The mechanics of semiflexible networks:

1. Elastomers, Networks, and Gels July 2005 The mechanics of semiflexible networks: Implications for the cytoskeleton Alex J. Levine

2. 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 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).

3. 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

4. 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.

5. 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:

6. 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

7. The collective elastic properties of semiflexible polymer networks Individual filament properties: Collective properties of the network:  u W 

8. 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

9. 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

10. 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

11. 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

12. 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.

13. 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

14. 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) 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

15. 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

16. 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.

17. 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:

18. 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)

19. Highest density 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.

20. 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.

21. 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

22. 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).]

23. If we take: 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.

24. 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.