Geometrical
RING
Optimization
Evangelos Coutsias
Dept of Mathematics and Statistics,
Univ. of New Mexico
Jointly with
Chaok Seok, Matthew Jacobson, and Ken Dill
Dept of Pharmaceutical Chemistry, UCSF
Michael Wester
Office of Biocomputing, UNM
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.


With the base Cn 1  Cn 1 and
the lengths of the two peptide
virtual bonds fixed, the vertex
C n is constrained to lie on
a circle.
Tripeptide Loop Closure
N

Cn
C
C'


Cn 1
Cn 1
Bond vectors
fixed in space
Fixed
distance
o
d min  4.7 
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).
o
d max  7.3 
Designing a 9-peptide ring
  111.6 o
  16.63o
  19.13o
  90
o
Given the span, the two consecutive peptide units are correlated
  90
o
2
1
This extends to the orientation of Cb
  111
o

A bimodal
example

  111
o
  110  10
o
Thetaperturbations are
not enough
o

  111
o
  19  5
o
o

  111
o
  19  5
o
o
1r69: Res 9-19 alternative
backbone configurations
Representation of Loop Structures
In the original frame
In the new frame
New View of Loop Closure
Old View
New View
6 rotations / 6 constraints 3 rotations / 3 constraints
x2
x1
x3
Follower
R4
1
G2
a2
Crank
R1
a1
G3
a3
1
a4
z4
1
s1
s4
Two-revolute, two-spheric-pair mechanism
z1
x



c



O
y
z
The 4-bar spherical linkage
  F ( ; , ,  , )
y
Transfer Function
for concerted
rotations
s1 

p

r
s2
d
R

r
R

p
z
x
t1

t2


 R  t 2 
1  cos   ( R  p ) cos 
  cos
  tan 
2
2 
 
 R  t1 



R

t

R

t
 1

2


1
z
y
R1
R2
1
2
x
L1
L2
4
3
A complete cycle through the allowed values for  (dihedral
(R1,R2) -(L1,R1) )and y (dihedral (R1,R2)-(L2,R2))
3
2
1
0
-1
t
-2
-3
2
4
6
8
10
flywheel equations
12
3
y1, y2
2.5
2
 .35*p
y2
0
1.5
1
r1=.81=r2
0.5
0
-3.5
-3
-2.5
-2
-1.5
y1
-1
-0.5
0
0.5
Derivation of a 16th Degree Polynomial
for the 6-angle Loop Closure
2
ri-1 ·ri = cos i gives Pi(ui-1, ui) =j,k=0
 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,
r
16
R16(u3) =  rj u3j
j=0
r1
2
Method of resultants gives an equivalent 16th degree
polynomial for a single variable zi , (i  1,2,3)
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).
The Minkowski sum of three squares,
of side a, b, c resp. Here a=2x, b=2y,
c=2z are the sizes of three scaled
Newton polytopes for
the three biquadratics
0,0, a  c
0, b  c,0
a  2x
b  2y
a  b, b  c, c  a
c  2z
a  b,0,0
There are at most 16 solutions: from first principles
V a, b, c   a  b b  c c  a 
 2abc  a 2 b  c   b 2 c  a   c 2 a  b 
 16 xyz  
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.
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
Loop Closure Algorithm
1. Polynomial coefficients are
determined in terms of the
geometric parameters on the
right.
2. u3 = tan(3/2) is obtained by
solving the polynomial
equation. 3, 1, and 2
follow.
3. Positions of the all atoms
are determined by
transforming to the original
frame.
1
2
3
General Chain Loop Closure
7-Angle Loop Closure
The continuous move
used in Monte Carlo
energy minimization
2b 
2
4
2a
3
1
1a
1
2
1b
1
4a
4
 3'
4b
4
5a
5
5b
5
5
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)
Longer Loop Closure in
Combination with an Existing
Loop Construction Method
Analytical closure of the two arms of a loop in the middle
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).
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
pep virtual bond
3-pep bridge
design triangle
C
9-pep
ring
cysteine bridge
1
2
Modeling
R. Larson’s
9-peptide
B
A
3
Designing a 9-peptide ring
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)
lmin  l1  lmax
C
l1
l2
B
l3
A
C
d2
d1
l1
l3
l2
B
A
d3
peptide virtual bond (3 dof for placement)x3=9
2-pep virtual bond (at most 8 solutions)
design triangle sides (3 dof )
8-2-4
4-2-4
Cyclic 9-peptide
backbone design
4-6-2
4-2-2
Numbers denote alternative
loop closure solutions at each
side of the brace triangle
Disulfide Loop Closure
• Start at C of the “last” Cysteine residue
• The dihedral angle  2 is a free variable:
vary continuously to get all possible conformations.
• Fix the bonds: N 9  C ,9 and C ,1  C1
• Note that a move rooted at the “first”
Cysteine must not fix N1  C ,1
but rather C  ,1  C ,1
Disulfide Bridge Loop Closure
PEP25:
CLLRMKSAC
4
8
16
24
32
40
9998 2319 -227.911 2.89 -234.104 2.66
9999
0 -214.445 2.07
-234.104 2.66
10000
0 -216.965 2.79
-234.104 2.66
---------------------------------------------------------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 22:24:29 2004
0.6 120 N 0
1.4 273 N 0
0.4 90 N 0
RMSD vs Time
0.25
0.20
RMSD(nm)
0.15
0.10
0.05
0.00
400
800
1200
1600
2000
2400
2800
3200
time (ps)
CLLRMRSIC
MD Calculation using GROMAX with explicit water
by Ilya Chorny, UCSF
3600
4000