2. Selforganization in Physics and Chemistry
The spontaneous emergence of spatio-temporal order has long
been thought
of to be a unique property of living organisms. Only the last few
decades it
has become clear that this is a very general property of dynamic
systems. The
process by which a highly organized structure evolves in such systems
out of an initial homogeneous or non-organized state is called
'selforganization'. This
phenomenon has been shown to occur in a variety of physical and chemical systems
and a theoretical basis has been established. A 'dynamic system' consists
of a large number of units (for instance molecules) interacting with
each other (i.e. by
exchanging properties like energy,
momentum etc.). An important
characteristic of these systems is the fact that the behaviour of the system
as a whole can be described
without referring to the behaviour of
the individual units. The state of the system depends only on the mean
behaviour of the units. As an
example, consider a physical system consisting of a volume
of oil contained in a
petri dish forming a thin layer of fluid. The units of
this system,
the oil molecules, interact with each other by exchanging energy
and momentum.
The state of the fluid, as characterized by its temperature,
density, viscosity etc., does not depend on the energy and momentum
of the individual
molecules. When the fluid is heated uniformly from below, the state of the
system alters.
A temperature and density gradient is forced upon the system and heat is being transported
along it. When the driving force (the
temperature difference) is small, the heat will be
transported by conduction (heat diffusion). When the
temperature difference is made larger, conduction no longer suffices
to transport all the heat and convection
sets in. At a critical
temperature difference, an
unexpected phenomenon will occur. The convection organizes itself as
a regular flow pattern
made of hexagonal cells, resembling a honeycomb
structure (cf.
Velarde and Normand, 1980). The cells are named
Benard cells
after their discoverer
(Benard, 1901). It was one of the first discoveries of
selforganization in a
dynamic system and a number of properties of the process of
selforganization can be
illustrated with it. First of all, it is important to realize that
the geometry of the pattern
which evolves is not forced upon the system from the
outside: the
driving force of the heat flow is uniform and isotropic. Secondly,
the temperature
difference must have at least a critical value: the heat flow through
the system
must have a minimal value, beyond which the selforganized structures
will always
be formed. When the pattern is established, this minimal heat flow
is needed for its
survival. Furthermore, the process is very reproducible: the size
and the shape of the cells
does not depend on the temperature difference (as long
as it is
in a certain range above the critical value) or on the size of the
petri dish, but only on
properties of the system itself like the viscosity, surface
tension, density etc. After
an externally applied local disturbance, the pattern reorganizes itself
into the same structure. This stability against disturbances is a general
property of selforganized structures.
Thermodynamics, a branch of science which is concerned with the behaviour of
dynamic systems, has provided a theory explaining the formation of
selforganizing structures in reaction-diffusion systems (Glansdorff and
Prigogine,1971; Nicolis and Prigogine,1977; Prigogine,1981).
The details of this theory
are beyond the scope of this paper, but the main ideas are the following.
Consider an 'open' thermodynamic system, that is a system which is able to
exchange energy and mass with its 'environment'. Energy flows through the
system and is partly 'dissipated' (degraded to heat by processes equivalent
to friction). Furthermore, mass transfer can take place and chemical
reactions. When the energy flow in a system reaches a critical value, then
under certain conditions the system becomes unstable. This means that
microscopical fluctuations in the state parameters (temperature, density
etc.) caused by the thermal
movement of the molecules are no longer damped, as is normally the case, but
are able to grow and cause a macroscopic observable effect: the state of the
system changes. As long as the
system is unstable it will keep changing under the influence of the growing
random fluctuations. Eventually the system reaches a state which is again
stable and settles down here. This stable state can be highly organized. For
instance, a spatial periodicity can arise, after the spatially homogeneous
state has become unstable, like in the case of the Benard cells. A
stationary state can become unstable, after which the system could embark on
a very stable oscillation. Finally, the system may become only "partially"
stable, while there still remains a permanent instability of some sort. The
behaviour of the system is now governed by two conflicting forces: a
stabilizing one, which prevents an "explosion"' and a destabilizing one,
which prevents the system from becoming either stationary or periodic. What
results, is a continuously changing, aperiodic behaviour, which has been
termed chaotic. In
Part 3
chaos will be explained in more
detail.
The order of the new
state, obtained after the system has become unstable, depends only on the
kinetics of the system, not on the forces exerted by the environment: the
order is intrinsic. The environment of the system only plays a role in
keeping the energy flow in the system on the required level.
After the theory of irreversible thermodynamics had predicted the occurrence
of selforganizing processes in chemical reaction systems, the first
demonstration of an oscillating chemical reaction was given by Belousov in
1958 and later it was shown that
a whole family of reactions could be made to oscillate (Zhabotinski,1964).
The reaction oscillates between a oxidized (red) and a reduced (blue) state
with a very constant frequency (0.5-1 cycle/minute) and amplitude. This
so-called Belousov-Zhabotinski reaction is also capable\of forming
spatio-temporal concentration patterns. The phenomenon was first described
by Zaikin and Zhabotinski in 1970. Later this
reaction system was shown to be capable of producing convoluting spiral
waves (Winfree,1972;1974), static mosaic-like patterns (Zhabotinski and
Zaikin, 1973), multi-armed vortices (Agladze and Krinski, 1982),
three-dimensional chemical waves (Welsh
et al., 1983; Winfree, 1984) and
spatial chaos (Agladze
et al, 1984). The reaction can be
made 'open' (with a continuous supply of reactants and removal of products)
in which case the bulk oscillations pertain indefinitely. In such open
conditions, a variety of 'time-structures' can be observed: transitions
among multiple steady states (Marek and Svobodova, 1975; Marek and Stulch,
1975), regular bursts (Sorensen, 1978),
'intermittent' behaviour (Pommeau
et al., 1981),
'chaotic' behaviour (Hudson and
Mankin, 1981; Smitz et al., 1977),
'period doublings' leading to chaos
(Simoyi et al., 1982).
The reaction mechanism of the BZ-reaction is known and its details are able
to explain the observed behaviour.
An autocatalytic step
produces the necessary
instability. Together with a
delayed negative feedback this causes the oscillations. Coupling of this
process to diffusion leads to traveling chemical waves and the static
concentration patterns. Computer simulations modeling the reaction mechanism
are able to explain most of the observed behaviour (Madore and Freedman,
1983).
An autocatalytic reaction is the chemical equivalent of a positive feedback mechanism. In general, positive feedback mechanisms are responsible for the onset of instabilities in dynamic systems. They result from certain kind of nonlinearities in the dynamics of the system. There are several ways in which such nonlinearities can enter into the dynamics of the system. One possibility is that the relation between thermodynamic "flows" and "driving forces" becomes nonlinear. This possibility is especially important for reaction-diffusion systems and has been extensively studied by Prigogine and collaborators. But even when this relationship is linear, there are several other ways for nonlinearities to arise (cf. foreword to the Dover edition of DeGroot and Mazur, 1962).When the influence of the positive feedback becomes more important than the other stabilizing processes, the system becomes unstable and the state of the system may change dramatically under the influence of blown-up local microscopical perturbations. What happens afterwards depends on the details of the kinetics. Thermodynamics cannot give a general answer to this question. As in the case of the Benard-cells and the BZ-reaction, the system undergoes a process of selforganization and settles down on a more organized state which is stable again. This state obviously is capable of dealing with the higher driving forces: the Benard-cells transport the heat more efficiently and therefore they can cope with the high temperature difference.
The best studied biochemical oscillator is the glycolytic pathway. The
intermediates have been observed to oscillate with a constant frequency and
amplitude. The allosteric enzyme phosphofructokinase has been proven to be
responsible for the oscillatory behaviour. This enzyme is activated by its
substrate, which leads to a positive feedback, and is inhibited by
intermediates which are formed later on, corresponding to a delayed negative
feedback. Boitteux and Hess (1980) showed that the reaction is capable of
producing
selforganized concentration patterns. By exposing a cell-free
yeast extract to a periodic glucose supply,
Markus
et al. (1984) were able to
demonstrate chaotic behaviour. As
the glycolytic pathway
is one
of the
most basic
metabolic pathways
common to
nearly all organisms, the importance of these findings cannot be
overestimated.
In conclusion, the
prerequisites for a process of selforganization
are:
•
A
dynamic system: a large set of similar interacting units.
•
Driving forces (a temperature or concentration gradient, reactants) which
invoke 'flows' (processes like diffusion, conduction, convection, chemical
reactions)
•
Positive feedback mechanisms which can threaten the stability of the system.
The most important
properties of selforganized structures are:
•
Stability against disturbances (other non-organized states are unstable).
•
Reproducibility: the geometry of the selforganized structures is a
consequence of the details of the kinetics of the system, which are constant
for a given system.
•
They can be switched on and off by changing the energy flow in the system.