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Fig. 6.7 Phase separation in mesenchymal tissues. High-density cultures of precartilage
mesenchymal cells from embryonic chicken wing and leg bud were grown contiguously
on a Petri dish. The upper panel shows the culture pair (wing on left, leg on right) after
six days of growth, ¬xed and stained for cartilage with Alcian blue (the darker areas).
The cartilage is uniformly dispersed throughout the wing culture but forms isolated
nodules separated by noncartilage mesenchyme (the less stained areas) in the leg
culture. The interface is convex from the leg side, indicating that leg tissue is more
cohesive than wing tissue, a result also obtained by different assays (Heintzelman et al.,
1978). The lower panel shows a higher-magni¬cation phase-contrast-microscope image
of a similar pair of cultures, again after six days™ growth. In this case both cultures are
living and unstained. There is a clear interface between the wing culture (on the left) and
the leg culture (on the right). No mixing appears to have occurred across the interface.
The total horizontal distance in the top panel is ∼10 mm. The bar in the lower panel is
0.5 mm long. (Reprinted from Downie and Newman, 1994, with permission.)

Mesenchymal condensation
As mentioned earlier, condensation occurs in a mesenchymal tissue
when cells suspended in an ECM move closer together at particular
sites. The distances traversed during this process are small -- usu-
ally less than a cell diameter. Cell condensation usually occurs in
two phases (Fig. 6.8). First, cells accumulate in regions of prospective

condensation, which are rich in one or more adhesive ECM glyco-
proteins, such as ¬bronectin, giving rise to mesenchymal aggregates;
second, cells undergo epithelialization, producing cell-surface adhe-
sive molecules such as N-CAM and the various cadherins, molecules
(with certain exceptions; see Sinionneau et al., 1995) not normally
expressed by mesenchymal cells (Hall and Miyake, 1995; 2000). Once
they are in proximity, it is physically straightforward for cells to form
direct adhesive associations. Explaining the aggregational phase of
condensation in physical terms represents more of a challenge.
A variety of cellular mechanisms has been suggested for the ini-
tial stage of condensation formation. The best evidence supports a
scenario based on an extended version of the differential adhesion hy-
pothesis (DAH: Steinberg, 1978; 1998; see Chapter 4). In this interpre-
tation, random cell movements occurring in a tissue mass in which
there are local patches of increased adhesivity drive cells into higher
density aggregates (Frenz et al., 1989a, b; Newman and Tomasek, 1996).
When test particles coated with the glycosaminoglycan heparin were
mixed with limb mesenchyme cells in culture, they accumulated at
sites of cell condensation in a fashion that depended on interactions
of the particle surface with ¬bronectin. This indicated that passive
movement, as the beads are buffeted by the surrounding cells, in
conjunction with adhesive gradients (i.e., haptotaxis) is suf¬cient to
translocate such particles (Frenz et al., 1989a). Limb mesenchymal cells
express heparin-like surface molecules (Gould et al., 1992) and are thus
subject to the same haptotactic forces (Frenz et al., 1989b).
Cell condensation is an example of cellular pattern formation, i.e.,
regulated changes in cell arrangement. This is a subject we will study
in more detail in Chapter 7. Interest in pattern formation has given
rise to a large number of models for these processes. These models

Fig. 6.8 Schematic representation of mesenchymal condensation. In the left-hand
panel a population of mesenchymal cells is depicted as being scattered throughout an
extracellular matrix (beige) that contains a region rich in condensation-promoting ECM
molecules such as ¬bronectin (brown). In the middle panel cells accumulate preferentially
and thus attain an elevated density in the ¬bronectin-rich region. In the right-hand panel,
cells in high-density foci establish broad cell“cell contacts, mediated by newly expressed
cell adhesion molecules (CAMs), and thus epithelialize. (After Newman and M¨ ller,

can be classi¬ed as continuous or discrete. In the former category,
space and time are considered continuous and pattern evolution is
described in terms of differential equations governing quantities (typ-
ically densities) that themselves vary continuously in space and time
(Murray, 2002). We used the continuous approach in Chapter 3 to
model cell differentiation and in Chapter 4 to describe gastrulation.
In discrete models, such as the cellular automaton model of neu-
rulation discussed in Chapter 5, variations in space and time occur
in ¬nite steps and thus patterns are formulated along grids or lat-
tices. Continuous and discrete models differ from each other mostly
in their initial formulations; while one or the other may be advan-
tageous for certain applications, some models draw on features from
both frameworks.
To describe cell condensation we introduce the discrete cellular
Potts model (CPM), which has been applied successfully to a num-
ber of biological phenomena including differential adhesion (Graner
and Glazier, 1992; Glazier and Graner, 1993; Mombach et al., 1995;
Jiang, Y., et al., 1998). Glazier and coworkers applied this model to the
initial phases of mesenchymal condensation (Zeng et al., 2003). The
reference data for simulations using this model were obtained from
in vitro experiments performed in planar culture dishes. The model
was therefore formulated in two dimensions.

The cellular Potts model and its application to
mesenchymal condensation
The two-dimensional CPM is a representation of a collection of N cells
distributed on the sites of a square lattice. With each lattice site, iden-
ti¬ed by a pair of indices (i, j), one associates a cell variable σ„ (i, j).
Here the index „ takes the value 1 or 0 depending on whether the
lattice site is occupied by a biological cell or by its acellular envi-
ronment (i.e., the culture medium). We set σ0 (i, j) = 0 for all lattice
sites; σ1 (i, j) can take on N discrete values (1, 2, . . . , N) (see Fig. 6.9). A
biological cell is de¬ned by a group of neighboring lattice sites with
the same value of σ1 . In the ¬gure, all the sites in a given cell have
the same color.
Cells interact with each other. We denote the energy cost
of forming a unit of contact area between neighboring cells by
J [σ„ (i, j), σ„ (i , j )]. (Note that J is de¬ned only for cells that contact
each other.) For biological cells, J re¬‚ects their surface adhesiveness;
this depends, typically, on the number of membrane-bound adhesion
molecules. Since only surface energies between distinct cells are in-
cluded, there is no energy cost incurred if the lattice sites belong
to the same cell: J = 0 for σ„ (i, j) = σ„ (i , j ). As a result of their
interaction and migration cells change their shape; this corresponds
to the deformation of domains with the same σ„ (Fig. 6.9) and can be
incorporated by updating the value of the cell variable at each lattice
Mesenchymal condensation is due to both cell--ECM and cell--cell
interactions. In the simple model described above it is assumed that

Fig. 6.9 Schematic illustration of the cellular Potts model in two dimensions. A cell is
represented by a set of squares (six in the ¬gure) of the same color, each corresponding
to the same value of the variable σ1 (as de¬ned in the text). The two panels in the ¬gure
show two possible con¬gurations of a cellular pattern with 13 cells arising in the course
of the Monte Carlo simulation. The uncolored region represents the tissue culture

J decreases linearly with the cell™s integrated exposure to ¬bronectin
(i.e., smaller values of J correspond to stronger, and thus energet-
ically more favorable, adhesion; see Eq. 6.5). Biologically this could
mean that binding to ¬bronectin causes cells to produce increased
amounts of cell adhesion molecules such as N-cadherin. Such adhesive
˜˜crosstalk” (Marsden and DeSimone, 2003; Montero and Heisenberg,
2003) leads to positive feedback: cells encountering high levels of
¬bronectin tend to stay longer in those regions, thus adding the
¬bronectin they produce to the local ECM. By the above-stated assump-
tion these same cells then produce more N-cadherin, making them
more adhesive. The linear size of cell clusters at these sites grows con-
tinuously until the cells are unable to undergo further movement.
With these ingredients, the CPM represents the total energy of
the condensing pattern as

E= J [σ„ (i, j), σ„ (i , j )] + » (A σ ’ A target,σ ) + µ C F (i, j).
(i, j),(i , j ) (i, j)


Here the ¬rst term on the right-hand side denotes the interaction
energy between cells as described above. The summation is extended
only over pairs of lattice sites belonging to neighboring cells. The
second term constrains the cells™ surface area and the sum is over all
cells. » is the compressibility of the cell™s material (larger » values
correspond to less compressible cells). A σ is the actual cell area and
A target,σ is the area of the cell in the absence of compression. The
third term describes the effect of preferential attachment of cells
to ¬bronectin and here the sum includes only sites that lie within
biological cells (i.e., those with „ = 1). C F (i, j) is the concentration
of ¬bronectin at site (i, j) and µ is the unit strength of ¬bronectin

It is clear from this analysis that the term containing µ is respon-
sible for the initial (i.e., aggregation) phase of condensation, whereas
the term containing J is responsible for the second, epithelialization,
The evolution of the cellular pattern is followed using stochastic
Monte Carlo simulations in the same manner as described in Chap-
ter 2 in connection with the Drasdo--Forgacs model. One starts with a
random distribution of cells within a circular area that corresponds
to the size of the ˜˜micromass” (small-diameter, high-density) culture,
which is approximately 3 mm. The length scale in the computational
representation is set by the pixel size (i.e., one elementary square in
Fig. 6.9). Each cell occupies a certain number of pixels (six in Fig. 6.9)
corresponding to its size (the linear size of a typical limb bud cell is
15 µm). At each step a lattice site is selected randomly and its cellular



Fig. 6.10 Patterns of cartilage nodules that formed in cultures plated at two different
densities, with, for comparison, simulations based on the cellular Potts model (Eq. 6.5).
(A, B) Leg-cell cultures were grown for six days and stained for cartilage with Alcian
blue, as described in Downie and Newman (1994) (see also Fig. 6.7 above). The initial
plating densities were higher in A than in B, but in both cases they were at con¬‚uency or
above, i.e., suf¬cient to cover the 5 mm diameter circular culture “spot” completely. The
unstained regions between the nodules in A and B contain cells that failed to undergo
condensation and therefore did not progress to cartilage. (C, D) Simulations of the in
vitro condensation process. Unlike living cells in high-density cultures, cells in the
two-dimensional simulations cannot readily move past one another unless there is space
between them. The simulations were therefore performed at subcon¬‚uent densities of
40% and 20% coverage. As in the experiments, the simulated cells form stripes and spots
at high density (C) and only spots at low density (D). The simulated cells are initially
white and become blue when the local ¬bronectin concentration exceeds a threshold
chosen to correspond to Alcian blue staining. (E, F) The distribution of ¬bronectin in the
simulations. The ¬bronectin concentration ranges from red (highest) to light blue
(lowest). High levels of ¬bronectin colocalize with the cell clusters in C and D. (Based on
Zeng et al., 2003; ¬gure courtesy of W. Zeng.)

variable σ„ (i, j) is changed to σ„ (i , j ), the value of one of its neigh-
bors, also selected randomly. The new con¬guration is accepted with
a certain probability P depending on the gain or loss in energy E ,
where E is de¬ned in Eq. 6.5, caused by this change: P = exp(’ E /F T )
if E > 0 and P = 1 if E ¤ 0. Here F T is the cytoskeletally driven
¬‚uctuation energy discussed in connection with the Drasdo--Forgacs
model near the end of Chapter 2. The system evolves towards the
minimum energy state. The results of the simulations and the corre-
sponding experiments are shown in Fig. 6.10. (For more details, see
Zeng et al., 2003).

Mesenchymal cells differ from epithelial and epithelioid cells by not
being directly attached to one another. This makes them suscepti-
ble to physically based morphogenetic mechanisms not applicable to
other tissue types. Mesenchyme-speci¬c mechanisms include those
based on the distinctive physical properties of the ECM, its ability
to ¬‚ow in micro¬ngering patterns or its ability to support the self-
organization of assembling ¬bers, for example. Because cells in a
mesenchymal tissue may be at variable distances from one another,
they can also undergo local rearrangements based on short-range
movement. Condensation is the most well described of these. De-
spite the differences between mesenchymal and epithelioid tissues,
certain physical descriptions and mechanisms emerge in common:
the rheological properties of tissues (describable by parameters such
as viscosity and elasticity), the ability of tissues to form compart-
ment boundaries by undergoing phase separation, and the applica-
bility of energetic considerations, which underlies the DAH and its
Chapter 7

Pattern formation:
segmentation, axes, and

In previous chapters we found that, by virtue of their internal dynam-
ics, cells could assume distinct states and thereby follow alternative
developmental pathways. This process ultimately generated various
types of terminally differentiated cells (Chapter 3). We also found that
tissues made up of multiple cells, linked together directly (Chapters
4 and 5) or via an extracellular matrix (Chapter 6), could undergo
alterations in shape and form (morphogenesis), leading to the de-


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