Download Locomotive force and angular momentum in 3D cell aggregates

Force & Angular Momentum in Cell Aggregates Antoine Fruleux (UoS, Sheffield)
(arXiv:1406.4820,tel-01095839)
Objectives
Experimental observations
Model
Migration of Dictyostelium Discoideum
Aims
● To explore the behavior of 3D Cell Aggregates
(multicellular systems).
Setup 1 : centrifugal force
Principe
Velocity (mm/h)
External force
Velocity (mm/h)
Ø= 108 μm L = 528 μm
Ø= 167 μm L = 967 μm
Direction of migration
, chemical gradient
Cell with cortical flow
Setup 2 : pressure difference
Migrating slug confined in a pipe
aggregate ("slug") with propagating cAMP wave
[K. Inouye et al, J. Cell Science 1980 ]
[K. Inouye et al, Protoplasma 1984 ]
, polarization of cells
Force
Ø= 69 μm L = 432 μm
Ø= 120 μm L = 380 μm
Force
Total active force
Inouye et al did experiments in which they confined a migrating slug into a thin pipe and opposed to its migration an external force. They observed the
migration velocity for various external conditions and they defined the total active force as the external force to apply to stop the slugs' migration.
●To shed light on the role of angular momentum
transfers in their macroscopic behavior.
During the migration stage of Dictystelium, individual cells aggregate into a
migrating slug. In the slug, a cAMP chemical wave propagates and the cells
tend to polarize along the chemical gradient. The polarized cells develop
cortical flows at their boundaries.
Observation 1
Total active force (E-3 N)
Naive picture:
Our approach
cancellation
of
cortical flow in the bulk
volume
By three steps: ● Coarse-graining of conserved fluxes
● Finding constitutive law for conserved fluxes
● Application to observations on Dictyostelium Discoideum
volume of slug (E-5 cc)
In the experimental observations, the total active force was
found to be proportional to the slugs volume.
Observation 2
Setup 3 : centrifugal force in an open geometry
Setup 1 : centrifugal force in an confined geometry
Velocity (mm/h)
Velocity (mm/h)
Ø= 69 μm L = 432 μm
Ø= 120 μm L = 380 μm
[M.Kitami. J. Cell Science 1982 ]
Force
But the naive consideration of the cancellations of the locomotive actions of the
cortical flows in the bulk of the aggregate, leads to a total active force
proportional to the contact area.
In a third experiment, Kitami exerted a centrifugal force on the migrating slug simply deposited on an agar substrate. As compared to the experiment of Inouye where the slug
was confined into a thin pipe, the velocity response to the external force is qualitatively different since in the open geometry the resistance of the slug to the external force
depends on the sens of the external force relatively to the sens of migration.
Aim
● To build a macroscopic description of the cell aggregate using our physical picture of the cell-cell interactions.
Force
Coarse graining of the linear/angular momentum conservations
●Linear/Angular momentum conservations,
We consider :
● Cell-cell interactions,
Action/Reaction law → Redundancies
Force distribution
at an interface
●Linea/Angular momentum fluxes,
, averaged number of neighbors.
relative position:
, cell density.
, average of
●Their distribution,
weighted by
.
To build a macroscopic description of our system, we consider cell-cell interactions such as the force and the torque exerted by the cells on their neighbors. We
consider also their distribution. Their distribution is submitted to redundancies relations imposed by the action reaction low. Those redundancies allows us to treat the
delicate cancellations of forces and torques in the aggregate to finally obtain the Linear/Angular momentum conservation in terms of averages of cell-cell quantities.
Constitutive laws
Geometry
Redundancy of viewpoint
We consider :
, the mean deviation of
the relative positions as the medium
evolves
●3-neighbor distribution function
Macroscopic
parameters
To establish the constitutive law for the conserved fluxes, we first give a geometrical description of the medium. We consider a three neighbor distribution describing
the statistics of the relative positions for triplets of neighbors. The redundancies of this distribution allow us to relate the mean deviation of the relative positions, as the
medium evolves, to the macroscopic parameters.
Mechanics
Mean field model of interactions
We consider :
● Mean deviations such as
Macroscopic
parameters
...
To relate the mean deviations to the macroscopic linear/angular momentum fluxes, we use a mean field model of interactions. Doing this, we can relate the cell-cell
interactions to the macroscopic parameters and finally obtain the macroscopic linear/angular momentum fluxes in terms of the Macroscopic parameters.
● Micro environment of a pair of cells
Results
Linear/Angular momentum fluxes :
Active term
Total active force :
Velocity response in the open/confined geometry:
Total active force (E-3 N)
Velocity (mm/h)
At fixed length :
Saturation of the total active force
→ Boundary layer effect
→ Thin sample :
Angular momentum play a role
The active force is proportional
to the volume
→ Thick sample :
Active force only from the
boundary
volume of slug (E-5 cc)
thickness
Force
thickness
Deviation of the polarization from the chemical gradient as a function of the
height for increasing values of the thickness (from left to right)
Intrinsic term
passive term
In our description, the total active force depends on the aspect ratio of the aggregate. For a given
length of the slug (dashed lines), we observe a saturation of the total active for higher
thicknesses. This is an effect of the boundary layer: The polarity and the flow velocity profiles
through the thickness of the aggregate shows a boundary layer for higher thicknesses.
Acknowledgements : Ken Sekimoto (U Paris Diderot / ESPCI)
Velocity response to the external force for increasing values of the aggregate thickness
(from left to right)
The asymmetric resistance of the slug to the external force seams also to be an effect of boundary layers. This behavior
appearing for higher thicknesses. It appears in the open geometry because of the higher thicknesses of the aggregates in this
geometry.