Modeling Chemotaxis, Cell Adhesion and Cell Sorting.

Examples with Dictyostelium

Eirikur PálssonDept of Biology, Simon Fraser University

•Throughout gastrulation and embryogenesis.

•In wound healing.

•Carcinoma cell invasion.

•Limb bud regeneration.

•Cell movement in Dictyostelium discoideum.

Examples of Processes Where Cell Movement Is Important:

Purpose of a Cell Movement model

• Visualization of cell movements in 3-D

• Understand how simple cell-cell interactions, signaling and adhesion lead to complex cell movements

• Simplification; Revealing the most important things

• Gives constraints. Suggests what behavior is possible

•Introduction

•Design of Model

•Results

•Conclusions & Future Work

Outline

•The basic unit of the model is an ellipsoidal cell

•Deformation of the cell depends on the history of the forces acting on it

•The cell conserves volume with variable ellipsoidal semi-axes

•The cell may adhere to other cells or to the substrate

•When the cell moves it sends out a pseudopod, attaches it to either a neighbor or the surface

•The cell responds to chemotactic signals

The Model.

A Representation of the Deformability of Each Axis

€

dridt

=κ1(Fi −ff)μ1(κ1 +κ2)

+dFidt

(κ1 +κ2)−ri

κ1κ2

μ1(κ1 +κ2)

€

rarbrc =Vellipse=const

di

dj

€

rr new =

∇γ (ˆ I − 2exp(−10η (r v ran

r v ran ))) if γ > γ thres

r a (ˆ I − 2exp(−3η (

r v ran

r v ran ))) if γ ≤ γ thres

⎧ ⎨ ⎩

€

dr a

dt= β (1− u)(

r r new −

r a )

du

dt= k2(γ − γ bac )(1− u) − k3u

Rotation

€

α =rcell

2(1di

+1dj

), x=(d

rcell

−b), b=mindist

r F =F

r r ijr r ij

Forces (static)

Force equations

€

r F i

stat=r F i( j /s)

act +r F ij

pass

j∈N(i )

∑ −r F ji

act

j∈N(i )

∑r F i

D =μsAis

A

r v i +μc

Aij

A(r v i

j∈N(i)

∑ −r v j )

€

r F i

stat=r F i

D =μs

Ais

Ar v i +μc

Aij

A(r v i

j∈N(i )

∑ −r v j)

Equation of motion

•All the neighbor cells are found

•The chemical gradient around each cell is calculated

•The cells orient towards the chemical gradient and apply an active force in that direction

•All the forces acting on a cell are determined. These are of two types; The passive and the active forces

•The cells are moved and deformed according to the equations of motion

•The chemical concentration is updated.

Temporal Evolution of the Model

R Foty 1996

Distance btwPlates

Surface Tension(dyne/cm)

Force(nN)

4.5 23 ± 2 575 ± 30

5.0 26 ± 2 440 ± 15

5.5 26 ± 2 340 ± 20

Sorting

Eaa ,Ebb > Eab

Eab > Eaa, Ebb

Ebb > Eab >Eaa

Separation

Random

Limb Bud

Pigm. Epith.

Heart

Liver

N. Retina

Green

Red

Yellow

Blue

Orange

AdhesionColorType

20.1

12.6

8.5

4.6

1.6R Foty 1996

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Sor

ting

Time

Sor

ting

Time

Sorting: With or Without Random Cell Movement

Sorting: Changing Cell Stiffness and Adhesion

Normal Cell

Stiffer Cell

Stiffer and more Adhesive Cell

Not Random

Random Motion

Sorting of Pre-spore and Pre-Stalk Cells

Takeuchi 1986

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Sorting due to specific Cell Adhesion

Dictyostelium discoideum Life Cycle

Camp Waves During Aggregation

1 mmK Lee Princeton U

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Aggregation (Firtel)

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Simulations of Dictyostelium discoideum Aggregation in Response to cAMP signals.

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Aggregation. Pacemaker Cells in Red

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Aggregation. 1 to 1 Pacemaker Cells in Red

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Aggregation,1-1. Reduced Diffusion

Bonner 1999

Simulations of 2-D slugs The Red Cells in the front

are cAMP Pacemakers

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Slug with cAMP wave

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Slug moving straight, Pacemaker graft

Cell Sorting.The Chemotactic force is 50 %

Larger in the Grey Cells

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Slug with 2 different Cell types, same adhesion

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Slug with 2 different cell types, specific cell adhesionGrey cells more adhesive than green

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Thicker Slug with 2 different cell types, specific cell adhesionGrey cells more adhesive than green

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Thicker Slug with 2 different cell types, specific cell adhesionGrey cells more adhesive than green (Cross section)

Conclusions

•The model reproduces well the observed behavior and properties of cell aggregates

•The chemotactic movement of cells in response to a cAMP wave are in qualitative agreement with experiments

New Findings

•Random movement, cell stiffness and cell adhesion affect the rate of cell sorting

•Cell specific adhesion enhances chemotactic sorting and may be necessary to achieve cell sorting in a timely manner

€

dridt

=κ1(Fi −ff)μ1(κ1 +κ2)

+dFidt

(κ1 +κ2)−ri

κ1κ2

μ1(κ1 +κ2)