Transcript On the deformation of semiflexible networks:

Elastomers, Networks, and Gels
July 2005
The mechanics of semiflexible
networks:
Implications for the cytoskeleton
Alex J. Levine
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).
Collaborators:
David A. Head
F.C. MacKintosh
The elasticity of flexible vs. semiflexible networks
B
C
Flexible Polymeric Gels
The red chain makes independent
random walks between cross-links
(A,B) and (B,C).
A
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).
A
B
C
Filament length can play a role in the
elasticity
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
MW=43k
F-actin
Keratocyte cytoskeleton
7 nm
The cytoskeletal network
found in the cortex associated
with the cell membrane.
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
Where:
defines the thermal persistence length
The thermal persistence length:
The mechanics of a semiflexible polymer:
Stretching Thermal and Mechanical
I. Thermal

ux, t 

Externally applied tension pulls out thermal fluctuations
Thermal modulus:
II. Mechanical
2a
F
F
Mechanical Modulus:
Young’s modulus for a protein
typical of hard plastics
Critical length
above which thermal modulus dominates
The collective elastic properties of semiflexible polymer networks
Individual filament properties:
Collective properties of the network:

u
W

Numerical model of the semiflexible network
We study a discrete, linearized model:
Cross links
Mid-points
Dangling end
• 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. T=0 simulation.
-actinin and filamin
A new understanding of semiflexible gels
Affine
Nonaffine
A rapid transition in both the geometry
of the deformation field
and the mechanical properties of the network
Summary
1.
We find that there is a length scale,  below which deformations become nonaffine.
2.
 depends on both the density of cross links and the stiffness of the filaments.
3.
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
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
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
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.
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
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
Bending dominated when:
and/or
The affine theory
is dominated entirely
by stretching
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
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.
Trends:
• As the cross link density goes up (lc ) the system becomes more affine
• As the bending stiffness goes up (lb ) the system becomes more affine
A scaling argument predicts this exponent to be:
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:
Creation of bending energy:
Bending correlation
length
The net energy change due to non-affine contraction of the end:
To maximize the reduction:
Typical number of crossing
filaments
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:
To minimize energy increase w.r.t.
the bend correlation length:
Comparing the two results:
Typical number of crossing
filaments
(This length should be the bigger of the two)
The correct asymptotic exponent?
Highest density
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.
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.
Experimental implications of the affine to nonaffine transition
Nonlinear Rheology: A Qualitative Difference
Nonaffine: Bending dominated
Affine Entropic: Extension dominated
Large linear response regime
Extension hardening
Experimental evidence of the nonaffine-to-affine cross-over
' (Pa)
Stress Stiffening
10
3
10
2
10
1
10
0
10
-2
10
-1
10
 (Pa)
0
10
1

No Stress Stiffening
G' (Pa)
10
0
10
-1
10
-2
10
There is an abrupt change in the
nonlinear rheology of actin/scruin
networks.
-3
10
-2
 (Pa)
10
-1
[M.L. Gardel et al, Science 304, 1301 (2004).]
Where is the physiological cytoskeleton with respect to the
affine/nonaffine crossover?
[Human neutrophil]
The cytoskeleton is at a high
susceptibility
point where small
biochemical
changes generate
large mechanical ones.
If we take:
Then:
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.