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
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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
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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
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..
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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
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
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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
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