Mechanically based generative laws of morphogenesis

Abstract

A deep (although at the first glance naïve) question which may be addressed to embryonic development is why during this process quite definite and accurately reproduced successions of precise and complicated shapes are taking place, or why, in several cases, the result of development is highly precise in spite of an extensive variability of intermediate stages. This problem can be attacked in two different ways. One of them, up to now just slightly employed, is to formulate robust macroscopic generative laws from which the observed successions of shapes could be derived. Another one, which dominates in modern embryology, regards the development as a succession of highly precise 'micropatterns', each of them arising due to the action of specific factors, having, as a rule, nothing in common with each other. We argue that the latter view contradicts a great bulk of firmly established data and gives no satisfactory answers to the main problems of development. Therefore we intend to follow the first way. By doing this, we regard developing embryos as self-organized systems transpierced by feedbacks among which we pay special attention to those linked with mechanical stresses (MS). We formulate a hypothesis of so-called MS hyper-restoration as a common basis for the developmentally important feedback loops. We present a number of examples confirming this hypothesis and use it for reconstructing prolonged chains of developmental events. Finally, we discuss the application of the same set of assumptions to the first steps of egg development and to the internal differentiation of embryonic cells.

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Eur. Phys. J. E (2013) 36: 132 DOI 10.1140/epje/i2013-13132-x
Morphogenesis can be driven by properly parametrised
mechanical feedback
L.V. Beloussov

DOI 10.1140/epje/i2013-13132-x
Regular Article
Eur. Phys. J. E (2013) 36: 132
THE EUROPEAN
PHYSICAL JOURNAL E
Morphogenesis can be driven by properly parametrised
mechanical feedback
L.V. Beloussov
a
Lab orato ry of Developmental Biophysics, Faculty of Biology, Moscow State University, Moscow 119992, Russia
Received 11 February 2013 and Received in final form 23 June 2013
Published online: 25 November 2013 –
c
 EDP Sciences / Societ`a Italiana di Fisica / Springer-Verlag 2013
Abstract. A fundamental problem of morphogenesis is whether it presents itself as a succe ssion of links
that are each driven by its own specific cause-effect relationship, or whether all of the links can be embraced
by a common law that is p o ssible to formulate in physical terms. We suggest that a common biophysical
background for most, if not all, morphogenetic processes is based upon feedback between mechanical
stresses (MS) that are imposed to a given part of a developing embryo by its other parts and MS that are
actively generated within that part. The latter are directed toward hyper-restoration (restoration with an
overshoot) of the initial MS values. We show that under mechanical constraints imposed by other parts,
these tendencies drive forth development. To provide specificity for morphogenetic reactions, this feedback
should be modulated by long-term parameters and/or initial conditions that a re set up by genetic factors.
The experimental and model data related to this concept are reviewed.
Introduction
The morphogenesis of multicellular organisms is an ex-
tremely compli cated (the most complicated in nature)
process that is enfolded in 3D space and time. Gener-
ally (although with some exceptions), morphogenesis is
directed towards diminishing the symmetry order (rota-
tional, reflectional and, in most cases, translational) of a
developing organism. Such a reduction serves as a rule
within more symmetric environments and should be pro-
vided for by internal instabilities [ 1]. In spite of their com-
plexity and substantial variability, which are most ob-
vious at earlier and intermediate stages, morphogenetic
processes and especially their end-results are perfectly re-
produced within an immense number of generations in a
given species.
Accordingly, remembering a famous definition of
physics due to Maxwell wh ich claims “Physical science is
that department of knowledge which relates to the order
of nature, or, in other words, to the regular succession of
events” [2], the reasons for attributing morphogenesis to
a list of p hysical problems are quite clear. To treat this
process in a way that is common in physics means to fo-
cus our efforts towards revealing dynamic laws that are as
broad as possible and that describe th e prolonged space-
time domains of morphogenesis. To this end, a specific
biological demand shou ld be added: these laws should be
applicable, even if in broad outlines, to the representatives
of different taxonomic groups.
a
e-mail: morphogenesis@yandex.ru
Meanwhile, in spite of few remarkable attempts [3–5]
and with few exceptions (i.e., the growth theories, see [6]),
such an approach has remained marginal. The opposing
idea, which splits the entire course of development into
small successive links in the hope of revealing some “magic
bullets”, or highly specific agents, each of which deter-
mines a single link (in physics, this would be identical
to ascribing a new motive force for each subsequent sec-
tion of a falling stone trajectory), has dominated. In most
cases, in classical embryology such agents were identified
as embryonic inductors, while, in modern developmental
biology, these agents included regulators of gene activity
and/or signaling pathways. In any case, a natural and fun-
damental question has remained and, in most cases, has
been neither formulated nor answered: why does each of
these agents act only at a given location and time period?
Moreover, recent progress in molecular and cell biology
has brought researchers towards u nexpected conclusions
on the lack of one-to-one relationships between a given
molecular factor and the higher level effects that look to
be determined by this very factor. For example, the prod-
ucts of the same genes an d /or signaling pathways trigger
quite different morphological events that have nothing in
common with each other. Many such examples have been
scattered throu ghou t the most frequently used text books
(see [7,8]; see also [9]). Thus, even if we knew all of the de-
tails regarding the space-temporal schedule of gene expres-
sion, we would not be able to d educe th e course or result
of development at the morphological level from this infor-
mation. Interestingly, the reverse statement is not true:

Page 2 of 16 Eur. Phys. J. E (2013) 36: 132
the development of homologous morphological structures,
even of taxonomically remote species (e.g., appendages of
the Insects and Vertebrates) is regulated by closely related
groups of genes [7].
It is true that th is situation has been widely acknowl-
edged and is often qu ali fied as “context-dependency” of
the genes and/or the end-results of standard signaling
pathways. However, this expression is a mere allegory that
confirms our ignorance, although it correctly points to-
wards the holistic nature of developmental reactions. Our
aim is to replace this allegory with a more concrete model.
In doing this, we shall follow a dualistic approach.
First, we will attempt to formulate a general scheme for
the dri vin g factors of development that are assumed to
be universal for at least all of the Metazoans and cor-
respond at the same time to the basic properties of the
active “soft matter” [10,11]. Next, we will suggest that, in
each concrete case, this scheme is modulated in the long
term by a set of independent factors that are of genetic
origin. This idea can be precisely formulated in terms of
a self-organization theory (SOT), which we believe to be
the main basis for apprehending morphogenesis. In SOT
terms, the universal factor that brings forth morphogen-
esis may be non-linear feedback between the sensing and
generation of mechanical stresses (MS) in embryonic tis-
sues, while the modulating factors may be the parame-
ters and/or initial conditions of this feedback. By defini-
tion, the parameters are the variables whose characteristic
time and space dimensions are an order of magnitude or
more greater than those of the traceable dynamic vari-
ables. Accordingly, in a biological context, the notion of
the parameters should be related to spatially and tem-
porally smoothed patterns (in the utmost case constant
throughout) rather than to precisely targeted regulatory
factors. Development begins from such smoothed patterns
(genetic, epigenetic or directly imprinted from the exter-
nal environment) as a rule, and one of our main tasks will
be to explore how they can provide what is usually called
a “specificity of morphogenesis”.
This paper will mostly deal with theoretical and com-
putational aspects of the proposed concept. Amongst
these aspects, the detailed analysis of scaling mechanisms
and the concept of parametric regulation of epithelial
buckling are presented for the first time. Meanwhil e, the
related experimental data are exposed in brief, an d the
interested reader is asked to address the appropriate ref-
erences [12–18].
Mechanics of morphogenesis: from ad hoc
introduced forces to stress-based feedback
More than a century has passed since the German
anatomist Wilhelm His advocated the leading role of me-
chanical forces in morphogenesis [19], but for most of
the intervening period morphogenesis has been treated as
nothing more than the blin d end-result of a complicated
chain of events governed by other factors, and has been
described mostly in terms of classical ch emistry. To arrive
at the idea of mechanically based self -regulation within
single cells and entire organisms, substantial transforma-
tions had to take place in biology and the physical sci-
ences. First, impressive progress in the molecular biology
of a cell led to the replacement of the old view of a cell as
an inflated balloon with a liqu id freely diffusing content by
a concept that described a molecularly crowded, intracel-
lular space with most of the water molecules being bound
to cytoskeletal structures [20,21], Moreover, a set of finely
tuned chemomechanical transducers generating precisely
directed pulling and p ushin g forces and being themselves
mechanosensitive were discovered [22–26]. Notably, these
forces were shown to retain their coherency far beyond
the single cell level, i.e. regular MS patterns that were
perfectly correlated with the given stage of morphology
in embryos of all studied species have been detected [27–
31]. In numerous studies, the effects of stress modulations
upon morphogenesis were also demonstrated [32–35]. It
was also understood that the forces that were involved in
morphogenesis were precisely balanced so that their vector
sums at each material point and time moment are prac-
tically equal to zero (acceleration period s are negligib ly
small) [12,36,37]. Consequently, as applied to morphogen-
esis, we must address stresses of tension, compression or
shear rather than free forces in the Newtonian sense. A
self-stressed stat e of a living cell is closely associated with
its geometry, and this was determined to be a necessary
condition for the very survival of cells [38].
On the other hand, application of the main SOT prin-
ciples to stressed bodies showed that even homogeneously
distributed forces could produce quite complicated geome-
tries due to instabilities [39] (see fig. 1A, B). In both de-
picted cases, the immediate driven force for shape forma-
tion was the tendency to move towards mechanical equilib-
rium and to equalize stress d istrib ut ion throughout an en-
tire body. Concurrently, owing to several constraints (e.g.
the initial geometry of a body and its mechanical param-
eters), a system may fall into some intermediate potential
wells long before achieving its “absolute” thermodynamic
equilibrium. The intermediate wells correspond to the re-
sulting shapes and may have a reduced symmetry order
(see also [40]). The main l esson from these models, which
correspond to some real situations in plant rudiments [41],
is that creation of a regular shape does not necessarily re-
quire some detailed pre-pattern that is associated with
uneven distri bu tion of some chemical substances. Rather,
a regular and complicated morphology could be set up by
the initial geometry of an entire object and its mechanical
parameters, which may be distributed uniformly in space
and time. However, the non-uniformity, e.g. a gradient dis-
tribution of any developmentally important parameters,
could be allowed as a special case of initial/border condi-
tions.
Instructive as they would be, both of the aforemen-
tioned models are far from being complete because their
initial shapes (in the first case, a rather complicated one)
have been taken ad hoc, and the morphogenetic events
that were derived from them were relatively poor. This
is because these models did not include any feedback be-

Eur. Phys. J. E (2013) 36: 132 Page 3 of 16
Fig. 1. Two examples of shaping regulated by uniformly distributed mechanical stresses. A) inflation of a flattened balloon
covered by elastic shell produces a series of vertical folds (solid slits) while its deflation makes horizontal folds (dotted) (see [39],
with the author’s permission). B) Bending patterns of laterally compressed elastic rods, connected with a firm substrate by
several vertical springs. In each case a resulted pattern corresponds to a minimal elastic energy state of an entire two-compo nents
(ro d s + springs) system and to a maximally homogeneous spread of this energy. Note that in no cases the bending wavelengths
fit the spring arrangement (see [41], with the authors’ permission, mo dified).
tween the successive shapes. Let us address, therefore, the
next generation of models including feedback.
In creating such models, a phenomenologically simple
but fundamental subdivision of MS into active and p assive
stresses should be introduced, which means that at every
moment of morphogenesis we will compare the mechan-
ical situation in different regions of the embryo, on e of
them strictly localized in space/time, while another spa-
tially/temporally more smo oth ed. We will define as active
the MS that are generated by chemo-mechanical transduc-
tion in the “given” (strictly localized) embryo region (A)
and within a “given” (rather brief) time period, and de-
fine as passive those MS generated outside A during more
prolonged (or less specified) times.
To my knowledge, the notion of active-passive stress
feedback was first employed in a morphogenetic context
in [42]. The authors suggested that the active tan gential
contraction of a cell or a group of cells within an epithe-
lial sheet by stretching the neighbourin g cells stimulates
those cells to actively contract. In such a way, the passive
stretching-active contraction relay should travel along a
cell layer. Meanwhile, this model cannot be directly re-
lated to morphogenesis because it does not reproduce its
main property, which is a capacity to form spatially re-
stricted, stationary multicellular structures, although it
describes the behaviour of mechanically excitable tissues
in broad outlines.
In the next set of models [43,44] , the tension produced
by cell contraction was regarded as an inhibitory rather
than a promoting factor. Accordingl y, a “+, −” feedback
loop was introduced that permitted modeling of the for-
mation of stationary cell domains. In particular, the Be-
lintzev et al. model (BM) described a self -organized (i.e.
lacking any pre-patterns) segregation of an embryonic ep-
ithelial layer into alternating domains of columnar and
flattened cells, which are observable e.g. during gastrula-
tion, neurul ation and the formation of sensory placodes.
While being unable to reproduce all of the important fea-
tures of morphogenesis an d requiring some principal addi-
tions (see below), the BM is still instructive and deserves
a d etailed analysis.
In its simplest form, the BM is expressed by a differ-
ential equation (1) describing the rate of columnarisation
(dp
i
/dt)ofthei-th cell belonging to a 1-dimensional (ex-
tended along the x-axis) epithelial layer, where p = h − w
(h is a cell’s height, and w is its width in the plane of
the layer). The rate is assumed to be determined by three
right-part members: 1) Bistability of each indi vid ual cell,
i.e. its inherent tendency to move towards two stable
states, columnar or flattened, where this value is expressed
by a non-linear 3rd-order point function f (p
i
) having two
stable nodules with an unstable nodule between them. 2)
Transmission of the “columnarisation tendency” from one
neighbouring cell to another, defined as the “contact cell
polarisation” (CCP), which is considered to be a diffusion-
like process (although “diffusion” of a physical state rather
than a chemical substance is allowed) and, therefore, de-
pends upon the second-order derivative of p along x.3)
Inhibition by a viscoelastic tangential tension T , where
k is the coefficient of viscoelasticity. Accordingly, for the
case of a 1-dimensional cell layer, the BM equation is as
follows:
dp
i
/dt = f(p
i
)+δ
2
p/δx
2
− kT. (1)
In the case of a layer with fi rmly fixed edges, which is
most common situation, the tangential tension is cre-
ated exclusively by cell columnarisation. As a result, feed-
back between this process and increasing tension is estab-
lished, which causes the entire process to be self-organized
(fig. 2A). It is simple to show th at under these conditions
T ∼p, where p is the cell columnarisation averaged
across the entire layer
p =
n

i=1
p
i
n
.
Correspondingly, we take T = qp, where q gives a mea-
sure of the tangential stress that is produced by the al-
gebraic sum of cell deformation. The biological meaning
of introducing a member (p) is to get a quantitative ex-
pression for the action of an indivisible “whole”. As a next
step, the dp
i
/dt value for each cell i is normalized accord-
ing to the p value by making it linearly proportional to
the difference (p−p
i
). As a result, we obtain a final
form of the BM equation for a mechanically closed system

Page 4 of 16 Eur. Phys. J. E (2013) 36: 132
Fig. 2. Belintzev et al. model for morphogenesis of epithelial layers. A) General scheme. Red arrows: active tangential contrac-
tion. Blue arrows: passive stretching. Green arrows: spreading of contact cell polarisation. Black points denote strict fixation of
a layer’s edges. B)-E) Modeled patterns under different initial conditions and perturbations. Horizontal axis: linear coordinates,
vertical axis: cell polarisation (p, see text). Starting configuration shown in blue and the final one in red. B) Unlimited spreading
of cell polarisation under free layers’ edges. C)-E) A stable segregation of a layer with fixed edges into the domains of highly
p olarised and flattened cells. Initial conditions in C) are presented by a gradient of cell polarisation, in D) by a single polarised
p erturbation (vertical arrow) and in E) by two closely arranged perturbations (arrows). In B)-E) Horizontal axis: length of
1-dimensional model cell layer, vertical axis: cell columnarisation [45].
(i.e. a cell layer with fixed ends in the absence of external
forces)
dp
i
/dt = f(p
i
)+δ
2
p/δx
2
− kq(p−p
i
). (2)
Note that the contribution due to the 3rd term is negative
for all cells with p
i
< (p) (these cells are more flattened
than average) and is positive for the more columnar cells.
Another function of the 3rd term is to increase the dis-
tance between the st able p nodules. This role makes the
general idea of the influence of a “whole” upon the final
states of the individual cells more concrete.
When considering the experimental basis of the BM,
one should n oti ce that the assumption of the bistability
of a cell, which is described by the first term in the ri ght
part of eq. (2), is in tune with the classical concept of
epigenetic landscape [45]. The next term that describes
the CCP also is experimentally supported [12]. The third
term, however, should be considered as a prediction of a
model that remains open for empirical verification. The
first group of supporting evidence is related to the BM
capacity to reprod uce a sharp and robust (stable to local
perturbations) segregation of an initially homogeneous cell
layer into columnar and flattened cell domains (fig. 2B-E).
A next group of evidences comes from the unique ability of
the BM to reproduce, without any additional assumptions,
a “scaling” property of morphogenesis, i.e. preservation
of the geometric similarity of the embryo structure un-
der its different absolute dimensions (this phenomenon,
discovered long ago but yet to be explained, is usually
called “embryonic regulation” [7]). Indeed, the following
phenomenological equation can be used:
p =
P
1
l + P
2
(L − l)
L
,
where L is the total length of the cell layer, l is the length
of one of the domains, and P
1
and P
2
are the final p values
that are established throughout each domain. The follow-
ing equation results
l
L
=
p−P
2
P
1
− P
2
.
As long as the right-hand side of the equation does not
contain any dimensional values, the proportions of the do-
mains should be kept intact under an infinite range of em-
bryo dimensions. No other model dating from and includ-
ing that developed by Turing [46] can reproduce scaling
without some additional assumptions.
In general, BM suggests a new approach for the es-
tablishment of embryonic patterns that is distinct from
those beforehand dominated. Indeed, as exemplified by
the famous Driesch statement “the fate of an embryo part
depends upon this part position within a whole” (see [7]),

Eur. Phys. J. E (2013) 36: 132 Page 5 of 16
Fig. 3. Testing Belintzev’s model with the use of statistical distributions of divergent-convergent cell movements (see for
details [49]). A) Detrended presentation of an empirical cell rates distribution showing overnormal excesses of moderately
diverged (to the left) and extensively converged (to the right) c ells. Horizontal axis: cell rates, µm/min. B) A linear ascending
function (dense) when plotted in det rended coordinates for normal distribution is transformed into S-shaped graph similar to
that shown in A). C) Graph of eq. (2), displaying a bistable point function (dotted curve O
1
OO
2
), a linear function kε(pi −p)
and the sum of the both (solid curve O
′
1
ZO
′
2
). Lilac horizontal arrow indicates the direction of a gradual shift of a linear function
as cells columnarisation proceeds. Vertical shadowed ellipse displays a concentrates set of highly columnarised cells in the vicinity
of a new right nodule O
′
2
and horizontal ellipse illustrates more smooth arrangement of flattened cells between zero point (Z)
and the new left nodule O
′
1
(connected by transparent arrows with the corresponding regions of empirical distribution). D) A
gradual decrease of the differences (∆, µm/10 min) between converging and diverging cell movements rates during first 2 h after
the start of experiments.
and due to the concept of positional information (PI) [47]
the patterns of developing embryos were regarded to be
settled down in the form of certain coordinate grids es-
tablished independently and prior to any cell activities
that they must govern. Such concepts cannot answer the
following fundamental questions: what is the origin of the
postulated coordinate grid; how does it determine the en-
tire course of development; in what moments of develop-
ment does this grid originate and why at the next moment
is it exchanged by another one? In particular, it has yet
to be explained how such a grid can emerge when devel-
opment is begun from a chaotic cell arrangement [48].
The advantage of the BM is in avoiding the idea of any
pre-existing coordinate grid or more detailed pre-patterns.
Instead, the mechanical activity of individual cells permits
them to scan the average MS within an entire embryo
body, or at least in a large part of it, and is considered a
necessary an d sufficient condition for integrated develop-
ment, including scaling capacities. Indeed, the biological
meaning of the term −kq(p−p
i
) is that each cell p erma-
nently senses (or explores) the averaged tangential tension
of a layer to which it belongs, and changes its shape with a
rate that is linearly proportional to the difference between
p and its own momentary shape. This phenomenon
is nothing more than the perception of tension that is
averaged throughout the entire layer, which is identical,
according to the BM, to a “feeling of a whole” that has
been postulated in a general form in classical embryology.
Recently new evidence in support of the existence of
a quasi-linear factor with the properties of the third term
of eq. (2) has been obtained [49]. Mutual shifts of sev-
eral hundred cell pairs belonging to Xenopus early gas-
trula tissues were traced by time-lapse filming and di-
vided into two categories: converging (cells appr oaching
each other) and di verging (cells moving apart). As long
as the cells were tightly bound to each other (i.e. un-
able to move freely), the converging movements indicated
tangential contraction and, hence, columnarisation, while
the diverging movements indicated tangential extension
(that is, flattening) of some cells located in between the
measured ones. Thus, in the long run, the dp/dt values
were measured. By comparing the distributions of con-
verging/diverging cell rates with the linearised normal
(Gaussian) distributions, we detected regular over-normal
excesses of the numbers of extensively columnarised and
moderately flattened cells (fig. 3A). Similar distributions
took place in the same coordinate system if an inclined,
linear function was introduced (fig. 3B). This evidence
was the first for the presence of an inclined, quasi-linear
component that corresponded, in BM terms, to the term
kε(p
i
−p)=−kε(p−p
i
). A more detailed scenario of
the participation of this component follows (fig. 3C).

Page 6 of 16 Eur. Phys. J. E (2013) 36: 132
Fig. 4. Active extension response to the over-threshold stretching of an epithelial sheet. A) General scheme. Strong enough
tangential contraction of t he left part cells may induce the active extension of the right part cells (double-head arrows) associated
with convergent movements of laterally located cells (oblique arrows). B) Trajectories of cell movements in explants of Xenopus
embryonic ectoderm stretched in horizontal direction, traced within first 20 min after the end of stretching. C)-E) Experimental
testing of TIAE. C) is a non-stretched sample.
Fig. 5. A-D. Generation of segmented patterns in the circular cell rows under different values of T parameter (shown by figures).
A) and B) illustrated the robustness of two domains pattern in spite of different T values. Under further T diminishment the
number of alternated domains increases. From [54], modified.
Initially, the entire cell population is settled along the p
axis in the immediate vicinity of a left stable nodule (O
1
),
which is determined by b i-stable dynamics. To trigger cell
columnarisation, a local perturbation is required to bring
up a small grou p of cells (up to one cell) along the p axis
to the attraction basin of the right-stable nodule, or O
2
(fig. 3C, blue arrow). As seen in fig. 3C, the presence of the
inclined linear component will enhance this basin, shift the
nodule further right (O
′
2
) and accelerate the shift of cells
along the p axis to this nodule. As a result, a nearly cell-
free area is expected to occur immediately to the right of
point zero (Z), and the cells condense in the vicinity of the
O
′
2
nodule (see the vertically shadowed ellipse in fig. 3C
connected by an arrow with the corresponding group of
cells in fig. 3A). At the same time, due to the position of
the linear component, the leftward shifts along the p axis
towards the flattened cell shapes should, at the start of the
process, be much smaller (horizontally shadowed ellipse).
Meanwhile, as cell columnarisation continues (p moves
to the right), the difference between the rates of move-
ment towards columnar and flattened cell shapes should
diminish, and this expectation is also confirmed (fig. 3D).
However, in spite of all of the advantages of the BM, it
describes no more than a restricted portion of the mor-
phogenetic pro cess and is not completely closed (self-
organized) because the initial perturbations are taken ad
hoc. Also, as shown experimentally, the segregation of a
cell layer into columnar and flattened domains is followed
by the generation of active pressure stresses within the
flattened domain, the pressure force being oriented in the
previous tension direction [14]. We define this value as the
tension-induced active extension (TIAE) (fig. 4A-F). A
wide-spread example of TIAE is a so-called “convergent
cell intercalation” [50–52].
To describe TIAE, we developed a model [53] that in-
cludes the main BM assumptions but adds the concept
of a threshold parameter Tch(0 < T ch < 1) taken to be
equal for all cells of a given sample: the threshold should
be exceeded for TIAE to take place. Accordingly, TIAE
reactions should be reduced under an increase in Tch so
that cell behaviour will approach that which was described
by the BM. However, when Tchvalues are low, TIAE can-
not be neglected and produces new effects. Namely, under
these conditions, a columnarisation of even a short cell file
should stretch the adjacent cells to a degree that is suffi-
cient to overpass Tch and, therefore, to exert a tangential
pressure to the adjacent area, thus promoting its cells to
become columnar. In this way, a single local perturbation
will b e sufficient for subdividing an initially homogeneous
cell layer into a number of fl attened and columnar seg-
ments, wh ose numbers increase with a decrease in Tch
(fig. 5). Under even smaller Tch values, the stationary
structures disappear and are exchanged by running waves
of cell columnarisation .

Eur. Phys. J. E (2013) 36: 132 Page 7 of 16
Fig. 6. Fairly precise preservation of scaling capacities within
the framework of TIAE model. Horizontal axis: cell number;
vertical axis: ratios of a flattened cell domain length to the
entire length of a model raw. Column T displays threshold
values for each graph. Under T ≤ 0.3 more than one colum-
nar/flattened domain appears and under further T decrease
the stationary domains are replaced by running waves. Under
very high T values and small cell number no stable patterns
app ear. For more details see [54].
Fig. 7. A robustness (preservat ion of proportions) o f cell rows
segregation to columnarised and flattened domains under dif-
ferent number and arrangement of initial perturbations (filled
squares). Lower row illustrates the arrangement characterized
by damping one of the initially perturbed cells (arrow). Thresh-
old value is 0.57.
Therefore, the introduction of a Tch parameter largely
extends the repertoire of the modeled structures when
compared to the BM proper. As for the most attractive
BM property, i.e. its ab solute scaling capacities, although
within the framework of the TIAE model the absoluteness
is lost, under realistic (not too small) numbers of cells,
the deviations from ideal scalin g were negligible (fig. 6).
TIAE models also showed a considerable robustness, so
that columnar domains of the same size were pr oduced
under rather different numbers and arrangements of initial
perturbations (fig. 7). Interestingly, some of the perturbed
cells, depending upon their relative positions (fig. 7, lower
row), returned to unperturbed shapes, which once more
indicated a holistic dependence of the reactions of indi-
vidual cells.
Integrating the observed cell reactions into a
common scheme of mechanical feedback
We will now describe an attempt to integrate the BM and
TIAE models into a common construction of the feedback
between active and passive MS, and this integration will
also include some oth er morphogenetic situations, in par-
ticular, those associated with the buckling of epith elial
layers. A central suggestion is that any piece of embryonic
tissue (from a single cell to whole embryos) responds to
MS changes that are caused by any external force (nor-
mally originati ng from other parts of the same embryo)
by the active generation of mechanical forces that are di-
rected toward restoration of the initial MS values but, as
a result, overshoot these to the opposite values. When-
ever such changes in stress are unevenly distributed, such
as a stronger stress in one place than in another, or are
anisotropic (stronger in one direction), the induced re-
sponses are assumed to be directed toward reducing (with
an overshoot) whichever deviations were gr eatest [12,28];
(for comments see [54]). This suggestion, which is illus-
trated by fig. 8, may be called a hyper-restoration (HR)
model.
Before addressing the morphogenetic applications of
the HR model, it should be pointed out that cell reac-
tions that are directed in the opposite direction to the
applied perturbations are widely presented. A standard
response of microfilaments and microtubules to stretching
forces is the self-assembly of those structures and, hence,
generation of pressure in the stretched direction; on the
other hand, mechanical relaxation leads to the disassem-
bly of these structures [55–59]. Similarly, the increase of
membrane tension promotes exocytosis and, therefore, re-
laxation, while in contrast, relaxation/compression stim-
ulates endocytosis [17,60–63]. Also, stretching promotes
growth of tension-loaded cell-cell and cell-matrix contacts,
while relaxati on/compression acts in the opposite direc-
tion [64,65]. In several crucial cases, overshoot effects at
the macroscopic level can also be demonstrated. Cases
of a tension-induced active extension (fig. 8A) are obvi-
ous without any further comments: the tension is trans-
formed into the similarly directed pressure. For the re-
verse cases of increased relaxation-promoted tension, con-
siderable overshoots that are associated with the for-
mation of columnarised cell domains have been demon-
strated [15,35].
We shall now demonstrate that the main cell reac-
tions that are postulated by the BM and TIAE model can
be reproduced, and some new models can be predicted,
within the HR model framework. We start from contact
cell polarization, which can be interpreted as follows: if
cell A is columnarised by an external perturbation, it will
stretch the neighboring cell B perpendicularly to the plane
of the layer, thus triggering the similarly directed active
extension within cell B. As a result, the next cell, C, will
be stretched, and this reaction, which may be called the
extension-extension (EE) relay, is identical to CCP.
The TIAE reaction can b e easily derived from the HR
model, which implies that a “+, +” feedback loop between
contraction and extension of the linearly arranged areas

Page 8 of 16 Eur. Phys. J. E (2013) 36: 132
Fig. 8. Model of hyperrestoration of mechanical stresses. A), B) Schemes of the responses to stretching and to relax-
ation/compression, correspondingly. Horizontal axis: mechanical stress (compression to the left, tension to the right). Vertical
axis: time. C) A typical way for a response to stretching (cell intercalation). D) Response to relaxation by tangential contraction
(columnarization) of some neighboring cells. Vertical bars: firmly fixed edges of a sample.
exists. Because of being able to start via instabilities and
small fluctuations out of a homogeneous state, this reac-
tion can be regarded as the main to ol for a “spontaneous”
break in translational symmetry, so often taking place in
embryonic development (fig. 9). We shall now define this
reaction as a contraction-extension (CE) feedback (see b e-
low for discussion).
A special type of CE feedback is of primary importance
for epithelial buckling and can be termed the curvature
increase (CI) feedback. This feedback is triggered by the
slight bending (that is, increase in the local curvature) of
an elastic cell layer by any external force (usually a lat-
eral p ressure). Such a bending will stretch th e convex and
shrink the concave surface (fig. 10A). The presence of the
corresponding stresses can be confirmed by the conver-
sion of a layer back to its initial shape if it were released
no later than a few minutes after bein g artificially bent.
However, if maintaining a sample of embryonic tissue in
a deformed state for several dozens of minutes, the oppo-
site reaction will occur, i.e. an imposed bending will be
slowly reinforced [16,17]. This occurrence fit s the expec-
tations of the HR model, which predicts that a convex
surface of a sample will (hyper)release its stretching by
inserting new material, whereas a concave surface tends
to engulf an excessive amount of its shrunken surface to
(hyper)restore its initial, tensed state. Both processes may
be accompanied by a flow of material from the concave to
convex side (fig. 10B, arrows) and are directed towards the
increase in curvature. In addition, the active bending will
initiate the oppositely directed bucklings onto its flanks
Fig. 9. A)-C) Successive steps of a contraction-extension (CE-)
feedback taking place within initially homogeneous tissue bulk.
Solid red arrows: active deformations, thin blue arrows: passive
deformations. A) Active contraction of a left part stretches the
right one. B) The active extension response of the right part
(mediated by convergence-extension) compresses the left part
initiating its subsequent active contraction. C) Both parts are
active, the contracted one extending transversely, while the
extended one narrowing.
(fig. 10B, dashed contours), and the entire series of folds
can commonly be formed. Lastly, if a curved part of the
epithelia presents a fold (or a “lip” by ubiquitous termi-
nology) by itself, the same tendencies will initiate a flow
of cells from the convex to the concave surface, which is
termed an “involution” (fig. 10C).

Eur. Phys. J. E (2013) 36: 132 Page 9 of 16
Fig. 10. A scenario of the active increase of an imposed curva-
ture. A) Passive convex side stretching and concave side com-
pression caused by an external force. B) Active extension of
a convex surface and contraction of a concave one directed
towards hyper-restoration of perturbed stresses and accompa-
nied by a flow of cell material towards a convex side (black
arrows). Dashed contours depict the inversely oriented curva-
tures expected to be formed on the flanks of the central one
(see text). C) Same mechanisms when attributed to a curved
lip will promote the involution of cells from the convex to the
concave side.
Another important HR model-based feedback is un-
folded in 3D space. This type of feedback deals with
toroidal bodies to which a number of embryonic struc-
tures and, first of all, the circular blastoporal lips, are
similar. As shown by incision experiments [16], the surface
of the lips is under tension. However, it is known from me-
chanics [66], that the meridional tensions on the surface
of a toroidal body twice-exceeds the equatorial tensions.
Under these conditions, the HR model predicts cell re-
arrangements that are directed towards (hyper)releasing
only meridional tensions. Such rearrangements may be re-
alized by equatorially oriented cell convergence followed
by cell intercalation (fig. 11A-C). As a result, the torus
will be transformed into a tube with a diminished diame-
ter (fig. 11D). Then, the maximal tensions will b e oriented
transversely to the tube axis, thus aiding it to transform
again into a toroidal structure. Such “swings” between
toroidal and tubular structures are observed, for example,
in the development of Echinodermata embryos.
Formalization of the HR model using Van der
Pol equations
Hyper-restoration responses can be reproduced with the
use of a two-stable-nodules version of the well-known Van
der Pol system of differential equations:
dx/dt = y − kx − B (k>0), (3.1)
εdy/dt = −(cy
3
+ ay + x) − D (a<0). (3.2)
The nodules (fig. 12, P and Q) separate the areas of
autonomous movements (red arrows) from those requir-
ing external perturbations (blue arrows). Accordingly, the
latter are thr esholds that should be overridden to jump
to another nodule. The n odules are stable only if the x-
zero isocline is inclined from the lower left to upper right.
Doing so provides the overshoots of the x-comp on ents of
the active responses as compared to those of the oppo-
sitely directed external perturbations: on the other hand,
if the x-zero iso cline were inclined to the opposite side,
the nodules would become unstable, that is, the entire
system would not exist. Hence, the excesses of the active
responses over the oppositely directed perturbations are
the natural consequences of Van der Pol dynamics. We
assume th at a slow variable x corresponds to MS (ten-
sions when x>0 and pressure when x<0) that have
relatively slow dynamics under normal conditions. A term
(−kx) from eq. (3.1) describes the self-inhibition of MS. At
the same time, we link the fast variable, y, with a rapid
switching between two alternative regimes (i.e. pressure
and tension-generating regimes).
Such a presentation of the HR model elucidates the
role of parameters and/or initial conditions in the sug-
gested feedback. It is plausible to identify the model pa-
rameters with the coefficients k, c, a, which regulate the
rates of the observed processes, and the initial conditions
with free members B and D. Next, many morphogenet-
ically important results can be achieved by altering the
linear x-zero iso clin e (eq. (3.1)) without perturbing the
more refined, non-linear dynamics (eq. (3.2)). Namely, by
shifting the x-zero isocline parallel to itself by varying the
B value from eq. (3.1), the threshold values for the al-
ternative regimes will be changed asymmetrically (one of
them will increase, while the opposite on e will decrease).
However, k variations will change both values equally. It
can be seen that the B and k modulations are identical
to the changes in the Tch values. For example, by dimin-
ishing the Tch values, more finely segmented patterns are
expected, and vice versa (see fig. 5).
The Van der Pol model can be also used to analyze
perpetual (auto) oscillations without stable points. These
cases will be considered in the next section.
Specifying modeled shapes by parametrising
curvature-increasing feedback
Most of what is deemed by morphologists to be stage-
or species-specificity of organic shapes is related to the
changes in the local curvatures that are produced by ep-
ithelial buckling. Accordingly, we shall explore whether
the parametric modulations of CI feedback, which we be-
lieve regulate the curvatures, can produce some biologi-
cally realistic shapes. As a main set of model examples,
we take hydroid polyps from the Thecate subfamily, whose
colonies are characterized by precise and species-specific
curvatures. Additionally, our constructions will be appli-
cable to other taxonomic groups. The Thecate polyps are
grown and shaped by finely regulated (at least partly by
Ca
2+
-dependent mechanisms) species- and stage- specific
pulses (growth pulsations, GP) of hydrostatic pressure

Page 10 of 16 Eur. Phys. J. E (2013) 36: 132
Fig. 11. Expected evolution of toroidal shapes. A) Cell convergence (followed by intercalation) towards torus meridians which
hyper-releases the maximal meridionally oriented tensions. B, C) Unrolled part of a torus surface showing its intercalation-driven
elongation which transforms a torus into a tube. D) Meridionally oriented cell convergence/intercalation towards meridians of
the arisen tube again result in a secondary deformation of the tube towards a toroidal shape (dashed contour).
Fig. 12. Phase portrait of Van der Pol equations with two stable nodules (P and Q) which illustrates HR response. Horizontal
axis: mechanical stresses (MS). Vertical axis displays mechanochemical switching taking place under overthreshold stress values.
Blue arrows show MS perturbations c aused by external forces and red arrows show the active responses of the affected tissues
which due to the inclined orientation of x-zero isocline exceed the perturbations and are directed to the opposite sides.
within vacuolar compartments of densely packed cells that
are located in the distal parts of the outgrowths (see [67]
for more details). At the height of each GP (i.e. under a
maximal pressure impulse), the curvature of a rudiment
becomes more differentiated (i.e. a number of the oppo-
sitely buckled zones is increased), while in the opposite
GP phase, the shapes are smoothed again; however, they
retain a part of the previously achieved curvatures. This
occurrence indicates the presence of elastic resistance to
deformations, which should be considered one of the pa-
rameters regulating the curvatures. Another parameter
should be the “bending rigidity” of a cell layer, which
was used in [41] for modeling morphogenesis of floral rudi-
ments: th e more folds are f ormed, the lower the rigidity.
A third par ameter (or set of parameters) should regulate
the time-amplitude relations within each GP, which play
a significant role in determining the specific shapes of hy-
droid rudiments.
The initial shapes used for modeling were as simple
as possible. For imitating shape formation in the lon-
gitudinal plane, we began from U-shaped configurations
(fig. 13A), and for doing the same in the transverse plan e,
we chose circles. In both cases, the model samples con-
tained the same number N of kinematically in dependent
units throughout the entire model time. The curvature
increase was modeled by ascribing a centrifugal vector that
was proportional to the local curvature created by three
neighbouring units. In the case of hydroid polyps, the cur-
vature increase was taken to be proportional to the lateral
cell-cell pr essure.

Eur. Phys. J. E (2013) 36: 132 Page 11 of 16
Fig. 13. Modeling morphogenesis of U-shaped rudiments under different pulsations regimes (displayed in the lower parts
of B-H frames; red bars corresp ond to centrifugal shifts and blue bars to centripetal ones). A: Starting shape, points are
kinematically independent elements. Shapes B-D are generated under the same pulsation patterns modulated by increasing
ratios of centripetal/centrifugal shifts. Shapes E-G are generated under different pulsation patterns and shape H is modeled
under constant pressure regime. All the modeled shapes except B and H are characterised by the presence of long-range order
(buckling perimeters are larger than the distances between neighbouring kinematically independent elements).
The parameters were kept constant for all of the points
of a model sample and during th e entire modeling time,
and were formalized as follows [68]:
1) Elasticity parameter W was taken within the range
0 <W <1. It provided backwards (centripetal) shifts
of imaging points that were proportional to the W val-
ues and followed each subsequent centrifugal displace-
ment.
2) The bending rigidity parameter R was proportional to
the minimal distance between two neighbouring points
that were able to shift in dependently from each other.
Hence, R ∼ 1/N , where N is the number of kinemati-
cally independent units (imaging points).
3) Parameters regulating time-amplitude GP patterns
were exp ressed as 5-figure sets (e.g. 2, 3, 8, 10, and 7)
that displayed the shifts that took place during con-
secutive time periods within each subsequent GP. Neg-
ative values corresponded to centripetal shifts.
For hydroid polyps, Van der Pol equations that lacked
stable nodules and described perpetual auto-oscillations
were considered. When the same biophysical meaning was
ascribed to the variables as before, th e W parameter,
which describes elastic tensions, was p lotted along the
horizontal axis towards the ri ght, and hydrostatic pres-
sure, which was the main force that increased the cur-
vatures, was plotted towards the left. The switching pa-
rameter was plotted along the vertical axis and, in this
case, could be identified as intracellular [Ca
2+
]. After this
procedure was performed, it was clear that the W param-
eter was most closely connected with the D member from
eq. (3.2), because the latter regulated the position of the
y-zero isocline along the pressure-tension coordi nate (i.e.
a rightward shift of the y-zero isocline would increase the
pressure/tension pr oportion, and vice versa).
The factors regulating GP time-amplitude relations
are to some extent connected with the a parameter
(eq. (3.2)), which regulates the shape of the y-zero isocline.
However, the main contribution in specifying GP patterns
is provided by member b from eq. (3.1), which regulates
the horizontal position of the x-zero isocline. This regu-
lation occurs because the more elevated the position of
x-zero isocline, the slower will be the movement along the
upper branch of y-zero isocline. Accordingly, the system’s
maintenance under highly pressurized state will be more
prolonged. In t his way, GP p attern s of a different “acute-
ness” can be established.
The main conclusions derived from the modeling re-
sults were the following.
For U-shaped and circular rudiments, the basic role
in the rough regulation of shape formation is played
by the W parameter, in that to reproduce biomorphic
shapes, this parameter shou ld be kept within a certain
optimal range. Under W val ues that are too small, the
initial shapes are transformed into tightly packed bun-

Page 12 of 16 Eur. Phys. J. E (2013) 36: 132
Fig. 14. A “morphospace” in N, W coordinates which is separated by wavy lines to the domains populated by shapes with
different number of lobes (shown by figures). Empty shapes covered by solid horizontal lines have at least one symmetry plane
and are absolutely stable under infinite numbers of iterations; those covered by dott ed lines are asymmetric and slightly unstable.
Filled shapes are permanently rotating.
dles (fig. 13B), and under W val ues that are too large,
smoothed shapes with flattened surfaces are produced
(fig. 13D).Within the intermediate W range (0.35 <W <
0.55), easily recognisable biomorphic shapes that resemble
different rudiments (from outgrowths of hydroid polyps to
the brain vesicles of vertebrate embryos) emerge (fig. 13C,
E-G). Such a role of the W parameter is not trivial, and ,
a priori, one could not expect that a mere change in the
proportions of the forward-backward shifts, which should
be kept constant for the entire modeling space/time, pro-
duces such profound morphological changes. The rea-
son may be th at by varying the W values, we change
the sizes of the short -range shifts of the imaging points
that occur between successive rounds of the curvature
“recalculations” within an entire r ud iment scale (long-
range mechanosensing). Therefore, we must conclude that
for producing biomorphic shapes, the r ou nd s of long-
range mechanosensing should be discrete and separated
by proper time intervals: if the latter are to o large, the
local activiti es dominate over the global activities, thus
producing densely entangled webs that lack a long-range
order (as in fig. 13B). However, under too small recalcu-
lation intervals, the arisen shapes are too smooth (as ob-
served under extensive diffusion). Therefore, n atur e must
utilise an intermediate regime.
Variations in the GP patterns under constant W values
produce more refined but nevertheless substantial changes
in the modeled shapes (fig. 13C, E-G), and these include
tri-dichotomy transformations (cf. fig. 13F and E, G).
Importantly, under completely constant growth regimes
(with no alterations in the shift values), the resulting
shapes do not exhibit long-range order (fig. 13H). This
finding confirms the role of alternating regimes in provid-
ing a holi stic order.
The main results of circular shape modeling are plot-
ted within the W/N coordinates of morphospace (fig. 14).
This graph is subdivided into oblique zones with station -
ary shapes (empty) that are alternated by areas occupied
by asymmetric and unstable shapes (filled). The number of
lobes in each of the stable shapes is smaller than the corre-
sponding Nvalue. Hence, in all cases, the symmetry order
of the obtained figures is reduced in relation to the max-
imally possible order which is permitted by the bending
rigidity. The main conclusion from this set of data is that
a reduced symmetry order of similar shapes can be gen-
erated under different combinations of N and W parame-
ters. Notably, the given number of lobes are kept constant
(i.e. they retain their structural stability) within relatively
large ranges of W values that are flanked by zones of struc-
tural in stabili ty (fig. 15), thus, 5-fold patterns are pre-
cisely reproduced within 0.385 <W <0.414 ≈ 3 × 10
−2
units and are abruptly exchanged by 4-fold patterns af-
ter a further W increase in only 10
−3
units. However,
such a change is unstable because, from W =0.381 to
W =0.425, the 5-fold shapes periodically return and com-
pletely disappears only out of these limits. The pr esence
of shape instability zones may be of interest in the context
of morphological evolution.

Eur. Phys. J. E (2013) 36: 132 Page 13 of 16
Fig. 15. A detailed evolution of a 5-lobe shape under smooth changes of W values. While a stable 5 · m shape is permanently
repro duced only within the densely outlined W range, it also emerges from time to time within a larger range (dashed contour),
for example under W =0.418 and W =0.381. On the other hand, 4-lobes shapes also appear within this range.
In general, the HR model, after being orchestrated by
a rather restricted set of parameters, demonstrates exten-
sive and realistic generative capacities. This finding con-
firms the presence of intrinsic, invariant laws for shape for-
mation. However, these laws, when taken in isolation, are
too general to provide all of the multiplicity and specificity
for real shapes. This work should be done using the afore-
mentioned parameters and/or initial conditions. Certainly
the latter, if taken independent of the context of the feed-
back in which they are involved, have no morphogenetic
meaning. However, due to the space-time constancy of the
parameters, there are no principal difficulties in relating
them to a more or less complicated set of genetic factors
and/or their epigenetic consequences. Accordingly, a task
that is in reverse to the standard f ormulations of genome-
morphological interactions can be formulated by begin-
ning from shapes (rath er than genes) and asking what the
genetic factors responsible for their formation might be.
As mentioned above, the pathways from morphogenesis to
genes may be more definite than the reverse pathways.
HR-based modeling of prolonged
morphogenetic successions
The main test for any model that claims to interpret em-
bryonic development is whether it can reproduce, at least
in broad outlines, a prolonged succession of developmen-
tal events rather than single, isolated steps. To prove that
the HR model passes this test, we shall briefly review the
earliest developmental periods of variou s Metazoans and
then reach blastulation and gastrulation, which are the
most crucial periods of development.
Hyper-restoration feedback in early development
The first gl obal mechanical transformation, which takes
place within a few minutes of fertilization, is an extensive
relaxation of tensions on the egg surface due to exocytosis
of cortical alveoli. In some species (e.g. nematodes) the
wave of relaxation, which spreads from the point of sperm
entrance, determines the anteroposterior axis of the future
organism. This step is followed by a contractile respon se of
subcortical actin, which provides a shift of morphogeneti-
cally active substances to the anterior egg region [69]. This
sequence of events falls into the category of CE feedback.
Fig. 16. De novo emergence of the main elements of an entire
embryo symmetry (animal-vegetal (An-Veg) and ventral-dorsal
(V-D) axes) in small explants of the blastocoel roof ( arrow 1)
and suprablastoporal area (arrow 2). While in the case 1 only
An-Veg axis is restored, in case 2 in addition the ventro-dorsal
p olarity is emerged, as indicated by the curved shape of a rudi-
mentary gastrocoel. Pointers show secondary emerged blasto-
p ores [14, 16].
The interactions between the actively contracted equa-
torial furrows and extending polar regions taking place
within each next cleavage division [70,71] belong to the
same category.
The next symmetry breaking event that occurs in am-
phibian eggs, i.e. cortical rotation, also has an essential
mechanical component [72]. It is initiated by an actively
elongated microtubule array that compresses more ani-
mal regions of the egg cortex and, hence, should trigger
(according to the expectations of the HR model) the con-
traction of these regions. Similar stress topology can be es-
tablished along any non-predetermined egg meridian as a
result of egg rotation in the gravity field. In both cases, to
equilibrate mechanical tensions within the entire egg scale
the cortical tensions should converge in the dorso-animal
direction and diverge in the ventro-vegetal direction. A
similar scheme has been proposed for fish eggs [30]. As
in early development of nematod es, mechanical patterns
precede the chemical patterns and are responsible for their
formation.

Page 14 of 16 Eur. Phys. J. E (2013) 36: 132
Fig. 17. A sketch of early development of Deuterostomia representatives, emphasizing the main instabilities, symmetry breaks
and morphomechanical feedback. A) An eccentric shift of the blastocoel (blc) caused by instability of its concentric position
triggers CE feedback (red arrows) which enhances the blastocoel shift and initiates blastopore trough (bp). Blue wavy line
indicates relaxation/compression of the lower part. This is the first symmetry break (establishment of animal-vegetal axis). B)
The active circular contraction of the blastopore caused by its toroidal geometry (red arrows along blastopore periphery) creates
a gradient of tensions around (blue converging arrows). C) Radial symmetry of a circular blastopore contraction is broken due
to action of CE feedback along its perimeter. At the same time, a new CE-feedback is established between the blastopore and
a suprablastoporal zone (SBA). This is a second symmetry break, establishing dorso-ventrality. D) Two successive stages of
the dorsal part differentiation into trunk and head regions, associated with formation of new CE feedback between these parts.
Now the translational symmetry of the dorsal part is broken. ant: anterior, pst: posterior embryo poles.
Morphomechanics of blastulation and gastrulation
The blastula is a spherical body with an internal closed
cavity (blastocoel). The blastocoel is under turgor pres-
sure caused by an influx of sodium/chloride, and this pres-
sure stretches its walls. In nearly all species, the blasto-
coel takes an eccentric position by subdividing the blas-
tula wall into a thin roof and a thick bottom. It is com-
monly thought that such a polar structure is irreversibly
determined already in early oogenesis. However, morpho-
logical polarity can be established de novo in small, ex-
planted pieces of the blastocoel roof (fig. 16, 1) where
it cannot be predetermined (for details, see [14]). The
same is true for the blastopore which, although being nor-
mally prelocalised by the aforementioned cortical rotation,
can be formed de novo as ru dimentary or perfectly devel-
oped ingressions (fig. 16, pointers). Thus, self-organisation
should play an important role during the blastulation-
gastrulation developmental period. We will now discuss,
in broad outlines, how this process can act within the
framework of the HR model.
First, we will show that in the context of the HR
model, the concentric position of the blastocoel is unsta-
ble. Indeed, any small shift of the pressurised cavity from
a precisely concentric position will in crease its stretching
load onto the thinner wall (future roof) more than to the
opposite one (future bottom). To (hyper)release the max-
imal stretching, the roof should be actively extended (by
means of cell-cell intercalation), which compresses the bot-
tom and leads to its active contraction. In other words, a
standard CE feedback will be established along the entire
blastula perimeter (fig. 17A). If allowed to proceed further,
the contraction of the bottom should lead to its invagina-
tion, i.e. to blastopore formation (fig. 17A, bl). This step
is the first developmental symmetry break caused by in-
stability. Second, if the blastopore is well expressed, its
circular edge (lip) can be considered as a half torus that
will be transformed into a narrow tube (see fig. 11 and
the corresponding comments). As a result, the embryonic
territory surrounding the blastopore should be stretched
in a gradient fashion (with tension “density” increasing
towards the blastopore) (fig. 17B). This stretching will

Eur. Phys. J. E (2013) 36: 132 Page 15 of 16
promote blastopore-directed movements of the surround-
ing cells (cell movements up the tension gradient towards
equalising tensions h ave been demonstrated elsewhere [18,
52]). Concurrently, the even contraction of the circular
blastoporal periphery should be unstable due to mechani-
cal competition of its parts, so that the entire lip should be
split into alternated and mutually enhancing contraction-
extension regions. If the blastopore is sufficiently wide,
this contraction should produce an n-order, radial p at-
tern (giving, for example, n tentacles). Meanwhile, when
the blastopore perimeter decreases or when it is initiated
from a narrow arch (such as in most vertebrate embryos),
a single contraction focus will arise that marks the d or-
sal side of the embryo. This step is the second symmetry
break, and its formation is associated with the establish-
ment of a new, powerful CE feedback between the domi-
nating contraction zone of the blastopore and the adjacent
parts of the blastopore surroundings, which are called the
suprablastoporal zone (SBA, fig. 17C). A third (transla-
tional) symmetry break that leads to the next CE feedback
is expected to be initiated at the dorsal emb ryo side due
to uneven stretching, i.e. the SBA is extended while the
more anterior region is contracted. The first region will be
transformed into the trunk, while the second region will
be converted into head structures.
It is now appropri ate to end our survey, although the
subsequent developmental processes (e.g. limb develop-
ment or the formation of sensory organs) are also promis-
ing in a morpho-mechanical analysis. Moreover, an impres-
sive set of recent data [73–75] indicates the applicability
of the same approach to cell differentiation events.
References
1. R.D. Mullins, Cold Spring Hrb Persp Biol 2, a003392
(2010).
2. J.C Maxwell, Matter and Motion (Dover, London, 1991).
3. A.G. Gurwitsch, Arch Entwmech Org. 51, 383 (1922).
4. A.G. Gurwitsch, A Theory of Biological Field (Sovetskaya
Nauka, Moskva, 1944) (in Russian).
5. D’Arcy Thompson, On Growth and Form (Cambridge Uni-
versity Press, Cambridge, 1961).
6. A.I Zot in, R.S. Zotina, Phenomenological theory of de-
velopment, growth and ageing (Nauka, Moskva, 1993) (in
Russian).
7. S.F. Gilbert, Developmental Biology (Sunderland, MA,
Sinauer Ass., 2010).
8. R Alberts et al., Molecular Biology of the Cell (Taylor &
Francis Group, 2003).
9. J.-J. Kupiec, The Origin of Individuals (Singapore, World
Scientific, 2009).
10. S. Ramaswamy, Annu. Rev. Condens. Matter Phys. 1, 323
(2010).
11. U.S. Schwarz, M.L. Gardel, J. Cell Sci. 125, 3051 (2012).
12. L.V. Beloussov, The Dynamic Architecture of a Develop-
ing Organism (Kluwer Academic Publishers, Dordrecht,
Boston, London, 1998).
13. L.V. Beloussov, Phys. Biol. 5, 015009 (2008).
14. L.V. Beloussov, N.N. Luchinskaia, A.S. Ermakov, N.S.
Glagoleva, Int. J. Dev Biol. 50, 113 (2006).
15. E.S. Kornikova, E.G. Korvin-Pavlovskaya, L.V. Beloussov,
Dev. Genes Evol. 219, 1 (2009).
16. E.S. Kornikova, T.G. Troshina, S.V. Kremnyov, L.V. Be-
loussov, Devel Dyn. 239, 885 (2010).
17. S.V. Kremnyov, T.G. Troshina, L.V. Beloussov, Mech. De-
vel. 129, 51 (2012).
18. A.N. Mansurov, A.A. Stein, L.V. Beloussov, Biomech.
Mo del. Mechanobiol. 11, 1123 (2012).
19. W. His, Unsere Korpers Form (Engelmann, Leipzig, 1878).
20. G.H. Pollack, Cells, Gels and the Engines of Life (Ebner
& Sons, Seattle, 2001).
21. H.X. Zhou, G. Rivas, A.P Minton, Annu. Rev. Biophys.
37, 375 (2008).
22. R. Fernandez-Gonzalez, J.A. Zallen, Sci. Signal 2,78
(2009).
23. G. Bao et al., Mol. Cell. Biomech. 3, 91 (2010).
24. B.D. Hoffman, C. Grashoff, M.A. Schwartz, Nature 475,
316 (2011).
25. J. Howard, Annu. Rev. Biophys. 38, 217 (2009).
26. J. Howard, S.W. Grill, J.S. Bois, Nat. Rev. Mol. Cell Biol.
12, 392 (2011).
27. L.V. Beloussov, J.G. Dorfman, V.G. Cherdantzev, J.
Embr. Exp. Morphol. 34, 559 (1975).
28. L.V. Beloussov, S.V. Saveliev, I.I. Naumidi, V.V.
Novoselov, Int. Rev. Cytol. 150, 1 (1994).
29. D.P. Kiehart, C.G. Galbraith, K.A. Edwards, W.L. Rick-
oll, R.A. Montague, J. Cell Biol. 149, 471 (2000).
30. E.V.Cherdantzeva,V.G.Cherdantzev,Int.J.Dev.Biol.
50, 157 (2006).
31. Ju.A. Kraus, Int. J. Dev. Biol. 50, 267 (2006).
32. R. Bellairs, D.R. Bromham, C.C. Wylie, J. Embryol. Exp.
Morphol. 17, 197 (1967).
33. V.G. Cherdantzev, Int. J. Dev. Biol. 50, 169 (2006).
34. N. Gjorevski, C.M. Nelson, Integr. Biol. 2, 424 (2010).
35. A.Ju. Evstifeeva, S.V. Kremnyov, L.V. Beloussov, Onto-
genez (Russ. J. Dev. Biol.) 41, 190 (2010).
36. G.F. Oster, G.M. Odell, Cell Motil. 4, 469 (1984).
37. B.C. Goodwin, L.E.H. Trainor, J. Theor. Biol. 117,79
(1985).
38. S. Chr Chen, M. Mrksich, Sui Huang, G.M. Whitesides,
D.E. Ingber, Science 276, 1425 (1997).
39. L.A. Martynov, Mathematical Biology of Development
edited by A.I. Zotin, E.V. Presnov (Nauka, Moskva, 1982)
(in Russian).
40. S. Svetina, B. Zeks, J. Theor. Biol. 146, 115 (1990).
41. P.B. Green, C.S. Steele, S.C. Rennich, Ann. Bot. 77, 515
(1996).
42. G.M. Odell, G. Oster, P. Alberch, B. Burnside, Dev. Biol.
85, 446 (1981).
43. A.K. Harris, D. Stopak, P. Warner, J. Embryol. Exp. Mor-
phol. 80, 1 (1984).
44. B.N. Belintzev, L.V. Beloussov, A.G. Zaraisky, J. Theor.
Biol. 129, 369 (1987).
45. C.H. Waddington, Organizers and genes (Cambridge Uni-
versity Press, Cambridge, 1940).
46. A.M. Turing, Philos. Trans. Roy. Soc. London, Ser. B 237,
37 (1952).
47. L. Wolpert, Tr ends Genet. 12, 359 (1996).
48. M. Spiegel, E.S. Spiegel, Amer. Zool. 15, 583 (1975).
49. T.G. Troshina, N.S. Glagoleva, L.V. Beloussov, L.V. On-
togenez, Russ. J. Dev. Biol. 42, 346 (2011).

Page 16 of 16 Eur. Phys. J. E (2013) 36: 132
50. R. Keller, P. Tibbett s P., Dev. Biol. 131, 539 (1989).
51. R. Keller, L. Davidson, A. Edlund, T. Elul, M. Ezin, D.
Sho ok, P. Skoglund, Philos. Trans. R. Soc. London, Ser. B
355, 897 (2000).
52. L.V. Beloussov, N.N. Luchinskaia, A.A.Stein, Dev. Genes
Evol. 210, 92 (2000).
53. L.V. Beloussov, V.I. Grabovsky, Comp. Methods Biomech.
Biomed. Engin. 8, 381 (2005).
54. L.A. Taber, Biomech. Model Mechanobiol. 7, 427 (2008).
55. O. Kaverina, O. Krylyshkina, K. Beningo, K. Anderson,
Yu-Li Wang, J.V. Small J., Cell Science 115, 2283 (2002).
56. M.K. Gardner, C.G. Pearson, B.L. Sprague, T.R. Zarzar,
K.Bloom, E.D. Salmon, D.J. Odde, Mol. Biol. Cell. 16,
3764 (2005).
57. A. Nekouzadeh, K.M. Pryse, E.L. Elson, G.M. Genin, J.
Biomech. 41, 2964 (2008).
58. S. Deguchi, M. Sato, Biorheology 46, 93 (2009).
59. T. Shemesh, A.D. Bershadsky, M.M. Kozlov, Biophys. J.
102, 1746 (2012).
60. H.R.W. Wirtz, L.G. Dobbs, Science 298, 1266 (1990).
61. J. Dai, M.P. Sheetz, Cold Spring Hrb. Symp. Quant. Biol.
60, 567 (1995).
62. C. Rauch, E. Farge, Biophys. J. 78, 3036 (2000).
63. N.C. Gauthier, M.A. Fardin, P. Roca-Cusach, M.P. Sheetz,
Pro c. Natl. Acad. Sci. U.S.A. 108, 14467 (2011).
64. O.J. Pletjushkina, A.M. Belkin O.J. Ivanova, T. Oliver,
J.M. Vasiliev, K. Jacobson, Cell Adhes. Commun. 5, 121
(1998).
65. D. Riveline, E. Zamir, N.Q. Balaban, U.S. Schwarz, T.
Ishizaki, S. Narumiya, Z. Kam, B. Geiger, A.D. Bershad-
sky, J. Cell Biol. 153, 1175 (2001).
66. L.D. Landau, L.M. Livshitz, Theory of Elasticity (Nauka,
Moskva, 1976) (In Russian).
67. L.V. Beloussov, Ju.A. Labas, N.I. Kazakova, A.G.
Zaraisky, J. Exp. Zool. 249, 258 (1989).
68. L.V. Beloussov, V.I. Grabovsky, Comp. Methods Biomech.
Biomed. Engin. 6, 53 (2003).
69. E. Munro, B. Bowerman, Cold Spring Hrb. Persp. Biol. 1,
a003400 (2009).
70. T.E. Schroeder, Ann. N.Y. Acad. Sci. 582, 78 (1990).
71. J.G. White, Ann. N.Y. Acad. Sci. 582, 50 (1990).
72. B.J. Cha, D.L. Gard, Dev. Biol. 205, 275 (1999).
73. E. Farge, Curr. Biol. 13, 1365 (2003).
74. R. McBeath, D.M. Pirone, C.M. N elson, K. Bhadriraju,
C.S. Chen, Dev. Cell 6, 483 (2004).
75. A.J. Engler, S. Sen, H.L. Sweeney, D.E. Discher, Cell 126,
677 (2006).