Non-living Dissipative Systems

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The main focus of the metaphysics presented on this website is the ontological study of Totalities. With a " Totality " I mean a uniform, but not necessarily homogenous, thing or being, and thus NOT an aggregate. Such a being is an intrinsic UNITY, and possesses something like an " it-self " It is (penultimately) generated by some dynamical system, and the dynamical law, governing that system, is interpreted as the (penultimate) Essence, or specific Identity of that being. For all Totalities, above the atomic level, the atoms are the ultimate constituents and harbor the ultimate dynamical law of such a Totality and consequently the ultimate Essence of that thing. (See the Essay on Being and Essence).
Several types of Totalities are already treated of in the foregoing Essays : Molecules and Crystals. Crystals -- See the Essay on Crystals -- are still somewhat similar to molecules and consist, just like these, entirely of a chemical bonding of microscopic constituents (See for the chemical bonding, the Essay on The Chemical Bond.
But the higher Totalities go beyond this. They consist of complexes of (generally large) molecules in the form of local accumulations of them and in the form of static and dynamical structures in which many molecules participate. A multitude of such structures together form an Organic Totality, i.e. a living being. In order to better understand such organic Totalities it is useful to study a number of dynamical systems that are, like organisms (but unlike crystals) dissipative structures, or maybe better expressed : dissipative dynamical systems that generate "dissipative structures". A dissipative system dissipates energy to the outside world and keeps on doing this, while it takes up matter and energy (of a certain sort) from that same outside world.
There are a number of such systems which can be interpreted as 'formal' (in contradistinction to 'historical') precursors of Organisms, and these are presently being studied intensively, in the form of computer simulations as well as in the form of real dissipative systems.
We shall study two of them (meant as a prelude to the study of Organic Totalities) :
  1. The Brusselator. This is a computer simulation of a chemical non-living dissipative system.
  2. The Belousov-Zhabotinsky Reaction. This is a real chemical non-living dissipative system.
Both systems contain an autocatalytic mechanism, and both generate MACROSCOPICAL PATTERNS.

Remark: In this Essay we will sometimes use powers of numbers. When a number is brought to the power of n, this means that that number must be multiplied n times with itself. When a particular power is indicated with a negative number, say -n, then that means also that the number must be multiplied with itself n times, but after this multiplication we must take the reciprocal value of the multiplication result. The resulting number is still positive (when the original number was positive). The convention adopted here on this webside for the notation of powers is the usual :
The number A to the n-th power is denoted as An.


In Classical Thermodynamics we have to do with equilibrium structures like for instance Crystals. In non-equilebrium thermodynamics other ways of organization (into wholes) emerge.
So when a dynamical system interacts with the outside world it can be forced into a far-from-equilibrium condition by continually taking in, and exporting matter and energy. In this way dissipative systems can be formed which exhibit special behaviors. Dissipative systems can lead to a form of supramolecular organization. While the parameters ( parameters are variable quantities which can be held constant during a certain process ), which describe the generation of crystal structures, can be deduced from the properties of the crystal components, especially from the range of their attraction and repulsion forces, dissipative structures are a reflection of the global situation of the condition of non-equilibrium which produces them. The parameters describing them are macroscopic. They are not in the order of 10-8 cm, like the distance between crystal constituents, but in the order of centimeters. Also the time scales differ. These do not correspond to molecular time intervals (as the periods of vibration of individual molecules, which can correspond to, say, 10-15 seconds) but with macroscopical time intervals : seconds, minutes or hours (PRIGOGINE & STENGERS, 1986, Order out of Chaos, p. 144).
For our purposes chemical dissipative systems are especially important. Such dissipative systems are, in contradistinction to crystallization processes, systems that find themselves in a wholly fluid (liquid or gaseous) condition, and keep staying in that condition.
In order to generate macroscopical structures in time (oscillations), or in space (constant spatial patterns), or in space and time (chemical waves), chemical reactions must find themselves in a state of far-from-equilibrium, which means that there must be a continual influx and efflux of matter and energy (into and out of the system). But this is not sufficient, because the stationary state, finally reached by the system, remains stable : perturbations are damped and the system remains in a homogenous and stable condition, without exhibiting a macroscopic pattern in time or space. It turns out that the capacity of generating macroscopical patterns is also dependent on the detailed mechanism of the chemical reactions. In contrast with (near-) equilibrium situations the behavior of a system finding itself far from equilibrium becomes highly specific. There is no universally valid law from which the behavior of the system could be deduced. Every such system is a special case. What we can say in general is the following : In order to become unstable -- through which the system can, as a result of even only a weak perturbation, switch to a new dynamical regime and generate macroscopical structures -- at least one autocatalytic step must be present in the reaction mechanism. This means that one of the products of the chemical reaction system is involved in its own synthesis. Because of that the chemical system becomes non-linear (i.e. that there is at least one term in the dynamical law in which a variable appears raised to a power greater than one).
An autocatalytic step involves a reaction step that could look like this :

A + 2X becomes 3X.

The kinetic equation, describing the variations of concentration of substance X, then looks like this :

dX / dt = kAX2.

This equation is accordingly not linear. Also without an autocatalytic step non-linearity of a kinetic equation could be the case, for example in the reaction :

2X becomes A

(See PRIGOGINE & STENGERS, 1989, p. 70), but that is barely sufficient for a possibility of generating macroscopic patterns.
Thus in the case of chemical reaction systems two conditions must be satisfied for them to be able to generate macroscopical structures :

  1. Far from equilibrium.
  2. The presence of an autocatalytic step.

What these systems are all about is the variation of concentration of one or more chemical substances participating in or produced by the reaction. We shall call the change in concentration of substance X in (the course of) time : dX / dt, and when this concentration remains constant (which would be the case when the system finds itself in a stationary state with respect to substance X), then dX / dt = 0 applies.
Let us first consider a simple chemical reaction ( different from the Brusselator, which we will consider hereafter and which is a model of a chain of reactions, one of which is an autocatalytic reaction step ) :

A + X becomes B + Y

This is a reaction equation.

Remark: The equation is in this case not specified. A real reaction equation could (for example) read as follows : NaCl + AgNO3 becomes AgCl + NaNO3.

(We shall confine ourselves to the reaction going to the right -- in most cases the reaction can proceed in the other direction as well, until equilibrium is reached. So in most cases we have to do with two reactions).
The reaction takes place by the collision of molecules. Only a small fraction of these collisions, for example 1/1000000, will result in a reaction and thus the production of B and Y from A and X. The reaction kinetics are described with differential equations which relate to the changes in concentrations, to be precise the rate of change of these concentrations. This rate of change is dependent on (i.e. proportional to) the concentration of the reacting substances. With respect to the above reaction equation the rate of change of the concentration of substance X, namely dX / dt, is thus proportional to the (arithmetical) product of the concentrations of A and X in the solution (i.e. the solution in which these substances find themselves, dissolved), and so we get :

dX / dt = -kAX

in which A and X now stand for the concentrations of substance A and of substance X, and in which k stands for the proportionality factor which relates to quantities as temperature and pressure and which provides a measure for the fraction of reactive collisions which do occur and which lead to the reaction

A + X becomes B + Y.

dX / dt = -kAX is a differential equation describing the kinetics of the reaction, and as such it is the Dynamical Law governing the reaction system.

Remark: In this case (and in likewise cases) this " dynamical law " is NOT the ultimate dynamical law, (ultimately) responsible for the formation of the macroscopic pattern (which we can loosely identify with a uniform thing, or Totality ). The ultimate law is located in the relevant properties of all the atoms that together build up the pattern, because we can consider the atoms as the ultimate constituents of the generated pattern, by reason of the fact that the atomic species do not change in such reactions, i.e. they are not changed into other species (See the Essay on the Determinations of a Substance, in the section on The Nature of the Elements ...). This ultimate dynamical law is identified with the Essence of the generated pattern.

But the equation

dX / dt = -kAX,

as given above, is a generic law. When A and X are specified we get a specific law, and only then it represents the dynamical law, i.e. the penultimate law, identifiable with the Essence of the generated pattern. The ultimate law resides on the level of atoms, as has been said.
k represents, in a way, the external conditions and co-determines the velocity of the reaction. We may stir the reaction mixture. When we don't do this, then we must take into account the diffusion rates of the substances involved.

The Brusselator

The Brusselator is a (numerical) model for a simple chemical dissipative dynamical system.
( See PRIGOGINE, 1980, From Being to Becoming. PRIGOGINE & STENGERS, 1986, Order out of Chaos. JANTSCH, 1980, The Self-organizing Universe. COVENEY & HIGHFIELD, 1995, Frontiers of Complexity.)
The set of reaction equations of the Brusselator reads like this :

A becomes X
2X + Y becomes 3X
B + X becomes Y + D
X becomes E

The second step is autocatalytic.
The product X, being produced from A and broken down to E is, via cross-catalysis, co-responsible for the production of Y.
X is being produced via a tri-molecular step from Y, but conversely Y is being formed in the reaction between X and a substance B.
In this reaction chain the focus of attention relates to the concentration dynamics of the (intermediary) substances X and Y. The concentration of A is held constant (because A is being consumed it must constantly be supplied to the system) and the behavior of the system is being investigated for increasing values of B (one value for each ' experiment '). Because of the supply of A and B (and removal of D and E) the system is held in a state of non-equilibrium. The initial and end products A, B and D, E are held constant during the reaction (i.e. during each individual ' experiment ') by means of the mentioned supply and removal, while the two intermediary components X and Y can have concentrations varying in time.
The kinetic equations for this system (corresponding to the above reaction equations) can (if we set the kinetic constants k, k', etc. equal to 1) be expressed in the form of a set of coupled differential equations which describe the changes in concentration of X and Y. They read :

dX / dt = A + (X2)Y - BX - X

dY / dt = BX - (X2)Y

This set of coupled equations thus represents the dynamical law governing the mentioned reaction system.
Because in the present case it concerns continuous variables (the concentrations of the participating substances) the dynamical law has the form of (a set of coupled) differential equations, in contrast with laws like Xn+1 = BXn, which describe a discrete process.

Remark: "Xn+1 = BXn" means the relation between successive states (process-stadia), always separated by a finite (in contrast with infinitely small) time interval : The state of X at time n+1 is B times the state of X at time n.

Differential equations relate to successive infinitely small increases or decreases (over infinitely small time intervals). When we want to deduce from these equations the behavior of the system, then all those infinitely small increases and/or decreases must be added together with the help of a technique called integration. Thus by integration of the dynamical law (in the present case the set of coupled differential equations) we find the behavior of the system.

Remark: The above is formulated in an epistemological way, i.e. relating to how to acquire knowledge (of the behavior). Ontologically [i.e. how things ARE, in themselves apart from being known] these equations -- as a dynamical law -- are 'existing' in a corresponding real dynamical system, but not necessarily in the mathematical form as presented by human investigators, whereby their correctness is of course presupposed, meaning that these formulated and mathematically expressed equations are equivalent to their ontological correlate.

When for example we want to know the concentrations of X and Y, prevailing in the Brusselator reaction system, when that system has evolved to a stationary state (= a state that does not change anymore), thus when dX / dt = dY / dt = 0 applies, then we substitute this value (the value 0) in the kinetic equations (the above differential equations), i.e. there we replace the expression dX / dt by 0, and dY / dt also by 0. Now we can solve the equations (integration) for X and Y and we so obtain : X = A, Y = B/A. These are accordingly the concentrations that prevail at a stationary state of the system.
If the identities of A, B, D, E, X and Y are explicitly determined, i.e. what substances are represented by A, B, etc., and then as such appear in the set of differential equations, then this set represents the specific dynamical law of the system in question. It then represents the Essence of the generated pattern (if such a pattern is indeed generated). With this the diffusion properties (expressed in diffusion coefficients) and also other relevant properties are directly implied and as such given ( The spontaneous faster or slower spatial expansion of some substance in a medium is called " diffusion " ). The diffusion itself also is a regular (= proceeding according to a law) phenomenon, it follows ' Fick's Law ', relating the concentration of a substance at a determined location in space with its change in time, for each substance present in the reaction mixture. The conversion factor in Fick's Law, which so relates, is the diffusion coefficient, which is a property of the chemical substance in question. All this accordingly appears in the equations when we completely specify A, B, D, E, X, and Y. Thus a term from Fick's Law has yet to be added to the dynamical law when we want to express the   s p a t i a l   events in the reaction mixture (See COVENEY & HIGHFIELD, 1995, Frontiers of Complexity, p. 163).
But let us first confine ourselves to a Brusselator reaction that is strongly stirred, in which we consequently do not have to account for (differences in) diffusion, and in which we accordingly confine ourselves to the generation of a macroscopical pattern in time, namely oscillation.
We then are finally able to formulate things in a way that serves our purposes (which is the search for the status of the dynamical law as Essence of the generated pattern -- the non-living dissipative structure, representing a uniform thing), whereby we imagine the Brusselator to be a real chemical system. In the following formulation the symbols A, B, etc. are thought to be completely specified in terms of the specific substances and their relevant properties. When we speak about "at a certain temperature and pressure", we mean : finding itself in a certain interval of temperature and pressure. Let's now formulate :

IF the (chemical) substance A and the substance B are present (in a solution), at a certain temperature and pressure, and if the (other, in addition to the latter) relevant external conditions result in   (1) the substances being (and remaining) well mixed (for example by stirring), and   (2) the concentrations of A and B remaining constant (effected by a compensating supply), and   (3) the concentrations of the final reaction products (implied as D and E by the chemism of A and B) also remaining constant (effected by compensating removal), THEN the reaction chain

A becomes X
2X + Y becomes 3X
B + X becomes Y + D
X becomes E

is implied,
AND THE DYNAMICAL LAW --- with respect to the concentration evolution of the intermediary substances X and Y (also implied by the chemism of A and B)---

dX / dt = A + (X2)Y - BX - X
dY / dt = BX - (X2)Y

(whereby the symbols A, B, X and Y denote the concentrations of these substances)

STARTS TO OPERATE, and the dynamical evolution of the system begins. (See the following NOTE (note 1.))

This set of coupled differential equations -- the dynamical law -- is the Essence of every macroscopical pattern generated by the Brusselator as specified above. Such a macroscopical pattern can represent a non-living uniform thing, a non-living Totality (meaning an intrinsic totality of constituents, and thus forming an intrinsic unity).
The concentrations of A and B together form the initial condition (start stadium) of the dynamical system, and they imply a start concentration of X and Y as soon as the reaction has begun, and because the concentrations of A and B are held constant during a run of the reaction, the concentrations of X and Y are the actual (the 'real') starting state (initial condition) of the system.

Remark :   The being, generated by the Brusselator is, of course, not (yet) a full-fledged being. It is a pattern, and as such a precursor of such a being. Dissipative systems, as discussed here, are a prelude to genuine beings, especially to organisms.

As long as the relation B > (1 + A2) -- with respect to the concentrations of A and B -- does NOT apply [" > " means : bigger than ] the system (the Brusselator ) will, under the conditions mentioned, evolve towards a stationary state, in which the concentrations of X and Y do not change anymore. In that case the relations

X = A and Y = B/A

hold. But, when we are now going to increase the concentration of B (in a second experiment) -- and that means that we push the system further from equilibrium -- then, when B > (1 + A2) applies, this stationary state will become unstable. Now, in case of the smallest perturbation the system will consequently leave the stationary state. It will end up in another dynamical regime. In the present case it will end up in a stable limit-cycle.This means that the concentrations of X and Y will start to oscillate in a regular way. Thus instead of remaining constant the concentrations of X and Y will swing with a constant period. When we imagine for a while that the substance X is blue and the substance Y red, then the system will first be, say, totally red, then totally blue, then totally red, etc.
That this ordering (this organization) would be the result of the activities of billions of molecules seems unbelievable -- it concerns after all just a model for a dissipative chemical reaction chain -- but such phenomena are indeed observed in real chemical reactions.
The phenomenon just described is a certain form of self-organization, in the present case a macroscopic pattern in time, namely an oscillation. Such a phenomenon is called a ' chemical clock '.
In addition to a pattern in time the system is also able to generate patterns in space, and in space and time. This happens when diffusion comes into play. The diffusion-reaction equations of the Brusselator reveal a baffling assortment of possible behaviors of the system. The diffusion of chemical substances finding themselves in a far from equilibrium condition can lead to new forms of instability. It is then possible that vibrations (oscillations) will occur in time as well as in space. These are waves in the concentrations of X and Y, periodically traversing the system (PRIGOGINE & STENGERS, 1986, p. 148). The condition of being far from equilibrium can also lead to the system going in the direction of a new stable stationary state with an inhomogenous distribution of X and Y in the Brusselator reaction.

Remark: Because the state is now stationary nothing changes in time. Patterns in space are formed.

It is possible to indicate a determined wavelength in the stationary concentration bumps which is directly proportional to the spatial extension of the system. With this the macroscopical character of the emergent order is clearly expressed. Somtimes even a polarity of the structure appears (a high concentration of X at one end of the container) when the system is subjected to a disturbance (Something like this happens in the development of an embryo from a homogenous cellmass : there appears a gradient)(See JANTSCH, 1980. PRIGOGINE & STENGERS, 1986). See next Figure.

Figure 1. Formation of a stable dissipative structure far from equilibrium in the chemical reaction system " Brusselator ". The concentration of X was calculated under the following assumptions : A = 2, B = 4.6, DX = 0.0016, DY = 0.0080. ( DX   and   DY   are the diffusion coefficients of substance X and Y respectively ). The higher concentration of X on the left side indicates spontaneous polarization. (The horizontal axis represents space in arbitrary units.)
( After Jantsch, 1980 )

If also the diffusion of the start-product A can no longer be ignored, dissipative structures can appear past a critical instability, whereby in this case the spatial organization (generated by the system) is limited to a determined volume (inside the system). Outside this volume the classical thermodynamical regime prevails. So in this case the Brusselator generates a pattern with intrinsic boundaries . See next Figure.

Figure 2. Formation of a localized dissipative structure in the chemical model reaction " Brusselator ". The concentration of X was calculated under the following assumptions : B = 26, DA = 0.197, DX = 0.00105, DY = 0.00526. ( DA  , DX   and   DY   are the diffusion coefficients of the substances A, X and Y respectively ). Outside the dissipative structure, the classical thermodynamic order reigns.
( After Jantsch, 1980 )

And here the size of the dynamical system is important with respect to the formation of dissipative structures (patterns). A system that is too small will always be dominated by boundary effects. Only above certain critical sizes the non-linearities will find a possibility to unfold their characteristics, and so bring with them novel structures (patterns), resulting in a certain autonomy of the system with respect to its environment. Not until then we can speak of the formation of a genuine Totality (distinguishing itself intrinsically from its environment, not until then it is an ' itself '). A dissipative structure (pattern) will only appear when a specific size is realized, but then there will be no difference in structure anymore between the case of the size (of the environment) being just sufficient and cases of much more larger sizes of the environment (except with respect to the 'life-span' of the system, being dependent on the environment with respect to 'food' in the form of free energy and a replenishment of reactants) (JANTSCH, 1980).
We clearly can see here -- in the embodiment of dissipative structures like (those generated by) the Brusselator, which is however just a model -- a formal forerunner of Organisms.

The Belousov-Zhabotinsky Reaction

Most phenomena obtainable from theoretical models (like the Brusselator ) are indeed also observed in real chemical systems. The most famous example is the Belousov-Zhabotinsky Reaction (BZ-reaction). This reaction comprises fairly many variants which indeed generate the structures (patterns) predicted by theory.
In the BZ-reaction an organic acid (Malonic acid) is oxidized by Potassium Bromate in the presence of an appropriate catalyst, for example Cerium, with Ferroin as an indicator (PRIGOGINE & STENGERS, 1986, BALL, 1994).
All kinds of structures can be generated like spiral waves, oscillations, etc., visible by different colors. All those changes in pattern or color represent changes of states of the system. A medium, sensitive to stimulation -- like the BZ reaction mixture -- is a medium which changes its state when subjected to a stimulus that exceeds a certain threshold value. After such an excitation such a medium becomes non-responsive, only to return to the original receptive state via a series of states which themselves also can be excited in turn. Everyday examples can be found in the form of spiral waves generated by heart muscles, the feeding activities of the Slime Mold ( NOTE 2 ), and the electrical activity in our brains.
Recently also the generation of static spatial patterns is accomplished in real experiments (See figure 4), like for instance a line of spots, a pattern simular to what could be generated with the Brusselator (which is, as had been said, just a theoretical model) (COVENEY & HIGHFIELD, 1995).
In order to understand something of the mechanism of reactions like the BZ-reaction it is useful to consider a hypothetical reaction far simpler than the BZ-reaction :
  1. A becomes B
  2. A + B becomes 2B
  3. B + C becomes 2C
  4. C becomes D
Step 2 and step 3 are competing autocatalytic steps : Step 2 produces B, whereby B catalyzes its own production. Step 3 produces C, whereby C catalyzes its own production (both steps are positive feedback reactions). But while step 2 produces B, step 3 consumes B, so these autocatalytic steps are competing steps. It is further assumed that substance A will constantly be supplied to the reaction, and D will constantly be removed. The reaction starts with A, which will spontaneously be transformed into B. Step 2 accelerates the formation of B, so the concentration of B rapidly increases. But such a high concentration of B will now give step 3 a chance, resulting in an increase of the concentration of C, and this will accelerate the formation of C because C catalyzes its own production. This production of C takes place at the expence of B, so the concentration of B decreases. And because there is now less B, step 2 will not be prominent anymore and so the production of B decreases. At first we saw a surge of B, now we have a surge of C, at the expence of B. But the concentration of C cannot rise indefinitely, because C will be consumed by step 4. The product D is assumed not to take further part in the reaction. It is constantly being removed and so prevented from clogging up the whole mixture. So in the course of time the concentration of C will eventually decrease because of step 4. And while the concentration of C is low again, not much B will be consumed anymore by step 3. So the concentration of B will rise again because of the production of B in step 2, i.e. this production of B will not be countered by a consumption of B by step 3. As a result of all this the concentrations of B and C will oscillate as long as the reaction is sustained (by supply of A and removal of D). When those two substances B and C are monitored by means of colored indicators we will see the corresponding color oscillation. Thus, the color of this simple four-step scheme oscillates under the influence of two autocatalytic but competing steps. What we have is an oscillation in time. But the color of the real BZ-reaction can be made to vary in space as well as time. To produce the   s p a t i a l   patterns we must simply ensure that the reactants are NOT well mixed. Then different diffusion rates can come into play. Because of the feedback loops, the reaction is very sensitive to the small, random variations in concentration that are likely to arise from place to place in an unstirred mixture -- they can induce a switch from the dominance of B to that of C, and vice versa. These imbalances spread out from their point of initiation in the form of colored chemical waves (See BALL, 1994, p. 306/8).

The Belousov-Zhabothinsky reaction itself is significantly more complicated than this idealized four-step process, but the basic principle remains the same. The reaction consists of two cyclic processes coupled together by the interconversion of one Cerium ion into another : Ce3+ and Ce4+. This interconversion can be monitored by the indicator Ferroin, which turns red in the presence of Ce3+ ions, and blue where Ce4+ ions dominate. The second of the two cyclic reactions just mentioned, is still not understood completely. The oscillations are caused by the alternating dominance of the two cyclic steps implying an alternation of the Cerium ions and thus an alternation of red and blue. The successive boost and inhibition of those reaction steps cause the chemical waves, visible as target patterns or spirals.

Figure 3. Target patterns formed by the Belousov-Zhabotinsky reaction
( After Coveney & Highfield, 1991, The Arrow of Time )

Chemical systems, related to the BZ-reaction, in which structures could form spontaneously in an autocatalytic system in which the reactant molecules diffuse at different speeds through the reaction medium -- are so-called reaction-diffusion systems (Also the BZ-reaction itself is such a system when the reaction is not stirred). In order to describe such systems I will quote a few passages from Philip BALL, 1994, Designing the Molecular World . Concerning reaction-diffusion systems he writes :
Specifically, if the faster-traveling reactants inhibit the reaction whereas the slower ones catalyze it, then within a certain range of diffusion rates of the reactants the system can transform suddenly from a uniform mixture into one in which the chemical composition varies in a regular manner from place to place. In effect, the system turns into a kind of crystal, with periodic structure -- but a very peculiar kind of crystal, because all the molecules within it are free to move around yet the patterned structure remains. In a normal crystal, this freedom of movement would destroy any periodicity.
In general, autocatalytic reactions like the BZ-reaction tend to produce moving chemical waves, not stationary structures. These latter are seen for the first time in 1990 by Patrick de Kepper and colleagues from the University of Bordeau. Let's quote BALL again :
De Kepper's group studied a variant of the BZ-process, called the chlorite-iodide-malonic acid (CIMA) reaction, in which the reactants were mixed within a gel to slow down the diffusion rates so that stable patterns might form. The researchers used a starch-based indicator to reveal the changes in chemical composition throughout the system : the indicator turns either yellow or blue, depending on whether the concentration of the ion I3- [ an ion consisting of three iodine atoms in such a way that the net electrical charge is one negative unit (equivalent to the charge of an electron ], an intermediate species in the reaction, is high or low. They observed a few rows of yellow dots within a certain region of the otherwise blue gel, which they identified as Turing structures [ In the 1950's Turing had predicted such structures on theoretical grounds ]. Harry Swinney and Qi Ouyang of the University of Texas at Austin were later able to "grow" large patches of these Turing structures. At first the reaction produces radiating target patterns like those of the BZ-reaction, but over a period of an hour or so these patterns break up into regular hexagonal arrays of yellow dots which slowly come to rest [ See Figure 4 ]. By changing the temperature of the system, Ouyang and Swinney were able to transform their dot pattern into a stripe pattern -- this kind of transformation too had been predicted by Turing's theory.

Figure 4. Two-dimensional arrays of stationary Turing structures can be generated in the CIMA reaction under appropriate conditions of temperature and reactant concentration. A hexagonal pattern at one temperature (left image) changes to a stripe pattern at another (right image).
( After BALL, 1994, Designing the Molecular World )

The investigation of such self-organizing chemical systems is still going on. All these chemical systems, mainly variants of the BZ-reaction, strongly point in the direction of corresponding biochemical processes in Organisms, for example the aggregation patterns of the Slime Mold. This mold consists of amoebae (unicellular animals) living on the forest floor. When starved (i.e. finding themselves in unfavorable conditions) they aggregate into a multicellular body which can then move in search of a more favorable habitat. The aggregation is triggered by the release of the compound cyclic adenosine monophosphate (cAMP) from some of the amoebae (See Figure 5).

Figure 5. Spiral waves in colonies of the mold Dictyostelium discoideum. These patterns are created when the mold forms aggregates in response to some external ' stress', such as a lack of moisture or nutrients.
( After BALL, 1994 )

Also static patterns are observed in the natural world. For example in larvae of the fruit fly. In the early development the embryo evolves stripes which represent the evolving segments of the insect. This development is thought to be related to the concentration of a protein called Bicoid, which increases gradually from one end of the developing egg to the other. The concentration of the Bicoid protein provides a signal that switches on genes at different points along the body axis, dividing the embryo into a sequence of segments which will develop into different parts of the body (BALL, 1994). (See figure 6)

Figure 6. In the early stages of development, fruit fly larvae develop striped patterns. The regions of different shade represent parts of the embryo that will subsequently follow different developmental pathways. A gradual variation in the concentration of a morphogen called the bicoid protein from one end of the embryo to the other provides the chemical signal that stimulates this patterning.
(After BALL, 1994 )

In the just discussed development of the larva of the fruit fly we in fact saw the development of metamers or sequential parts. The potential metamers will develop partly into qualitatively the same or almost the same actual metamers, partly into qualitatively different metamers. So the final result is not just a linear repetition of identical structural elements, thus not something like a one-dimensional crystal, but is - what we earlier called - a tectological structure. And all this seems to be accomplished by certain dissipative subsystems (within the overall dissipative system that is represented by the whole developing organism in its environment). So the actions of such systems (including diffusion phenomena) can indeed account for the organism's  
s p a t i a l   structure -- here a tectological structure - especially, I think, because all organisms enjoy a more or less viscous (i.e. jelly-like) consistency by reason of their semi-liquid state, that supports slow diffusion of certain substances from one place of the body to others.
The same can be said with respect to the overall   b a s i c   s y m m e t r y   that the organism will acquire. In the case of the fruit fly - and most, but by far not all, other organisms - it is its bilateral symmetry, that can as such geometrically be represented by half a rhombic pyramid (See Figure 7) : This hemipyramid has three non-equivalent ideal axes perpendicular to each other, one of which is homopolar, representing the organism's right-left axis, while the other two are heteropolar, respectively representing the organism's back-belly axis and the head-tail axis, the latter being its main axis. We can group all the organisms, having this ideal Stereometric Basic Form, together, by classifying them as Eudipleura. They are called Eudipleura, because they consist, in addition to a sequence of (differentiated) metamers, of two antimers or counterparts, i.e. two almost identical sides (pleura). (When those sides are totally different, as we see for example in flatfishes, we call them Dysdipleura). The next Figure shows the geometric body representing the Eudipleura.

Figure 7. One half of a rhombic pyramid, as such indicating the axial relations of the Eudipleura (bilateral s.str. organisms).
The left image emphasizes the only mirror plane present. The right image names the poles of the three axes (the latter are indicated in red).

Many other organisms, like radiolarians, and medusae, or stages of them, like pollen grains and eggs, show different ideal stereometric forms, that can be represented geometrically by spheres, regular or irregular endospheric polyhedrons, or by ellipsoids, cones or by all kinds of pyramids, and hemipyramids.
The study of these ideal stereometric basic forms (of natural objects, and especially of organic individuals) can be called Promorphology. It proceeds on the basis of the mentioned tectological structure of organisms, and should be the first stage of an overall scientific organic Morphology.
All these global   s p a t i a l   aspects can indeed, as it seems, be generated by complex organic dissipative systems and subsystems, and then be fine-tuned by still subtler subsystems.
And all this clearly indicates that organisms, as living dissipative structures, really have the nature of dynamical systems. We shall discuss them as such extensively further down.

So we see that, when studying non-living dissipative systems, we automatically end up at a contemplation of analogues in the organic world.
The patterns we see in reactions like the BZ-reaction can also be simulated in a computer by means of Cellular Automata (CA)(See Figure 8). These are simple, but of course abstract, dynamical systems governed by a certain rule by applying that rule again and again, i.e. the rule is recursively applied to its output (See for CA's the Essay on Cellular Automata).

Figure 8. CA-analogue of the BZ-reaction. The pattern (which is dynamic) is generated by the rule ZHABO.JC in the CA-simulator CELLAB of Rudy Rucker and John Walker.


The systems considered in this Essay generate patterns. Those patterns are generally macroscopic, they are assemblies of molecules forming regions of different concentrations of several substances, or, when we look beyond the boundary of the non-living, to the living, even assemblies of biological cells, forming intricate patterns. The penultimate constituents are thus molecules or larger units. But the ultimate constituents are atoms. These do not change into other atomic species during these processes. So the ultimate dynamical laws governing such generated patterns and structures reside in the collections of the relevant atomic species. As I pointed out in the Essay on The Determinations of a Substance such a collection harbors a multitude of such laws, and when a certain (macroscopic) pattern is generated then -- with hindsight -- one of those laws had started to operate. The possibilities for generating patterns are almost unlimited, even already in the inorganic world. They make our life on this planet worthwile. Chemistry is the most important science to study those patterns.
To conclude this Essay on non-living dissipative structures (which will be follwed of course by an Essay on living dissipative structures -- organisms) let us quote some interesting remarks of Philip Ball from his stimulating book Designing the Molecular World -- Chemistry at the Frontier , 1994:
Today's chemistry is as mathematical as you could like (some would say more so). But the " reductionist " approach to many-molecule systems, which tries to provide a description based on a few simple equations representing the interactions between molecules, becomes rapidly intractable as the numbers grow larger. The world therefore abounds with systems that have in the past been regarded as far too complex to permit any obvious means of realistic mathematical analysis. One of the astounding revelations of recent decades, however, is that complexity does not necessarily imply disorder, intractability or general messiness. Rather, it has become apparent that complexity can itself give rise to the abrupt manifestation of order, often in the form of patterns of startling richness that are a far cry indeed from the geometric sterility generally exhibited by simpler systems. The real surprise is that there may be nothing in the fundamental elements of the mathematical description to warn us that these structures might appear. A reductionist sees only the most prosaic of interactions between the system's individual components, but the " holist " who considers the system as a whole discovers within it an unguessed capacity for intricate organization.
Many of the patterns that arise out of complexity resemble the delicate and often beautiful " organic " forms of the natural world. Moreover, similar patterns may be found in systems that are ostensibly unrelated. There is a common thread, however : these systems tend to be undergoing processes of rapid or unstable transformation. This has lead to hopes for a unified description of pattern formation in systems that are far from attaining equilibrium.

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