Simulação de Processos Biológicos
Morphogenesis: an interdisciplinary scientific field
stable solutions of the non-linear
reaction-diffusion equations. From
the point of view of chemistry and
biology, stable patterns are in
principle associated to specific
chemical or biochemical processes.
Rui Dilão and Joaquim Sainhas
Grupo de Dinâmica Não-Linear,
Instituto Superior Técnico,
Av. Rovisco Pais,
1049-001 Lisboa
1. Introduction
Culm. 2inating the work of a half
century of the naturalist D´Arcy
Thompson
(1917),
merging
knowledge from biology, physics and
mathematics, Alan Turing (1952)
developed the basis of the
mathematical
theory
of
morphogenesis
(La
Recherche,
special issue on patterns and
morphogenesis, Feb. 1998).
According
to
Turing,
the
development of forms, structures and
functions in biological systems
should emerge from the principles
that govern the motion and
interactions of atoms and molecules.
Consequently,
the
meaningful
microscopic mechanisms leading to
morphogenesis should be diffusion
and chemical reactions.
In diffusion, heavier molecules in a
medium move according to the (non-
reactive) collisions of lighter atoms or
molecules. Chemical reactions occur
when colliding molecules bind,
forming a new chemical species. Due
to the coupling between diffusion
and reaction, the time evolution of
the concentrations of reactive
systems in extended media is
described by a partial differential
equation
with
the
non-linear
contribution
of
the
reaction
mechanisms.
From the experimental point of view,
the
Turing
ansatz
becomes
meaningful if a chemical system in an
extended medium can be found with
the property of generating patterns.
The
autocatalytic
BelousovZhabotinski reaction is an example of
such a system, leading to the
spontaneous
formation
of
propagating circular and spiral
waves, Fig. 1. From the theoretical
point of view, it is important to find
the generic mechanisms leading to
Figure 1. Spontaneous chemical patterns with the Belousov-Zhabotinsky reaction in
extended media. Experiment performed in the laboratory of the Non-Linear Dynamics
Group (IST).
2 Boletim de Biotecnologia
The Non-Linear Dynamics Group of
IST has a multidisciplinary research
activity in the theoretical and
experimental aspects associated with
reaction-diffusion systems. This work
involves questions of mathematical
analysis and computability, analysis
of
biological
and
chemical
mechanisms and optics of waves in
excitable media.
2. Non-linear reaction
diffusion
systems:
computation of solutions.
Reaction-diffusion partial differential
equations are mathematical objects
describing the time evolution in
space of quantities characterising
physical, chemical and biological
systems. They appear in quantum
mechanics, in optics, in astrophysical
problems, and transport phenomena
in chemistry and biology. However,
their mathematical properties are
poorly understood due to the fact
that, in the most interesting
situation, they are non-liner. This
means that, if we have two
independent outcomes as solutions
of the equation, and if we prepare
simultaneously the conditions that
lead to them, the outcome of the
combined system is not related with
the independent outcomes. This is an
unpleasant situation, both from the
experimental
and
from
the
mathematical point of view. In fact,
no predictability exists in the sens
usually associated to a linear
mathematical description,
In order to study the patterns
associated to reaction- diffusion
mechanisms, we convert the nonlinear equation into an equivalent
simpler equation, in a form suitable
Simulação de Processos Biológicos
for computation. The simplest
method is to approximate derivatives
by finite differences, converting the
initial continuous equation into a
discrete equation. However, the
discrete equation obtained has a
different set of solutions when
compared with the solutions of the
initial reaction-diffusion equation,
and, we are not describing the same
physical, biological or chemical
system.
There are two ways of handing this
difficulty. One way, is to make a
prototype of the real system, as it is
generally done in systems involving
motion of fluids or, in chemistry, with
the construction of prototype
question-answer experiments. A
second way, more difficult, more
inexpensive and more general, is to
perform the bench-marking between
the non-linear and the discrete
approximated equations describing
the system. In this case, the benchmarking of the reaction-diffusion
equation is a mathematical problem
that compares the (theoretical)
solutions of the non-linear reactiondiffusion partial differential equation
with the (computable) solutions of
the discrete equation.
This is one of the problems our group has
been studying and we have found
the precise mathematical conditions
for the solutions of reaction-diffusion
non-linear equations to coincide with
the computable solutions of the
associated discrete equations (Dilão
& Sainhas; 1998).
From the practical point of view, we
can now obtain solutions of nonlinear
reaction-diffusion
partial
differential equations up to an
arbitrary precision. This enables the
analysis of non-linear models with
the same degree of certitude as in
real experiments, opening the way for
the calibration and validation of
mathematical
models
with
experiments. This issue is of critical
importance, with a strategic value in
engineering.
In Fig. 2, we show a spiral and a
Turing pattern obtained with a model
for
the
Belousov-Zhabotinski
reaction, calculated with our
methods. A close observation shows
that spurious lattice symmetries
underlying discretization methods are
absent and pattern symmetries are
well preserved.
This technique can also be used to
process information from image
analysis (Alves-Pires et al., 1998).
3. Morphogenesis with
reaction-diffusion
systems
Morphogenesis is the ensemble of
processes through which extended
systems develop their spatial form
and shape. Independently of the type
of natural system, a limited family of
patterns arises. This universality
suggests the existence of a general
theory of pattern formation.
Based on the experimental results of
Brakefield, et al. (1996), we have
constructed a model for the formation
of butterfly eyespots patterns
through a sequence of cascading
kinetic reactions. The elements
involved in the kinetic mechanism
represent processes, more or less
complicated, which are called
Boletim de Biotecnologia
3
Simulação de Processos Biológicos
morphogenes. Morphogenes can be
chemical species, simple or complex,
or complete processes. Some of the
morphogenes are allowed to diffuse,
and a system of reaction-diffusion
partial differential equations has been
derived for the whole process
(Sainhas & Dilão, 1999b; Dilão &
Sainhas, 1999a).
Based on the mathematical theory we
have developed previously, we were
able to predict some of the
phenotypic features of butterflies
wing eyespots, as a function of the
spatial
distribution
of
the
morphogene responsible for the
initiation of the cascading reaction. In
Fig. 3, we depict the stable patterns in
the wings of the butterfly Lopinga
achine (Satyridae family), as well as
the stable patterns obtained with our
model.
The mathematical properties of this
model suggest general mechanisms
for the formation of segmentary
structures in the development
process of living organisms.
4. Wave optics in reactiondiffusion systems
Pattern development is a propagation
phenomena described by non-linear
reaction-diffusion partial differential
equations. However, these equations,
can only support and propagate
coherent
signals
under
very
restrictive assumptions. Technically,
these signals are solitons, presenting
a very restrictive balance between the
non-linearity of reactions and the
dispersion effect of the media.
Therefore, it was not expected that
optical effects occured in active
extended
media.
However,
Zhabotinsky et al., (1993) reported
the experimental observation of
phenomena similar to refraction at the
interface of two chemically active
media. These phenomena present
striking similarities with refraction
and reflection phenomena of linear
optics.
We have shown (Sainhas & Dilão,
1998a) that the non-linear reactiondiffusion equation has solutions
4 Boletim de Biotecnologia
obeying a Snell’s law of
refraction,
provided
the
solitonic signal arriving at the
interface between the two
media encounters the second
media in a chemically
homogeneous
unstable
steady state. Moreover, a new
phenomenum called back
refraction has been predicted
and simulated. This paper was
reviewed by Physical Review
Focus and is available at the
web
site
http://publish.aps.org/FOC
US/v1/st19.html.
Back refraction occurs in the
incident medium and is
characterised by the fact that,
if the solitonic wave has an
incidence angle greater than
the Brewster angle, the
incident angle has a jump
transition to the critical angle.
In the second media a new solitonic
wave is created, perpendicular to the
interface. If the angle of incidence is
smaller than the Brewster angle,
refraction occurs, as we know from
optics.
Under the conditions of this
generalised Snell’s law, it is possible
to obtain a lens effect in chemically
active media. In Fig. 4, we show
simulations of refraction, back
refraction, and the lens effect, after
the interaction of a coherent chemical
signal with the interface between the
two media.
The back refraction phenomenon
introduces a new mechanism of
pattern formation in morphogenesis,
due to the possibility of triggering
differentiation by local fluctuation of
reaction rates in biological tissues.
References
Alves-Pires, R., Dilão, R. Neves, H.,
Parreira, L. and Sainhas, J. (1998).
Anisotropy-free Laplacian filters,
contour detection, and 3D image
reconstruction
for
confocal
microscopy imaging. Proceedings of
the 10 th Portuguese Conference on
Pattern Recognition – RECPAD’98
247-251, 1998.
Brakefield, P. M., Gates ,J., Keys, D.,
Kesbeke, F., Wijngaarden, P. J.,
Monteiro, A., French, V. and Carrol,
S. B. (1996). Development, plasticity
and evolution of butterfly eyespot
patterns. Nature 384, 236-242.
D’Arcy Thompson (1917). On growth
and form. (An abridged edition edited
by Bonner, J. T., 1995). Cambridge
University Press, Cambridge.
Dilão, R. and Sainhas, J. ( 1998).
Validation and calibration of models
for
reaction-diffusion.
Int.
J.
Bifurcation and Chaos 8, 1163-1182.
Dilão, R. and Sainhas, J. ( 1999a). A
mathematical
model
for
the
development
of
eyespots
in
butterflies, pre-print.
Sainhas, J. and Dilão, R. (1998).
Wave optics in reaction-diffusion
systems. Phys. Rev. Lett. 80, 52165219.
Sainhas, J. and Dilão, R. ( 1999b).
Modeling butterfly wing eyespot
patterns, pre-print.
Turing, A. (1952). The chemical basis
of morphogenesis. Philo. Trans. Roy.
Soc. Lond. Ser. B237, 5-72.
La Recherche, special issue
morphogenesis, Feb. 1998.
on