Download The Dynamics of Microscopic Filaments

The Dynamics of Microscopic Filaments
Christopher Lowe
Marco Cosentino-Lagomarsini (AMOLF)
The Dynamics of Microscopic Filaments
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
Example, tying a knot in Actin
Accounting for the fluid
At its simplest, resistive force theory
Ff    v 
Ff   ||v||

v
  || are respectively the perpendicular and
parallel friction coefficients of a cylinder
Gives good predictions for the swimming speed
of simple spermatozoa
Why might this not give a complete picture?
A simple model, a chain of rigidly connected
point particles with a friction coefficient 
F
Vf
Why might this not give a complete picture?
A simple model, a chain of rigidly connected
point particles with a friction coefficient 
F
Vf
Ff = - (v-vf)
v
Vf
The Oseen tensor gives the solution to the inertialess
fluid flow equations for a point force acting on a fluid

 


 
 r 
1 F
v f (r ) 
 F r 3
8  r
r 


These equations are linear so solutions just add

 

 

F

 r 

j

Fif (ri )  vi 
 F j  rij 3


8 i  j  rij
rij 


Approximate the solution as an integral. For
a uniform perpendicular force.

F
 s(1  s)  
F f ( s)    v 
ln 
 
2
8b  b


•s = the distance along a rod of unit length
•b = is the bead separation
Approximate the solution as an integral. For
a uniform perpendicular force.

F
 s(1  s)  
F f ( s)    v 
ln 
 
2
8b  b


•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
Ff
Fx
Ft
Fb
Fb - bending force (from the bending energy for a
filament with stiffness G)
Ft - Tension force (satisfies constraint of no relative
displacement 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


4L
 2 2 
4L

ln ln
L / b L / b 
||

||
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


4L
 2 2 
4L

ln ln
L / b L / b 
||

||
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 the
filament travel before re-orientating itself?
Is this experimentally relevant?
•For a microtobule, Sed ~ 1 requires F~1 pN. This is
reasonable on the micrometer scale.
•For sedimentation, no. Gravity is not strong enough.
You’d need a ultracentrifuge
•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 flexible
filaments taking into account the non-local
nature of the filament/solvent interactions
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
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