Exact Analytical Formulation

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Exact Analytical Formulation
for coordinated motions in
Polypeptide Chains Vageli
Coutsias, Chaok Seok and Ken Dill

with applications to
Fast Exact Loop Closure in Homology
Modeling Monte Carlo Minimization for
Conformational Search Small Peptide
Ring Efficient Conformational Search
Department of Mathematics and Statistics,
University of New Mexico Department of
Pharmaceutical Chemistry, UCSF

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Study of localized motions in a polypeptide chain
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Ca2
N2
N3
C2
N1
C1
C3
Ca1
Ca3
d
LOOP CLOSURE find all configurations with two
end-bonds fixed The angle between the planes
N1-Ca1-Ca3 and Ca1-Ca3-C3 is given, the
orientation of the two fixed bonds (N1-Ca1 and
Ca3-C3) wrt the plane Ca1-Ca2-Ca3 can assume
several values (at most 8 solutions are possible)
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Peptide the elemental unit
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1.
119
1.47
123
3.80
1.32
114
1.53
122
1.24
A Canonical Peptide unit (trans configuration) in
the body frame (after Flory)
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The pep-2 capstone
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Tripeptide Loop Closure
With the base and the
lengths of the two peptide virtual bonds fixed,
the vertex is constrained to lie on a
circle.
Bond vectors fixed in space
Fixed distance
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Given the distance and angle constraints, three
types of virtual motions are encountered in
the body frame
In the body frame of the three carbons,
the anchor bonds lie in cones about the fixed
base.
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Ca2
N2
N3
C2
N1
C1
C3
Ca1
Ca3
d
Transferred motions in the body frame of
three contiguous Ca carbon units In this
frame the Ca carbons resemble spherical
4-bar linkage joints
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Motion type 1 peptide axis rotation With the two
end carbons fixed in space, the peptide unit can
rotate about the virtual bond
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1
Motion type 2 Coordinated rotation at junction
of 2 rotatable bonds (the angle between the two
bonds remains fixed as each rotates about its own
peptide virtual axis).
2
4
3
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Crank
Follower
Two-revolute, two-spheric-pair mechanism
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The general RSSR linkage
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The 4-bar spherical linkage
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y
d
x
z
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y
are constrained to lie on the circles
fixed
d
x
z
The use of intrinsic coordinates distinguishes
our method from other exact loop closure methods
(Wedemeyer Scheraga 00, Dinner 01)
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Brickard (1897) convert to polynomial form via
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L2
L1
A complete cycle through the allowed values for j
(dihedral (R1,R2) -(L1,R1) )and y (dihedral
(R1,R2)-(L2,R2))
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Differential equations for the reciprocal angles,
s and t.   Fixed angle between the two bonds,
Ca-N and Ca-C  
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t
a .35p
r1.81r2
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a .35p
r1.81r2
r1.4,r2.81
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A stressed peptide in the body frame of the
virtual bonds P(n-1)P(n)
Motion type 1 Peptide axis Rotation (rigid)
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Motion type 2 Coordinated rotation at
junction of 2 rotatable bonds
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Definitions
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Solution
Closure requires
Label
Branch present if
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8 real solutions at most Numerical evidence only
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The transformation 3
coupled polynomials
Common (real) zeros give feasible solutions.
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Method of resultants gives an equivalent 16th
degree polynomial for a single variable
Numerical evidence that at most 8 real solutions
exist. Must be related to parameter values
the similar problem of the 6R linkage in a
multijointed robot arm is known to possess
16 solutions for certain ranges of
parameter values (Wampler and Morgan 87 Lee
and Liang 89).
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Methods of determining all zeros (1) carry out
resultant elimination twice derive
univariate polynomial of degree 16 solve
using Sturm chains and deflation (2) carry out
resultant elimination once convert matrix
polynomial to a generalized eigenproblem
of size 24 (3) work directly with trigonometric
version use geometry to define feasible
intervals and exhaustively search. It is
important to allow flexibility in some degrees
of freedom
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Timings for loop closure by reduction to 16th
degree polynomial zero localization via Sturms
method. Successively solve loop closure by
successively removing the two peptide units
adjacent to each Ca carbon in a chain of known
conformation. Loop closure should
reproduce original, however off canonical
structures do abound. Zero solutions indicate
that the closure was not possible with
canonically configured backbone, i.e. there was a
deformation of some bond angles or w dihedrals
in the original strucure.
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Timings for loop closure via reduction to 24x24
generalized eigenproblem.
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Application to loop sampling
Analytical closure of the two arms of a loop in
the middle
Comparison 10 residue loop sampling (Matt
Jacobson)
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1r69.pdb
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1r69 Res 9-19 alternative backbone
configurations
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The 3 fixed points/3 virtual axes transform can
be found among any three Ca atoms, anywhere along
the chain
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Motion type 3 Dihedral rotation (actual
move, length changing not limited to -
type dihedrals)
(quaternion notation)
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Altering an internal dihedral leads to a nearby
loop closure problem. A sequence of small changes
results in a continuous family of deformations
(shown here as applied to the deformation of a
disulfide bridge).
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Refinement of 8 residue loop (84-91)
of turkey egg white lysozyme
Native structure (red) and initial structure
(blue)
Baysal, C. and Meirovitch, H., J. Phys. Chem. A,
1997, 101, 2185
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The continuous move given a state assume
D2b, D4a fixed, but D3 variable
tau2?sigma4 determined by D3 (1) tau1?sigma2,
tau4?sigma5 trivial (2) alpha1, alpha5 variable
but depend only on vertices as do lengths
(lengths 1-2, 1-5, 4-5 are fixed)
Given these sigma1?tau1,
sigma5?tau5 known
(sigma1?tau5 given) (3)
Dihedral (2-1-5-4) fixes remainder
alpha2, alpha4 determined
(sigma2?tau2, sigma4?tau4 known)
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pep virtual bond
3-pep bridge
C
design triangle
9-pep ring
cysteine bridge
1
2
Modeling R. Larsons 9-peptide
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Designing a 9-peptide ring
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3 peptide units are placed at the vertices of a
triangle with random orientations, and they are
connected by exact loop closure. The max and min
values of the 3-pep bridge set the limits for the
sides of the triangle.
Designing a 9-peptide ring
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In designing a 9-peptide ring, the known
parameters of 2-pep bridges (and those of the S2
bridge, if present) are incorporated in the
choice of the foundation triangle, with vertices
A,B,C (3 DOF)
C
B
A
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C
B
A
peptide virtual bond (3 dof for placement)x39
2-pep virtual bond (at most 8 solutions)
design triangle sides (3 dof )
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4-6-2
8-2-4
4-2-4
4-2-2
Cyclic 9-peptide backbone design
Numbers denote alternative loop closure solutions
at each side of the brace triangle
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The 3 fixed points/3 virtual axes transform can
be used as a means of enforcing constraints
(such as loop closure). It can be used to
generate minimum- Distortion moves for Monte
Carlo energy minimization. Generalizations where
one pair is disjointed are also possible with a
simple solution as well.
Using backbone kinematics in combination with
efficient (clever) placement of sidechains can
be used in a rational approach for
exploring conformation space.
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Visits to UCSF where much of the work was
performed supported in part by
a NIH grant to Ken Dill Cyclic peptide modeling
inspired by conversations with
Michael Wester, R. Larson
Animations Raemon Gurule, Carl Mittendorff,
Heather Paulsen and Marshall
Thompson (math. 375,
Spring 02 class project)
THANK YOU!
References Analytical loop closure Wedemeyer
and Scheraga J Comput Chem 1999 Go and
Scheraga Macromolecules 1978 Dinner
J Comput Chem 2000 Bruccoleri and
Karplus Macromolecules, 1985 Coutsias, Seok,
Jacobson and Dill (preprint) 2003 Mechanisms
Hartenberg and Denavit 1964 Hunt Oxford
1990 Duffy 1980 Numerical Methods
Manocha, Appl. of Comput. Alg. Geom., AMS,1997
Wampler and Morgan Mech Mach Theory 1991
Lee and Liang Mech Mach Theory 1988
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Proline
Ramachandran regions
Glycine
General