Fig. 2.11 Schematic sequence of cell divisions in holoblastic cleavage. The upper panel

shows an idealized situation where the daughter cells occupy the original volume of the

zygote. The individual daughter cell from the eight-cell stage shown in the lower panel

has extremely sharp bends and thus high bending energy. As a consequence, it

experiences forces (denoted here by F 1 and F 2 ) that tend to smoothen the sharp edges.

Depending on the bending rigidity of the cell membrane, the rounding of each cell results

in a cell con¬guration similar to the last diagram in the lower panel (for large bending

rigidity) or the last but one (for small bending rigidity). The requirement that a change in

cell shape leaves the total cell mass, or the cell volume, unchanged leads to a net

displacement of the cell mass away from the center.

taking place during individual cell divisions, discussed above, co-

operative effects involving groups of cells become increasingly more

important. It is this cooperativity that ¬nally gives rise to the ¬rst

nontrivial morphogenetic structure, the blastula. In what follows we

describe a model of blastula formation (Drasdo and Forgacs, 2000)

for the case of perfect, radial, holoblastic cleavage, such as occurs in

the sea urchin. The model combines a physical mechanism for cleav-

age at the single-cell level, with physical interactions between cells

(occurring on a multicellular scale). Thus, for the ¬rst time, we will

be dealing with a phenomenon involving several length scales, as is

typical of most biological processes. The salient features of the model

are the following.

1. Cells begin spheroidal and remain so during cleavage. Strong

bends or kinks in the material of their membrane are therefore

energetically disfavored. Thus, it is assumed that large changes

in the local geometry (i.e., the curvature) of the membrane gen-

erate a restoring force, which tends to ¬‚atten the variation in

geometry. As Fig. 2.11 illustrates, such an assumption requires

that during cleavage the cell mass shifts outward, away from

the center of the embryo.

2. Cells of the sea urchin embryo bind to an extracellular (˜˜hya-

line”) layer. Hence the mechanical properties of the extracel-

lular layer in¬‚uence those of the blastula (Davidson et al.,

1999).

2 CLEAVAGE AND BLASTULA FORMATION 45

3. Cells in the early embryo are polar (i.e., spatially nonuniform)

and, as a consequence of the inhomogeneous distribution of

their adhesion molecules (see Chapter 4), form cell--cell contacts

in special regions of their membrane, resulting in preferred cell

con¬gurations (Newman, 1998a; Lubarsky and Krasnow, 2003).

These con¬gurations are assumed to correspond to local min-

ima in the (free) energy. Deviations from preferred cell shapes

and con¬gurations increase the energy and thus are unfavor-

able. The energy of a cell con¬guration contains the following

contributions.

(i) An interaction energy Vi j of neighboring cells with both at-

tractive and repulsive contributions. Attraction is due to

the adhesive bonds between cells (see Chapter 4). If the cells

in isolation are spheroidal, contact between them leads

to the stretching of their membranes. Furthermore, cells

are composed mostly of water and therefore have small

compressibility. Membrane deformations and limited com-

pressibility give rise to repulsion. The manifestation of lo-

cal physical interactions (i.e., between individual cells) at a

larger scale (i.e., the blastula) is quite insensitive to the de-

tailed shape of the corresponding interaction energy (Odell

Vij

d0

d ij

Fig. 2.12 The interaction energy V i j in the contact region between two neighboring

cells i and j as a function of the distance di j between the cell centers in the

Drasdo“Forgacs model. The shape of V i j re¬‚ects the limited compressibility of the cells

and incorporates the entropic contributions of their membranes. Limited compressibility

implies that di j cannot be smaller than a minimal distance d 0 , at which a cell would lyse.

To prevent this, V i j is set to in¬nity at d 0 . Furthermore V i j contains contributions from

direct cell“cell adhesion and from elastic changes resulting from any alterations in cell

shape. The parameter δ is the range of interaction between neighboring cells. It

determines the distance over which a cell can be stretched or compressed, as well as the

interaction range of cell-adhesion molecules. The parameter µ characterizes the intensity

of the elastic and direct cell“cell interactions. To ensure that cells remain in contact

during blastula formation and gastrulation, the value of V i j at d 0 + δ is set to in¬nity.

46 BIOLOGICAL PHYSICS OF THE DEVELOPING EMBRYO

et al., 1981; Drasdo et al., 1995). Therefore the combina-

tion of the attractive and repulsive energy contributions is

modeled by the interaction energy Vi j shown in Fig. 2.12.

The physical interactions responsible for the competing

energy contributions have a characteristic range δ and

magnitude µ. These parameters are determined by the

deformability of the cell and its immediate molecular mi-

croenvironment (the ˜˜glycocalyx,” see Chapter 4), the prop-

erties of the hyaline layer and the nature of the adhesion

molecules that bind the cells to each other and to the hya-

line layer.

(ii) A curvature energy. Polarized cells often form two-

dimensional single-cell layers or sheets. The preferred

geometries of the layer and the shape of the cells within

the layer depend on the location of the cell-adhesion

molecules, as shown schematically in Fig. 2.13. In analogy

with polymer membranes, the preferred shape of the sheet

at the position of the ith cell has a local spontaneous cur-

vature (see Eq. 2.5). In the minimum-energy state the local

average curvature equals the spontaneous curvature of the

sheet. Any bending distorts the membrane, resulting in a

deviation of the curvature from the spontaneous curvature

and, consequently, in the increase of the membrane™s bend-

ing energy, which according to Eq. 2.5 depends on κ, the

membrane™s bending rigidity. Whereas the preferred shape

of the individual cell (characterized by the angle β0 in Fig.

2.13A) exclusively determines the spontaneous curvature,

once the cell is part of a tissue layer the actual local cur-

vature of the layer depends also on the positions of the

cells™ neighbors (see Fig. 2.13B, C). In the idealized situ-

ation, when all cells in a given arrangement are identical

(of the same type, at the same phase of the cell cycle, etc.),

their spontaneous curvature is the same.

4. Active cell movement characterizes early morphogenesis. This

movement, which leads to the appearance of new forms, on the

one hand must satisfy the constraints imposed by the activity of

the maternal and zygotic genes and on the other hand should

proceed according to the governing physical mechanisms,

which exert forces on the cells. These forces depend on the ex-

plicit form of the interaction energy and curvature energy dis-

cussed above, as well as on many other factors, and should even-

tually lead to con¬gurations with minimal mechanical stresses

(at least temporarily). In order to incorporate additional factors,

it is assumed that, during cleavage, extracellular components

provide a friction-like resistance to the displacements and ori-

entational changes of the cells within the developing tissue

sheet. The model incorporates friction and other biological and

chemical processes, such as metabolism, intra- and extracellu-

lar transport, movements of the cytoplasm, and reorganization

2 CLEAVAGE AND BLASTULA FORMATION 47

A

Apical

0

=0 0

0

Basal

C

2 2

B

2

3

1 2

d 23 r2

Fig. 2.13 Illustration of the curvature energy of a cellular layer. (A) The preferred

individual cell shapes, denoted by the broken contours, which depend upon the location

of adhesion molecules (¬lled areas). In this two-dimensional representation, the angle β0

uniquely determines the preferred shape of the cell and thus the local spontaneous

curvature of a cellular layer. The circles within the broken-line contours demarcate the

simpli¬ed shape used to represent cells in the computer simulations of the

Drasdo“Forgacs model. The optimal con¬guration of a cellular layer, containing only cells

with preferred shapes in the left-hand panel (β0 < 180—¦ ) or right-hand panel (β0 > 180—¦ )

is a closed surface with the basal side, as de¬ned here, oriented respectively towards the

interior (see panel C) or the exterior. (B) The preferred cell shape when β0 = 0 results

in an optimal con¬guration with an open, planar, cell sheet and an equal distance

between the centers of the cells. (C) A deviation from the optimal con¬guration shown

in panel B. For a cell type with β0 = 0 any bend (characterized by the existence of a

¬nite local radius of curvature r and deviation angle β) increases the bending energy.

Here the radius of curvature and the deviation angle are shown only for cell 2. Note

that, for a cell type with β0 < 180—¦ (shown in A), the con¬guration illustrated in panel C

is optimal if β j = β0 , where j denotes any cell in the sheet.

of the cytoskeleton, by imposing an additional stochastic force

on the cells.

5. A cell™s mobility, its geometric environment, and its interac-

tion with its neighbors affect the observed cell-cycle time. The

cell-cycle time also depends on the cell™s intrinsic properties,

which are incorporated by introducing the intrinsic cell-cycle

time „ , a quantity that is in¬‚uenced by the chemical environ-

ment (e.g., by growth and inhibitory factors). For an isolated cell

not affected by physical interactions with neighboring cells, „

is the average cell-cycle time. If interactions are present, the

48 BIOLOGICAL PHYSICS OF THE DEVELOPING EMBRYO

R

a(t)

+

R

R

Fig. 2.14 Cell division as simulated in the Drasdo“Forgacs model. The cell is

deformed by decreasing its instantaneous radius in randomly chosen small steps from the

√ √

original radius, 2R . The parameter ζ < 2 denotes the cumulative effect of these

steps in an intermediate stage of cell division. As the radius decreases, the axis a(t)

increases to keep the total area of the cell constant during one division cycle, which is

guaranteed by the dumb-bell shape. Area conservation also requires the two daughter

cells to have radius R .

observed cell-cycle time is typically larger than the intrinsic

cell-cycle time.

6. The described friction-limited stochastic dynamics is simulated

using the Monte Carlo method (Metropolis et al., 1953). The sim-

ulation chooses a cell randomly and modi¬es its state by (i) a

shape deformation in accordance with the adopted mechanism

of cell division (see Fig. 2.14), and (ii) a small random displace-

ment. After each division a chosen cell deforms in small steps

by decreases in its instantaneous radius and simultaneous in-

creases in the distance between the centrosomes. After each

modi¬cation in the state of an individual cell the total energy

of the cell con¬guration is recalculated. If the modi¬cation in-

creases the energy, the cell returns to its original shape with

probability P = 1 ’ e ’ V /F T , otherwise it stays in its new po-

sition. Here V = V ’ V , V and V being the total energy of

the cell assembly respectively before and after the change in

state of the randomly chosen single cell. The quantity F T is

a reference energy analogous to the thermal energy discussed

in Chapter 1. (The characteristic energy scale for the motion

of ¬‚uid particles is the thermal energy, whereas the character-

istic energy scale for cellular motion must have its origin in

the cytoskeletally driven cell-membrane ¬‚uctuations fueled by

metabolic energy.) The value of F T during blastula formation in

the sea urchin is not known but F T was estimated for chicken

embryonic cells by Beysens et al. (2000). The above exponential

form of the probability is the most typical one used in Monte

Carlo simulations.

Monte Carlo simulations were performed in two dimensions, for

a representative cross section of the approximately rotationally sym-

metric sea urchin embryo, using experimentally measured or derived

2 CLEAVAGE AND BLASTULA FORMATION 49