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Transcript of Modeling of Biofilaments: Elasticity and Fluctuations Combined D. Kessler, Y. Kats, S. Rappaport...
![Page 1: Modeling of Biofilaments: Elasticity and Fluctuations Combined D. Kessler, Y. Kats, S. Rappaport (Bar-Ilan) S. Panyukov (Lebedev) Mathematics of Materials.](https://reader035.fdocuments.us/reader035/viewer/2022062305/56649e9d5503460f94b9e123/html5/thumbnails/1.jpg)
Modeling of Biofilaments: Elasticity and Fluctuations Combined
D. Kessler, Y. Kats, S. Rappaport (Bar-Ilan)
S. Panyukov (Lebedev)
Mathematics of Materials and MacromoleculesIMA, Minneapolis, October 3, 2004
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Stretching of helical springs
Overview
1. Motivation
2. Ribbons: geometry, elasticity, fluctuations
3. Computer simulations: Frenet algorithm
Stretching of filaments
Twisting dsDNA
CyclizationDistribution functions
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Polymers – objects with atomic thickness (1 A) and arbitrary length
Atomic resolution
Quantum mechanics
RIS models
Coarse grained description
Statistical mechanics
Random walks
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What sort of objects are described by this model?
N
n
nnn l
P1
2
21exp)(
RRR
This is the probability distribution of a random walk!
Beads connected by entropic springs
The standard model of polymers:
nn-1
nRspring constant= kT/l2
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Random walks are not lines!
s
R(s)
0
L
1|/| dsdR
Continuous curve:
1d f
Inextensible line
Random walk:
2d f
arbitrarydsd |/| R
Extensible fractal
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What about nano-filaments: thickness 10-100 A?
1 Intrinsic shape 2 Resistance to change of shape (bending, twist)
Biofilaments: DNA, actin and tubulin fibers, flagella, viruses …
Synthetic filaments: organic microtubules, carbon nanotubes, …
But: thermal fluctuations are still important!
Theory of elasticity of fluctuating filaments with arbitrary intrinsic shape
New elements:
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Bending elasticity of inextensible lines
Modeling dsDNA at large deformations
Bustamante et al., Science 265, 1599 (1994)
The first step:
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dsDNA under stretching and torque
1. Cannot twist lines 2. Lines have no chirality
degree of over/unwinding
Strick et al., Science 271, 1835 (1996)
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Geometry of space curves:
s
s’t
tn
n
b
b
,nb
ds
d,bt
n ds
dn
t ds
d
Frenet eqs: generate curve by rotation of the triad bnt ,,
- curvature, - torsion This is not a physical twist !
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Helix
p
2r22
22
2
r
p
Straight line
0
0
/1
r
Circle
r
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Ribbons (stripes)
t2(s)
t1(s)
Physical triad: t1, t2, t3
n(s)
b(s))(s
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3
2
1
12
13
23
3
2
1
0
0
0
t
t
t
t
t
t
ds
d
Generalized Frenet eqs. – rotation of physical axes
Ribbon - principal axes ; tangent 21, tt 3t
cos1 sin2 ds
d 3
Configuration of the ribbon – uniquely defined by )(sior by )(),(),( sss
rate of twist
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Mechanics: Linear Elasticity
Deviations from stress-free state: kkk 0
Elastic Energy k
L
kkel dsbU0
2
2
1
kb - rigidity with respect to bending and twist
Small local but arbitrarily large global deviations from equilibrium configuration!
k0Equilibrium shape defined by spontaneous curvatures
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Stretching a helical spring
pitch > radius, bending rigidity > twist rigidity4 turns,
minimize ))(,,()(2
100
2
0sFRsdsbE ii
L
ii
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pitch < radius, bending rigidity < twist rigidity
Phys. Rev. Lett. 90, 024301 (2003)
The energy landscape E(R) has multiple minima with depths and locations that vary with F
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Stretching helical ribbons of cholesterol:
Smith, Zastavker and Benedek, Phys. Rev. Lett. 87, 278101 (2001)
Mechanical noise-induced transitions?
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Stretching transitions and hysteresis in chromatin ?
Y. Cui and C. Bustamente, PNAS 97, 127 (2000).
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Correlation functions )'()( ss ji tt for ribbons with arbitrary spontaneous
shape and rigidity!
0)( si )'()'()( 1 ssass ijiji
i - random Gaussian variables
L
iiiel ads
kTU
0
2
2Fluctuation energy:
Thermal Fluctuations
ia - persistence lengths
Phys. Rev. Lett. 85, 2404 (2000) Phys. Rev. E 62, 7135 (2000)
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01
31
21
1 aaa
Weak fluctuations of a helix:
skijk
sjiij
sjiji
RR esesetst
)sin()cos()0()( 0
0
002
0
00
20
00 1
e1
t3(s )t2(s )
t1(s )
s
e3 ( )
e2 ( )
Persistence lengths > helical period
,001 ,002 003 20
200 frequency
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Ribbon with spontaneous twist – model for dsDNA?
20 10 Lk
,10Lk0 10,100,1 321 aaa
)(1000);(350);(50/ cbakTFlf Europhys. Lett. 57, 512 (2002)
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Buckling under torsion: stability diagram
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Frenet-Based Computer Simulations
)'()'()( ssb
kTss ij
iji
1. Generate random numbers i
2. Integrate Frenet eqs. to generate configurations
3. Excluded volume, attractive interactions – Boltzmann weights
Direct simulation of fluctuating lines!
Phys. Rev. E 65 020801 (2002)
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Rectilinear rod 12321 bbb
L=2
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Does twist affect conformation?
is independent of twist !2R
Exact result: if there is no spontaneous curvature -
WLC model ok ?
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321 105.7,75.0 aa
Rectilinear ribbon
Twist affects conformation!
J. Chem. Phys. 118, 897 (2003)
L=2
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What about objects with spontaneous curvature?
Consider small deformations of a planar ring
y
x
2/0
rss /)(0
00
2r
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Twist and bending fluctuations – always decouple, but:
for curved filaments – twist is not simply rotation of cross-section!
Example: small fluctuations of a planar ring
andTwist rigidity - coupling between (rotation) (conformation)
0ta zero-energy modesrds
d
Out-of-plane fluctuations diverge!
(vanishes for )r
222
rds
da
ds
d
rds
dadsE tb
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Euler Angles
)0()( s )0()( s
)0()( s )0()( s
s/r
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Open Ring
1
4321 10,1 bbb
510
310
110
10
Pro
babi
lity
T=
Fluctuation-induced shape transitions – at fixed local curvature!
elastic moduli
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0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6-0.05
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35Y
Axi
s T
itle
X Axis Title
k0=0
k0=1
k0=2
k0=3
k0=4
Length L=1.5
Effect of spontaneous curvature on cyclization
Pro
babi
lity
of
R
End-to-end distance Rcyclization
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0.1 0.15 0.2 0.25 0.3
1E-3
0.01
log
(P(r
|r<
R)
log(R)
r3
Fundamental Exponent
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2 4 6 8 10 12 14 160.00000
0.00001
0.00002
0.00003
0.00004
k0=0
Yamakawa
P0
L
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1 22 3 44
1E-4
1E-3
0.01
k0=0
k0=1
k0=2
k0=3
k0=4
log
(P0)
L
Effect of constant spontaneous curvature
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0 1 2 3 4 5 6 7 8
0.0000
0.0001
0.0002
0.0003
0.0004
0.0005
0.0006
0.0007
Y A
xis
Titl
e
X Axis Title
sequence 1,2<3
k0=0
Effect of random spontaneous crvature
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0 1 2 3 4 5 6 7 8
0.0000
0.0001
0.0002
0.0003
0.0004
0.0005
0.0006
0.0007Y
Axi
s T
itle
X Axis Title
sequence 1 sequence 2 k
0=1.5
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1 2 3 4
0.005
0.010
0.015
0.020
0.025
P0
L
b3=0.01 b3=100 b3=1
40
Effect of twist rigidity on cyclization of curved filaments
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Stretching fluctuating filaments
Unbiased sampling of configurations – works only for small f
f
How are fluctuations affected by the force?
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Large-scale fluctuations are suppressed by stretching
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MS approximation breaks down for short filaments with L<a (neglect orientational effects)!
L=6.28
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All orientations are equally probable
Flexible chain Rigid filament
No Wall
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f=1 f=10f=2 f=3
y
xEnd fluctuations of stretched filaments:simulation results
Experiments: short dsDNA segments (ca 1000 bp)actin filaments
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Take home message:
Bending rigidity is not enough!
New generation of models of biofilaments that account for :
• intrinsic shape (spontaneous curvature and twist)
• twist rigidity