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.