The Dynamics of Microscopic Filaments

Christopher LoweMarco Cosentino-Lagomarsini (AMOLF)

Christopher LoweMarco 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

Example, tying a knot in Actin

Accounting for the fluid

At its simplest, resistive force theory

||||vFf vF f

v

are respectively the perpendicular and parallel friction coefficients of a cylinder

||

Gives good predictions for the swimming speedof simple spermatozoa

Vf

F

Why might this not give a complete picture?

A simple model, a chain of rigidly connectedpoint particles with a friction coefficient

Vf

F

Why might this not give a complete picture?

A simple model, a chain of rigidly connectedpoint particles with a friction coefficient

Ff = -(v-vf)

v Vf

The Oseen tensor gives the solution to the inertialessfluid flow equations for a point force acting on a fluid

These equations are linear so solutions just add

38

1)(rrrF

rFrv f

ji ijijj

ij

jiiif r

rrFrF

vrF 38)(

Approximate the solution as an integral. Fora uniform perpendicular force.

2

)1(ln8

)(bss

bFvsF f

•s = the distance along a rod of unit length•b = is the bead separation

Approximate the solution as an integral. Fora uniform perpendicular force.

2

)1(ln8

)(bss

bFvsF f

•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

Numerical Model

Fb

Ft

Fx

Ff

Fb - bending force (from the bending energy for afilament with stiffness G)

Ft - Tension force (satisfies constraint of no relativedisplacement along the line of the links)

Ff - Fluid force (from the model discussed earlier,with F the sum of all non hydrodynamic forces)

Fx - External force

Solve equations of motion (with m << L / v)

Advantages

•Simple (a few minues CPU per run)•Gives the correct rigid rod friction coefficient in the limit of a large number of beads

bLL/ln

42 || bLL/ln

42 ||

if the bead separation is interpreted as the cylinder radius

Advantages

•Simple (a few minues CPU per run)•Gives the correct rigid rod friction coefficient in the limit of a large number of beads

bLL/ln

42 || bLL/ln

42 ||

if the bead separation is interpreted as the cylinder radius

Disadvantages

•Only approximate for a given finite aspect ratio

What happens?

Sed = FL2/G = ratio of bending to hydrodynamic forces

Sed = 10

Sed = 100

Sed = 500

Sed = 1, filament aligned at 450

How many times its own length does thefilament travel before re-orientating itself?

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 isreasonable on the micrometer scale.

•Microtubules are barely charged, we estimate an electric field of 0.1 V/m for Sed ~ 1

Conclusions

•We have a simple method to model flexiblefilaments taking into account the non-localnature of the filament/solvent interactions

Conclusions

•We have a simple method to model flexiblefilaments taking into account the non-localnature of the filament/solvent interactions

•When we do so for the simplest non-trivial dynamic problem (sedimentation) the response of the filamentis somewhat more interesting than local theories suggest

Conclusions

•We have a simple method to model flexiblefilaments taking into account the non-localnature of the filament/solvent interactions

•When we do so for the simplest non-trivial dynamic problem (sedimentation) the response of the filamentis somewhat more interesting than local theories suggest

•It’s just a model, so we hope it can be tested against experiment