Eukaryon, Vol.13, March 2017, Lake Forest College
News and Views
The Turing Mechanism: How did the leopard get his spots?
where:
C(x,t) is a vector function of chemical concentration depending on time (t)
and position (x).
D is the diagonal matrix of diffusion coefficients.
f is the vector of reaction kinetics
Joshua Dan Spreng
Department of Biology
Lake Forest College
Lake Forest, Illinois 60045
According to Philip Ball, the “... proof that animal pigment patterns are
indeed Turing patterns would [be] require to identify the morphogens
involved.” (Ball, 2012). He furthermore mentions that despite no one has
been able to do that until this day there “...are other types of biological
pattern for which we do seem to be closing in on the likely biochemical
agents underlying the process” (Ball, 2012)
Although Turing most likely did not encounter a free living leopard - certainly
not in Manchester - his studies of formation of patterns and appearance in
nature also included the spotted pattern of the leopard. In their paper the
scientists R. T. Liu, S. S. Liaw and P. K Maini describe their process of
generating the spotted pattern of a leopard’s fur based on the parameter
Introduction
It is possible that you are reading those lines on a computer or
on one of the various mobile devices that are available today. However, if
you read this article in printed form I can ensure you that it is still linked
to a computer as this text was written and formatted on a computer as
we know it today. Although being much more evolved, today’s computers
can be seen as “...incarnations of a Turing machine” (wikipedia source).
This brings us to the man about whom this article is about, Alan Mathison
Turing. You might say: “I know Turing but how does he fit in the topic of this
journal and what is it about the leopard?” Do not worry we are getting to
that.
Alan Turing, born in 1912, was an English computer scientist,
mathematician, crypt analyst and theoretical biologist. Despite his short and
tragic live - prosecution for homosexual activities in 1952. Homosexuality
was still a crime in England, following “corrective” hormone therapy and
his mysterious death - Alan Turing did not only set the ground stone for
the field of computer science, his achievements shaped world history and
saved thousands of lives. One of Turing’s best known accomplishments
was the breaking of the German cryptography device Enigma during the
Second World War. His encryption was therefore a tremendous help in the
process of successfully defeating an ongoing war crime regime. A second
milestone consisted of Turing’s concepts of algorithms and computations
as well as the invention of the Turing machine - the blueprint of an electric
computer.
Shortly after the end of the Second World War, Alan Turing
joined the Max Newman’s Computing Machine Laboratory at the University
of Manchester where he developed a series of stored - program electric
computers known as the Manchester computers. During this time, the
scientist became interested in the field of Mathematical Biology. After
1950, Turing worked on “... a new mathematical theory of morphogenesis”
which was based on the “... consequences of non-linear equations
for chemical reaction and diffusion” (Stanford, 2002). Turing was the
first to use computers for this type of work and predicted the oscillating
chemical reactions such as the Belousov -Zhabotinsky reaction which
was discovered a decade later. Turing was fascinated by the fact that
identical cells can differentiate into cells with various different forms and
functions as well as of the presence of a “plan” for this -morphogenesis.
Turing published his findings in a single paper where he stated that his
intention is to show that it a combination of physical elements is sufficient to
explain biological pattern formation (Kondo and Miura, 2010). His concepts
and thoughts have shaped the fields of chemistry as well as of biology.
Various scientists have applied Turing’s theory experimentally and were
able to reproduce patterns that occur in nature. Turing’s theory is today
seen as the “... candidate for explaining a variety of puzzles in biological
development, from the spontaneous differentiation of some tissues to
the formation of pigmented markings and the patterns of leaves on plant
stems” (Ball, 2012). It is, as Shigeru Kondo and Takashi Miura, write “…
the best-known theoretical model[s] used to explain self-regulated pattern
formation in the developing animal embryo” (Kondo and Miura, 2010).
Turing’s reaction – diffusion model consists of two substances
that are diffusible and that interact in order to describe the formation of
patterns in an embryo. Turing discovered that such a construct would allow
an autonomously generation of spatial patterns. Furthermore, Turing’s
model introduces a reaction that creates morphogens (Kondo and Miura,
2010). According to Kondo and Miura, if the process of diffusion is solely
at work then “… local sources of morphogens are needed to form the
gradient” (Kondo and Miura, 2010). If these morphogens are coding for
the creation of pigments the this could be responsible for the markings that
can be seen on animals (Maini, 2014).
In mathematical terms Turing’s model can be described as
following (Maini, 2004):
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Eukaryon, Vol.13, March 2017, Lake Forest College
References
News and Views
Liu, R. T., Liaw, S. S. Maini, P. K. Two-stage Turing model for generating
pigment patterns on the leopard and the jaguar , Physical Review.
2006.
Kondo, Higeru, Miura, Takashi. Reaction-Diffusion Model as a Framework
for Understanding Biological Pattern Formation, Science. 2010.
Ball, Philip. Turing patterns, Chemistry World. 2012.
N/a. Alan Turing, Stanford Ecyclopedia of Philosophy. 2002.
Maini, P.K. The Impact of Turing’s Work on Pattern Formation in Biology,
Mathematics Today. 2004.
Figure 1: Fur pattern of a leopard at 2 days old, 8 weeks old and as
an adult (Images by Liu et al.).
tuning of a reconstructing a Turing reaction - diffusion model. The scientists’
state that the pattern of the leopard’s fur changes as the animal growths:
the markings change from spots to rosettes (Liu et al., 2006). Figure 1
shows the fur patterns of a leopard at different age stages.
The scientists write that Turing models are mainly used because
they are effective models to recreate patterns found on mammals, insects,
fish and even bacteria colonies. Although, it is not yet completely clear
whether this patterns are results of Turing reaction diffusion processes,
there exist evidence about the essential components of Turing models
namely the morphogens. Furthermore, studies revealed the occurrence of
Turing patterns in Chemistry (Liu, Liaw, Maini, 2006).
Based on phylogenetic analysis that showed that flecks on the
fur individuals of Felidae are the primitive pigmentation patterns of more
complex patterns including rosettes and blotches that arose from them
Liu and his colleagues applied a simple Turing model and were able to
generate the fur patterns of leopards at different age stages. In a first step
the scientists recreated the spotted pattern as it occurs in young leopards.
In a next step, the model was altered: Three parameters were changed.
It wwas crucial to change the parameters in sequence to in order to “…to
enhance, to shift, and to stabilize spatial modes consistent with the pattern
transformation observed during the growth of the animals” (Liu, Liaw,
Maini, 2006). Although previous studies demonstrated how Turing models
can be used to generate patterns of animals, the study of Liu et al. is one
of few that focus of how patterns change during the growth of an individual.
With their study Liu et. al. showed that Turing reaction-diffusion models
are capable of reproducing temporal spatial patterns that are consistent
with those that occur in nature. With that it contributes to the support of
the involvement of morphogens in development biology. However, as the
scientists write in the end, it is important to note that this is not sufficient
to fully prove it. Further research in this direction is necessary to better
understand the development of patterns on animals.
Portrait of Alan Turing By Joshua Dan Spreng
Note: Eukaryon is published by undergraduates at Lake Forest College,
who are solely responsible for its content. The views expressed in Eukaryon
do not necessarily reflect those of the College.
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