The mechanics of semiflexible networks:

Implications for the cytoskeleton

Alex J. Levine

Elastomers, Networks, and GelsJuly 2005

Collaborators:

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 deformationModes 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 networksPRL 91, 108102 (2003).

Jan Wilhelm and Erwin Frey Elasticity of Stiff Polymer Networks PRL 91, 108103 (2003).

The elasticity of flexible vs. semiflexible networks

A

B C

The red chain makes independent random walks between cross-links

(A,B) and (B,C).

Flexible Polymeric Gels

Semiflexible Polymeric Gels

The green chain tangent vectorbetween cross-links (A,B) is strongly

correlated with the tangent vector between cross-links (B,C).

A

BC

Filament length can play a role in the elasticity

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

F-actin

7 nm

G-actin, a globular protein of MW=43k

Semiflexible networks in the cell

Keratocyte cytoskeleton

The cytoskeletal networkfound in the cortex associatedwith the cell membrane.

The mechanics of a semiflexible polymer: Bending

There is an energy cost associatedwith bending the polymer in space.

Where:

Consequences in thermal equilibrium:

Bending modulus

The thermal persistence length:

Exponential decay of tangent vector correlationsdefines the thermal persistence length

The mechanics of a semiflexible polymer: Stretching Thermal and Mechanical

Externally applied tension pulls out thermal fluctuations

txu ,

2a

F F

II. Mechanical

I. Thermal

Thermal modulus:

Critical lengthabove which thermal modulus dominates

Mechanical Modulus:

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:

W

u

Numerical model of the semiflexible network

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.

Cross linksMid-pointsDangling end

-actinin and filamin

A new understanding of semiflexible gels

Nonaffine

Affine

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 LimitF.C. MacKintosh, J. Käs, and P.A. Janmey PRL 75, 4425 (1995). Affine deformations

A rapid transition in both the geometry of the deformation field

and the mechanical properties of the network

Summary

Three lengths characterize the semiflexible network

Example network with a crosslink density L/lc = 29 in a shear cell of dimensions

W●W and periodic boundary conditions in both directions.

A small example:

There are three length scales:

Rod length:

Mean distance between cross links:

Natural bending length:

• Zero temperature• Two-dimensional• Initially unstressed

For a flexible rod

2a

The shear modulus of affinely deforming networks

Consider one filament in a sea of others:

Under simple shear it stretches from L to L:

Averaging over angles 0 to and multiplying by the number density of the rods:

The total increase in stretching energy of the rod is:

N = rods/area

Freely rotating cross-links implies no bending energy in affinely deformed networks

A pictorial representation of the affine-to-nonaffine transition:Energy stored in stretch and bend deformations

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) (b) (c)

Sheared networks in mechanical equilibrium. lb/L = 2x10-3 with network densitiesL/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

(a) (b) (c)

A pictorial representation of the affine-to-nonaffine transition:Energy stored in stretch and bend deformations

The affine theory is dominated entirely

by stretching

Bending dominated when:

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)

L/lc = 29.09

Fraction of stretching energy

More dense networks: More affine More stiff filaments: More affine

Data collapse for affine transition

Direct measure of nonaffinity vs. length scale

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

We compute the nonaffine measure:

Under shear:

The connection between mechanics and geometry

?

What is the length scale for affinity?

The system attempts to deform nonaffinely on lengths below

Potentialnon-affine domain

From numerical data collapse:

A scaling argument predicts this exponent to be:

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

When filaments are long and stiff they enforce affine deformation: A competition between and L.

One filament

The length scale for non-affine deformations: Relaxing stretch by producing bend

Extensional stress vanishes near the ends over a length:

Extension direction

Reduction of stretching energy:

But segment is displaced by:

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:

To maximize the reduction:

Why do these bend and not just translate? They are tied into thelarger network, which must also be deforming as well!

The net energy change due to non-affine contraction of the end:

Typical number of crossingfilaments

Typical number of crossingfilaments

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?

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 scalingdue the proximity of the rigidity percolation point at lower densities.

Highest density

Attempted datacollapse with:

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

Nonaffine: Bending dominated

Large linear response regime

Affine Entropic: Extension dominated

Extension hardening

Nonlinear Rheology: A Qualitative Difference

Experimental evidence of the nonaffine-to-affine cross-over

Stress Stiffening

10-3 10-2 10-110-2

10-1

100

G' (

Pa)

(Pa)

No Stress Stiffening

10-2 10-1 100 101

100

101

102

103

' (

Pa)

(Pa)

There is an abrupt change in thenonlinear rheology of actin/scruin

networks.[M.L. Gardel et al, Science 304, 1301 (2004).]

Where is the physiological cytoskeleton with respect to the affine/nonaffine crossover?

If we take:

Then:

The cytoskeleton is at a high susceptibility

point where small biochemical

changes generate large mechanical ones.

[Human neutrophil]

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.