Force & Angular Momentum in Cell Aggregates      

Antoine Fruleux (UoS, Sheffield)

Principe

Experimental observationsObjectivesAims

Our approach

● Coarse-graining of conserved fluxes● Finding constitutive law for conserved fluxes● Application to observations on Dictyostelium Discoideum

Geometry

Results

Linear/Angular momentum fluxes : Total active force : Velocity response in the open/confined geometry:

● To explore the behavior of 3D Cell Aggregates (multicellular systems).

●To shed light on the role of angular momentum transfers in their macroscopic behavior.

By three steps:

Model Migration of Dictyostelium Discoideum

, chemical gradient

aggregate ("slug") with propagating cAMP wave

, polarization of cellsCell with cortical flow

Migrating slug confined in a pipe

Direction of migration

External force

Setup 1 : centrifugal force Setup 2 : pressure difference

Velocity (mm/h) Velocity (mm/h)

Ø= 108 μm L = 528 μmØ= 167 μm L = 967 μm

Ø= 69 μm L = 432 μmØ= 120 μm L = 380 μm

[K. Inouye et al, J. Cell Science 1980 ][K. Inouye et al, Protoplasma 1984 ] Force Force

Total active force

Observation 1Total active force (E-3 N)

volume of slug (E-5 cc)

volumecancellation

of cortical flow in the bulk

Naive picture:

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.

Observation 2

In the experimental observations, the total active force was found to be proportional to the slugs volume.

Setup 3 : centrifugal force in an open geometry

Ø= 69 μm L = 432 μmØ= 120 μm L = 380 μm

Force

Setup 1 : centrifugal force in an confined geometryVelocity (mm/h)Velocity (mm/h)

Force

[M.Kitami. J. Cell Science 1982 ]

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.

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.

Coarse graining of the linear/angular momentum conservations

Aim

● To build a macroscopic description of the cell aggregate using our physical picture of the cell-cell interactions.

We consider :

● Cell-cell interactions,

●Their distribution,

relative position:

Action/Reaction law → Redundancies

●Linear/Angular momentum conservations,

●Linea/Angular momentum fluxes,

, averaged number of neighbors.

, cell density.

, average of 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.

Redundancy of viewpoint

, the mean deviation of the relative positions as the medium evolves

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.

Force distribution at an interface

Mechanics

We consider :

We consider :

●3-neighbor distribution function

● Mean deviations such as

Mean field model of interactions ...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.

Intrinsic term passive term

Total active force (E-3 N)

volume of slug (E-5 cc)

At fixed length :Saturation of the total active force→ Boundary layer effect

→ Thin sample :Angular momentum play a roleThe active force is proportional to the volume→ Thick sample :Active force only from the boundary

Velocity (mm/h)

Force

thickness

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)

Velocity response to the external force for increasing values of the aggregate thickness (from left to right)

Active term

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.

Constitutive laws

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.

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.

Acknowledgements : Ken Sekimoto (U Paris Diderot / ESPCI)

(arXiv:1406.4820,tel-01095839)

● Micro environment of a pair of cells