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

Study of localized motions in a polypeptide chain

C1

C2

C3

N3N2

N1

C’1

C’2

C’3

δ

LOOP CLOSURE: find all configurations with two end-bonds fixedThe angle between the planes N1-C1-C3 and C1-C3-C’3 is given,the orientation of the two fixed bonds (N1-C1 and C3-C’3) wrt the plane C1-C2-C3 can assume several values (at most 8 solutions are possible)

Peptide: the elemental unit

 

nP1.32

114

1231.47

1.53

3.80

1nC

122

119

1.24

1.

nN

nH

'1nC

1nO

A Canonical Peptide unit (trans configuration) in the body frame (after Flory)

nC

31.2

70

07.1

54.1

nn

onnn

OH

CCN

HC

CC

o2.13o2.22

50.2

82.2

)0(

1

1

nn

nn

HH

HO

N

1

'C

C

C

H12

nP

1nC

1nC

nn

1nP

nC

The pep-2 “capstone”

1nC

1nC

nC

With the base andthe lengths of the two peptidevirtual bonds fixed, the vertex is constrained to lie ona circle.

11 nn CC

nC

Tripeptide Loop Closure N

C

'C

Bond vectorsfixed in space

Fixed distance

 

1nC

nC

1nC

In the body frame of thethree carbons, the anchor bonds lie in cones about the fixed base.

C

Given the distance and angle

constraints, three types of virtual

motions are encountered in the

body frame

C1

C2

C3

N3N2

N1

C’1

C’2

C’3

δ

Transferred motions in the body frame of three contiguous C carbon units: In this frame the C carbons resemble spherical 4-bar linkage joints

nC

z

y

1nC

Motion type 1: peptide axis rotationWith the two end carbons fixed in space, the peptide unit can rotate about the virtual bond

C

nC

1nC

x

y

z

nn

p

p

R

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).

4

3

2

1

R

1nC

1

1

1

1a2a

3a4a

3x

2x

1x1R

4R

1z

4z 1s

4s

3G

2GCrank

Follower

Two-revolute, two-spheric-pair mechanism

ts

cB

A

AO

BO

O

Q

g

f

yx

z

p

The general RSSR linkage

c

O

y

x

z

The 4-bar spherical linkage

z

d

R

R

p

p

x

y

1t

2t

1s2s

r

r

),,,;( F

22

21

1

1

21 cos)(coscostan

tRtR

pR

tR

tR

z

d

R

R

p

p

x

y

1t

2t

1s2s

r

r

),,,;( F

-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.50

0.5

1

1.5

2

2.5

3

RR , are constrainedto lie on the circles

d fixed

The use of intrinsic coordinates distinguishes our method from otherexact loop closure methods (Wedemeyer & Scheraga ‘00, Dinner ‘01)

sinsincoscoscossinsin

sinsincossincossinsin

coscoscos

Brickard (1897): convert to polynomial form via

2

1tan

2

1tan

v

u

  

y

z

x

R1 R2

4

3

2

1

 

A complete cycle through the allowed values for (dihedral(R1,R2) -(L1,R1) )and (dihedral (R1,R2)-(L2,R2))

L1

L2

Differential equations for the reciprocal angles, and .  Fixed angle between the two bonds, CN and CC’: 

)cossinsincos(cos

)sinsin(sincos

21

1221

RRF

1,

,,

222/122

dt

d

dt

dFF

F

dt

dF

dt

d

0 2 4 6 8 10 12-3

-2

-1

0

1

2

3

y1, y

2

=.81=2

t

-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.50

0.5

1

1.5

2

2.5

3

y1

y 2

flywheel equations

0 2 4 6 8 10 12-3

-2

-1

0

1

2

3

y1, y

2

0 2 4 6 8 10 12-4

-3

-2

-1

0

1

2

3

y1, y

2=.81=2

=.4,2=.81

A stressed peptide in the body frame of the virtual bonds P(n-1)—P(n)

1nPnC

1nN

1nH

nC

nO

1n

n

)( 111

)0(111

nnnnn

nnnn

NCCCdih

nn

np

n)0(1n

nz

1nz

ny

1ny

1n1np

1nC

Motion type 1:Peptide axis Rotation (rigid)

22

21

1

1

21 cos)(coscostan

tRtR

pR

tR

tR

nC

1nC

x

y

z

nn

p

p1

R

Motion type 2: Coordinated rotation at junctionof 2 rotatable bonds

Definitions

,,,;:cos

tan

cos)(cos

;;

sincos)(cos

sincoscoscos

sincoscoscos

,sin

,sin

,1,1

1

1

2221

21

21

21

121

121

FD

C

A

B

pRC

BADtRBtRA

tRtRpRRR

ttprpR

ssprpR

spses

tptet

pp

ppepp

Z

Z

Z

Solution

1,0;cos

1,0;cos

1,0;cos

2

3331

333

223

2221

222

112

1111

111

0,110,1

Closure requires: 313

Label: 123 24 l

Branch present if 1,, 321

8 real solutions at most

Numerical evidence only

The transformation 3 coupled polynomials:2

tan 1iz

3,2,1,),( 1

2

0,

)(,1

izzpzzP k

iji

kj

ikjii

Common (real) zeros give feasible solutions.

14 zz

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).

)3,2,1(, izi

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

input coeff sturm coord tot

8 0.067 0.253 3.814 0.442 4.576

0 0.084 0.253 0.141 0.478

4 0.085 0.252 6.513 0.218 7.068

4 0.088 0.252 3.392 0.228 3.960

0 0.066 0.296 0.138 0.500

2 0.066 0.293 0.356 0.115 0.830

2 0.085 0.253 0.411 0.124 0.873

4 0.067 0.253 1.957 0.227 2.504

4 0.067 0.252 0.582 0.219 1.120

2 0.067 0.251 1.803 0.114 2.235

2 0.066 0.253 0.438 0.141 0.898

6 0.067 0.257 2.041 0.321 2.686

6 0.066 0.254 2.131 0.322 2.773

2 0.066 0.251 0.336 0.115 0.768

2 0.067 0.252 1.726 0.115 2.160

2 0.068 0.250 0.332 0.115 0.765

2 0.067 0.251 1.678 0.115 2.111

0 0.067 0.251 0.138 0.456

4 0.066 0.252 6.360 0.218 6.896

4 0.068 0.253 1.870 0.219 2.410

Timings for loop closureby reduction to 16th degreepolynomial; zero localizationvia Sturm’s method. Successively solve loopclosure by successivelyremoving the two peptide units adjacent to eachCcarbon in a chain ofknown conformation. Loop closure should reproduceoriginal, however offcanonical structures doabound. Zero solutionsindicate that the closure was not possible with canonicallyconfigured backbone, i.e.there was a deformation ofsome bond angles or dihedrals in the originalstrucure.

input matr gen_e coord tot

8 0.067 0.069 1.096 0.431 1.663

0 0.067 0.069 3.748 3.884

4 0.067 0.068 2.036 0.221 2.392

4 0.067 0.069 2.741 0.223 3.100

0 0.067 0.069 3.735 3.871

2 0.067 0.068 3.690 0.118 3.943

2 0.067 0.069 3.711 0.119 3.966

4 0.067 0.068 3.656 0.220 4.011

4 0.066 0.069 2.032 0.225 2.392

2 0.068 0.068 3.206 0.119 3.461

2 0.065 0.069 3.710 0.118 3.962

6 0.067 0.068 1.984 0.329 2.448

6 0.066 0.069 1.710 0.326 2.171

2 0.067 0.073 3.215 0.118 3.473

2 0.069 0.068 3.700 0.117 3.954

2 0.068 0.069 2.789 0.116 3.042

2 0.067 0.069 2.785 0.119 3.040

0 0.067 0.068 3.760 3.895

4 0.067 0.068 2.373 0.219 2.727

4 0.067 0.069 2.033 0.220 2.389

Timings for loopclosure via reduction to24x24 generalizedeigenproblem.

Application to loop sampling

No. of loops Numerical closure Analytical closure

generated

(best RMSD)

151 (1.61) 40,000 (1.23)

accepted

(best RMSD)

1 (6.75) 1,374 (1.58)

Analytical closure of the two arms of a loop in the middle

Comparison: 10 residue loop sampling (Matt Jacobson)

1r69.pdb

1r69: Res 9-19 alternative backbone configurations

The 3 fixed points/3 virtual axes transform can be found among any three Ca atoms, anywhere along the chain

n

~

l

l~

Motion type 3:Dihedral rotation(actual move,length changing;not limited to - type dihedrals)

~

~

,~ cqcqc

b

bq 2/sin,2/cos

,~~1 cbal

al

al

cl

cl~

~~cos,~~

~~~

cos

c

ba

c~

(quaternion notation)

2 2~

1

2

3

length

fixed

23321 ,,,,

:

changed

Altering an internal dihedral leads to a“nearby” loop closure problem. A sequenceof small changes results in a continuous family of deformations (shown here as applied to the deformationof a disulfide bridge).

1l

1

~l

22

~

3~ 3

11

~

3l

2l

321321321

32123

22

21

sinsinsin

sin2

llllll

lllll

2.0

1.0

3

3

2

2

1

1

32

2

1

2

3

2

21

sinsinsin

2sin

lll

ll

lll

2.0

,~ cqcqc

b

bq 2/sin,2/cos

,~~1 cbal

al

al

cl

cl

1

13

1

12 ~

~~cos,~~

~~~

cos

1

a2

b2

3 4

a4

b4

5

a5 b5a1 b1

1

2

4

51

2

4

5

'3

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

The continuous move: given a state assume D2b, D4a fixed, but D3 variable tau2sigma4 determined by D3

(1) tau1sigma2, tau4sigma5 trivial

(2) alpha1, alpha5 variable but depend only on vertices as do lengths (lengths 1-2, 1-5, 4-5 are fixed) Given these sigma1tau1, sigma5tau5 known (sigma1tau5 given)

(3) Dihedral (2-1-5-4) fixes remainder: alpha2, alpha4 determined (sigma2tau2, sigma4tau4 known)

21

3

BA

C

Designing a 9-peptide ring

pep virtual bond

3-pep bridge

design triangle

cysteine bridge 9-pepring

ModelingR. Larson’s9-peptide

o

d 7.4min

o

d 3.7max

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

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)

1l 2l

3lB A

Cmax1min lll

3d

1l 2l3l

B A

C

1d2d

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 4-6-2

4-2-4 4-2-2

Cyclic 9-peptide backbone design

Numbers denote alternativeloop closure solutions at each

side of the brace triangle

Using backbone kinematics in combination with efficient (clever) placement of sidechains can beused in a “rational approach for exploringconformation space.

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 Carloenergy minimization. Generalizations where one pairis disjointed are also possible with a simple solution as well.

THANK YOU!

Visits to UCSF where much of the work was performed supported in part by a NIH grant to Ken DillCyclic 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)

ReferencesAnalytical 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

General

Proline

Glycine

Ramachandran regions