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## 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

David A. Head

F.C. MacKintosh

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).

The elasticity of flexible vs. semiflexible networks

C

B

A

A

B

C

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

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

The thermal persistence length:

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:

The mechanics of a semiflexible polymer: Stretching Thermal and Mechanical

Thermal modulus:

I. Thermal

Externally applied tension pulls out thermal fluctuations

II. Mechanical

2a

F

F

Mechanical Modulus:

Critical length

above which thermal modulus dominates

Young’s modulus for a protein typical of hard plastics

The collective elastic properties of semiflexible polymer networks

Individual filament properties:

Collective properties of the network:

u

W

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. 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

- 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

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 x10-5 (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

and/or

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

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?

- 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:

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.

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:

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)

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.

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

Large linear response regime

Affine Entropic: Extension dominated

Extension hardening

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).]

Then:

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.

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.

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