010

000100010001

Ti+1 =

z

x

y

1000cb-sb00100d

0ct-st0sbcb000100

0stct000100010

000100010001

Ti+1 =

y

i+1

Ti+1

z

x

t

i-1

d

i

b

i-2

z

x

y

1000cb-sb00100d

0ct-st0sbcb000100

0stct000100010

000100010001

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

T1

0

1

T2

-1

-2

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

- 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)

Why?

Root group of 3 atoms

p atoms 3p -6 parameters

T0

world coordinate system

Root group of 3 atoms

p atoms 3p -6 parameters

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

f

C

C

N

Ca

3.8Å

w: Ca Ca

f: C C

y: N N

w = p

w

peptide group

side-chain group

C

C

N

Ca

f=0

- 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

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

ψ

φ

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

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

- 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

Computational errors may accumulate

x’i1j1k1txx

y’i2j2k2tyy

z’i3j3k3tzz

100011

=

- 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

R(r,q)

R(r,q+2p)

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

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

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)