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.
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
Van der Waals radii in Å
Probe of 1.4Å
Probe of 5Å
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
The atomistic model does not encode this kinematic structure( algorithms must maintain appropriate bond lengths)
What is the potential problem with homogeneous coordinate matrix?
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
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
Tk(i) = Tk…Ti+2 Ti+1 position of atom k in frame of atom i
In physiological conditions:
They assign probabilities to φ-ψ pairs based on frequencies in known folded structures
Rotatable bonds along the backbone define the f-y torsional degrees of freedom
Small side-chains with c degree of freedom
Caf-y-c Linkage Model of Protein
0 to 4 c angles: c1, ..., c4
Computational errors may accumulate
y’ i2 j2 k2 ty y
z’ i3 j3 k3 tz z
1 0 0 0 1 1
=Drawback of Homogeneous Coordinate Matrix
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)