dsy
A 1S
Keywords:
Dictyostelium
Ellipsoidal cells
hem
nts
g. Th
s on
, ex
are calculated from equations of motion using the total force acting on each cell, ensuring that forces are
balanced. The simulations show that the sorting patterns of prestalk and prespore cells, emerging
during the slug stage, depend critically on the type of cell adhesion and not just on chemotactic
in sin
r orga
amoebae (Bonner, 1967; Konijn et al., 1967). Under starvation
nto a
d its
ment is arguably the most important universal feature of
ARTICLE IN PRESS
Contents lists availabl
.e
Journal of Theor
Journal of Theoretical Biology 254 (2008) 1– 13development (Bowers-Morrow et al., 2004). There are a number

Corresponding author. Tel.: +1604 2914808; fax: +1604 2913496.conditions, excitable cells produce and relay circular waves of
cyclic adenosine-3
0
,5
0
-monophosphate (cAMP) (Alcantara and
adhesion in development’’ (Bowers-Morrow et al., 2004). Inves-
tigating cell contact phenomena important for Dd development
can provide clues to universal mechanisms required for transition
from single-celled to multicellular organisms, and metazoan
of molecules known to affect adhesion and cell sorting in Dd
including (a) cadA/DdCAD-1, a Ca
2þ
dependent cadherin like
E-mail address: epalsson@sfu.ca
URL: http://www.sfu.ca/epalsson0022-51
doi:10.1what role cell deformability plays.
Life of the cellular slime mold, Dd, begins with free ranging
1.1.1. Cell adhesion
‘‘The dynamic balance between cell adhesion and cell move-(Bonner, 1967). In this paper, I use an improved version of an
individual cell-based model developed earlier (Palsson and
Othmer, 2000) to explore how differences in cell adhesion
between cell types influence the chemotactically driven cell
sorting in the slug stage of Dictyostelium discoideum (Dd), and
responsible for organizing the transformation of the mound i
slug, the migration of the slug over the substratum, an
culmination into a fruiting body (Siegert and Weijer, 1992).movement can lead to cell type segregation and the formation
of specific structures such as occurs during gastrulation and
wound healing (Alberts et al., 1994; Elul et al., 1997), cancer cell
invasion into tissues (Takeichi, 1993), limb bud regeneration
(Gilbert, 1991), and development of Dictyostelium discoideum
types (Jermyn et al., 1989; Maeda et al., 2003) with slugs having
10–30% prestalk cells (Rafols et al., 2001). Prestalk cells differ-
entiate in a position-independent fashion and then move
individually or as a group towards the anterior of the slug
(Yamamoto, 1977; Odell and Bonner, 1986; Buhl and MacWilliams,
1991; Thompson et al., 2004). It is believed that cAMP waves areDeformable cells
Prestalk
1. Introduction
1.1. Biological background
Cell movements play a vital role
Amoeba, and also in multicellula93/$ - see front matter & 2008 Elsevier Ltd. A
016/j.jtbi.2008.05.004cells from passing the slower cells. The patterns are distinctively different when the prestalk cells are
more or less adhesive than the prespore cells. These simulations suggest that sorting is not solely due to
differential chemotaxis, and that differences in both adhesion strength and type between different cell
types play a very significant role, both in Dictyostelium and other systems.
& 2008 Elsevier Ltd. All rights reserved.
gle cell organisms like
nisms where relative
Monk, 1974; Gross et al., 1976). The chemotactic migration of
cells towards the signaling center is responsible for the formation
of a mound of cells that rises upward and eventually topples over
to form a cigar-shaped mass called the slug, containing up to 10
6
cells (Bonner, 1967). During late aggregation and early slug
differentiation, cells differentiate into prespore and prestalk cellDifferential cell adhesion
Pattern formation
differences between cells. This occurs because cell size and stiffness can prevent the otherwise fasterA 3-D model used to explore how cell a
sorting and movement in multicellular
Eirikur Palsson

Department of Biology, Simon Fraser University, Burnaby, British Columbia, Canada V5
article info
Article history:
Received 26 June 2007
Received in revised form
6 May 2008
Accepted 6 May 2008
Available online 15 May 2008
abstract
A three-dimensional mat
signalling on cell moveme
and to visualize cell sortin
where movement depend
from the neighboring cells
journal homepage: wwwll rights reserved.hesion and stiffness affect cell
stems
6
atical model is used to determine the effects of adhesion and cell
during the aggregation and slug stages of Dictyostelium discoideum (Dd)
e building blocks of the model are individual deformable ellipsoidal cells,
internal parameter state (cell size and stiffness) and on external cues
tracellular matrix, and chemical signals. Cell movement and deformation
e at ScienceDirect
lsevier.com/locate/yjtbi
etical Biology

molecule, (b) csA/gp80, a glycoprotein first expressed during
the streaming stage, (c) pspA, a prespore-specific glycoprotein,
(d) lagC/gp150, a heterophilic glycoprotein expressed predomi-
nantly in prestalk cells, (e) ampA, a regulatory protein that affects
other adhesion molecules (Siu et al., 1983, 2004; Ponte et al., 1998;
Wang et al., 2000; Bowers-Morrow et al., 2002; Wong et al., 2002).
csA-null cells sort out from aggregating wild-type cells and many
fail to enter the multicellular phase of development (Ponte et al.,
1998) and mutant cadA prestalk cells do not sort out to re-occupy
the anterior zone when mixed with wild-type prespore cells in a
slug (Bowers-Morrow et al., 2002; Siu et al., 2004). In this article, I
focus on the role that cell adhesion, coupled with individual cell
chemotaxis, has on the collective motion of a multicellular
system. There are two types of adhesion: (1) differential cell
adhesion strength as a result of differences in number of adhesion
rea
1975; Sulsky et al., 1984; Honda, 1983; Mochizuki et al., 1996;
Umeda and Inouye, 1999) explore cell sorting where the cell
aggregate is considered a mixture of two fluids. Cell size is
infinitesimal and the sorting time depends on the ‘‘tempera-
ture’’ (randomness) of the fluid mixture. The strength of these
fluid-type models is that it is easy to model a large number of
cells, and viscoelastic fluid models provide better descriptions
of the system. A limitation of these models is that first, it is
very hard to put the active force generation by the cells into the
stress tensor in a physically correct manner and that second,
cell individuality, size and stiffness, is often not considered.
 Models based on Voronay tessellation or polygons (Odell et al.,
1981; Jacobson et al., 1986; Weliky and Oster, 1990) simulate
the cell shape and boundaries very well and often include
forces to model the cell–cell interactions. They give a very
biologically realistic description for 2-D in vivo situations. A
drawback is that in 3-D it is very difficult to find cell
boundaries, and computational costs become prohibitive for
a large number of cells.
 Models using a cellular automata or cellular Potts approach
include Graner and Glazier (1992), Glazier and Graner (1993),
ARTICLE IN PRESS
Fig
of p
(b)
wh
The
E. Palsson / Journal of Theoretical Bi2molecules, (2) specific/preferential adhesion to molecules of the
same type, whereby the adhesion molecules are homotypic and
cell-type specific.
1.1.2. Cell characteristics and forces
There are various biochemical and biophysical aspects of cells
that affect movement (Small, 1989; Elson et al., 1999; Sheetz et al.,
1999) such as their degree of stiffness (Bray, 1992; Eichinger et al.,
1996), cell adhesion strength (surface tension) (Foty et al., 1996),
and their locomotive force (Usami et al., 1992; Lee et al., 1994;
Inouye and Takeuchi, 1980; Guilford et al., 1995)(Table 1). Cells
can deform and exert forces onto other cells, and they respond to
external chemical and mechanical cues. The external forces that
act on a cell arise from cell adhesion and cell resistance to
deformation, from the active locomotive force that a cell applies
either to a neighbour cell or the substrate, and finally from the
drag exerted by the extracellular fluid or substrate on a moving
cell. When subjected to a force, a cell initially resists deformation,
an elastic response, but under sustained force, the actin network
in the cytoskeleton breaks and re-forms, producing a viscous
response (Chien, 1984; Evans, 1985b). External forces can affect
cell shape: in suspension a cell is spherical, when resting on a
surface it is often slightly flattened, and when it moves, it
elongates in the direction of movement (Taylor et al., 1982).
When a Dd amoeboid cell moves, either randomly or in
response to a chemotactic signal, it sends out pseudopods. The
extensions of pseudopods and the retraction of the rest of the cell
body requires active force generation. When the cell is moving on
a surface, the pseudopod attachment is onto the surface and the
applied force is transmitted directly to the surface (Fig. 1(a)).
However, when the cell is inside a multicellular aggregate it must
attach the pseudopod to another cell (Fig. 1(b)). It is important to
Table 1
An overview of experimentally observed parameters
Parameter Typical
value in
model
Experimental justification
Cell stiffness, k
1
: k
2
ðnN=mmÞ
50:0:5 0.01–50 (Bray, 1992; Skalak et al., 1984;
Eichinger et al., 1996)
Cell damping, m
1
(dyne s=mmÞ
6000 10
3
Estimate (Skalak et al., 1984)
Active force, F
act
ðnNÞ
10
3 8 10
3
dyne leukocytes (Usami et al.,
1992). Cells in Dd slug (Inouye and Takeuchi,
1980)
Cell adhesion, a
(10
3
dyne)
1 Estimated from surface tension
measurements (Foty et al., 1996)
Cell–surface viscosity,
8 10
3
Estimated from active force and 20mm=minm
s
(dyne s=mmÞ cell velocity (Alcantara and Monk, 1974)
Cell–cell viscosity, m
c
(dyne s=mm)
25 10
3
EstimateA complete model of cell movement should balance all forces,
this is especially important in 3-D aggregates, e.g. in upward
movement of cells. Dd cells can only pull towards an attachment,
they cannot push off, therefore any active upward movement can
only be achieved by crawling up other cells. Neglecting the
reaction force can create artifacts that violate physical laws.
1.2. Models of cell movements
1.2.1. Brief literature review
There are many ways of modeling cell movements in multi-
cellular systems and depending on the situation, some models are
better than others as discussed below
 Fluid type models: Odell and Bonner (1986), Vasiev and Weijer
(2003) and Umeda and Inouye (2004), model slug movement
or mound formation in Dd as a fluid. Several models (Mostow,exp
Bonlize that when a cell applies force to a neighbour, the other cell
eriences an equal force in the opposite direction (Odell and
ner, 1986).
1
2
12
F
21
. 1. (a) The crawling of a cell on a surface by pseudopod extension: (i) extension
seudopod, (ii) attachment at the front, (iii) retraction of the back and (iv) rest.
In a multicellular aggregate F
12
is the reaction force, that pulls cell 1 forward
ile F
21
is the corresponding active force from cell 1 that pulls cell 2 backwards.
vertical bars indicate the adhesive attachments for cell 1.i) ii) iii) iv)
Fology 254 (2008) 1–13Savill and Hogeweg (1997), Jiang et al. (1998), Maree et al.
(1999) and Kafer et al. (2006). Each cell is represented by a
number of smaller automatons. These models are fluid-like in