Untangling Complex Systems, page 52
nanometric dimensions (their radius depends on the ratio R
O]/[AOT])) and behave like
w/s = ([H2
nanoreactors because the reactants of the BZ reactions are polar and they partition within the
aqueous droplets. By playing with the ratio R
w/o = [water]/[oil], it is possible to have either isolated
nanodroplets wandering through the oil phase by Brownian motion (when R is low), or droplets
w/o
that coalesce into water channels (when R is high). The formation of Turing patterns is favored
w/o
by a low value of R and when two distinct conditions of mass transport are satisfied. First,
w/o
when the BZ reaction begins, apolar intermediates, notably the inhibitor bromine, are produced
within the nanodroplets and diffuse through the oil phase. Second, the polar species, includ-
ing the activator HBrO , diffuse together with the entire water droplet. The isolated droplets
2
move around much slower than single molecules, and when they collide, they mix their contents
through a fission-fusion mechanism. The average time between collisions is about 1 millisecond
(ms), which is much shorter than the period of the oscillations. Hence, the medium can be treated
as macroscopically continuous.
Since the movement of apolar single Br molecules, which play as an inhibitor, occurs at rates
2
much faster than that of the nanodroplets containing the activator HBrO , Turing patterns can
2
emerge. If the microemulsions are sandwiched between a pair of glass plates (Vanag and Epstein
2001a) separated by an 80-mm-thick Teflon gasket, the Turing structures are bi-dimensional.
On the other hand, if the microemulsions are placed in a cylindrical quartz capillary with an inner
diameter (0.3–0.6 mm) that exceeds the wavelength of the patterns, the Turing patterns are three-
dimensional. These experiments are remarkable because they show that Turing patterns, persisting
for one hour or more, can also be obtained in closed systems. They are transient because they are
not sustained like those in an open system. Previously, a few cases of transient Turing patterns in
closed systems have been found. For instance, the CDIMA reaction performed in the presence of
starch at 4°C in a Petri dish (Lengyel et al. 1993) giving rise to Turing patterns of mixed spots and
stripes or network-like structures that remain stationary for 10–30 minutes. Another example is the
polyacrylamide-methylene blue-sulfide-oxygen reaction carried out in a Petri dish and originating a
variety of spatial patterns such as hexagons and stripes (Watzl and Münster 1995).
TRY EXERCISES 9.11, 9.12 AND 9.13
9.5 TURING PATTERNS IN NATURE
The brilliant idea contained in the paper titled “The chemical basis of morphogenesis” written by
Turing in 1952 did not emerge into the spotlight until two decades later. In 1972, two developmen-
tal biologists, Hans Meinhardt and Alfred Gierer at the Max Planck Institute for Virus Research in
Tübingen (Germany), proposed a theory of biological pattern formation that paralleled that described
by Turing. In Turing’s, Gierer’s, and Meinhardt’s model, the spontaneous formation of patterns occurs
when two morphogens or generic ingredients interact non-linearly. One must be an autocatalyst.
256
Untangling Complex Systems
Ecology;
Sociology;
Economy;...
Geomorphology
Biology
(formation of
(Development of
dunes; erosion;...)
embryos; animal
markings;
regeneration;
phyllotaxis;...)
Turing
patterns
in nature
FIGURE 9.5 Sketch that shows the broad applicability of the Reaction-Diffusion model.
The other must be a self-inhibitor. The autocatalyst is also an activator of the self-inhibitor. On the
other hand, the self-inhibitor inhibits the autocatalyst (see Table 9.1).5 Crucially, the two species must have different rates of diffusion, the inhibitor being faster. Such model, named as Reaction-Diffusion
(RD) model, does not need to be limited to discrete molecules as interacting elements, and diffusion
is not the only mode of transmission. In fact, Turing’s RD model has been having a profound impact
on vast range of disciplines, such as physiology, ecology, botany, chemistry, as well as geomorphol-
ogy, social sciences and economy (see Figure 9.5). Turing’s RD model is mathematically easy and
effective in extracting the nature of many phenomena in Complex Systems, although it omits many
details of them. As Turing said, his original model is a simplification and an idealization of Complex
Systems. Therefore, it could be falsified, in agreement with the view of the epistemologist Popper.6
9.5.1 biology: The develoPmenT of embryos
The Really Big Question that Turing raised in his paper “The chemical basis of morphogenesis”
is this: “How is it possible that fertilized eggs give rise to such complex forms as are the liv-
ing beings?” A fertilized egg, named zygote (from the ancient Greek word ζυγωτo′ ς that means
“joined”), has spherical symmetry, but, in the end, it gives rise to an animal with well-defined axes.7
5 We must remind also the Schnackenberg’s model that involves different relations between the self-activator and the self-inhibitor.
6 Karl Popper (Vienna, 1902–London, 1994) is considered one of the greatest philosophers of science of the twentieth century.
According to Popper, scientists are “problem-solvers;” the growth of human knowledge proceeds from our problems and
from our attempts to solve them. These efforts involve the formulation of theories that can never be proven, but they can be falsified, meaning that they can and should be scrutinized by decisive experiments. According to Popper, the advance of scientific knowledge is an evolutionary process similar to the biological evolution. To respond to a given problem, some tentative theories are proposed. These theories are, then, checked for error elimination. The error elimination procedure performs a similar function for science that natural selection plays for biological evolution. The theories that survive the process of refutation are not truer, but they fit better than the others to the data available. Just as the biological fitness of a species does not assure unlimited survival, neither does rigorous testing protect a scientific theory from refutation in the future. According to the Popper’s view, the evolution of theories reflects a certain type of progress towards greater and greater problems in a process very much akin to the interplay between genetic modifications and natural selection.
7 There are two main axes in almost all animals that are called bilateria. There is the antero-posterior axis that defines the
“head” and the “tail” ends. But there is also the dorso-ventral axis that is at right angle to the former. For instance, human faces are ventral, whereas the back of human heads are dorsal.
The Emergence of Order in Space
257
Zygote
Eight-cell body
Blastula
FIGURE 9.6 Sketch that represents the development of a zygote into a blastula.
How does it happen? Development begins when the fertilized egg divides into two cells. Then, there
is a second cleavage at right angle with respect to the first, and a third cleavage again at right angle
producing eight cells. After many more cleavages, the embryo becomes a hollow ball, called blas-
tula, with the cells arranged as a spherical sheet (see Figure 9.6). At this stage, there is no indication of the asymmetric animal into which the blastula develops. Then, the gastrulation starts. The cells
of the blastula rearrange so that the front and back, the top and the bottom of an animal become
evident, and the basic body plan is laid down. It is only after the gastrulation that the form of the
animal begins to emerge. This is the reason why Wolpert (2008) stated that gastrulation is the truly
important event in our life.
During gastrulation, movement and folding of cell sheets form the basis of the early develop-
ment of many structures as diverse as the heart, lungs, and brain. Small changes in how fast and
how far the cell contraction spreads have profound effects on the forms. How do cells know where
and when to specialize, change shape, or move? It is evident that there exists positional information
(Wolpert 2011) and a generative program somewhere within the zygote. The spatial distribution of
specific chemicals encodes the positional information. The instructions for molding the embryo are
served into the DNA that works as if it were a memory of a modern electronic computer. A memory
of a computer having the Von Neumann architecture stores both data and instructions (remember
what we learned in Chapter 2). The same can be said of the DNA of the nucleus. All the cells have
the same DNA and the same sequence of genes. Within the DNA, there are the instructions for
making all the proteins in the cell and the program that controls their synthesis.8 How do the cells differentiate and specialize? Animals are made up of different types of cells, such as nerve cells,
muscle cells, blood cells, germ cells, skin cells, and so on. Humans have about 350 different types of
cells (Wolpert 2008), while lower animals have less. The function of a cell depends on the proteins
it contains. Proteins perform either structural or catalytic functions (as we learned in Chapter 7).
There are proteins that are common to most cells and are needed to carry out basic functions, such
as the production of energy or the synthesis of key molecules. But there are also proteins that are
present only in certain types of cells. For example, albumin is peculiar to liver cells; hemoglobin
is only within red blood cells; insulin belongs to pancreas cells, keratin is expressed in skin cells,
the contractive actin and myosin are synthesized within muscle cells, et cetera. The proteins that
are made within each cell depend on the cells receiving positional information, which is the mutual
communication between the nucleus and the cytoplasm and the extracellular signals. A useful pic-
ture to describe cell diversification is that of a ball on top of a mountain. The ball can slide along
different downhill pathways ending on distinct valleys. The ball represents an undifferentiated cell
that can transform into a specialized one, depending on the epigenetic landscape (read Box 9.2
of this chapter). In the case of humans, there are so much as 350 possible valleys! Which factors
rule the selection of a particular branch? The development of a zygote into an organism is truly an
astonishing phenomenon that is still under investigation (Wolpert 2008). Scientists have unveiled
only few scenes but not the entire film. We are aware that different mechanisms are involved into
8 The mathematician Gregory Chaitin (2012), in his book Proving Darwin. Making Biology Mathematical, wherein he looks for a mathematical demonstration of Darwin’s theory of evolution, asserts that the DNA of living beings is a software. It is a particular software, because it evolves, relentlessly.
258
Untangling Complex Systems
the embryonic development. One of these is the Turing’s Reaction-Diffusion model. For example,
the proteins Nodal and Lefty appear to work as an activator-inhibitor pair during the induction
of the mesoderm that is one of the three primary layers of germ cells9 in the very early embryo
of bilaterian animals (Nakamura et al. 2006). Turing’s RD models have been also proposed for
the limb development when digits emerge from the undifferentiated limb bud (Raspopovic et al.
2014), and for embryonic feather branching in birds (Harris et al. 2005). It is reasonable to expect
that the Turing’s RD model will be used to interpret other events in embryonic development, in the
next future. But it is not the only mechanism to produce shapes and structures in an embryo. For
example, it has been found that another important morphological mechanism is that based on gra-
dients of chemicals. Thomas Hunt Morgan (1866–1945 AD), an American evolutionary biologist,
geneticist, embryologist, who, for his discoveries concerning the role played by the chromosome
in heredity, was awarded the Nobel Prize in 1933, clearly proposed how gradients could control
patterning. The combination of genetic and embryological studies allowed for the identification
of the regulatory genes that control patterning in the early embryo of the fruit-fly Drosophila.
One of the most important is the gene bicoid, which is involved in patterning the anterior end of
embryo. The polarity of the embryo depends on which end will become the head, and which end
the tail, is defined into the egg. A special chemical composition of the cytoplasm is located at the
future anterior end. In this special portion of cytoplasm there is the message for synthesizing the
protein coded by the bicoid gene. When the egg is laid by the mother fly, the bicoid protein begins
to be synthesized at the anterior end and diffuses along the egg setting up a concentration gradient.
The largest value of the bicoid protein is at the front end. This gradient controls the position of the
boundary between the head and the thorax and also activates other genes involved in patterning the
posterior end of the embryo (see Figure 9.7). If there is not the right gradient, the bicoid proteins are not synthesized at the anterior end and the embryo develops into a larva lacking both head and
Head
Thorax Abdominal segments
FIGURE 9.7 Structure of the Drosophila (at the bottom) and its embryo (on top).
9 Animals with bilateral symmetry (also called plane symmetry), having one symmetry plane that divides an organism into roughly two mirror-image halves with respect to the external appearance only, produce three primary layers of germ cells within their embryos. The ectoderm is the external one from which the epidermis and the nervous system will develop.
The endoderm is the internal one; it will give rise to the digestive system. The mesoderm is the intermediate layer; it will give rise to the muscular system, the heart, blood and other internal organs.
The Emergence of Order in Space
259
thorax. The resulting abnormal egg can be rescued by injecting the chemicals of a normal anterior
cytoplasm into the anterior region of the egg. After restoring the right gradient, normal development
takes place. If, on the other hand, the chemicals of the normal anterior cytoplasm are injected into
the middle of the egg, the head develops in the middle of the embryo.
TRY EXERCISE 9.14
Also, the gradient mechanism is reasonably involved in the migration of cells. In fact, there is no
doubt that some cells exhibit chemotaxis, meaning, cells receive signals from the surrounding tis-
sues that direct them along the appropriate developmental pathway. During gastrulation, it is the
difference in cell adhesion that guides the cells. In fact, change in cellular adhesiveness is another
essential mechanism in the developmental program. The adhesiveness of a cell depends on specific
proteins that are embedded in the cell’s surface membrane and have one portion that sticks out
and binds to a similar or complementary molecule on adjacent cells. The crucial point is that cells
express different Cell Adhesion Molecules (CAMs) at various stages in the development. CAMs
play a pivotal role in the spatial arrangements of cells. The spatial organization of the different types
of cells is essential for the formation of organs. Consider our arms and legs. Our arms and legs con-
tain the same types of cells, such as muscle, tendon, skin, bone, and so on, yet they are different. The
explanation lies in how these different types of cells arrange spatially. The differentiation of cells
occurs through either environmental signals or unequal distribution of some special cytoplasmic
factors at cell division.
Finally, even chemical oscillations can play a role in the development of an embryo and the
formation of specific patterns. For example, in vertebrates, shortly after gastrulation, the brain
can be seen forming at the anterior end of the embryo. Behind the brain, there are the somites
that are blocks of tissue that will develop the vertebrae and the muscles of the back. The somites
look like two lines of paving stones (see Figure 9.8). Somitogenesis is an example of a dynamic
embryonic process that relies on precise spatial and temporal control of gene expression. In fact,
somitogenesis involves the oscillation of the expression of certain genes, ruled by an internal
clock (Goldbeter and Pourquié 2008). The oscillations of the clock are converted into separated
blocks of tissue. One can think of each cell as having a clock and those behind them a clock that
is set ticking a little later. When such cells synchronize, they form a somite (Baker et al. 2006;
Tsiairis and Aulehla 2016).
Cranial
neuropore
Somites
Caudal
neuropore
FIGURE 9.8 Schematic structure of a human embryo at roughly the 29th day. The top of the cranial neuro-
pore corresponds to the terminal lamina of the adult brain and the posterior neuropore (or caudal neuropore)
