Geometrical RING Optimization

1
Geometrical RING
Optimization
Evangelos CoutsiasDept of Mathematics and
Statistics, Univ. of New Mexico
2
Jointly withChaok Seok, Matthew Jacobson, and
Ken DillDept of Pharmaceutical Chemistry, UCSF
Michael Wester Office of Biocomputing, UNM
3
Abstract
In previous work, we considered the problem of
loop closure, i.e., of finding the ensemble of
possible backbone structures of a chain molecule
that are consistent geometrically with preceding
and following parts of the chain whose structures
are given. We provided a simple intuitive view
and derivation of a 16th degree polynomial
equation for the case in which the six torsion
angles used for the closure belong in three
coterminal pairs. Our work generalized previous
results on analytical loop closure as our torsion
angles need not be consecutive, and any rigid
intervening segments are allowed between the free
torsions. We combined the new scheme with an
existing loop construction algorithm to sample
protein loops longer than three residues and
used it to implement a set of local moves for
Monte Carlo minimization. Here we present an
application to the sampling of S2-bridged
9-peptide loops and discuss the implementation of
the local moves as a Metropolis Monte Carlo
scheme for the uniform sampling of
conformational space.
4
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
5
The triangle formed by three consecutive Ca
atoms Given the span, d, there are constraints
on the orientation of the middle Cb, the side
chain and the two coterminal peptide units about
the virtual bonds between the Ca (green
circles).
Designing a 9-peptide ring
6
(No Transcript)
7
Given the span, the two consecutive peptide units
are correlated
8
This extends to the orientation of Cb
9
A bimodal example
10
Theta-perturbations are not enough
11
(No Transcript)
12
(No Transcript)
13
1r69 Res 9-19 alternative backbone
configurations
14
Representation of Loop Structures
In the new frame
In the original frame
15
New View of Loop Closure
Old View 6 rotations / 6 constraints
New View 3 rotations / 3 constraints
16
Crank
Follower
Two-revolute, two-spheric-pair mechanism
17
The 4-bar spherical linkage
18
y
Transfer Function for concerted rotations
d
x
z
19
 
 
 
L2
L1
A complete cycle through the allowed values for j
(dihedral (R1,R2) -(L1,R1) )and y (dihedral
(R1,R2)-(L2,R2))
20
(No Transcript)
21
t
a .35p
r1.81r2
22
Derivation of a 16th Degree Polynomial for the
6-angle Loop Closure
2
ri-1? ri? cos ?i gives Pi(ui-1, ui) ? pjk
ui-1j uik , where ui tan(?i/2). Using the
method of resultants, the three biquadratic
equations P1(u3, u1), P2(u1, u2), and P3(u2, u3)
are reduced to a polynomial in u3, R16(u3) ?
rj u3j
j,k0
r2?
r1?
16
j0
23
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).
24
The Minkowski sum of three squares, of side a, b,
c resp. Here a2x, b2y, c2z are the sizes of
three scaled Newton polytopes for the three
biquadratics
25
There are at most 16 solutions from first
principles
By the Bernstein-Kushnirenko-Khovanski theorem
the total number of isolated solutions cannot
exceed the mixed volume of the Minkowski sum of
the Newton Polytopes of the consitutive
polynomial components. That is, the number 16 is
generic for this problem.
26
Methods of determining all zeros (1) carry out
resultant elimination derive univariate
polynomial of degree 16 solve using Sturm
chains and deflation (2) carry out resultant
elimination but 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
27
Loop Closure Algorithm

• Polynomial coefficients are determined in terms
  of the geometric parameters on the right.
• u3 tan(?3/2) is obtained by solving the
  polynomial equation. ?3, ?1, and ?2 follow.
• Positions of the all atoms are determined by
  transforming to the original frame.

?1
?2
?3
28
General Chain Loop Closure
29
7-Angle Loop Closure
30
The continuous move used in Monte Carlo energy
minimization
31
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)
32
Longer Loop Closure in Combination with an
Existing Loop Construction Method
Analytical closure of the two arms of a loop in
the middle
33
Coutsias, Seok, Jacobson and Dill, J Comp Chem
25(4), 510 (2004) Jacobson, et al, Proteins,
2004. Canutescu and Dunbrack, Protein Science,
12, 963 (2003).
34
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
35
pep virtual bond
3-pep bridge
C
design triangle
9-pep ring
cysteine bridge
1
2
Modeling R. Larsons 9-peptide
3
Designing a 9-peptide ring
36
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
37
C
B
A
peptide virtual bond (3 dof for placement)x39
2-pep virtual bond (at most 8 solutions)
design triangle sides (3 dof )
38
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
39
Disulfide Loop Closure

• Start at of the last Cysteine residue
• The dihedral angle is a free variable
• vary continuously to get all possible
  conformations.
• Fix the bonds and
• Note that a move rooted at the first
• Cysteine must not fix
• but rather

40
Disulfide Bridge Loop Closure
41
(No Transcript)
42
(No Transcript)
43
(No Transcript)
44
(No Transcript)
45
(No Transcript)
46
(No Transcript)
47
(No Transcript)
48
(No Transcript)
49
(No Transcript)
50
PEP25 CLLRMKSAC
51
(No Transcript)
52
(No Transcript)
53
4
54
8
55
16
56
24
57
32
58
40
59
9998 2319 -227.911 2.89 -234.104
2.66 0.6 120 N 0 9999 0
-214.445 2.07 -234.104 2.66 1.4
273 N 0 10000 0 -216.965 2.79
-234.104 2.66 0.4 90 N 0
--------------------------------------------------
-------- n_trial, n_accept, ratio 10000 2319
0.2319000 ---------------------------------------
------------------- min E -234.104 rmsd
2.664 num saved pdb 40 Number of total
energy evaluations 1788596
--------------------------------------------------
-------- Total User Time 10004.895 sec
0 dy 2 hr 46 min 44.895 sec Final
time Apr 10 222429 2004
60
(No Transcript)
61
(No Transcript)
62
(No Transcript)
63
CLLRMRSIC MD
Calculation using GROMAX with explicit water
by Ilya Chorny, UCSF
64
(No Transcript)