Simulating the mammalian blastocyst--molecular and mechanical interactions pattern the embryo.

Abstract

Mammalian embryogenesis is a dynamic process involving gene expression and mechanical forces between proliferating cells. The exact nature of these interactions, which determine the lineage patterning of the trophectoderm and endoderm tissues occurring in a highly regulated manner at precise periods during the embryonic development, is an area of debate. We have developed a computational modeling framework for studying this process, by which the combined effects of mechanical and genetic interactions are analyzed within the context of proliferating cells. At a purely mechanical level, we demonstrate that the perpendicular alignment of the animal-vegetal (a-v) and embryonic-abembryonic (eb-ab) axes is a result of minimizing the total elastic conformational energy of the entire collection of cells, which are constrained by the zona pellucida. The coupling of gene expression with the mechanics of cell movement is important for formation of both the trophectoderm and the endoderm. In studying the formation of the trophectoderm, we contrast and compare quantitatively two hypotheses: (1) The position determines gene expression, and (2) the gene expression determines the position. Our model, which couples gene expression with mechanics, suggests that differential adhesion between different cell types is a critical determinant in the robust endoderm formation. In addition to differential adhesion, two different testable hypotheses emerge when considering endoderm formation: (1) A directional force acts on certain cells and moves them into forming the endoderm layer, which separates the blastocoel and the cells of the inner cell mass (ICM). In this case the blastocoel simply acts as a static boundary. (2) The blastocoel dynamically applies pressure upon the cells in contact with it, such that cell segregation in the presence of differential adhesion leads to the endoderm formation. To our knowledge, this is the first attempt to combine cell-based spatial mechanical simulations with genetic networks to explain mammalian embryogenesis. Such a framework provides the means to test hypotheses in a controlled in silico environment.

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Simulating the Mammalian Blastocyst - Molecular and
Mechanical Interactions Pattern the Embryo
Pawel Krupinski1, Vijay Chickarmane2, Carsten Peterson1,3*
1Computational Biology & Biological Physics, Department of Astronomy and Theoretical Physics, Lund University, Lund, Sweden, 2Division of Biology, California Institute
of Technology, Pasadena, California, United States of America, 3Lund Strategic Research Center for Stem Cell Biology and Cell Therapy, Lund University, Lund, Sweden
Abstract
Mammalian embryogenesis is a dynamic process involving gene expression and mechanical forces between proliferating
cells. The exact nature of these interactions, which determine the lineage patterning of the trophectoderm and endoderm
tissues occurring in a highly regulated manner at precise periods during the embryonic development, is an area of debate.
We have developed a computational modeling framework for studying this process, by which the combined effects of
mechanical and genetic interactions are analyzed within the context of proliferating cells. At a purely mechanical level, we
demonstrate that the perpendicular alignment of the animal-vegetal (a-v) and embryonic-abembryonic (eb-ab) axes is a
result of minimizing the total elastic conformational energy of the entire collection of cells, which are constrained by the
zona pellucida. The coupling of gene expression with the mechanics of cell movement is important for formation of both
the trophectoderm and the endoderm. In studying the formation of the trophectoderm, we contrast and compare
quantitatively two hypotheses: (1) The position determines gene expression, and (2) the gene expression determines the
position. Our model, which couples gene expression with mechanics, suggests that differential adhesion between different
cell types is a critical determinant in the robust endoderm formation. In addition to differential adhesion, two different
testable hypotheses emerge when considering endoderm formation: (1) A directional force acts on certain cells and moves
them into forming the endoderm layer, which separates the blastocoel and the cells of the inner cell mass (ICM). In this case
the blastocoel simply acts as a static boundary. (2) The blastocoel dynamically applies pressure upon the cells in contact
with it, such that cell segregation in the presence of differential adhesion leads to the endoderm formation. To our
knowledge, this is the first attempt to combine cell-based spatial mechanical simulations with genetic networks to explain
mammalian embryogenesis. Such a framework provides the means to test hypotheses in a controlled in silico environment.
Citation: Krupinski P, Chickarmane V, Peterson C (2011) Simulating the Mammalian Blastocyst - Molecular and Mechanical Interactions Pattern the Embryo. PLoS
Comput Biol 7(5): e1001128. doi:10.1371/journal.pcbi.1001128
Editor: Denis Thieffry, Ecole Normale Supe ´rieure, France
Received August 26, 2010; Accepted March 31, 2011; Published May 5, 2011
Copyright: ? 2011 Krupinski et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits
unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: This work was funded by the Swedish Foundation for Strategic Research; Senior Individual Grant A3 06:215. The funders had no role in study design,
data collection and analysis, decision to publish, or preparation of the manuscript.
Competing Interests: The authors have declared that no competing interests exist.
* E-mail: carsten@thep.lu.se
Introduction
How a complete embryo emerges starting from a single
fertilized egg is an intriguing process in developmental biology,
understanding of which has important clinical implications [1].
Recent advances in live imaging have allowed for the tracking of
single cells as they grow and divide and subsequently form
different tissues of the embryo [2]. Using fluorescent labeling one
is able to monitor in real time the expression levels of key
transcription factors in single cells as they move and divide. Recent
experiments have shown significant correlations between the
individual cell fates and specific gene expression patterns [3,4].
Studies with respect to early events in the morphogenesis of the
mammalian embryo suggest that, although the combined interplay
between gene expression and cell polarity perhaps determine the
cell division rules, the mechanical properties of cells which may
also depend on gene expression, collectively organize cells into
different tissues [3–5].
The first developmental phase occurs when some of the cells
from the morula differentiate to become part of the trophectoderm
(TE) lineage, forming an outer layer surrounding the inner cell
mass (ICM) [6] (Figure 1). After the TE layer is formed, cells
secrete a fluid, which coalesces and expands as a single entity, the
blastocoel [7]. The latter gradually pushes all ICM cells to one end
of the protective outer envelope, the zona pellucida (Figure 1). At
this stage a second developmental event occurs – the formation of
the primitive endoderm (PE). This is the covering which separates
the ICM from the blastocoel. The analysis of molecular and
mechanical processes, which ensure the robust patterning of these
layers of cells [3], is the subject of this work.
Previous studies have identified specific gene expression with the
three lineages, ICM, TE and PE. The inner cells which ultimately
give rise to the three germ layers are pluripotent and express the
well known embryonic stem cell transcription factors Oct4, Sox2
and Nanog amongst several others [8]. The cells forming the
trophectoderm exclusively express Cdx2, whereas, the cells which
are part of the endoderm lineage express Gata6. There are several
mutually antagonistic interactions between these key transcription
factors. Cdx2 represses Oct4 and vice versa [9]. In addition, these
transcription factors are also positively auto-regulating (see [10,11]
and references therein), ensuring that once turned on, they remain
stably expressed.
Recent work suggests that stochasticity is instrumental in the
patterning process, in which key genes are initially expressed in a
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fluctuating manner and only later in the development does a
pattern emerge [12]. When cells have decided upon particular
lineages, the positive auto-regulation ensures that only the
trophectoderm cells, which are the outer cells express Cdx2
(simultaneously suppressing Oct4), whereas the inner cells express
Oct4 (suppressing Cdx2) thereby enforcing mutually exclusive
expression of lineage genes [13]. Similarly, Gata6 and Nanog are
very likely mutually antagonistic [14], the former is expressed in
PE whereas Nanog, which is part of the trio of embryonic
transcription factors, is expressed in the epiblast cells. In [11] a
computational model, based upon several interactions of these key
genes, was developed for the genetic circuit which determines cell
fate, i.e., TE, epiblast or PE. The main conclusion from [11] was
that the network dynamics exhibited a switch-like behavior, as a
function of an external signal. The question of interest in this work
is how the circuit dynamics of these various components regulate
cell fate, as cells become part of the PE, ICM and TE.
After the fertilized egg has undergone three rounds of division
(Figure 1), the outer cells get polarized along the apical-basal
direction. If an outer cell undergoes a symmetrical division, both
daughter cells retain the cell polarity, but if the division is
asymmetrical, the inner cell looses polarity. Cell polarity and Cdx2
expression have been implicated to feedback onto each other [15],
thereby making the polarized cells increase Cdx2 expression. This
ensures that the outer cells express Cdx2. However, this begs the
question as to which factor determines symmetrical/asymmetrical
divisions. In [15], the authors suggest that CDX2 levels themselves
affect the division pattern. Cells, which express higher levels of
CDX2, divide symmetrically, whereas for lower levels of CDX2, cells
divide asymmetrically, such that upon division, the inner cell gives up
most of its Cdx2 mRNA to the outer cell. Although the mechanism
by which cells choose their division plane by reading out the levels
of CDX2 is not known, it can be classified as implementing the
‘‘cell lineage determines cell position’’ rule. An alternative is the
‘‘cell position determines cell lineage’’ rule, which is thought to be
connected to nuclear localization of YAP, which is a cofactor of
Tead, a transcription factor upstream of Cdx2. In [16], the authors
suggest that cells that are outside lack a signal, which necessarily
allows YAP to be localized to the nucleus. In this way, the outside
cells automatically express Cdx2, whereas the inner cells do not
express Cdx2. In both of the above hypotheses, it can be assumed
that the Oct4- Cdx2 mutual antagonism gradually fine tunes any
small discrepancies in their levels, once they are determined to be
expressed in cells.
The next stage of development is the PE formation. The
transcription factor Gata6 is expressed by the PE cells, which
ensures through the mutual repressive interactions with Nanog, that
Figure 1. Schematic view of morphological and lineage specification steps during the early mouse embryonic development. Starting
from fertilization (E0), after three rounds of cleavages (E1–E2.5), the blastomers undergo compaction and polarization (E3). Then the trophectoderm
outer layer starts to separate from the inner cell mass (ICM) followed by the expansion of the blastocoel and the localization of the ICM to one part of
the embryo (E3.5). After this stage the endoderm is formed as a layer separating epiblast from blastocoel (E4.5). After 4.5 embryonic days, the
preimplantation embryo contains more than 100 cells.
doi:10.1371/journal.pcbi.1001128.g001
Author Summary
We elucidate by computational means the processes by
which the development of the mammalian embryo during
its first four to five days occurs, as it is transformed from a
single stem cell into hundreds of cells of different tissue
types. We are interested in understanding the fundamen-
tal processes of how gene expression dynamics within
each cell is coupled to the mechanical forces between
cells, such that cells move to take up their positions as part
of different tissues depending on the genes they express.
Recent experiments which track single cell movement and
division in conjunction with their gene expression
dynamics suggest various hypotheses as to how this
coupling functions to pattern the embryo. We have
developed a computational model which can test these
hypotheses. The model consists of dividing cells, interact-
ing with each other through mechanical forces, within a
confinement of embryo boundary. Each cell contains a
genetic network of specific genes which influence cell
adhesion properties and cell division plane directions. We
explicitly simulate the formation of the trophectoderm and
endoderm layers of cells which illuminates the principles
by which the embryo is robustly patterned.
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the embryonic genes are shut off. However, initially, Gata6 and
Nanog seem to be expressed in spatial ‘‘salt and pepper’’ pattern
[12]. From this initial distribution the pattern changes such that,
the cells occupying the outer layer of the ICM, which face the
blastocoel, must express Gata6. How cells get patterned in this
manner, has been the subject of ample research [3]. Three
different processes are thought to occur [17]. If the cells, which
express Gata6 have slightly different adhesive properties from the
cells expressing Nanog, the two populations of cells can get sorted
out. However, for some of the Gata6 cells, which must move out
from deeper layers of the ICM to occupy the outer layer, there
could be some type of external ‘‘homing’’ signal. It is interesting to
speculate if the fibroblast growth factor FGF signaling, which plays
role in the endoderm development [3], might provide such a cue.
Finally, cells expressing Gata6, which are unable to move through
the deeper layers and emerge to the outer layer, can undergo
apoptosis, and be hence removed from the entire population of
cells. These processes can be combined to give a robust
‘‘movement’’ of GATA6 cells [18], thereby implementing the
‘‘cell lineage determines cell position’’ rule.
One of the aspects of embryo development is the formation of
the embryonic-abembryonic (eb-ab) axis. An early proposal was
that the eb-ab axis position was correlated to the first division
plane of the fertilized egg [3]. Each of two cells would then
contribute to different tissues (TE and epiblast) of the embryo.
However, significant cell movement of individual cells occurs and
the entire mass of cells can also rotate [2]. This makes it difficult to
follow and assess the clonal expansion of cells. Another hypothesis
is that the emergence of the eb-ab axis is entirely due to
mechanical constraints [19]. The pellucid zone (ZP) is usually
elliptical, and hence it is possible that cells move into one end of
the long axis to minimize the elastic energy.
We propose a mathematical framework which takes into
account growing and proliferating cells, interacting through
physical forces to understand the patterning of the blastocyst into
the trophectoderm, epiblast and the primitive endoderm. Two
hypotheses which we explicitly explore are, (1) gene expression
determines the geometry of division (2) gene expression determines
cellular motion through modification of cellular adhesion
properties. The gene expression itself is determined by an
underlying genetic network, which is coupled to both division
and spatial location within the embryo.
Faced with the complexity of the processes described above, we
believe that such a computational approach, in which each
hypothesis is simulated explicitly, provides the means to bridge
intuition with understanding. Further, it provides novel predictions
which can subsequently be tested.
Model
The cell based model presented here aims at describing
morphogenesis of the mammalian embryo in the early phases of
development in terms of simplified mechanical interactions
between blastomeres, which are coupled to gene network
dynamics within each cell. The network dynamics feeds back on
(i) division patterns and (ii) adhesive properties of cells.
The mechanical part of our model is inspired by Dallon and
Othmer [20], who analyzed cell movement in the Dictyostelium
discoideum slug. We model cells as elastic spheres interacting with
each other and constrained by the pellucid zone. Spherical
geometry faithfully reproduces the shape of the cells in early
mammalian embryogenesis except for the flattened trophectoderm
cells where the oblate elliptical shape is more appropriate. The
blastomeres are treated as incompressible elastic bodies, whose
mechanical response is confined to three orthogonal axes. This is
equivalent to cells being represented as membranes whose
deformation is restricted by springs in three orthogonal directions.
These directions correspond to the principal directions of the stress
tensor and all the external forces acting upon the cell are resolved
to these coordinates. Cells come into contact and mechanically
interact with each other (Figure S1). Keeping track of the
attachment points of the forces on the surface of the cell allows
the model to follow the changes in translative, compressive and
tensile forces. The forces acting on the cell are summed up over all
the neighbors of the cell. The neighborhood relation itself is
determined from the Voronoi diagram of the cell centers, which is
dynamically updated at each step of the simulation. Assuming that
due to the low Reynolds number of the cell movement we can
neglect accelerations in the dynamics we write the equations of
motion for individual cells i in the form,
mi
dxi
dt~Felastic
i
zFdrag
i
zFadhesion
i
z?F Fi:
ð1Þ
In Eq. (1) we account for three types of mechanical forces:
elastic interaction between cells (Felastic), adhesive drag force of
cells sliding against each other (Fdrag) and an attractive adhesion
force (Fadhesion). The ?F Fi represents forces due to the active
movement of cells or pressure from blastocoel, which do not
originate from intrinsic mechanical interactions between the cells.
The factor mistands for the viscosity coefficient and xiis the three-
dimensional position vector of cell i (see Text S1 for details).
Each blastomere is defined by the genetic network (see Text S1
for details), which evolves the mRNA and protein concentrations
of the cell, a set of parameters (Table S1) which define its
mechanical properties, a polarization vector and the cell cycle
length. These parameters can be different for different cell types
(i.e. TE, PE and ICM).
Cell division in our model (see Text S1 for details) is a discrete
event where a single mother cell is replaced by two daughter cells.
The cells undergo division when the time elapsed from the last
division exceeds the cell cycle length. The latter is randomized
among the blastomeres according to a normal distribution [21].
The total cell volume is conserved during this step and initially
overlapping daughter cells occupy the space inside the mother cell.
During division, in addition to the obvious change of geometry,
cells also need to partition their content to the daughter cells. This
is accomplished in the model using two different recipes: (i) random
symmetrical, where the direction of division is random and the
content is distributed symmetrically, and (ii) polarized asymmetrical
division, where the direction of division is correlated with the cell
polarization vector and the content is partitioned asymmetrically.
Note that the division gives rise to opposing forces for both
daughter cells, in a direction perpendicular to the division plane.
The mechanical equations of motion as well as genetic networks
equations for the entire system are solved numerically using a fifth
order Runge-Kuta differential equation solver, which makes
adaptive time steps based on the requirement of keeping the error
less than a given threshold. A typical movie of the resulting
dynamics can be found in Video S1.
Results
Our results discuss how the two important developmental stages
of the embryo, namely the formation of the trophectoderm and
endoderm, are dependent on the processes of cell division, gene
expression and the mechanical interactions between cells within a
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confined space. We begin by considering purely mechanical
interactions between cells, and how cell divisions affect their
positioning within the embryo. Later we use this as a substrate
upon which we add gene expression, by including a genetic
network within each cell and further by allowing cells to interact
with the external environment.
Embryonic axes align due to the mechanical interactions
We first analyze the correlation between the orientation of the
blastocyst embryonic-abembryonic (eb-ab) axis and the axis of the
two-cell embryo. Our motivation is to test the hypothesis that this
phenomena occurs from the alignment of both of these directions
with the long axis of elliptical pellucid zone (ZP) due to the
mechanical constraints. This hypothesis has been analyzed
experimentally in several studies providing data both in favor
and against it [17,22–24]. The inconsistent results could be due to
different strains of specimen, different experimental techniques, or
difficulty in tracking cells given the considerable cell mobility in
the early embryo and the embryo inside the ZP as a whole.
However, this discrepancy could also be a result of different
mechanisms that are involved in the formation of the eb-ab axes
and the cleavage pattern of the two cell embryo. Since the shape of
the ZP is not perfectly spherical, it provides a directionality, which
could influence the orientation of the cells in the developing
embryo. Here we test whether the mechanical constraints arising
from the ZP geometry could be the underlying cause of the
orientation of the embryo, both at two-cell and blastocyst stages.
In [25] the authors developed a computational model of the
embryo comprised of cells with a blastocoel, surrounded by the ZP
to study mechanical effects on the eb-ab axis. They concluded that
the cells would acquire the configuration of minimal energy
orienting themselves along the corner of the ZP long axis. Hence,
mechanical interactions would determine the eb-ab axis. Although
their model demonstrates this interesting result, it does not take
into account cell proliferation. Within our model, we are able to
analyze the joint effects from mechanical interactions on
orientation of both two cell embryo and the eb-ab axes together
with cell proliferation. In our model, the ZP acts as a static barrier
elastically repelling the spherical blastomeres in contact, while the
cells interact mechanically with each other and favor configura-
tions which minimize the elastic energy of the system (Figure 2b).
Based on the fact that isolated blastomeres attain spherical shapes,
the elastic blastomere energy in the model increases with the
overlap of the spheres representing their native shape. Therefore
the two first blastomeres will position themselves along the long
axis of ellipsoidal pellucid zone minimizing their overlap or
deformation. In the model we assumed that there are no frictional
forces between ZP and blastomers, since our analysis of the motion
of the cells in experiments and test simulations including friction
suggest that such interaction has a marginal effect (see Text S1 for
details). In such case, even tiny differences (v1%) in the axes
length of the ellipsoid were sufficient to provide a positional cue for
the two cell embryo. As expected, due to the increase in
blastomere overlaps, the dynamics of the alignment process is
faster and more robust with increasing difference of axes length.
The random division of outer cells, which are close to the ZP, can
create a torque, which rotates the entire cell mass (Video S2). This
rotation is also observed in experimental movies [2], which
reaffirms that the mechanics within a confined region plays an
important role in the arrangement of cells within the blastula.
As the development of the embryo progresses, the interactions
between blastomeres become more complex due to both their
increased number and changes in their mechanical properties.
Around the 32-cell stage, with the trophectoderm well defined, the
fluid filled blastocoel cavity begins to form with secretion of
intracellular vacuoles which coalesce. We model the blastocoel in a
simplified manner, as a slowly expanding spherically shaped
region inside the ICM, aiming to capture the behavior of spatially
restricted ICM cells. The adhesion strengths of the cell-cell
interactions are deduced by what is qualitatively known for
different cell types [12,26]. Compacted ICM cells exhibit strong
self-adhesion, trophectoderm cells adhere to each other through
tight junctions and have decreased adhesion to ICM cells. While
we do not expect any adhesion-like force between the cells and the
blastocoelic fluid. The simulations are initialized with a small
spherical blastocoel volume in the center of the ZP at the 32-cell
stage that later expands to (20%–30%) of the whole embryo.
Depending upon the degree of ZP elongation, we observe
preferential localization of the blastocoel to the one end of the
ZP long axis. To ensure that our results do not depend upon the
initial 32-cell configuration, for each simulation we used a different
initial template obtained from a single blastomere by five rounds of
cleavages with stochastic time and direction of the cell division (see
Text S1 for details). A 10% difference in the length of axes of the
ellipsoidal ZP provides alignment of ab-eb axis to the long axis of
ellipsoid (Figure 3, Video S3) in 76% (n=50) of the simulations,
suggesting that this configuration is mechanically preferred and
that the oblate shape of the ZP could influence alignment of both
the two-cell embryo and the blastocyst axes. We consider two axes
aligned, for the purpose of the simulation, if the angle between
them is less than 10 degrees and we define embryonic-
abembryonic axis as the line passing through blastocoel center
and center of mass of ICM cells. One should mention that in vivo,
the ZP is not essential for blastocyst formation [2]. In our model
we can also form a blastocyst without the ZP. In that case we do
not observe the alignment of its axis with the axis of two cell
embryo (Video S4). We conclude that in cases where the ZP is
present, its shape affects the relative position of ICM and
blastocoel in agreement with findings from the model in [25].
The same mechanical constraint aligns axis of two cell embryo
and, as cells with the same lineage tend to occupy nearby positions,
Figure 2. Examples of geometrical and mechanical effects in
embryogenesis. (a) The number of inner cells in the simulations at
the 32-cell stage (E3.5 in Figure 1) as a function of the blastomeres sizes.
As the diameter of the cells increases, geometrical constraints force
more cells to position themselves inside the cluster. (b) Average elastic
deformation energy of the blastomeres (energy of the springs
simulating elastic response of the cells) as a function of time in a
typical simulation. The sharp peaks in energy correspond to cell division
events. We observe a decline of the deformation energy perceived by
blastomeres as the number of cells increase. Particularly large drop in
energy takes place at 4- to 8-cell transition. The 32-cell stage shows the
lowest deformation energy in this picture. Interestingly, these are the
stages of development when morula compaction and blastocoel
formation happen.
doi:10.1371/journal.pcbi.1001128.g002
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it influences lineage allocation to trophectoderm and ICM. This
we confirm by lineage tracking in our simulations.
Mechanics and geometry play important roles in embryo
development
In relation to the orientation of eb-ab axis and blastocyst linage
formation, timing and orientation of two- to four-cell divisions has
been studied by several groups [21,27,28]. In particular, certain
patterns of cleavages, meridional-equatorial (M-E) together with
reversed equatorial-meridional (E-M) order, were found to occur
more often (80% of cases) and were associated with specific
tetrahedral arrangement of blastomeres in the blastula. In our
model we find that the configuration of blastomeres in four-cell
embryos depends upon the size of the blastomeres relative to the
ZP and upon the ZP shape even more than upon the division
pattern. We characterize the size of four-cell embryo blastomeres
with respect to the size of the initial spherical zygote, R0. After two
rounds of cleavages and conserving the cell volume we obtain
blastomeres of the radius R2~R0=22=3. As a maximal radius of the
zygote we consider the radius of the sphere of the same volume as
the ZP. In simulations we used R0in range 0:8Rmax
varied the elongation ratio of the ellipsoidal ZP within 20%. In the
case of spherical ZP, due to the symmetry, cells always attain
tetrahedral configuration in four-cell embryo simulations. How-
ever, even a slight deviation (*5%) from spherical ZP symmetry
causes blastomeres to prefer different configurations minimizing
the total elastic energy. By decreasing the blastomere sizes at this
point, their mobility is increased since the drag force decreases and
the elastic interaction between them is lowered. This is sufficient to
rescue the tetrahedral arrangement at some point (*0:9Rmax
The exact numbers depend upon simulation parameters but the
trends are robust. In the regime of large blastomeres, when they
tend to depart from a tetrahedral configuration, their mobility is
lowered and specific patterns of cleavages become increasingly
0
to Rmax
0
and
0
).
important for their final configuration. Our results suggest that
geometrical factors like size of the blastomeres and specific shapes
of the ZP, ignored in studies of blastocyst lineage so far, may be
influencing positions of the blastomeres in the blastula.
Similarly we analyzed the number of cells located inside and
outside at the 32-cell stage as a function of the individual
blastomere size. We found that the number of inner cells decreases
when lowering the cell size (Figure 2a). While the spherical
approximation may not accurately describe the flattened shape of
the trophectoderm cells and we cannot expect to reproduce
observed ratio of inner to outer cells in this way, the result again
confirms that mechanical and geometrical constrains very likely
play prominent roles in blastocyst development.
Early mammalian embryo formation is characterized by a
sequential order of morphological events, such as morula
compaction or blastocoel expansion, which take place at precise
stages of development. Even if these events are under the genetic
control of processes inside each blastomere, the exact mechanism
governing them is unknown and it is possible that cues other than
genetics can contribute to triggering those events. Our model
offers the possibility to analyze mechanical interactions taking
place during embryogenesis. We have measured the average
energy per cell of elastic deformation of identical blastomeres, as a
function of time, from 2- to 32-cell stage (Figure 2b). High peaks in
this energy are observed during cell division, because just after a
division, the daughter cells are highly deformed from their native
spherical shape. More interestingly, we find differences in the
stress perceived by blastomeres at different stages. We see an
overall decrease of average deformation energy as the simulation
progresses. This is expected since as blastomeres are getting
smaller during cleavages, they can fit the pellucid zone shape with
less overlap on average. In Figure 2b we observe larger differences
in deformation energy between 4- to 8-cell and 16- to 32-cell
stages than between 2- to 4-cell and 8- to 16-cell stages. Also note
that the average cell deformation energy is lowest for the 32-cell
stage. Since morula formation and compaction of blastomeres
happen precisely at the 8-cell stage and the secretion of vacuoles
forming blastocoel takes place approximately at the 32-cell stage,
these results raise the intriguing question: Can sensing of
mechanical signals provide triggers for some of the important
events during embryogenesis?
Is trophectoderm formation determined by cell position
or cell division pattern?
Trophectoderm is the first occurring specialized tissue distin-
guished from the embryo mass. Despite remarkable progress in
our knowledge about this process, the exact mechanism of
trophectoderm formation is still an area of active research. Here
we evaluate the robustness of two conceptual models of Cdx2
expression pattern during this process. In the first, ‘‘position-based
model’’ the position of the cell in the embryo dictates the Cdx2
expression level, with inner cells having lower expression levels of
Cdx2 than the outer cells [13]. We have implemented this model
with a simple switch-like genetic network based on the mutual
repression between Cdx2 and Oct4 in each cell (Figure S2, Figure
S3, Text S1, Table S3) [8]. We assumed that the outer cells receive
additional signal from polarity genes which enhances Cdx2
expression in those cells (Figure 4a). Starting with small and
random CDX2 levels at the 4-cell stage, we evolve the system to 32
cells. Since the relation between position and CDX2 level in the
cell is defined in a straightforward manner in this model, we expect
the distribution of CDX2 concentration to be clearly separated
between the inner and outer cell populations. Indeed, we
consistently observe (Figure 4b,c, Table S2) significantly higher
Figure 3. Alignment of the two-cell embryo symmetry axis and
embryonic-abembryonic axis with the long axis of ellipsoidal
pellucid zone. In the case of the two-cell embryo constrained by the
elliptical pellucid zone (a) the positioning of the cells is consistently on
opposite sides of the longest axis of ellipsoid by purely mechanical
interactions and independent of the direction of the first division. The
second polar body (a small cell with little genetic content positioned on
the side of blastomers) is a product of the last meiotic division of the
oocyte and marks polarization of the maternal mRNA. The blastocoel,
which is modeled as a slowly expanding sphere inside of the embryo
(b), positions itself preferentially at one end of the same axis. The
frequency of this effect depends upon the elongation ratio of the
pellucid zone and parameters of the model. However, it is consistently
above 50% suggesting that orientation of the embryonic-abembryonic
axis is mechanically biased as well.
doi:10.1371/journal.pcbi.1001128.g003
Simulating the Mammalian Blastocyst
PLoS Computational Biology | www.ploscompbiol.org5May 2011 | Volume 7 | Issue 5 | e1001128