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

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 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 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

4. 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 network found in the cortex associated with the cell membrane.

5. The mechanics of a semiflexible polymer: Bending There is an energy cost associated with bending the polymer in space. Where: Consequences in thermal equilibrium: Bending modulus  The thermal persistence length: Exponential decay of tangent vector correlations defines the thermal persistence length

6. The mechanics of a semiflexible polymer: Stretching Thermal and Mechanical Externally applied tension pulls out thermal fluctuations 2a FF II. Mechanical I. Thermal Thermal modulus: Critical length above which thermal modulus dominates Mechanical Modulus: 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: W u  

8. 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 links Mid-points Dangling end  -actinin and filamin

9. 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 Limit F.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

10. Three lengths characterize the semiflexible network Example network with a crosslink density L/l c = 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

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: 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

12. A pictorial representation of the affine-to-nonaffine transition: Energy stored in stretch and bend deformations Sheared networks in mechanical equilibrium. L/l c = 29.09 with differing filament bending moduli: l b /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)

13. Sheared networks in mechanical equilibrium. l b /L = 2x10 -3 with network densities L/l c = 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

14. 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/l c = 29.09 As predicted by E. Frey, K. Kroy, J. Wilhelm (1998) L/l c = 29.09 Fraction of stretching energy More dense networks: More affineMore stiff filaments: More affine

15. 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 r2r2 r1r1 We compute the nonaffine measure: Under shear: The connection between mechanics and geometry ?

16. What is the length scale for affinity? The system attempts to deform nonaffinely on lengths below Potential non-affine domain From numerical data collapse: A scaling argument predicts this exponent to be: Trends: As the cross link density goes up (l c ) the system becomes more affine As the bending stiffness goes up (l b ) the system becomes more affine When filaments are long and stiff they enforce affine deformation: A competition between and L. One filament

17. 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 :

18. 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 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 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. 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 scaling due the proximity of the rigidity percolation point at lower densities. Highest density Attempted data collapse with:

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 Nonaffine: Bending dominated Large linear response regime Affine Entropic: Extension dominated Extension hardening Nonlinear Rheology: A Qualitative Difference

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. 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]

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.