The mechanics of semiflexible networks:

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The mechanics of semiflexible networks
Elastomers, Networks, and Gels July 2005

• Implications for the cytoskeleton

Alex J. Levine
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For more information A. J. Levine, D.A. Head,
and F.C. MacKintosh Short-range deformation of
semiflexible networks Deviations from continuum
elasticity PRE (2005). A. J. Levine, D.A. Head,
and F.C. MacKintosh The Deformation Field in
Semiflexible Networks Journal of Physics
Condensed Matter 16, S2079 (2004). D.A. Head,
A.J. Levine, and F.C. MacKintosh Distinct regimes
of elastic response and dominant
deformation Modes of cross-linked cytoskeletal
and semiflexible polymer networks PRE 68, 061907
(2003). D.A. Head, F.C. MacKintosh, and A.J.
Levine Non-universality of elastic exponents in
random bond-bending networks PRE 68, 025101 (R)
(2003). D.A. Head, A.J. Levine, and F.C.
MacKintosh Deformation of cross-linked
semiflexible polymer networks PRL 91, 108102
(2003). Jan Wilhelm and Erwin Frey Elasticity of
Stiff Polymer Networks PRL 91, 108103 (2003).
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The elasticity of flexible vs. semiflexible
networks
Flexible Polymeric Gels
The red chain makes independent random walks
between cross-links (A,B) and (B,C).
Semiflexible Polymeric Gels
The green chain tangent vector between
cross-links (A,B) is strongly correlated with the
tangent vector between cross-links (B,C).
Filament length can play a role in the elasticity
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Semiflexible networks in the cell

• Eukaryotic cells have a cytoskeleton, consisting
  largely of semi-flexible polymers, for structure,
  organization, and transport

G-actin, a globular protein of MW43k
F-actin
Keratocyte cytoskeleton
7 nm
The cytoskeletal network found in the cortex
associated with the cell membrane.
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The mechanics of a semiflexible polymer Bending
There is an energy cost associated with bending
the polymer in space.
Bending modulus ?
Consequences in thermal equilibrium
Exponential decay of tangent vector
correlations defines the thermal persistence
length
Where
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The mechanics of a semiflexible polymer
Stretching Thermal and Mechanical
I. Thermal
Externally applied tension pulls out thermal
fluctuations
II. Mechanical
2a
F
F
Mechanical Modulus
Critical length above which thermal modulus
dominates
Youngs modulus for a protein typical of hard
plastics
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The collective elastic properties of semiflexible
polymer networks
Individual filament properties
Collective properties of the network
?
u
W
?
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Numerical model of the semiflexible network
Cross links Mid-points Dangling end
We study a discrete, linearized model

• Mid-points are included to incorporate the
  lowest order bending modes.
• Cross-links are freely rotating (more like
  filamin than ?-actinin)
• Uniaxial or shear strain imposed via boundary
  conditions (Lees-Edwards)
• Resulting displacements are determined by Energy
  minimization. T0 simulation.

?-actinin and filamin
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A new understanding of semiflexible gels
A rapid transition in both the geometry of the
deformation field and the mechanical properties
of the network
Summary

• We find that there is a length scale, ? below
  which deformations become nonaffine.
• ? depends on both the density of cross links and
  the stiffness of the filaments.
• We understand the modulus of material in the
  affine limit.
• K. Kroy and E. Frey PRL 77, 306 (1996). E. Frey,
  K. Kroy, and J. Wilhelm (1998). Bending Limit
• F.C. MacKintosh, J. Käs, and P.A. Janmey PRL 75,
  4425 (1995). Affine deformations

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Three lengths characterize the semiflexible
network
A small example
Example network with a crosslink density L/lc
29 in a shear cell of dimensions W?W and
periodic boundary conditions in both directions.

• Zero temperature
• Two-dimensional
• Initially unstressed

There are three length scales
Rod length
Mean distance between cross links
2a
Natural bending length
For a flexible rod
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The shear modulus of affinely deforming networks
Consider one filament in a sea of others
Under simple shear it stretches from L to ?L
Freely rotating cross-links implies no bending
energy in affinely deformed networks
The total increase in stretching energy of the
rod is
Averaging over angles 0 to ? and multiplying by
the number density of the rods
N rods/area
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A pictorial representation of the
affine-to-nonaffine transition Energy stored in
stretch and bend deformations
(a)
(b)
(c)
Sheared networks in mechanical equilibrium. L/lc
29.09 with differing filament bending
moduli lb/L 2 x 10-5 (a), 2 x 10-4 (b) and 2 x
10-2(c). Dangling ends have been removed. The
calibration bar shows what proportion of the
deformation energy in a filament segment is due
to stretching or bending.
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A pictorial representation of the
affine-to-nonaffine transition Energy stored in
stretch and bend deformations
(a)
(b)
(c)
Sheared networks in mechanical equilibrium. lb/L
2x10-3 with network densities L/lc 9.0 (a),
29.1 (b) and 46.7 (c). Dangling ends have been
removed. The calibration bar shows what
proportion of the deformation energy in a
filament segment is due to stretching or
bending. Line thickness is proportional to total
storaged energy in that filament
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The mechanical signature of the transition Shear
Modulus of the filament network
L/lc 29.09
As predicted by E. Frey, K. Kroy, J. Wilhelm
(1998)
More dense networks More affine
More stiff filaments More affine
Fraction of stretching energy
L/lc 29.09
The affine theory is dominated entirely by
stretching
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The connection between mechanics and geometry
A purely geometric measure of affine deformations
Note Affinity is a function of length scale
We use the deviation of the rotation angle ?
between mass points in the deformed network from
its value under affine shear deformation.
Applied shear
r2
r1
? ?
Data collapse for affine transition
Under shear
We compute the nonaffine measure
Direct measure of nonaffinity vs. length scale
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What is the length scale for affinity?
From numerical data collapse
Potential non-affine domain
?
The system attempts to deform nonaffinely on
lengths below ?
One filament
When filaments are long and stiff they enforce
affine deformation A competition between ? and L.
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The length scale for non-affine deformations
Relaxing stretch by producing bend
Extensional stress vanishes near the ends over a
length
Reduction of stretching energy
But segment is displaced by
Extension direction
The displacement of the segment by d causes the
cross-linked filaments to bend
Induced curvature
Bending correlation length
Creation of bending energy
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The net energy change due to non-affine
contraction of the end
Typical number of crossing filaments
To maximize the reduction
Why do these bend and not just translate? They
are tied into the larger network, which must also
be deforming as well!
The net energy change due to non-affine
contraction of the end
Typical number of crossing filaments
To minimize energy increase w.r.t. the bend
correlation length
Comparing the two results
(This length should be the bigger of the two)
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The correct asymptotic exponent?
Attempted data collapse with
At higher filament densities the z 1/3 data
collapse appears to fail. z 2/5 may be high
density exponent and there are corrections to
this scaling due the proximity of the rigidity
percolation point at lower densities.
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Proposed phase diagram Rigidity percolation and
the Affine/Non-affine cross-over
Rigidity Percolation D.A. Head, F.C.
MacKintosh, and A.J. Levine PRE 68, 025101 (R)
(2003).
There is a line of second order phase transitions
at the solution-to-gel point.
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Experimental implications of the affine to
nonaffine transition
Nonlinear Rheology A Qualitative Difference
Nonaffine Bending dominated Large linear
response regime
Affine Entropic Extension dominated Extension
hardening
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Experimental evidence of the nonaffine-to-affine
cross-over
Stress Stiffening
No Stress Stiffening
There is an abrupt change in the nonlinear
rheology of actin/scruin networks. M.L. Gardel
et al, Science 304, 1301 (2004).
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Where is the physiological cytoskeleton with
respect to the affine/nonaffine crossover?
Human neutrophil
The cytoskeleton is at a high susceptibility poin
t where small biochemical changes generate
large mechanical ones.
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Summary
Semiflexible networks allow a more rich range of
mechanical properties

• The Affine-to-Nonaffine cross-over is a
  simultaneous abrupt change in the geometry of the
  deformation field at mesoscopic lengths, form of
  elastic energy storage, as well as the linear and
  nonlinear rheology of the network.
• Can reconcile previous work in the field K.
  Kroy and E. Frey (Bending/Nonaffine deformation)
  vs. F.C. MacKintosh, J. Käs, and P.A. Jamney
  (Stretching/Affine deformation)
• In the vicinity of the cross-over both the
  linear and nonlinear mechanical properties of the
  network are highly tunable.
• Simple estimates suggests that the eukaryotic
  cytoskeleton exploits this tunability.