1. The Dynamics of Microscopic Filaments Christopher Lowe Marco Cosentino-Lagomarsini (AMOLF)

2. Christopher Lowe Marco Cosentino-Lagomarsini (AMOLF) Why we’re interested: Flexible filaments are common in biology New experimental techniques allow them to be imaged and manipulated It’s fun The Dynamics of Microscopic Filaments

3. Example, tying a knot in Actin

4. Accounting for the fluid At its simplest, resistive force theory are respectively the perpendicular and parallel friction coefficients of a cylinder

5. Gives good predictions for the swimming speed of simple spermatozoa

6. VfVf F Why might this not give a complete picture? A simple model, a chain of rigidly connected point particles with a friction coefficient 

7. VfVf F Why might this not give a complete picture? A simple model, a chain of rigidly connected point particles with a friction coefficient  F f = -  (v-v f ) v VfVf

8. The Oseen tensor gives the solution to the inertialess fluid flow equations for a point force acting on a fluid These equations are linear so solutions just add

9. Approximate the solution as an integral. For a uniform perpendicular force. s = the distance along a rod of unit length b = is the bead separation

10. Approximate the solution as an integral. For a uniform perpendicular force. s = the distance along a rod of unit length b = is the bead separation If the velocity is uniform the friction is higher at the end than in the middle

11. Numerical Model FbFb FtFt FxFx FfFf F b - bending force (from the bending energy for a filament with stiffness G) F t - Tension force (satisfies constraint of no relative displacement along the line of the links) F f - Fluid force (from the model discussed earlier, with F the sum of all non hydrodynamic forces) F x - External force Solve equations of motion (with m << L  / v)

12. Advantages Simple (a few minues CPU per run) Gives the correct rigid rod friction coefficient in the limit of a large number of beads if the bead separation is interpreted as the cylinder radius

13. Advantages Simple (a few minues CPU per run) Gives the correct rigid rod friction coefficient in the limit of a large number of beads if the bead separation is interpreted as the cylinder radius Disadvantages Only approximate for a given finite aspect ratio

14. What happens? Sed = FL 2 /G = ratio of bending to hydrodynamic forces

15. Sed = 10

16. Sed = 100

17. Sed = 500

18. Sed = 1, filament aligned at 45 0

19. How many times its own length does the filament travel before re-orientating itself?

20. Is this experimentally relevant? For sedimentation, no. Gravity is not strong enough. You’d need a ultracentrifuge For a microtobule, Sed ~ 1 requires F~1 pN. This is reasonable on the micrometer scale. Microtubules are barely charged, we estimate an electric field of 0.1 V/m for Sed ~ 1

21. Conclusions We have a simple method to model flexible filaments taking into account the non-local nature of the filament/solvent interactions

22. Conclusions We have a simple method to model flexible filaments taking into account the non-local nature of the filament/solvent interactions When we do so for the simplest non-trivial dynamic problem (sedimentation) the response of the filament is somewhat more interesting than local theories suggest

23. Conclusions We have a simple method to model flexible filaments taking into account the non-local nature of the filament/solvent interactions When we do so for the simplest non-trivial dynamic problem (sedimentation) the response of the filament is somewhat more interesting than local theories suggest It’s just a model, so we hope it can be tested against experiment