1 statistics and minimal energy comformations of semiflexible chains gregory s. chirikjian...

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Statistics and Minimal Energy Comformations of Semiflexible Chains

Gregory S. ChirikjianDepartment of Mechanical EngineeringJohns Hopkins University

2

Overview of Topics

My BackgroundKinematic analysis

Equilibrium conformations of chiral Equilibrium conformations of chiral semi-flexible polymers with end semi-flexible polymers with end constraintsconstraints

Probabilistic analysis Conformational statistics of Conformational statistics of

semiflexible polymerssemiflexible polymers

3

Simulations from the PhD Years

4

Hardware from the PhD Years

5

Equilibrium conformations of chiral Equilibrium conformations of chiral semi-flexible polymers with end semi-flexible polymers with end constraintsconstraints

6

Inextensible Continuum Model

Elastic potential energy:

Inextensible constraint

ωbωω

ω

TT

L

BU

dssUE

2

1

,))((0

ss arclength respect toh locity witangular ve : )( where ω

L

dssAL0

3)()( ea

)(sA

)0( sA

)(sa

)3(SOAg x

yz

x

y

z

7

The general representation of U

KP model: c=0

000

00

00

0

0

B

0

0

0

b

Yamakawa model:

0

0

0

00

00

00

B

00

00

0

b )(

2

1 200

200 c

MS model:

v

v

0

00

02

B

0

0

0

v

b 202

1 vc

A General Semiflexible Polymer Model

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Definition of a Group

A group is a set together with a binary operation o satisfying:

Associative: a o (b o c) = (a o b) o c Identity: e o a = a Inverse: a-1 o a = e

Binary operation o: a o b G whenever a,b GExamples: {R, +} where e=0; a-1 =-a; rotations; rigid-body motions

9

Definition of Rotational Differential Operators

Let X be an infinitesimal rigid-body rotation. Then

XR can be thought of as the right directional derivative of f in the direction X. In particular, infinitesimal rigid-body rotation in the plane are all combinations of:

0

t

tXR

dt

gedfgfX

000

001

010

000

100

000

000

000

100

321 XXX

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Euclidean Group, SE(3)

An element of SE(3):

Basis for the Lie Algebra: Small Motions

10T

aAg

0000

0010

0100

0000

~1X

0000

0000

0001

0010

~3X

0000

0001

0000

0100

~2X

0000

0000

0000

1000

~4X

0000

1000

0000

0000

~6X

0000

0000

1000

0000

~5X

11

Lie-group-theoretic Notation

Coordinates free no singularities

(3) ofelement basis:

0

(3)))(),(()(For

1

6

1

1

seX

A

AAgg

AAAXgg

SEtAttg

i

T

T

T

TT

iii

v

ω

aξ

0

a

a

(3) ofelement basis :

,)3()(For 3

1

soX

AA

XAA

SOtA

i

T

iii

T

ω

)(tg

Space-fixed frame

Body-fixed frame

ω

v

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Extensible Continuum Model

We can extend inextensible model by adding parameters such as stretching stiffness, shear stiffness, twist-stretch coupling factor, etc.

This model, and the inextensible one, do not include self-contact, which can be included by adding another potential function.

)3(,2

1

,))((0

seKU

dssUE

TT

L

v

ωξξkξξ

ξ Note: no constraints

)3(),( SEAg a

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Variational Calculus on Lie groups

Given the functional and constraints

one can get the Euler-Poincaré equation as:

where

2

1

2

1

)(,);;( 1t

tkk

t

t

dtghCdttgggfJ

,)(11,

m

lll

Ri

n

kjj

kij

ki

hfXCff

dt

d

6

1

0

,

))exp(()(

kk

kijji

ti

Ri

XCXX

tXgfdt

dgfX

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Explicit Formulations

Inextensible

Can be solved iteratively with I.C. (0) = and given , together with

Position a(s) is determined by the constraint.

Extensible

where

Can be solved iteratively with I.C. (0), together with

0

)( 1

2

eλ

eλ

bωωω A

A

BB T

T

0ξkξξ )(KK

3

1

)(i

ii XsAA

12

1212

2

2

1

1

vω

vvωω

v

ω

v

ω

3

1

)(i

ii Xsgg

s

dAs0

3)()( ea

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How to get to the desired pose

To reach the desired position and orientation , we need an inverse kinematics.

Let be the vector of undetermined coefficients

((0) for extensible case), and denote the distal frame for a

given as

Let

Define an artificial path functions which satisfy

Use Jacobian-velocity relation and position correction term.

)(tg p

.)1(,)0( 0 dpp gggg

dg

TTT ],[ λμη

),( Lgg η

guess. initialan is where,),( 000 ηη Lgg

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Inverse Kinematics – Graphical Explanation

C o n fo rm a tio nre su ltin g fro m

in itia l g u e ss

g p (0 )

g p (1 ) = g d

C o n fo rm a tio nsa tis fy in g e n d

c o n s tra in ts

D e s ire d r ig id -b o d ytra je c to ry , g p ( t)

g (tk)

g p( tk)

g (tk + 1 )

A c tu a l r ig id -b o d ytra je c to ry , g (t)

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Graphic Explanation – Cont’d

Initial conformation

Final conformation

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Example – histone binding DNA

F re e se c tio n

B in d in g se c tio n

P itc h

D ia m e te r

Swigon, et al., Biophysical Journal, 1998, Vol. 74, p.2515-2530.

F. D. Lucia, et al. J. Mol. Biol. 289:1101, 1999.

N: number of base pairs, varying from 351 to 366.

w: wrapping of DNA around the cylindrical histone molecule, 1.40 or 1.75.

hb: helical repeat length in bound section = 10.40 [bp/turn]

Pitch=2.7 nm, diameter=8.6 nm

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Simulation Results

N [bp] w Lk Wr E [kcal/mol]

353 1.4 33 -0.8729 5.2214

354 1.75 33 -1.2559 10.496

356 1.4 33 -0.9422 4.4306

358 1.75 33 -1.4987 7.2721

361 1.4 34 -0.7874 6.5229

362 1.75 34 -0.6601 11.5992

363 1.4 34 -0.8549 5.2834

366 1.75 34 -1.3865 8.833

N [bp] w Lk Wr E[kcal/mol]

353 1.4 33 -0.9381 4.6305

354 1.75 33 -1.5655 7.0435

356 1.4 33 -0.9466 4.4347

358 1.75 33 -1.5829 6.3217

361 1.4 34 -0.9289 4.7634

362 1.75 34 -1.5601 7.3264

363 1.4 34 -0.9348 4.5247

366 1.75 34 -1.5775 6.4216

00 b 00 4.2 cb

• N: number of base pairs, w: number of wraps, Lk: linking number, Wr: Writhe, E: elastic energy of the loop.

• Experimental data from F. D. Lucia, et al. J. Mol. Biol. 289:1101, 1999.

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Simulation Results - Conformations

(a) N=353, w=1.4, Lk=33 (b) N=354, w=1.75, Lk=33

(c) N=356, w=1.4, Lk=33 (d) N=358, w=1.75, Lk=33

(e) N=361, w=1.4, Lk=34 (f) N=362, w=1.75, Lk=34

(g) N=363, w=1.4, Lk=34 (h) N=366, w=1.75, Lk=34

• Red line: isotropic

• Black line: anisotropic

• Blue line: histone-binding part

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Conclusions for Part I

A new method for obtaining the minimal energy conformations of semi-flexible polymers with end constraints is presented.

Our method includes variational calculus associated with Lie groups and Lie algebras.

We also present a new inverse kinematics procedure.

Numerical examples are in good agreement with the experimental results published.

Extensible model can be used to do the same if all parameters are known.

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Conformational statistics of Conformational statistics of

semiflexible polymerssemiflexible polymers

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Elastic Energy of an Inextensible Chiral Elastic Chain

LUdsE

0csssU TT )()()(

2

1ωbBωωwith

Total arc length

Stiffness matrix

Chirality vector

Spatial angular velocity

A General Semiflexible Polymer Model

L

B

b

(s)

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Model Formulation

Potential energies of bending and twisting of a stiff chain (e.g. see [Yamakawa])

Path integral over the rotation group

TT

L

0

bB2

1)(U

,ds))s((UE

)A,A(U)(UxAAx T

))((),(exp)()();,(00

)(

)0(

sADAAUdssuLaLaAFLLALA

IA

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Model Formulation

Apply the classical Fourier transform w.r.t. a

Treat the inner most integrand as j times a Lagrangian with

Calculates the momenta and Hamiltonian

))(()(exp);,(ˆ0

)(

)0(

sAddsUukjLkAFLALA

IA

Bj2

1T T uk)b(jV

ukpBbpBpjHVT

p TT

kk

11 )2

1(

)(

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Model Formulation

Get the Schrödinger-like equation corresponding to H and quantization, pi = -j XR

i ,

Apply the classical Fourier inversion formula

FHL

Fj ˆ

ˆ

0~~~~

2

1 3

16

3

1,

FXXdXXDL l

RRll

lk

Rk

Rllk

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A diffusion equation describing the PDF of relative pose between the frame of reference at arc length s and that at the proximal end of the chain

),,()~~~~

2

1(

),,(6

3

1

3

1,

sfXXdXXDs

sf R

l

Rll

lk

Rk

Rllk Ra

Ra

Initial condition: f(a,R,0)= (a) (R)

1][ BD lkD bBd 1][ ldDefining

A General Semiflexible Polymer Model

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Differential operators for SE(3)

6,5,4for

3,2,1for~

3 iA

iXX

iaT

RiR

i

6,5,4for

3,2,1for)(~

3

3

1

ia

ia

eeaXX

i

k kki

Li

Li

00 ))~

(())(())((~

titiRi XtIHf

dt

dtHHf

dt

dqHfX

00 ))~

(())(())((~

titiLi HXtIf

dt

dHtHf

dt

dqHfX

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Fourier Analysis of Motion

Fourier transform of a function of motion, f(g)

Inverse Fourier transform of a function of motion

G

dgpgUgfpffF ),()()(ˆ)( 1

dpppgUpftracegffF )),()(ˆ()()ˆ(1

where where g g SE(N)SE(N) , , pp is a frequency parameter, is a frequency parameter, U(g,p)U(g,p) is a matrix representation of is a matrix representation of SE(N),SE(N), and anddg dg is a volume element at is a volume element at gg..

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Propagating By Convolution

dddddrdrdgG

2

0 0

2

0 0

2

0 0

2 sinsin

dhLghfLhfLLgfG

),(),(),( 21

121

dLLgfLLrf ),(),( 2

2

0

2

0 0

2

0

1

0

21

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Operational Properties of Fourier Transform

)(ˆ),~

(

)(),()(),~

exp(

)(,)~

exp()(

)(),())~

exp((~

10

),(),(),(

01

)~

exp(

10

2121

pfpX

hdphUhfpXtUdt

d

hdphXtUdt

dhf

gdpgUXtgfdt

dfXF

i

G

ti

pgUpgUpggU

G

ti

Xtgh

G

tiRi

i

),~

()(ˆ~pXpffXF i

Li

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Entries of (Xi , p) for i=1,2,3

0)),~

(exp(),~

(

tii pXtU

dt

dpX

mmllmlml

mmll

lmmmll

lmmlml

mmll

lmmmll

lmmlml

jmpX

cj

cj

pX

ccpX

,,3,;,

,1,,1,2,;,

,1,,1,1,;,

''''

''''''

''''''

),~

(

22),

~(

2

1

2

1),

~(

l

lk

lkm

s

mlmlAUamlspmlpgu )()](,|,|,[),( ''

,;, ''

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Entries of (Xi , p) for I = 4,5,6

llmm

smlllmm

smlllmm

s

ml

llmm

smlllmm

smlllmm

s

mlmlml

jpjpjp

jpjpjppX

i

i

,11,,,1,,,11,,

,11,,,1,,,11,,4,;,

'''''''

'''''''''

222

222),

~(

llmm

smlllmm

smlllmm

s

ml

llmm

smlllmm

smlllmm

s

mlmlml

ppp

ppppX

i

i

,11,,,1,,,11,,

,11,,,1,,,11,,5,;,

'''''''

'''''''''

222

222),

~(

llmm

smlllmmllmm

s

mlmlmljp

ll

smjpjppX i ,1,,,,,1,,6,;, '''''''''

)1(),

~(

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Solving for the evolving PDF

rrr

ds

dfB

f ˆˆ

where B is a constant matrix.

),,()~~~~

2

1(

),,(6

3

1

3

1,

sfXXdXXDs

sf R

l

Rll

lk

Rk

Rllk Ra

Ra

rsr esp Bf ),(ˆ

dpppUpfsfr rl rl

l

lm

l

lm

rmlml

rmlml

2

||' ||

'

''0 ,;','',';,2

);,()(ˆ2

1),,(

RaRa

A General Semiflexible Polymer Model

Applying Fourier transform for SE(3)

Solving ODE

Applying inverse transform

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Numerical Examples

2

1

0.50.1

36

Numerical Examples

0.80.30:5HW

0.10.1:3HW

0.10.5:2HW

5.05.2:1HW

5.0

00

00

00

00

00

HW5

HW2

HW3HW1

KP

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The Structure of a Bent Macromolecular Chain

xd1,xp2

b

zd2

yd2

xd2

zp1 yp1

xp1

zp2 yp2

yd1

zd1

Subchain 1 Subchain 2

1) A bent macromolecular chain consists of two intrinsically straight segments.

2) A bend or twist is a rotation at the separating point between the two segments with no translation.

A General Algorithm for Bent or Twisted Macromolecular Chains

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The PDF of the End-to-End Pose for a Bent Chain

),)(**(),( 321 RaRa ffff

•f1(a,R) and f3(a,R) are obtained by solving the

differential equation for nonbent polymer.•f2(a,R)= (a)(Rb

-1R), where Rb is the rotation

made at the bend.

2) The convolution on SE(3)

)3(

1 )()()())(*(SE jiji dffff hghhg

A General Algorithm for Bent or Twisted Macromolecular Chains

1) A convolution of 3 PDFs

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Computing the Convolution using Fourier Transform for SE(3)

)(ˆ)(ˆ)(ˆ)(ˆ123 pppp rrrr ffff

where

bb imb

lnm

imll

ll

SE

rmlml

r

mlml

ePe

ddpUfpf

)(cos)1(

);,(),()(ˆ

,'

',)'(

)3( ',';,,;','2

aRRaRa

A General Algorithm for Bent or Twisted Macromolecular Chains

))(())(()))(*(( 1221 ggg fFfFffF

1) An operational property

2) Fourier transform of the 3-convolution

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Two Important Marginal PDFs

1) The PDF of end-to-end distance

0

200,0;0,0

2

0

2

0 )3(2

2 sin)(ˆ2

sin),(2

)( dpppa

papf

adddf

aaf

SO

RRa

2) The PDF of end-to-end distance and the angle between the end tangents

0

20

||0,;0,

2

0

2

0

2

0

2

02

2

)())(cos)(ˆ(2

sin

sin),(8

sin),(

dpppajPpfa

ddddfa

af

r rll

rll

Ra

A General Algorithm for Bent or Twisted Macromolecular Chains

41

1. Variation of f(a) with respect to Bending Angle and Bending Location__KP Model

Examples

42

2. Variation of f(a) with respect to Bending Angle and Bending Location__Yamakawa Model

Examples

43

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.5

0

0.5

1

1.5

2

2.5

3

3.5

4 Marko-Siggia model with =v=0.5, =0.5, 0=2, L1=L2=0.5

a

f(a)

b=

b=3/4

b=/2

b=/4

b=0

3. Variation of f(a) with respect to Bending Angle and Bending Location__MS Model

Examples

44

Conclusions for Part II

A method for finding the probability of reaching any relative end-to-end position and orientation has been developedIt uses the irreducible unitary representations of the Euclidean motion group and associated Fourier transformThe operational properties of this transform convert the Fokker-Planck equation into a linear system of ODEs in Fourier space.The group Fourier transform can be used to `stitch together’ pdfs of segments joined by joints or at discrete angles.

45

1) J. S. Kim, G. S. Chirikjian, ``Conformational Analysis of Stiff Chiral Polymers with End-Constraints,’’ Molecular Simulation 32(14):1139-1154. 2006

2) Y. Zhou, G. S. Chirikjian, ``Conformational Statistics of Semiflexible Macromolecular Chains with Internal Joints,’’ Macromolecules. 39:1950-1960. 2006

3) Zhou, Y., Chirikjian, G.S., “Conformational Statistics of Bent Semi-flexible Polymers”, Journal of Chemical Physics, vol.119, no.9, pp.4962-4970, 2003.

4) G. S. Chirikjian, Y. Wang, ``Conformational Statistics of Stiff Macromolecules as Solutions to PDEs on the Rotation and Motion Groups,’’ Physical Review E. 62(1):880-892. 2000

E. References

46

Acknowledgements

This work was done mostly by my former students: Dr. Yunfeng Wang, Dr. Jin Seob Kim, and Dr. Yu ZhouThis work was partially supported by NSF and NIH