modeling of biofilaments: elasticity and fluctuations combined d. kessler, y. kats, s. rappaport...
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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
Stretching of helical springs
Overview
1. Motivation
2. Ribbons: geometry, elasticity, fluctuations
3. Computer simulations: Frenet algorithm
Stretching of filaments
Twisting dsDNA
CyclizationDistribution functions
Polymers – objects with atomic thickness (1 A) and arbitrary length
Atomic resolution
Quantum mechanics
RIS models
Coarse grained description
Statistical mechanics
Random walks
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
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
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:
Bending elasticity of inextensible lines
Modeling dsDNA at large deformations
Bustamante et al., Science 265, 1599 (1994)
The first step:
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)
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 !
Helix
p
2r22
22
2
r
p
Straight line
0
0
/1
r
Circle
r
Ribbons (stripes)
t2(s)
t1(s)
Physical triad: t1, t2, t3
n(s)
b(s))(s
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
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
Stretching a helical spring
pitch > radius, bending rigidity > twist rigidity4 turns,
minimize ))(,,()(2
100
2
0sFRsdsbE ii
L
ii
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
Stretching helical ribbons of cholesterol:
Smith, Zastavker and Benedek, Phys. Rev. Lett. 87, 278101 (2001)
Mechanical noise-induced transitions?
Stretching transitions and hysteresis in chromatin ?
Y. Cui and C. Bustamente, PNAS 97, 127 (2000).
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)
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
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)
Buckling under torsion: stability diagram
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)
Rectilinear rod 12321 bbb
L=2
Does twist affect conformation?
is independent of twist !2R
Exact result: if there is no spontaneous curvature -
WLC model ok ?
321 105.7,75.0 aa
Rectilinear ribbon
Twist affects conformation!
J. Chem. Phys. 118, 897 (2003)
L=2
What about objects with spontaneous curvature?
Consider small deformations of a planar ring
y
x
2/0
rss /)(0
00
2r
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
Euler Angles
)0()( s )0()( s
)0()( s )0()( s
s/r
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
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
0.1 0.15 0.2 0.25 0.3
1E-3
0.01
log
(P(r
|r<
R)
log(R)
r3
Fundamental Exponent
2 4 6 8 10 12 14 160.00000
0.00001
0.00002
0.00003
0.00004
k0=0
Yamakawa
P0
L
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
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
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
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
Stretching fluctuating filaments
Unbiased sampling of configurations – works only for small f
f
How are fluctuations affected by the force?
Large-scale fluctuations are suppressed by stretching
MS approximation breaks down for short filaments with L<a (neglect orientational effects)!
L=6.28
All orientations are equally probable
Flexible chain Rigid filament
No Wall
f=1 f=10f=2 f=3
y
xEnd fluctuations of stretched filaments:simulation results
Experiments: short dsDNA segments (ca 1000 bp)actin filaments
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
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