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First published online March 21, 2008
doi: 10.1242/10.1242/dev.014555
,
,*
1 Hyogo University, Kakogawa, Hyogo 675-0195, Japan.
2 Department of Developmental Biology, Max-Planck Institute of Immunobiology,
Freiburg D-79108, Germany.
3 Physics Department, Kyushu Kyoritsu University, Kitakyushu 807-8585,
Japan.
4 Institute of Statistical Mathematics, Tokyo 106-8569, Japan.
* Authors for correspondence (e-mails: hihonda{at}hyogo-dai.ac.jp; hiiragi{at}mpimuenster.mpg.de)
Accepted 6 February 2008
 |
SUMMARY |
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 TOP SUMMARY INTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONREFERENCES |
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Key words: Computer simulation, Mammalian development, Asymmetry
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INTRODUCTION |
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TOPSUMMARYINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONREFERENCES |
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;
Rossant and Tam, 2004).
Blastocyst morphogenesis is unique to mammals, resulting in the formation of
the extraembryonic structures. The mammalian blastocyst displays an obvious
asymmetry, with the inner cell mass (ICM) at one end and the blastocyst cavity
surrounded by the trophectoderm (TE) at the other. The position of the ICM in
the blastocyst, i.e. the embryonic pole, provides a basis for further
embryonic patterning in that the embryonic-abembryonic (Em-Ab) axis should in
principle give rise to the dorsoventral axis of the mouse embryo
(Beddington and Robertson,
1999; Rossant and Tam,
2004). Central questions include how this apparently initial
asymmetry is established in the early mouse embryo, and whether the underlying
mechanism in this process is comparable to that operating in non-mammalian
`model' organisms, in which localized (`prepatterned') determinants play an
essential role. Whereas recent reports
(Gardner, 1997;
Gardner, 2001;
Piotrowska et al., 2001;
Piotrowska and Zernicka-Goetz,
2001; Piotrowska-Nitsche et
al., 2005; Torres-Padilla et
al., 2007) claim the presence of `prepatterning' in the mouse egg
and preimplantation embryos reminiscent of non-mammals, our studies
(Hiiragi and Solter, 2004;
Motosugi et al., 2005) and
those by others (Alarcon and Marikawa,
2003; Alarcon and Marikawa,
2005; Chroscicka et al.,
2004; Kurotaki et al.,
2007; Louvet-Vallee et al.,
2005) provide evidence supporting a `regulative' and `mechanical
constraint' (Alarcon and Marikawa,
2003; Kurotaki et al.,
2007; Motosugi et al.,
2005) model of blastocyst morphogenesis. In this model, the first
embryonic polarity, the Em-Ab axis, is not established until the blastocyst
cavity is localized at one end (the abembryonic pole), and its eventual
position is determined by the spatial constraints imposed by the zona
pellucida (ZP) in conjunction with the epithelial seal in the outer cell
layer. This model does not depend on any localized determinants, leading us to
apply mathematical and physical modeling not only to examine the feasibility
of the mechanical constraint model, but also to elucidate the essential
cellular properties necessary to achieve blastocyst morphogenesis. |
MATERIALS AND METHODS |
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TOPSUMMARYINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONREFERENCES |
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). The
vertices obey the equation of motion:
 | (1) |
i the nabla differential
operator. The left side of Eq. (1) represents a viscous drag force
proportional to the vertex velocity dri/dt with a
positive constant 
(an analog of the coefficient of viscosity). Vertices
do not have mass (inertia), so that the motion of the vertices and cells is
completely damped. The right side of Eq. (1) represents a potential force
(driving force), i.e. minus the gradient of the potential U. The
potential U includes various terms related to cell surface where
ri is a 3D-positional vector of vertex
i, and i the nabla differential operator.
The left side of Eq. (1) represents a viscous drag force proportional to the
vertex velocity dri/dt with a positive constant
(an analog of the coefficient of viscosity). Vertices do not have mass
(inertia), so that the motion of the vertices and cells is completely damped.
The right side of Eq. (1) represents a potential force (driving force), i.e.
minus the gradient of the potential U. The potential U
includes various terms related to cell surface area, cell volume, and
potential energy under restriction of the ZP, that are all expressed by vertex
coordinates. Hence, U is a function of the vertex positions.
In the present study, the potential U contains terms of surface
energy (US), elastic energy (UEV,
UEI) and boundary energy due to restriction by the ZP
(Ubound):
 | (2) |
 | (3) |
The first two terms in Eq. (3) represent the interface energy between
neighboring cells and the surface energy between cells and the outside, where
nS and nO are the numbers of polygons
facing an adjacent cell and facing the outside, respectively.
Sk is the surface area of a polygon k facing
adjacent cells and 
S is its interface energy per unit area.
Ok is the surface area of a polygon k facing the
outside and O is its surface energy per unit area. The third
term in Eq. (3) represents the interface energy between cells and the internal
cavity (blastocyst cavity), where nI is the number of
polygons facing the blastocyst cavity. Ik is the surface
area of a polygon k facing the blastocyst cavity and
I is its interface energy per unit area. The potential
UEV contains two terms, the elastic energy of compression
and expansion of the cells and that of the blastocyst cavity:
 | (4) |
V and VI are the elastic constant of
volumes of the cells and the blastocyst cavity, respectively, and n
is the total number of cells. V
and
VI are the volumes of cell and of the blastocyst
cavity. Vstd and VIstd are the volumes
of relaxed cells and of the relaxed cavity. Vstd is
defined as the average volume of all cells. Eq. (4) imposes a constraint on
the volumes of the cells and the blastocyst cavity. The potential
UEI denotes the elastic energy of compression and
expansion of the surface area of a polygon k facing the blastocyst
cavity:
 | (5) |
I is the elastic constant of the surface of a polygon
k facing the blastocyst cavity. Istd is the
relaxed surface area of the cells facing the blastocyst cavity. Eq. (5)
provides the cell surface constraint of a polygon k facing the
blastocyst cavity. Restriction of the embryo by the ZP is described as an
additional term Ubound:
 | (6) |
 | (7) |
f(ri)}], where /4 is the
slope of the logistic function at
f(ri)=0, and
wbound is the strength of the potential. Thus, Eq. (1)
takes the form: 
In order to reduce the parameter number without loss of generality, we
introduce a new length unit R0 and rewrite Eq. (8) using
new dimensionless quantities ri',
i'
(=
/ri'),
Sf', Sm',
V', Vstd' and
z' as
follows:
Thus, Eq. (8) takes the
form:
in which the new dimensionless quantities are defined as
follows:
We take R0
(=Vstd
) =1, so that
Vstd'=1. Eq. (11) lacks explicit parameters
corresponding to and S. Thus, without loss of
generality, we can describe cell behaviors using the dimensionless parameters
O', I',
V', VI',
I', w'bound,
', a', b' and c'.
Below, cell motions are measured in terms of the new length unit
R0=Vstd and the new
time unit /S, which are the characteristic length scale
and time scale of the cell aggregate, respectively. Hereafter, we omit primes
(') on the rescaled quantities in Eq.
(11).
which is equivalent to Eq. (13) and Eq. (16) in the Results.
In addition to the equations of motion, our model involves an elementary
process of reconnecting neighboring vertices
(Honda et al., 2004). When the
length of an edge connecting two neighboring vertices becomes short (less than
a critical length 
), the relationship of the neighboring vertices
changes and the neighbors reconnect themselves
(Fig. 1B).
Computer simulations
Vertex dynamics
Cell boundary surface areas (Sk, Ok,
Ik), cell volumes (V), and the
volume of the blastocyst cavity (VI) are calculated by the
vertex coordinates, providing the potential (U) and its partial
derivatives with respect to xi, yi, and
zi (i U). The Runge-Kutta method
(Ohno and Isoda, 1977) with
step size h is applied to numerically solve the simultaneous
equations of motion (Eq. 12). Thus, we obtain the movement of all vertices.
After each Runge-Kutta time step, we apply the elementary process of
reconnection (Honda et al.,
2004) (Fig. 1B) to
edges shorter than the critical length .
Initial structure of the cell aggregate and its development in computer simulations
An initial structure of a cell aggregate consisting of 31 or 40 cells
(n=31, 40) is produced according to the method described
(Honda et al., 2004). To make
an internal space (blastocyst cavity) in the cell aggregate, we choose a
vertex in the cell aggregate and replace the vertex by a small tetrahedron
(Fig. 1C). To simulate
development to the expanded blastocyst, the volume of the tetrahedron
(Istd) is made to increase to half the initial volume of
the cell aggregate (Istd=15.5 or 20) during
t=0-500 (Fig. 1D),
while keeping the same conditions for the cell aggregate. In this way, the
total volume of the cells slightly decreases (e.g. -11.8%), while the entire
embryo volume significantly increases (e.g. +45.7%) accompanied by the cavity
expansion (e.g. 0 to 19.09). This increase in the entire embryo volume during
the blastocyst cavity formation is confirmed by time-lapse recording of
preimplantation mouse embryos (data not shown) (see
Motosugi et al., 2005). This
period of linear increase in the cavity volume is followed by a plateau period
(Fig. 1D; t=500-3500),
which is technically necessary for computer simulation in order to obtain the
stabilized structure with the minimum potential energy after the forced change
in condition.
Parameter values used in the simulations
At present, accurate values for parameters related to the cell properties
are not available. Thus, the present study investigates the general behavior
of polyhedral cells in typical cell aggregates using Eq. (12) with
dimensionless parameter values. The parameters O,
I, V, VI,
I and wbound are values relative to the
coefficient of interface energy between the cells (cell-cell boundary energy),
S. The smaller the coefficient of the cell volume elasticity
(V), the easier the change in the cell volume and the more
unstable the system becomes; when V is larger, the shape of
the cell aggregate hardly changes. The V
(V=12) in this study allows both flexibility and rigidity.
The coefficients of surface energy are chosen to be smaller
(o=I=0.7) than that of interface energy
between the cells, S, so that the internal space smoothly
expands. The coefficient of volume elasticity of the internal space,
VI, is considered to be smaller than the coefficient of cell
volume elasticity (V). However, when VI is
too small, the internal space does not expand although the standard volume of
the internal space (Istd) is forced to increase in the
simulations, leading us to choose VI=6. In addition, we
introduce the coefficient of surface elasticity of the cells facing the
internal space, I. The cell surfaces facing the internal
space are expected to minimize the elastic energy (deviation from the relaxed
surfaces). Without this elasticity, some vertices produce complicated
tessellation patterns and the process freezes. We also confirm that the
surface shows elastic properties when the relaxed area
Istd is between 0.2 and 1. The process freezes when the
elastic constant of the surface of polygons facing the blastocyst cavity
I is set at 0.2, suggesting that it is too large and leading
us to choose Istd=0.433 and I=0.1. The
last term on the right side of Eq. (12) expresses hindrance of outward motion
of a vertex from the ZP. Its parameters wbound and
allow a wide range of values. When wbound is too
large (e.g. wbound=50), the space between the outer
surface of the blastocyst and the ZP becomes too large. is a
parameter indicating the slope of the logistic function, which imitates a step
function. In this simulation we use wbound=1 and
=50. The critical length for reconnection, , is the lower
cut-off length in the model and should be small for a detailed description,
although a value too small in complicated tessellation patterns freezes the
process. Here we use =0.05, where the relaxed cell volume
(Vstd) is 1. The step size h for the Runge-Kutta
method is 0.005 to enable smooth changes in the process.
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Partial digestion of the zona pellucida of the mouse embryo
Collection of the mouse embryo and time-lapse recordings were carried out
essentially as described (Hiiragi and
Solter, 2004; Motosugi et al.,
2005), except that embryos were photographed only once at the
2-cell stage, and were then time-lapse recorded from the morula to the
blastocyst stage to minimize light exposure (see
Fig. 3C and Movie 1 in the
supplementary material). The ZP was partially digested by incubation with 0.5%
pronase (Sigma, P8811) in H-KSOM-AA (KSOM with amino acids and 21 mM Hepes) at
37°C in 6% CO2 for a few minutes with repeated microscopic
observations, followed by several washes with H-KSOM-AA.
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RESULTS |
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TOPSUMMARYINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONREFERENCES |
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). Briefly (see Materials and methods for details), an
aggregate of multiple cells is considered as a 3D space tessellation composed
of polyhedra without gaps or overlaps (Fig.
1A). The interfaces (cell surface) and volumes are expressed by 3D
vertex coordinates, and the position of the vertex is rearranged according to
an equation of motion so that the total potential energy of the cell
aggregate, composed mainly of surface energy and elastic energy, should be
minimal. (Ass. 2) All cells have equivalent mechanical properties. (Ass. 3) A
cavity is created inside the cell aggregate by replacing one vertex with a
small polyhedron with equivalent surface properties.
Accordingly, the potential energy of the aggregate, U, consists of
three surface energy terms and two elastic energy terms. These terms are
expressed by the positional vectors ri
(i=1,..., nv) of nv vertices
of the polyhedra. The vertex obeys the equation of motion:
 | (13) |
i is a nabla differential operator with respect to
ri. The vertices move according to Eq.
(13) so that the potential energy U becomes smaller
(Honda et al., 2004). In
addition to the equation of motion, our model involves an elementary process
of reconnection of neighboring vertices
(Honda et al., 2004). When the
distance of the two neighboring vertices is shorter than a critical length,
, their relationship changes, including reconnection
(Fig. 1B).
The potential energy U is:
 | (14) |
; the volume of the relaxed cell is assumed to be
1) and of the blastocyst cavity (VI; the volume of the
relaxed blastocyst cavity is VIstd). Parameters,
O, I
(O=I), V and
VI are weights applied to the respective terms.
In this study, an aggregate of 40 cells is considered to simulate the
morphology of the mid- to late-stage expanded blastocyst
(Chisholm et al., 1985;
Dietrich and Hiiragi, 2007;
Gueth-Hallonet et al., 1993;
Smith and McLaren, 1977). The
number of cells is constant in this simulation, and the calculation is carried
out to obtain the most stable (i.e. eventual) structure of the cell aggregate
that has the minimum potential energy. An additional simulation is carried out
under exactly the same conditions with an aggregate of 31 cells, equivalent to
the nascent blastocyst stage (Chisholm et
al., 1985; Dietrich and
Hiiragi, 2007; Smith and
McLaren, 1977), which in essence produces the same results (see
Fig. S1 in the supplementary material) as described below. Therefore, the
following study is presented only for the aggregate of 40 cells. A cavity is
created in the aggregate by replacing one vertex with a small tetrahedron
(Fig. 1C; the length of the six
edges is 0.1). The volume of the tetrahedron (VIstd),
which subsequently forms the polyhedron, is forced to increase during
t=0-500 (Fig. 1D) to
half the total volume of the initial cell aggregate
(VIstd=20), reaching the expanded blastocyst stage. The
simulation includes an additional phase, after the increase in the volume
(VIstd) stops (t=500), until the actual cavity
reaches the maximal volume (which is not necessarily equivalent to that at the
t=500 time point and is usually delayed owing to the elastic nature
of the structure; see Fig. 4A)
and the structure reaches a stable state, depending on the conditions. The
blastocyst cavity is an intercellular space in the mammalian embryo, created
by the directional fluid secretion of the outer layer cells
(Aziz and Alexandre, 1991;
Calarco and Brown, 1969;
Wiley and Eglitis, 1981). This
secretion most likely occurs in essentially all outer blastomeres with
apicobasal epithelial polarity, thus initiating multiple small cavitations
(Motosugi et al., 2005), and
these multiple cavities coalesce with each other to form one large cavity. The
current simulation calculates the most stable structure with the minimal
potential energy composed of 40 cells and one cavity that should correspond to
the eventual blastocyst morphology (see Discussion).
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I 
k
(Ik-Istd)2, is included in Eq. (14),
in which Ik is the surface area of the cells facing the
blastocyst cavity, Istd is the surface area in relaxed
condition, and I is an elastic constant of the surface area.
Thus, the potential energy U takes the form:
 | (15) |
This modified Eq. (15) allows the computer simulation to successfully
produce a structure recapitulating the mouse blastocyst
(Fig. 2A,B). In fact, this
additional elastic energy of the cell surface facing the blastocyst cavity has
the effect of minimizing deviation from the relaxed cell surface area, and
suggests that there should be some structure lining the blastocyst cavity with
a tendency to keep its surface area closer to the relaxed condition. This
result thus suggests the presence of the basement membrane at the basal side
of the TE facing the blastocyst cavity, and/or possibly the difference in cell
surface elastic properties or adhesive properties between the TE/primitive
endoderm and the epiblast. Indeed, we find past
(Nadijcka and Hillman, 1974;
Thorsteinsdottir, 1992) and
recent (Klaffky et al., 2001)
studies clearly showing the presence of the basement membrane at the surface
of the TE cells facing the blastocyst cavity, and the difference in its
molecular components between the TE and the ICM. Thus, the basement membrane
in this region may account for the altered surface properties required by the
model and might be essential for maintaining the integrity of the cavity
during blastocyst morphogenesis.
Under this simulation condition (Fig.
2A,B), however, the embryonic axis, i.e. the location of the inner
cell cluster, is never fixed and remains variable during the process of
calculation (data not shown), indicating that there is no certain orientation
in which potential energy U in Eq. (15) becomes minimal. It is
important to note, however, that there is no problem in forming the blastocyst
structure itself, consistent with the fact that the ZP is not essential for
the mouse embryo to develop to the blastocyst stage
(Motosugi et al., 2005;
Kurotaki et al., 2007). This
simulation result rather predicts that, without the ZP, the embryonic axis of
the mouse blastocyst can be oriented to any direction at an equal probability
in terms of the potential energy (see below). This result leads us to include
an additional element in the equation: (Ass. 5) the embryo is enclosed by a
capsule (corresponding to the ZP in vivo) that constrains movement of the
cells. This is expressed in the potential energy U as a restriction
energy by a boundary, f(ri), in
which f(ri) is a quadratic
function, i.e. f(ri)=1 expresses
a quadratic surface (e.g. sphere or ellipsoid). Thus, Eq. (15) now takes the
form:
In an initial attempt in which the surrounding capsule is assumed to be spherical, the axis of the embryo again remains variable (Fig. 2C). However, when the capsule is ellipsoidal (1.2: 1) with its long diameter vertical (Fig. 2D) or parallel to the z-axis (see Fig. 4), the position of the inner cell cluster, i.e. the axis of the blastocyst, always coincides eventually with the long axis of the ellipsoidal capsule. This indicates that the potential energy U of the cell aggregate becomes minimal when the position of the inner cell cluster is localized at one end of the long axis of the ellipsoidal capsule, thus eventually stabilizing the embryonic axis parallel to the long axis of the ellipsoidal ZP. This is independent of the initial position of cavity formation, regardless of whether it is initiated from a center (black circle in Fig. 2A; the outcome in Fig. 2B-D) or from a peripheral point (blue circle in Fig. 2A; the outcome in Fig. 2E,F).
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). In 13 embryos in which the ratio was approximately
1.2, the Em-Ab boundary tended to be aligned with the shortest ZP axis
(Fig. 3A,B, Embryo Group I;
P=0.018, 
2 analysis), thus confirming one of the two
predictions of the computer simulation (i.e. when the ZP is ellipsoidal; see
Fig. 2D-F and
Fig. 4) as well as the
mechanical constraint model for blastocyst morphogenesis proposed earlier
(Alarcon and Marikawa, 2003;
Kurotaki et al., 2007;
Motosugi et al., 2005). The
other prediction (when the ZP is absent or spherical, i.e. when there is no
mechanical influence or bias by the ZP) was again experimentally verified by
partially digesting the ZP to create a larger and spherical ZP that would
exert essentially no mechanical or spatial constraints on the embryo
(Fig. 3C and see Movie 1 in the
supplementary material; showing the same experimental embryos). Note the
spherical shape of the blastomeres of the 2-cell embryos
(Fig. 3C, 47:10 post-hCG),
which indicates the absence of mechanical constraints [see Motosugi et al.
(Motosugi et al., 2005) for
the ellipsoidal blastomeres of the normal 2-cell embryo]. In these
experimental embryos with a spherical ZP, the tilt angle between the shortest
ZP axis and the Em-Ab boundary (Fig.
3A) is randomly distributed
(Fig. 3B, Embryo Group II;
n=55, P=0.64, 2 analysis), which confirms
the other prediction of the computer simulation when the ZP is spherical (see
Fig. 2C). The calculation process to find a stable state is exemplified in Fig. 4A (and see Movie 2 in the supplementary material). In this example, the aggregate is surrounded and restricted by an ellipsoidal (1.2:1) capsule, corresponding to an ellipsoidal ZP in vivo. A vertex (solid circle in Fig. 4A at t=0), replaced by a tetrahedron (see Fig. 1C), is enlarged until its volume reaches half the initial total volume (at t=500; see Fig. 1D). The embryonic axis (Em-Ab axis) keeps changing with respect to the outer capsule until the embryonic pole, the position of the ICM, is localized at one end of the long axis of the ellipsoidal capsule (t=2000). The structure is then `stabilized', remaining unchanged thereafter at least for a substantial time (t=2000-3500) equivalent to the time required to stabilize the structure (t=0-2000). Tracing of cells (Fig. 4A, marked in green, red, orange and blue; and four cells surrounding the orange cell at t=1000) further demonstrate that cells and the cavity gradually change their position relative to each other during the process; those cells marked either with a black circle or a black square at some point acquire a cell at their interface with the orange cell, and those marked with red circle and square disappear from the plane of observation (see Discussion). The enlarging cavity eventually becomes surrounded by an outer single-cell layer, with its position stabilized at one end of the long axis, while a cluster of cells forms on the opposite side (Fig. 4B,C and see Movie 3 in the supplementary material). Thus, the simulation recapitulates well the blastocyst morphology in vivo, with the outside cells corresponding to the TE and the inner population to the ICM.
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; Dietrich and
Hiiragi, 2007; Kelly et al.,
1978), corresponding to those of the in vivo embryo. Intriguingly,
despite the assumed equivalence in mechanical properties of the 40 cells in
various simulations (Ass. 2), enlargement of the cavity enhances the
difference in volume of the inner versus outer cells
(Fig. 5), with the inner cell
volume stabilizing at significantly lower levels (0.702±0.0088) than
that of the outer cells (0.744±0.0136, average cell volume±s.d.;
P<0.001, Student's t-test). This prediction based on the
computer simulations is again confirmed by a recent study
(Aiken et al., 2004)
demonstrating that in vivo the average cell volume of the outer cells is
significantly larger than that of the inner cells. |
DISCUSSION |
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TOPSUMMARYINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONREFERENCES |
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; Kurotaki et
al., 2007; Motosugi et al.,
2005). The computer simulation in the present study first confirms
that the mechanical constraint imposed on the embryo is indeed sufficient to
orient the blastocyst axis. Furthermore, it illustrates the cellular
mechanical properties minimally required for creating the blastocyst
morphology, thereby allowing us to gain a deep insight into blastocyst
morphogenesis from a mechanical point of view.
When the dynamics of the cell aggregate are restricted by the capsule of a
deformed sphere, equivalent to the ZP in vivo, the inner cell cluster (ICM)
consistently localizes to one end of the long axis of the ellipsoidal capsule.
The simulated blastocyst contains the ICM formed by 11 or 12 cells, thus
recapitulating well its composition in vivo. During the calculation process,
most of the outside cells contribute to the TE, while the inside cells
contribute to the ICM, except for one cell in this simulation that moves from
inside to the outside TE (data not shown). However, a quantitative description
of the contribution of cells to the ICM or TE awaits further studies with
more-complex simulations (see below). The volume of the inner cells becomes
significantly smaller than that of the outer cells during the simulation
process, again similar to the difference observed in the blastocyst in vivo
and despite the assumption of component cell equivalence in the simulations.
Cellular polarization at the 8-cell stage and subsequent asymmetric division
play a major role in the generation of asymmetry between inside and outside
populations (Dietrich and Hiiragi,
2007; Johnson and McConnell,
2004; Ralston and Rossant,
2008). The difference in cell size in those populations, for
example, can be initiated by asymmetric divisions, and may be further enhanced
by mechanical constraints, as seen in this study. Thus, the inside-outside
asymmetry essential for blastocyst morphogenesis
(Tarkowski and Wroblewska,
1967) may emerge autonomously. This idea contrasts with the major
role assigned to localized determinants in developmental mechanisms in
non-mammalian `model' organisms. Whereas some reports
(Niwa et al., 2005;
Smith, 2005;
Strumpf et al., 2005) claim a
crucial role for the differential expression of Cdx2 and Oct4 (Pou5f1)
molecules in specifying the fate of TE and ICM, recent studies, consistent
with our present one, suggest that the initial difference might emerge as a
result of cellular polarization (Ralston
and Rossant, 2008), possibly includes stochastic mechanisms in the
subsequent asymmetric divisions (Dietrich
and Hiiragi, 2007), and is ultimately stabilized to accommodate
intrinsic (gene expression pattern) and extrinsic (mechanical and structural
integrity) cues.
Several issues remain to be addressed, largely owing to the limitations of
current knowledge and simulation techniques. First, this simulation is not
designed to recapitulate the developmental process of blastocyst morphogenesis
because it lacks cell division and initial multiple cavities. This study
allows us, however, to conclude that for an aggregate of 40 intrinsically
equivalent cells with one cavity that has half the entire cell volume
(conditions equivalent to the embryo at the mid- to late-blastocyst stage),
the calculated structure has the minimal potential energy, regardless of the
actual developmental process. Namely, the present computer simulation predicts
that, even if the total cell number increases and multiple cavities are
created during the developmental process, the eventual expanded blastocyst
(composed of around 40 cells and most likely one cavity of half the entire
cell volume) will have the calculated structure in terms of mechanical
properties. Based solely on the mechanical properties, it is most likely that
such an aggregate will eventually form a structure containing an inner cluster
of cells localized to one end of its long axis, if the surrounding capsule is
ellipsoidal. The process of the calculation (as shown in
Fig. 4A) may thus appear less
dynamic than development leading to blastocyst, as we observe only minor
rearrangement of the cavity and the surrounding cells. The complete simulation
to recapitulate blastocyst morphogenesis would certainly require integration
of cell division and multiple cavity formation to provide dynamic aspects of
the developmental process. Secondly, the TE:ICM cell volume ratio is 1.06 in
the present simulation, whereas the actual cell volume ratio in the blastocyst
in one study was reported as being 1.67
(Aiken et al., 2004). This
discrepancy might be partly due to less heterogeneity in the cellular
population of the present simulation. Since embryonic cleavage in the mouse
preimplantation stage is not precisely synchronous, the embryo at the
blastocyst stage contains cells of various sizes and there may be dynamic
changes in their relative position. Our simulation lacks this heterogeneous
aspect, focusing on the eventual structure of the simplest population.
Furthermore, the present simulations consider the cell membrane as a flat
interface, without curvature, which may additionally account for the reduced
difference in the TE:ICM cell volumes in the simulation. Thirdly, we assigned
the surface energy coefficient of the outer surface and of the blastocyst
cavity surface (O and I, respectively) at
0.7, and that of the interface energy between the neighboring cells at 1.0,
based on the assumption that the surface tension of the single-membrane layer
should be less than that of the double layer. More-complex simulations await
measurement of the surface tension of the actual cell membrane in different
regions of the embryo.
The present simulation also provides an insight into the mechanism by which the inside cells form a distinct cluster (ICM) instead of a continuous cell layer beneath the TE. This would indeed be the case only if the entire structure of the embryo were precisely radially symmetrical, and cavity formation were initiated at the very center of such a structure. However, in biological systems, such absolute precision is essentially impossible, and once there is the slightest deviation from the central location, the cavity never returns to the center but stabilizes at the periphery, forming a cluster of inside cells, the ICM. Taken together, the present findings using computer simulation reveal several unique features of early mammalian development, and will provide a conceptual basis for understanding the molecular mechanism of early embryonic patterning.
Supplementary material
Supplementary material for this article is available at
http://dev.biologists.org/cgi/content/full/135/8/1407/DC1
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ACKNOWLEDGMENTS |
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Footnotes |
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Present address: Department of Molecular Life Science, School of Medicine,
Tokai University, Kanagawa 259-1193, Japan 
Mammalian Development Laboratory, Max-Planck Institute for Molecular
Biomedicine, Muenster D-48149, Germany
Institute for Integrated Cell-Material Sciences (iCeMS), Kyoto University,
Kyoto 606-8501, Japan
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