1

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

8

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

10

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

12

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

13

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

14

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

15

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

16

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)

17

Graphic Explanation – Cont’d

Initial conformation

Final conformation

18

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.

20

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

21

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.

22

Conformational statistics of Conformational statistics of

semiflexible polymerssemiflexible polymers

23

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)

24

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

25

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(

)(

26

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

27

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

28

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

29

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..

30

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

31

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

32

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

,;, ''

33

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(),

~(

34

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

35

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

37

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

38

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

39

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

40

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