Geometric and Kinematic Geometric and Kinematic Models of Proteins Models of Proteins From a course taught firstly in Stanford by JC Latombe, then in Singapore by Sung Wing Kin, and now in Rome by AG…web solidarity. With excerpta from a course by D. Wishart. LECT_4 8 th Oct 2007
Geometric and Kinematic Models of Proteins. LECT_4 8 th Oct 2007. From a course taught firstly in Stanford by JC Latombe, then in Singapore by Sung Wing Kin, and now in Rome by AG…web solidarity. With excerpta from a course by D. Wishart. (x 4 ,y 4 ,z 4 ). (x 5 ,y 5 ,z 5 ). - PowerPoint PPT Presentation
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Geometric and Kinematic Geometric and Kinematic Models of ProteinsModels of Proteins
From a course taught firstly in Stanford by JC Latombe, then in Singapore by Sung Wing Kin, and now in Rome by AG…web solidarity. With excerpta from a course by D. Wishart.
LECT_4 8th Oct 2007
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)
Composition of two transforms represented by matrices T1 and T2 : T2 T1
Building a Serial Linkage ModelBuilding a Serial Linkage Model
Rigid bodies are:• atoms (spheres), or• groups of atoms
Building a Serial Linkage ModelBuilding a Serial Linkage Model
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 ModelBuilding a Serial Linkage Model
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
Atom: -2
-1
0
Building a Serial Linkage ModelBuilding a Serial Linkage Model
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 angle2. Introduce each additional atom in the sequence
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
The sequence of N-Cα-C-… atoms is the backbone (or main chain)
Rotatable bonds along the backbone define the -ψ torsional degrees of freedom
Small side-chains with degree of freedom
Cα
Cβ
--ψψ-- 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 (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 In linkage model a small local change may have big global effecthave big global effect