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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)(
r
rrF
r
Frv f
ji ij
ijjij
jiiif
r
rrF
r
FvrF
38)(
Approximate the solution as an integral. Fora uniform perpendicular force.
2
)1(ln
8)(
b
ss
b
FvsF 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(ln
8)(
b
ss
b
FvsF 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 || bL
L
/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 || bL
L
/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
