Self-Organization, Symmetry and Morphomechanics in Development of Organisms

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Self-Organization,
Symmetry and Morphomechanics in
Development of Organisms
Lev V. Beloussov
Department of Embryology, Faculty of Biology Moscow State University
Russia
1. Introduction
This chapter may look strange for a text-book. While the usual text-books expose firmly
established facts and theories, the main aim of this essay is to tell about what we do not
know and do not understand and to show that this “dark area” is probably greater than the
elucidated one. No less strange may look that our arguments are based to a great extent
upon the data obtained long ago and for many times described but, as we try to argue, up to
now adequately non-interpreted. On the other hand, the main pathway which we suggest to
move along, that is the application of a self-organization theory to developmental events, is
missed in conventional text-books. Taking into consideration a strange genre of this essay,
the author have to apologize the potential readers for its inevitable shortages: some points
may be discussed too briefly, while others too much emphasized. Nevertheless, my goal will
be achieved if just single readers will realize that in the science about organic development
much more than some small details are unknown and unexplained; and that the young
generation of researchers has ahead a fascinating field for further studies.
2. Do we understand development?
Being an aged Professor of Moscow State University, within several decades I am reading
Embryology lectures for a large class of Biology students. I was a witness of an exciting
transformation of this science (which, in the hope to look more modern changed its
traditional name to “Developmental Biology”) from a minor and poorly known affiliation of
zoology or histology to a powerful and highly respectable branch of life sciences, closely
linked with genetics and molecular biology and becoming an indispensable part of stem cell
research, regenerative medicine, and so on. At the first glance, everybody even to a small
extent related to this science should be proud of its achievements. But nevertheless, several
times during my lecture course I feel myself uneasy with my students, as if I do not tell
them the whole truth. And the truth is that, in spite of all the technological achievements,
we the specialists do not understand the development of organisms not so much in details,
as in its main outlines. Yes, we can produce by our willing in artificial conditions some types
of cells and multicellular structures, but we have to take these and other results as given,
without really explaining them. Actually, we cannot answer a question which looks naïve,

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but is actually very deep: why in the course of normal development a given stage (that is, a given
set of embryonic structures) is exchanged by another one, no less definite; or why, what looks even
more miraculous, a variable set of structures comes towards quite a definite end-result (Fig. 1).
Fig. 1. Two examples of developmental successions. A-M: succesive stages of sea-urchin
development.In this case the structure of each next stage is strictly determined. N-Q:
development of a hydroid polyp from a cleaving egg to larva stage.The early and
intermediate stages (N-P) have quite a variable structure, but the end-stage Q is the same in
all the cases (From Beloussov, 2008).
True, a response to a question “why” which can satisfy us is itself in no way definite and
unambiguous, especially in biology. If you ask, why a given embryonic structure
is appeared at this time and location, at least three different kinds of “explanation” can
be given.
First, some people will be satisfied by claiming that a given structure is arisen here and at
that time moment because this is required for fulfilling its subsequent physiological
functions, and/or obtaining some selective advantages, and so on. All such statements,
which exchange the question “why” by “for what purpose” belong to so called teleology – a
view which is looking into future for finding the reasons for what has happened just now or
in the past. Teleology cannot be completely withdrawn from life sciences –for a biologist to
look for goals is a respectable business. However, if we want to follow the main way taken
by other natural sciences, we have to search the answers to the “why” questions in the
immediate past of a given event, rather than in its future.

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Just this idea became a basis for a classical causality, which is often called Laplacian, because
it was formalized by the great French mathematician Pierre Simon Laplace at the beginning
of XIX century and was considered for a long time to be the only one compatible with a real
science. By this approach, the main aim of a science is to analyze the observed world to such
an extent that it could be presented as (being split to) a chain of one-to-one cause-effects
links, a single cause being able to produce no more than one event. The main task of
investigator is to compile a complete list of the causes. If fulfilling this task, the surrounding
world will become completely predictable: nothing new (unexpected) can happen in it.
Paradoxically, the Laplacian approach became, in the course of time, much more deeply
rooted in biology, than in physical sciences, which Laplace had into mind. In particular, it
has been introduced in embryology by the German embryologist Wilhelm Roux already at
the end of XIX century. Up to now it remains to be the leading ideology of this science
(although most of experimenters do not even suspect this).
Meanwhile, in physics since Galileo and Newton times another approach, which may be
called law-centered one, took the leading positions. In a certain sense, it is opposite to the
causal one, although the both developed hand by hand. While classical causality is directed
towards detalization (by splitting a world into a set of as detailed as possible cause-effects
relations), the law-centered approach tends to generalize, by establishing invariable relations
between as much as possible events. For example, if two physical bodies are moving along
different trajectories, this approach invites us to formulate a common law describing the
both movements, while if following the classical causality we should search the specific
causes for each of the movements, and even for the small parts of the trajectories. It was the
law-centered approach who gave to the physical sciences a predictive power, that is, any
power at all. Our main question will be - should we use this approach in developmental
biology, or we are completely satisfied by a classical causality? The answer will depend
mainly upon whether the successions of developmental events are underlain by perfect
causal chains, determined in all their links. Let us look, whether this is the case. In doing
this, we shall explore the possibilities of two mostly used versions of a causal approach. The
first one claims, that the main causes of the developmental events are genes, while another
ascribes a leading role to the influences of the earlier arisen embryonic structures upon the
subsequent ones.
A genocentric approach seems, at the first glance, firmly substantiated, because the genes,
or, if speaking more precisely, so called signaling pathways, that is the relays of protein-
protein interactions, triggered by so called ligands (in most cases, products of genes activity)
and switching on other genes are indispensable participants of virtually all the biological
processes, including developmental ones. If blocking (knockouting) certain genes and/or
signaling pathways, many developmental events will be abolished and distorted. Does it
mean however that there exists one-to-one relation between a gene/signaling pathway on
one hand and a given embryonic structure on the other?
Many years ago the biologists believed, that this was just the case. One of the milestones of a
first half of XX century biology was a claim: “One gene – one character” (a character was
taken at that time as something static, related to an adult state). More recently, however,
when the amazing technical progress permitted to trace the expression of single genes in the
course of development, quite unexpected results have been obtained: it turned out that the

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products of activity of the same or closely homologous genes and/or the same signaling
pathways were involved in quite different developmental events and vice versa. Text-books
in developmental and cell biology are full of such examples. Here are just few of them:
- “The interactions betweeen msx-1 and msx-2 homeodomain proteins characterize the
formation of teeth in the jaw field, the progress zone in the limb field, and the neural
retina in the eye” (Gilbert, 2010).
- The transcription factor Pax-6 is expressed at different times and at different levels in
the telencephalon, hindbrain and spinal cord of the central nervous system; in the lens,
cornea, neural and pigmented retina, lacrimal gland and conjunctiva of the eye; and in
the pancreas (Alberts et al., 2003).
- In Drosophila embryos a gene Engrailed is involved in segmentation of a germ band,
development of intestine, nervous system and wings. In mouse same gene participates
in brain and somites development. In Echinodermata it takes part in skeleton and
nervous system development (Alberts et al., 2003).
- Delta-Notch signaling pathway regulates: neuro-epithelial differentiation in insects,
feather formation in birds,fates of blastomeres in Nematodes, differentiation of T-
lymphocytes etc (Alberts et al., 2003).
- Hunchback gene is involved at the early stage of Drosophila development as one of so-
called gap genes and at the later stages participates in development of neural system.
By summarizing: if we know everything about the genes/signaling pathways being in work
in the given space/time location, we can tell nothing about what embryonic process is going
on, and vice versa. This is enough for concluding that the genes/signaling pathways in spite
of all their importance cannot be considered as “causes” of development; much better to say
that they are tools, which can be utilized by a developing organism for quite different
purposes. Certainly, the tools deserve to be studied, and such studies can be very important
and useful, but they do not help us to answer our main question. Accordingly, the results of
our studies will have no predictable power – we are doomed to investigate each next
experimental point separately.
Let us pass to the second version of the causal approach, ascribing the main role to the
interactions between embryonic rudiments. Just this version was used by Roux and his
followers.
The first task which Roux decided to solve by his approach was a long standing controversy
between two general views upon development. The first of them, called preformism,
claimed that each structure of an adult has its own material representative from the very
beginning of development, the latter being localized somewhere inside an egg or
spermatozoon. By this view, from the very beginning of development an embryo is no less
spatially complicated than the adult organism. The alternative view, called epigenesis,
negated this idea, suggesting that an early embryo is less complicated than the advanced
one, and may be even homogeneous. Roux attempted to resolve this alternative by dividing
an embryo into parts: if the preformism were true, an isolated part of embryo will produce,
under subsequent development, nothing else than that set of organs, which will be normally
produced from this very part; if, meanwhile, a part, after its isolation, will produce another
or, moreover, the larger set of organs, preformism should be rejected. Roux himself
performed this procedure by killing one of two first cells (blastomeres) of a frog egg with a

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heated needle. As a result, the remaining blastomere produced roughly a half of embryo. It
looked, as if preformism was true. However, such a situation lasted less than for a decade.
Several years later another German embryologist, Hans Driesch, separated the blastomeres
of another animal, sea urchin, by more delicate technique: by using sea water lacking
calcium ions he separated the blasstomeres, but kept all of them alive. The result was quite
another: each of two first blastomeres, and even each of the first four ones gave rise to entire,
almost normal embryos (although of a correspondingly diminished sizes), rather than to the
parts which had to be normally developed from the isolated blastomeres (Fig. 2). This effect
was called embryonic regulations.
Fig. 2. A scheme of Driesch’s experiment demonstrating embryonic regulations: so-called
plutei larvae developed from the single blastomeres separated at 4-cell stage have roughly
the same structure (at the diminished size) as the normal larva (from Gilbert, 2010,
modified).
A similar result was obtained in another set of Driesch experiments, in which the
blastomeres were rearranged (changed their neighbors). In spite of rearrangement, the
subsequent development was going in a normal way. Since Driesch times, hundreds of such
experiments have been performed at the different animal species and stages of
development, obtaining in general (if omitting details, unnecessary in out context) quite
similar results.
What should we derive from these experiments as related to the concept of one-to-one
causal chains?

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If we continue to assume that each structure of an adult or of an advanced embryo possesses
its own causal chain traceable from the very beginning of development, embryonic
regulations enforce us to conclude that each one of say four blastomeres contains at the
same time ¼, ½ or a full set of the causal chains and, moreover, the portions of these sets
contained in the same blastomere may be different in the different experiments. Taking into
mind, that a blastomere “do not know” in advance, whether he will be isolated or not, and
what neighbors will he have, we have to accept that embryonic regulations make the idea of
one-to-one causal chains contradictory and absurd: at least during the developmental period
when the regulations are taking place, any causal chains should be smoothed and lost.
Besides embryonic regulations, there are also other arguments against the existence of
one-to one causal chains. One of the main ones is a so-called equifinality, illustrated by
Fig. 1N-Q. It is the attainment of the same end-result of development by quite various,
sometimes purely stochastic developmental pathways. A stochasticity of embryonic
processes firstly emphasized already a century ago by the Russian biologist Alexander
Gurwitsch, looks to be a background of many morphogenetic events, first of all those
associated with branched rudiments (blood vessels, lungs, leaf veins). These structures are
of a fractal nature and are hence generated in chaotic regimes, to which the notion of
specific causes is completely inapplicable.
Now let us look, what conclusions from his experiments made Driesch himself.
He expressed them in a laconic statement, known as Driesch law: “A fate of a part of
embryo depends upon its position within a whole” [let us add: rather than upon its
internal properties].
By this formulation Driesch wanted to interpret embryonic regulations in the following way.
At the first step, the shape of a normal early embryo is in rough outlines and in diminished
size restored. Next, each cell of a regulated embryo “recalculates” its position according to
its coordinates within a new “whole” and develops according to this recalculated position,
rather than follows its normal destiny. Formally such interpretation may be true, but several
important questions remain unanswered. First of them is: by what means a roughly normal
shape of an early embryo is restored? This process is not explained by Driesch law.
Moreover, well after Driesch it was shown that a normal shape can be restored from the
cells arranged in a completely chaotic manner (Fig. 3). The second question is: what are the
Fig. 3. Normal sea-urchin larvae can arise from completely random aggregations of
embryonic cells.

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reference points for the recalculation procedure? Driesch formulation – “according to a
whole”- is too vague, although, as we’ll see later, such a vagueness has its own justifications.
Meanwhile, in a new and a most popular version of Driesch law – a concept of “positional
information” (PI) (Wolpert, 1996) - the answer was another: cell positions are referred to
certain special predetermined points, often defined as “sources” and “sinks” of some
diffusible substances, the morphogenes. But it is easy to demonstrate that the existence of
such predetermined points is incompatible with embryonic regulations. The matter is that
under either partial removal or rearrangement of embryonic material all of its elements
(including those which are suggested to be the reference points) take the positions,
geometrically non-homologous to those occupied by the same points in normally
developing embryos (Fig. 4). Moreover, so far as the early embryos are capable to
Fig. 4. Embryonic regulations are incompatible with the assumption of any prelocalized
specific material elements (say, P andQ), regarded as the sources of a positional information
(PI). After the dissection of an embryo part upper from a dotted line shown in A and closure
of the wound (B) the positions of the elements P and Q (as well as all the others) will become
geometrically non-homologous to the same elements’ positions in A. As a result, any points
of an embryo which occupy in A and B homologous positions (say, a and a1, b and b1, c and
c1) will perceive quite different PI signals which is incompatible with embryonic regulations
(from Beloussov, 1998).
regulations after the removals and rearrangements of quite different embryonic areas, any
predetermined elements will take in different experiments quite different (but each time
geometrically non-homologous) positions. Consequently, we have only two formal
possibilities to “save” a concept of predetermined reference points: either to suggest that
such a role is passed each time to the elements, occupying geometrically homologous
positions, or to assume that all the elements of a partial or normal embryo play a role of the
reference points. However, the both versions (out of which the second one looks more
consistent) imply that either only the reference points or even all the embryonic elements
should somehow “feel” the shape of a whole. Here we see that a vagueness of the Driesch
formulation had its reasons: intuitively he felt that the regulations which he discovered
cannot be explained without implying the action of something related to irreducible whole
to the minor elements. In his time, when the Laplacian determinism still was in full power,
such a claim looked as something inappropriate for a real science. Today it is impossible to

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negate such things; moreover, the verbal expressions like “top-down causation”, “emergent
behavior” and “context-dependency” have been coined for describing them. However, mere
words are not enough; what we need is a coherent law-centered theory explaining the
arising of complexity, holistic regulations and so on. Remarkably, such a theory has been
emerged already several decades ago quite outside of biology as a result of convergence of
several branches of mathematics and physics. This is a self-organization theory (SOT).
Before addressing to SOT directly, we must however get some knowledge of another
topics – a symmetry theory (even irrespective to SOT it is very useful for any biologist).
3. Elements of a symmetry theory as related to development
In short, a symmetry theory is dealing with invariable transformations of geometric
bodies, that is, with those kinds of movements which superpose a body with itself. The
emphasis upon invariability makes this theory closely related to the very essence of law-
centered approach.
We’ll restrict ourselves to the elementary part of a symmetry theory, which is dealing with
three kinds of movements: rotations, reflections and translations and explore some simple
examples. It is easy to see, for example, that a rectangle superposes with itself under
rotations around its center to 900, 1800, 2700 and 3600 (that is, in four positions), a regular
triangle do the same under rotations to 1200, 2400 and 3600 (3 positions), while a disc
superposes with itself under rotation to an infinite (∞) numbers of angles. A number of
positions superposing a body with itself is defined as an order of symmetry (in the above
cases it is a rotational symmetry). If, as in the case of a disc, the number of such rotations is
infinite, one speaks about a power of a symmetry order. Passing from 2-dimensional disc to
3-dimensional sphere, we obtain a rotational symmetry power ∞/∞, what means that the
rotations may go around an infinite number of central axes intersected at any angles.
The bodies possessing any order of a rotational symmetry may have or not have reflection
(mirror) symmetry, its plane denoted as m. Thus, a combined rotation/reflection symmetry
of a sphere is ∞/∞· m. The bodies having no mirror symmetry at all can exist in two mutually
reflected modifications, which can be arbitrarily defined as left and right.
The translational symmetry is that of linear repeated patterns. Its order is characterized by
a smaller linear shift n, which superposes a shifted pattern with non-shifted one. If a
pattern is homogeneous along the shift direction (n is infinitesimal), the translational
symmetry order is ∞.
After learning these definitions, we can easily see that the development of an egg towards
the adult state is associated with a stepwise reduction of symmetry order (or a series of
symmetry breaks, as is often told). Thus, an egg before the establishment of its polar axis has
a symmetry order of a sphere (∞ / ∞ · m), after the axis establishment it is reduced up to ∞ ·
m, while after determination of a saggital plane (into which the antero-posterior axis of a
future organism is located) it becomes 1 · m (we ignore a right-left asymmetry, which is of a
molecular origin and seems to persist throughout the entire life cycles without any
fundamental perturbations). The development of advanced embryos is mostly associated
with reduction of a translational symmetry order, that is, with establishment of the finite
(rather than infinitesimal) n values. Most obvious examples are the formation of
mesodermal somites out of a roughly homogeneous cell mass, or a subdivision of an
initially smooth neural tube into brain vesicles.

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We pay so much attention to symmetry breaks, because they are closely associated with the
entire problem of causation. This linkage has been formulated by a classical principle
claimed by the French physicist Pierre Curie exactly at the time when Driesch made his
regulations experiments (although the both scientists did not know anything about each
other). In his principle, Curie gave for the first time a strict definition of an effect and its
cause. By his idea, any observable event is associated with the reduction of a symmetry
order (by his words, “This is a dissymmetry, which creates an event”). Next, by Curie
principle, no symmetry break can take place spontaneously, that is, without a somewhere
located “dissymmetrizer”, an object with the already reduced symmetry order. It is a
dissymmetrizer, which fits a notion of a “cause”.
By applying this concept to developmental events, we have to conclude that any step of the
above mentioned symmetry breaks, according to Curie principle, demands a
dissymmetrizer, located either outside or inside of an entire egg/embryo. Let us start from
the earliest developmental events. At the first glance, they require external dissymmetrizers.
For example, an egg polarity in the eggs of brown algae can be established by a directed
illumination of an egg and the polarity of many animal eggs by the surrounding structures
of an ovary. The position of a saggital plane in amphibian eggs is determined by the point of
a sperm entrance, and so on. However, very accurate observations have shown, that the
external agents are not necessary: the algae eggs acquire polarity under absolutely isotropic
illumination and amphibian eggs can select a plane of saggital symmetry out of an infinite
bunch of planes even in the absence of a spermatozoon (parthenogenesis), or if it was
inserted accurately into the egg pole (where it cannot act as a dissymmetrizer).
Even less are the chances to find any dissymmetrizers for the events taking place in more
advanced embryos. Here, as known from embryology text-books, in very many cases one
rudiment plays a role of a so called inductor which triggers the development of another one,
and in most cases this process is directly or indirectly mediated by chemical agents, emitted
by inductor. Usually the inductors are regarded as the “causes” of the induced organs
formation, but is it so in the terms of Curie principle? It is easy to show, that virtually in all
these cases the symmetry order (as a rule, translational) of an induced morphological
structure is considerably reduced in relation to that of an inductor; for the cases of purely
chemical induction this is obvious without any comments. In the terms more customary for
biologists this means that the morphological structure of an induced rudiment cannot be
derived in one-to-one manner from that of an inductor: certain factors, increasing the
complexity of the induced organ and non related to inductor itself should be involved.
In general, both embryonic regulations and symmetry breaks without dissymmetrizers
leads us to conclude, that in the course of development more complicated (less symmetric),
although if perfectly ordered entities are emerged from less complicated (more symmetric)
ones. This is incompatible with a classical causal approach, but perfectly fits to what is
called self-organization. Is such a process unique for the living beings?
4. Self-organization in inanimate matter
Already more than century ago the first examples of the similar events proceeding in non-
biological systems has been described by the French physicist Benard. This was the
formation of cell-like structures (Benard cells) from a homogeneous viscous liquid,

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intensively heated from below (Fig. 5A, D). These structures immediately disappeared after
heating was stopped, or became less pronounced. As proved later by one of SOT founders,
Ilya Prigogine, these structures appeared because under enough intense flow of energy the
convection streams (upward shifts of heated liquid particles and downward shifts of the
cooled ones) pass from a random to so-called coherent regime, characterized by collective
movements along some common trajectories which became now energetically more
advantageous than the random movements along individual tracks (Fig. 5B, C). What we
see here is a real emergence of an ordered complexity from a homogeneous state or, in other
words, a spontaneous (non-embedded from outside) reduction of a symmetry order: the
initial infinite order of a translational symmetry is reduced up to that of n order, where n is a
Benard cell diameter.
Fig. 5. Benard cells. A: general view from the top. B, C: schemes Of coherent convection
streams. D: evolution of Benard cells patterns under constant heating (from left to right)
While the phenomenon of Benard cells formation did not pay much interest and was not
considered as a breakthrough event, quite another was a public reaction to the occasional
discovery of a fluctuating chemical reaction by the Russian chemist Boris Belousov in
1950ieth. Although firstly it was rejected by the editorial board of a scientific journal (the
referee wrote that it violates the second law of thermodynamics and hence should not exist)
very soon an entire research team from the Institute of Biophysics, Russian Acad. Sci.
extensively elaborated this reaction, transformed it into space-unfolded “autowaves” and
gave its complete theory. Because of its vividness, the reaction became very popular
throughout the world: everybody could see that within a couple of minutes a series of ever
complicated spiral waves appear from “nowhere”, that is from a completely homogeneous
state (Fig. 6).

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Fig. 6. Successive structures (1-8) arisen during Belousov-Zhabotinsky chemical reaction.
5. A theory of something emerged from nothing
Now we’ll give a very brief and elementary review of SOT principles (for much more
complete, but still popular SOT account see Capra, 1996; for a developmentally related
account see Beloussov, 1998). Let the readers only slightly familiar with math be not afraid:
the math will be minimal. As other great specialist in this field, Rene Thom said – “this is not
the math, this is a mere drawing”. Our drawings will be also minimal – most will be
expressed by words.
The first point to be noticed is that contrary to classical mathematics, SOT is about a real
world, which is full of so called unexpected perturbations, or a noise. Without noise none of
the effects, predicted and described by SOT, will take place. For us biologists this is quite
obvious: all the organisms are living in a very noisy world, which they have to resist and/or
assimilate, preserving their individual, or a species-specific way of living. Such a property of
a dynamic, or functional (not static!) resistance is also one of the main components of a self-
organization. It is called robustness. All the natural systems are to a certain extent robust –
otherwise they would not exist at all. However, robustness always has its limits, and when
they are exceeded, a system abruptly passes into another state, which is as a rule also robust.
Let us express the above said by mathematical symbols. We shall see that such a
transformation will very much clarify what was told before. Our main tool will be
differential equations, firstly one variable linear, and then two variables nonlinear ones.
Why is it necessary? The matter is, that even the simplest differential equation like
dx/dt = kx (1)
has the following properties, lacking, say, in algebraic equations:

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1. It describes a process, rather than a static state;
2. It contains a feedback loop: not just the right part variables affect the left part ones, but
vice versa as well. The feedback may be either positive, or negative, or, in the case of
two variables equations, positive-negative (±). The latter is mostly useful for self-
organization.
3. Most important in our context: differential equations combine the values of quite a
different order - dx/dt is an infinitely small part of x. Thus, x represents a whole, while
dx/dt its small part. Correspondingly, the action of x upon dx/dt is the action of a whole
upon its parts. Just formally, differential equations imply a holistic causation, which we
beforehand derived from experiments. .
Let us now add a free member “-A” to eq. (1), obtaining
dx/dt = kx – A (1a)
and make a graph (Fig. 7A), depicting by arrows the directions of dx/dt . We’ll get what is
called the vector field (in this case 1-dimensional). Owing the presence of a free member, we
have a stationary point dx/dt = 0 at x = A/k, from which the vectors dx/dt are diverged.
Correspondingly, if we reverse the signs of the right part members, getting
dx/dt = - kx + A (1b)
the vectors will be converged towards the point with the same coordinates (Fig. 7B). This is
enough for coming to the main notions of SOT: those of a dynamic (or Lyapunov)
stability/instability. The solution (stationary point) in eq (1a) is unstable, because any
infinitesimal shift from this point will bring us away without any chance to return back. On
the contrary, in the framework of (1b) equation after any shift we’ll come back to the
stationary point, which is unlimitedly stable. We call this kind of stability/instability
dynamic because it relates to the variable x, which dynamics is just traced in the equations.
Besides these dynamic variable(s), in all the equations another kind of values is always
present and plays a leading role: those are so called parameters, which either do not change
at all their values, or change them in an order more slowly than the dynamic variables. A
distinction between dynamic variables and parameters is very important, because it relates
to the fundamental concept of the structural-dynamic levels. This concept, belonging to so
called systems theory, claims that a surrounding world (both animate and non-animate) is
stratified into a number of more or less discrete levels distinguishing from each other by
characteristic times (Tch) and characteristic dimensions (Lch) of the related events; the both
hierarchies are, as a rule, roughly parallel to each other. In this language, the parameters, at
least by Tch criteria, should be attributed to a much higher level than the dynamic variables.
As concerning Lch it is crucial that in the developing organisms the dynamic variables are
always the collective entities: all the developmental events are based upon the action of
many cells, or many molecules, occupying different positions. As a rule, all the members of
this collective share the same parameters values (otherwise this would not be a common
system). Correspondingly, the area of the parameters action (the parameters Lch) is also
greater than Lch for each one dynamic variable.
In eq (1a, 1b) the parameters are represented by k and A values. Even in these simplest
equations they are playing the main role and, remarkably, do it in quite a robust manner.
Namely, there are the signs (+ or -) rather than the absolute values of the both parameters

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which decide whether the solution will be stable or unstable: it is easy to see, for example,
that only eq (1b) rather than (1a) has stable solutions. At the same time, in the immediate
vicinity of k = 0 just very small shifts of k values are enough for switching a solution from
stable to unstable, and vice versa. In other words, in relation to the shift of, say, parameter k
eq (1) is unstable at k≈0 and stable in all the other areas. This is another kind of
stability/instability, which is called parametric, or structural. A notion of a structural stability
very adequately represents such biological realities as, for example, a morphology of a
taxon, because it reflects at the same time a preservation of a general “Bauplan” and some
considerable, but nevertheless limited fluctuations.
The notions of stability/instability (both dynamic and parametric) and their regulation are
of a primary importance for understanding the developmental transformations and their
relations to causality. When we notice, as mentioned above, that at least some of the
symmetry breaks look to be proceeded “spontaneously”, this actually means that the
preceded symmetry order has lost its dynamical stability and hence can be broken by
negligibly small perturbations (of a noise intensity), to which a developing organism is
insensible during stable periods. By the way, this means that Curie principle formally keeps
its validity during instability periods as well, but at that time the “causes” are so small that
cannot be distinguished from the ever presented noise.
Fig. 7. Vector fields and solutions of some simple differential equations. A, B: linear
equations. In A the solution is unstable (empty circle, vectors diverging), while in B it is
stable (filled circle, vectors converging). C: under k < 0 there is only one stable solution x =
0, while under k > 0 this solution becomes unstable while two stable solutions appear in
exchange. A transition from negative to positive k values corresponds to that from a single
non-differentiated to a differentiated state (colored scheme to the right). For more details
see text.

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Under these conditions, it is meaningless to look for the “causes” in their classical sense.
What we need to know instead is why at the given moment an embryonic structure has lost
its previous stability. Meanwhile, we know already, that the loss (or acquiring) of stability is
provided by the changes of the parameters values. This conclusion is of a direct
methodological usage: instead of splitting the studied systems into ever smaller parts in
pursuit of ever escaping causes, we should concentrate our interest onto macroscopic levels,
to which the parameters belong.
For illustrating the role of parametric regulation in more details, we have to go from linear
to non-linear equations. This shift is not just formal: it means that we pass from
independently acting elements to interacting ones (those which either enhance, or inhibit
each other, or make the both things together). In other words, non-linearity means
cooperative interactions, most of all important for developmental events. A simple example:
suggest that N identical elements affect the event A. If these elements are independent of
each other, their action upon A is proportional to N, while if N elements enhance each other,
it is proportional to N(N-1) ≈ kN2.
We miss a quadratic non-linearity and come directly to the 3-rd order one, as providing one
of the best illustrations of developmental processes. Consider the equation
dx/dt =kx – k1x3( k1 > 0) (2)
which describes a first order positive feedback between kx and dx/dt and 3-rd order
negative feedback between k1x3 and dx/dt . It can be easily tested, that under k < 0 eq (2) has
only one rational solution x1 = 0 which is stable, while under k > 0 this solution becomes
unstable, while two new stable symmetric solutions appear:
x2, x3 = ± √ k/k1
(Fig. 7C). The main property of this model is that when k parameter in his rightwards
movement reaches positive values, a number of stable solutions increases from one to two.
Accordingly, it is a simplest model of the complexity increase, or of the reduction of a
symmetry order (under k < 0 the sole stable solution x1 = 0 is the axis of a rotational and
reflection symmetry, while under positive k none of the stable solutions x2, x3 can play this
role). In biological language this is just what we define as differentiation. The most
important lesson from this model is that such a crucial step is under a full parametric
control. That does not mean that the dynamic variables play no role at all, but this role is
parametrically dependent. Namely, only under k > 0 the dynamic variables can select one of
two vacant stable solutions. But they do it in quite a robust manner, without being obliged
to take precise values: under any x > 0 the positive solution is selected, while under any x <
0 the negative one. Therefore:
(1) there are the parameters which determine the number and the values of stable solutions,
that is, stable states, potentially achievable by a system; (2) among these, the actual states are
selected by the dynamic variables in quite a robust manner: each stable state is a “basin of
attraction” of infinitesimal number of the dynamic variables values.
By the way, these conclusions undermine a myth of an extremely precise organization of the
living beings: the main condition for surviving and keeping their individuality is a small

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number of potentially achievable and highly robust stable states, rather than a precise
arrangement of the dynamic variables, which never exists.
Remarkably, most of the above said can be easily translated into embryological language.
One of the most important notions of embryology is indeed that of a competence: briefly
speaking, this is a capability of a given embryo region at a given stage to develop into more
than one direction. Now we may see that in SOT language it corresponds to that region of
the parameters values, which has more than one stable solution. Therefore, the existence or
the absence of a competence should be regulated parametrically. The next step after
reaching the region of competence will be to come into the “attraction basin” of a definite
solution. This event corresponds to what is defined in embryology as determination, and we
can conclude that it is a matter of dynamic regulation. Same will be true for the final
reaching of a stable state; in embryological language this is differentiation.
At the end of this section, let us briefly describe, even missing formulae, some more
complicated self-organizing systems. Biologically very important class of systems is
described by 2-variables (X and Y) non-linear differential equations, where Tch for Y are in
an order smaller than for X: therefore, if including the parameters, these systems are at least
three-leveled. In addition, the variables are interconnected by (+, -) feedbacks: a slower
variable X inhibits a fast variable Y, while the latter enhances the first one. As a result, in a
wide range of the values of a single controlling parameter we get so called autooscillations,
that is, non-damped regular fluctuations of the both variables values. Complementing this
system by a linear dependence between Y and dx/dt, we transform ever persisting
autooscillations into a so-called trigger regime with two stable states, exchanging each other
after finite perturbations of one of the variables. The arisen structures may be either only
time-dependent, or in addition space-unfolded. In the latter case one has to assume that at
least one of the variables is diffusing through space (it may exemplify not only a chemical
substance, but also a certain physical state). In any case, all of these either purely temporal,
or spatial-temporal structures are able to create, under a proper range of controlling
parameters, quite stable patterns out a completely homogeneous state; note however that
the patterns are stable until the supply of dynamic variables will continue.
6. Application of SOT to embryonic development
The first person to be mentioned here is Conrad Waddington, a British scientist who, even
before SOT emerged in its present form, suggested a very stimulating allegory of
development, that of a mountain landscape, consisting of valleys (which symbolize stable
developmental trajectories) and crusts (imaging unstable states between valleys) (for recent
account see Goldberg et al., 2007). There is also a tale that it was Waddington who asked a
famous mathematician Alan Turing whether it is possible to construct a model generating a
macroscopic order out of a completely homogeneous state. Turing did so postulating
feedback interactions and diffusion of two reagents (Turing, 1952). His model became quite
famous, even if it had no relations to any real biological process. An entire series of models,
aiming to imitate biological realities have been constructed later on by Gierer and Meinhardt
(Meinhardt, 1980). In general, the models postulated feedback interactions between two
chemical substances, one of them (the activator) stimulating the development of a certain
structure, while another (the inhibitor) suppressing the activator. Necessary was also the

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inequality of the both components diffusion rates: the inhibitor should diffuse much more
rapidly than the activator.
These models permitted to reproduce a number of biomorphic patterns, mostly periodic
ones and in particular those related to surface designs. Also, they introduced an important
principle: “short-range activation – long range inhibition” - which seems to be a wide-
spread tool for pattern formation, although not necessarily connected with diffusible
chemical substances. Meanwhile, the authors were fully satisfied by reproducing some
single steps of development, without expressing any interest to model more or less
prolonged chains of events. As a result, the initial conditions and the relations between
postulated chemical substances and morphological structures which they assume to
“activate” or “inhibit” had to be taken each time in quite an arbitrary way.
7. Mechanically-based self-organization (morphomechanics)
Much closer to biological realities and less connected with arbitrary assumptions became
another class of models, emerged since 1980ieth. The main acting agents in these models
were mechanical stresses (MS), generated by embryonic cells. Even a priori MS looked to be
good candidates for being involved into regulatory circuits by the following reasons at least:
- they belong to universal (largely non-specific) natural agents;
- they are acting at the same time on quite different structural levels, from molecular to
that of whole organisms;
- MS create very effective feedbacks with geometry of stressed bodies: any changes in MS
pattern affect geometry in a well-predicted way, and vice versa. As D’Arcy Thompson
told in his classical book “On Growth and Form” (last edition: Thompson, 1961) “Form
is a diagram of forces”.
As discovered during several last decades, embryonic tissues of all the studied animals,
from lower invertebrates to human beings are mechanically stressed (same, even to a greater
extent is true for plants). Embryonic MS are of different origin. In early development the
main stressing force is turgor pressure in embryonic cavities (blastocoel, subgerminal
cavity), which is born due to ion pumping and which stretches the surrounding cell layers.
At the advanced stages most of stresses are caused by collective movements of many dozens
of cells. Cell proliferation also contributes to MS. It is of a particular importance, that MS are
arranged along ordered patterns, remaining topologically invariable during successive
developmental periods and drastically changing in between. They never are uniformly
spread throughout the developing embryos, but are generated in a certain part and
transmitted by rigid structures to others.
Already several decades ago the German anatomist Bleschmidt described a large set of MS
patterns emerging in human development, and claimed that “the general rules… that are
applicable to man … have much in common with the rules of the developmental
movements that take place in animals and even in plants” (Bleschmidt and Gasser, 1978). In
advanced embryos he distinguished 8 different kinds of MS fields which participate in
development of practically all the organs. Some of them are depicted in Fig.8A-C.
Modulations of MS patterns (relaxation, reorientation, changes in MS values) in amphibian
and chicken embryos lead to grave developmental anomalies. A number of fetus
pathologies are also mechano-dependent.

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Fig. 8. Some examples of “biokinetic” schemes of human embryos anlagen by Blechschmidt
and Hasser (1978). A: a rudiment of a finger; B: heel pad of 5 months fetus; C: a somite with
surrounding tissues. Diverging and converging arrows depict stretching areas and those
resisting to stretch, correspondingly. The main idea is that all the anlagen have their own
patterns of mechanical stresses.
Most important, self-generated MS affect each other, creating feedbacks. Harris and
coworkers (1984) evidenced the presence of such feedbacks by observing cell cultures
seeded onto highly elastic substrates which the cells were able to stretch by their own
contractile forces (Fig. 9A, B). As a result, homogeneously seeded cells became rearranged
into regular clusters (Fig. 9C). This is a real self-organization (reduction of symmetry order)
created by a feedback between short range adhesive interactions, tending to clump cells
together into a tight cluster and long range stretching forces which extend the substrate and
hence decrease cell density. Within the model framework, the adhesive forces correspond to
short range activation, while the stretching forces to the long range inhibition of Gierer-
Meinhardt models. Therefore, mathematics is roughly the same, but physics quite another –
mechanics instead of chemistry! Quite similar, although if independently developed
approach has been used in Belintzev et al. (1987) model, aiming to reproduce a segregation
of initially homogeneous epithelial layers into the domains of columnar and flattened cells.
In this model a role of short range activation was played by so called contact cell
polarization (CCP) – cell-cell transmission of a tendency to become columnar. At the same
time, long range inhibition, similarly to Harris et al. model, was provided by mechanical
tension, arisen in the epithelial layer with fixed ends just because of CCP. Hence, again we
have here a mechanically based (+, -) feedback. This model is of a special interest, because
(unexpectedly to the authors) it became able to reproduce some main properties of
embryonic regulations, namely preservation of proportions under different absolute

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dimensions of a layer. This became possible, because the model equation contains a
member, referring to holistic (independent from individual elements) property of a layer:
this is the average cell polarization throughout the entire layer. Thus, the model can be
considered as a mathematical expression of the Driesch law.
Fig. 9. Formation of regular cell clusters onto an elastic substrate. A: a single crawling cell
shrinks the underlain substrate and hence stretches that located outside. B: a large cluster of
adhered cells (to the left) stretches the cells located outside (to the right). C: a regular cell
pattern arisen in the dermal layer of chicken embryo under the similar mechanical
conditions (from Harris et al., 1984, with the authors’ permission).
The described models made the first steps away from a purely static view upon development:
we began to understand, why a given stage is exchanged by the next one. However, the
modeled chains were too short and very soon abrupt. Is it possible to use a mechanically-based
approach for reproducing much more prolonged developmental successions, including the
above models as particular cases? Such an attempt has been performed by our research group
about two decades ago (at the initial stage of this enterprise very important contribution was
made by Dr. Jay Mittenthal from Illinois University, USA).
Our main idea was very simple. It is well known that any organism, deviated by any
external perturbation (including certainly mechanical forces) from its normal functioning,
tries to diminish the results of perturbation up to their complete annihilation. We modify
this almost trivial statement by adding that any part of a developing organism affected by a
mechanical force (coming normally from another part of the same embryo) not only tends to
restore its initial stress value, but do it with a certain overshoot. This assumption, called the
hypothesis of MS hyper-restoration (HR), permitted to make several predictions, opened for
experimental and model testing (Beloussov and Grabovsky, 2006; Beloussov, 2008).
For example, according to this model, a stretching of a tissue piece by an external force
should produce the active reaction which firstly diminishes stretching and then, as a part of
HR response, generates the internal pressure force, directed along a previous stretching (Fig.
10A). As a rule, this is done by so called cell intercalation, that is, cells insertion between
each other in the direction, perpendicular to stretching (Fig. 10C). Accordingly, if a tissue
piece is relaxed or, the more, compressed, its cells should actively contract in the direction of
relaxation/compression, tending to produce tension is this very direction (Fig. 10B, D). If
applying these predictions to a cell sheet bent by external force, we should expect that its

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concave (compressed) side will be actively contracted, while the convex (stretched) one
extended. As a result, their cooperative action will actively increase the folding, just
triggered by external force.
Importantly, these reactions are connected by feedbacks with each other. Among them, one
of the main can be called “contraction-extension” (CE) feedback. As any other self-
organizing event, it starts from a fluctuation – in this case, of a stretching/compression
stress along cell layer. For example, if a part A of a layer is stretched slightly more than a
neighboring part B and the layers edges are firmly fixed, at the next time period part A will
be actively extended and hence compresses the part B. The latter will respond to this by
active contraction, even more stretching A, and vice versa. The modeling showed
(Beloussov & Grabovsky, 2006), that the results of these interactions crucially depend upon
one of the parameters, a so called threshold stretching stress (TSS), that is the minimal
stretching stress required for generating the internal pressure. If TSS is taken large enough, a
layer will be segregated into single alternated domains of columnar and flattened cells. Just
this situation corresponds to Belintzev et al. model. Under TSS decrease the number of
alternative cell domains is increased while under very small TSS no stationary structures are
produced: instead, a series of running waves is generated. This exemplifies a parametric
dependence of morphogenesis.
Fig. 10. Model of hyperrestoration of mechanical stresses. A, B: schemes of the responses to
stretching and to relaxation/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.
Now let us reproduce in very broad outlines a more or less prolonged (but still uncomplete)
chain of morphogenetic events. We start from so called “idealized blastula” stage, a
spherically symmetric body with the walls of equal thickness which surround a concentric

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cavity and are stretched by the turgor pressure within the latter. Why during normal
development a blastula stage embryo will not stop at this stage but passes instead towards a
more complicated (less symmetric) form? By our suggestion, this is because, according to
CE-feedback, a spherical symmetry of blastula is unstable: even small local variations in its
wall thickness will produce the corresponding differences of tensile stresses: thinner parts
will be stretched to a greater extent and hence produce the greatest internal pressure,
actively extending themselves and compressing the resting part(s) of the cavity wall. At the
next step the mostly compressed part will generate, according to above said, the active
contraction force. This delimits the start of the next stage, called gastrulation. In general,
such a contraction can be achieved by different ways. The first of them is emigration of some
cells from the compressed part inside the blastula cavity. This is typical for some lower
Invertebrates (Cnidaria). Another one is the folding of a compressed part of a cell sheet; it is
more elaborated type of gastrulation, called invagination. However, the folding itself may
go in different geometric ways: the extreme ones are exemplified by a creation of a straight
slit (Fig. 11A), and a circular fold (Fig. 11E). Take a sheet of paper and try to reproduce each
of them. You will see how easy is to make a slit-like fold, while to make a circular one is
virtually impossible – so much radial folds are arisen around! In order to smooth out the
folds, the excessive cells should be removed (emigrated) from the folded area. In principle,
CE-feedback can provide such a mechanism, but it is to be well tuned. On the contrary, a
slit-like folding does not demand so refined regulation.
Nature employed the both ways: the first one is typical, for example, to Annelides and
Arthropoda (belonging to a large group called Protostomia), while the second one for
Echinodermata and Chordata (belonging to so called Deuterostomia). Interestingly, some
lower Invertebrates, belonging to the type Cnidaria, took a variable intermediate way (Fig.
11B). In any case, the geometry of gastrulation very much affects subsequent development.
The laterally compressed slit-like Protostomia blastopores should actively elongate themselves
along the slit axis, compresses their polar regions (which later on transform to the oral and
anal openings) but to a very small extent affects mechanically the rest of embryo (Fig. 11C, D).
On the contrary, a gradually contracted hoop-like blastopore of Deuterostomia embryos
creates around it a diverged radial tensile field, being extended over the entire embryonic
surface and thus involving it into a coordinated morphogenesis (Fig. 11F).
Meanwhile, a uniform mode of a circular blastopore contraction is also unstable: similarly to
what took place at the blastula stage, even small local irregularities in contraction rate along
the blastopore periphery should subdivide it into the compression and extension zones. As a
result, an ideal circular symmetry will be sooner or later broken and the blastopore together
with its surroundings will acquire either a radial symmetry of n order, depicted by a symbol
n·m, or a mirror symmetry 1·m. The latter mode of symmetry means formation of dorso-
ventrality, still rudimentary in Echinodermata but fully expressed in Chordata. The dorsal line
is that of a maximal active extension of embryonic body. Along this line another CE-feedback
is created: its posterior part becomes extended, while the anterior one relaxed/compressed.
This latter region transforms into a transversely extended head (Fig. 11H, h).
Later on the body of Vertebrate embryos becomes segregated into more or less
independently developing territories (“fields of organs”) into which the similar events are
taking place in diminished scales. It will be a fascinating and as yet almost untouched field
of studies to construct self-organizing models for all of them.

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Fig. 11. Formation and mechanical role of slit-like and circular blastopores. A: typical
blastopore of Protostomia. B: irregular blastopores of Cnidarian embryos. C, D: tensile fields
in the vicinity of slit-like blastopores. E: circular blastopore of amphibian embryo. F-H:
transformation of a radially symmetric tensile field around a circular blastopore (F) into 1·m
symmetry field with a dominating dorsal axis (G, H). H is a view from the left. d: dorsal
side, h: head region.
8. Few words in conclusion
The main message of this essay is that the classical causal approach, continued to be used
(even if unconsciously) by overwhelming majority of investigators cannot explain the main
properties of development – the arising of more complex entities from less complex and even
homogeneous ones and the associated “top-down causation” – influence of an irreducible
whole upon its parts. As a result, a present-day developmental biology looks, even in the best
cases, as a list of separate “instructions” of how to make this or that structure rather than a
science with its own laws and predictive power. On the other hand, SOT, one of the leading
branches of the modern knowledge, turned out to be quite suitable for promoting
developmental biology to perform a desired transformation. To the present time, however,
only the first steps on this way have been made. What we need now, is not so much a
construction of specific models, but a general understanding of the nature of feedbacks,
parameters and dynamic variables mostly involved in development. To a great extent this is
related to a widely used notion of “genetic information”. We have already seen, that it is far
from being specific (an old concept “one gene – one morphological structure” is fully
incompatible with modern data). Meanwhile, SOT opens a much more rational way for qualify
it more properly: so far as genomes belong to the most constant components of the life cycles
and the genetic factors are dealing, first of all, with the rates of molecular processes, it looks
reasonable to attribute to genetic factors the role of the highest order parameters (those having
the largest Tch and Lch). Within such a framework, the role of “genetic information” is quite
powerful, but non-addressed: knowing the parameters’ values but being non-informed about
the structure of the feedback loops into which they are involved, we cannot tell anything about
the role which is played by the first ones (as a rule, these roles should be quite multiple). The

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parameters themselves, if taken out of the context of an associated equation or, at least, of a
feedback contour are, so to say, blind – they have no definite meaning at all. If continuing to
apply the notion of information to developmental events (although this never can be done
with a desired degree of precision), we have to conclude, that it is smoothed between several
structural levels: not only DNA and proteins, but also morphological structures and embryo
geometry bear an essential amount of “information”, irreducible to other levels events. Or, in
other words, the biological information is embedded in wide contexts, rather than in single
elements. By believing that the solution of all the developmental mysteries can be reached by
splitting embryos in ever diminished parts, we may miss the very essence, which is resided
onto meso- and macroscopic, rather than microscopic level.
Our last question will be of a utilitarian nature: why do we need to pay any efforts by
transforming biology into a law-centered science, if already in its present-day state it gives so
many results, useful for medicine and biotechnology? True, as a great physicist Boltzman told,
nothing is more practical than a good theory. But is this citation applicable to biology? Nobody
can be sure of it, but the attempt is worth to be performed. In any case, one can hardly be
content to see the science about the most complicated, ordered and aesthetically perfect
natural processes using a methodology not very different from that of a medieval alchemy.
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