Antichaos and evolution

I n biological evolution more than just natural selection could have played a role. Computer models show that complex systems can exhibit a tendency towards self-regulation. This casts a new light on the theory of evolution.

Developments in the area of mathematics invite us to another way of thinking over the way order arose in evolution.

Living beings are by definition to a high degree ordered systems: their internal structure is maintained and doubled by a very precise choreography of chemical reactions. Since Darwin natural selection has usually been seen as the only source of order.

But Darwin could not have imagined that something like self-organisation exists. This phenomenon has only recently been discovered and appears to be an ‘acquired’ property of some complex systems. It could be that biological order partly originates in this spontaneous order, which natural selection has preserved.

Selection supplied the form, but wasn’t the inventor of the coherence to be found in ontogeny, biological development. The ability to change and to adapt may even be the result of evolution.

Research in this area is still at an early stage, but what can be predicted from mathematical models on biological systems with self-organisation, appears to agree with the actual properties of organisms.

During recent decades much has happened in the research of complex systems, both in the natural as well as the social sciences. This is a new field of research; many questions have not yet been answered but some aspects have become clear - the term ‘chaos’ has penetrated to various popular books. It is known that systems, which were originally orderly, eventually could become totally chaotic and thereby unpredictable. Circumstances, which are initially equal, could lead to completely divergent results. In meteorology we speak of the ‘butterfly effect’: the idea that the flapping of its wings by a butterfly in Rio de Janeiro eventualla could change the weather in Chicago.

Chaos is only part of the behaviour of complex systems. There is also a ‘counter-intuitive’ phenomenon, which you could call anti-chaos: orderless systems can spontaneously ‘crystallise’ into very ordered systems. This could well have played a major role in biological development and evolution.

The discovery of antichaos in biology happened thirty years ago with the mathematical study of the differentiation of an inseminated egg cell into a large number of cell types. Since then much has happened.

Biology is full of complex systems:

• the thousands of genes which regulate each other’s operations within the cell,
  
• the network of cells and molecules which provide the body’s immune reaction,
  
• the billions of neurones in their networks which cause behav-iour and the learning process,
  
• the networks of relations within ecosystems which are full of co-evolving species.

The example of the self-regulating network of a genome is especially useful for showing how anti-chaos controls development.

The genome of a higher organism such as a human is coding for about 35,000 different proteins. The cells in an organism, whether they be liver cells, nerve cells or skin cells, differ in that different genes are active in each type, although the cells have the same genes. The difference lies in the genetic activity, not in the genes themselves.

Each genome works like a parallel processing computer or a network where genes regulate each other’s activity, directly or through their products. The co-ordinated activity of this system causes cell differentiation. Research tries to comprehend the logic and the structure of this system.

Mathematical models can help us gain insight into the characteristics of such complicated systems. Each complex system has so-called local characteristics: these describe the way in which individual elements are related and how they influence each other. The elements of the genome are the genes. Each gene is influenced by a small number of other genes, and these interactions follow certain rules.

For every set of local characteristics one could construct a larger whole (a class) of all the other systems which correspond to it. A new type of statistical calculation can identify the average characteristics of all the systems in such a class. The individual systems can be very different, but the statistically typical behaviour and structure form the best hypothesis to forecast the properties of each system.

Boole functions

One starts by idealising the behaviour of each element - each gene in our case - as a simple binary (“on” or “off”) variable. To study the behaviour of thousands of elements, which are coupled to each other, we use a so-called Boole network (named after George Boole, an English mathematician).

In a Boole network every variable is regulated by other variables which serve as input. The dynamic behaviour of each variable depends on a logical rule (a Boole function). This indicates the rules governing whether a variable is on or off. For example, the ‘OR” function: this switches on when one of the inputs is active; the “AND” function switches on when all inputs are on. If you know how many inputs there are you can calculate how many combinations are possible.

Each combination has a network-condition (STATE) attached to it. Through the predetermined way of being influenced, we know which STATE follows on which. There is thus a fixed order of STATES. This we call the trajectory of the network. Because the number of elements is limited a series of STATES will follow each other in a fixed pattern which will repeat itself after a period of time, giving a cycle - such a cycle is called an attractor - and the system lands by itself at such an attractor; there could be one or more attractors. After a disturbance the same cycle will often return, but it could also be that after a period of time a new cycle arises. Some networks can be very sensitive to disturbances and others less so. A structural change for example would be switching from an OR to an AND function. This could change the whole system.

If one has a system where each element can influence every other element, 2 to the power of 200 STATES are possible (about 10 with 60 zeroes after it); each change would affect each following change. This is therefore chaos. It is surprising that the number of possible cycli is only 74. As long as the number of elements that regulate an element is larger than 3, this sort of chaotic system persists, but if there are only two, order arises all at once. A network of 100,000 elements, but each with 2 inputs could attain 2 to the power of 100 000 STATES (a 10 followed by 30 000 zeroes), yet spontaneous order would arise: the system would remain in one of the 370 possible cycli.

Such a network which has landed in such an ordered cycle is relatively insensitive to disturbances. Thus a sort of homeostasis has arisen and homeostasis is exactly characteristic of all life.

Why do you find such permanent order in such a network? This seems to result from ‘frozen’ networks within the network, which form mutually related ‘walls’ to which elements remain attached.

So one could design networks which are quite rigid and others which are quite chaotic and everything in-between - compare this to material which could be solid, liquid or gaseous.

In a complex network parts could be ‘solid’ and parts ‘liquid’. Networks, which are sensitive to mutations, could play a role in evolution. Here cascades of mutations could have relatively easily occurred by which quick adaptations are possible. A network, which is in the ‘solid’ state, will not change as quickly again. In computer simulations with such networks it appears they evolve to a border between order and chaos, precisely the condition that makes natural selection possible.

If we view the genome in this way we will see the following: genes are usually influenced directly by a small number of other genes, perhaps at most ten. The Boole diagram is thus fairly simple, with few inputs. Genes are mostly controlled by one gene, which can only switch on or off (OR function). Such a system is stable (many ‘frozen elements’), and mutations only give modest changes to the system.

This system could also possibly explain something about the different cell types with their different system of gene-expression. A genome consisting of 30 000 genes can in theory exhibit something like 10 to the 10 000^th different patterns of gene-expression. A stable cell type allows a certain portion of the genes to come to expression. This could well be such a stable cycle (an attractor). A cell with 30 000 genes can have about 250 – 300 different cycli. An eukaryote cell needs 1 to 10 minutes to come into action, thus such a cell would need something like 250 to 2500 minutes to examine all cycles. This agrees with real times that are necessary for different processes in development. The length of a cell cycle is approximately proportional to the root of the quantity of DNA - which also agrees approximately with reality.

On the basis of this it is possible to predict how many different cell types there possibly could be in an organism: this would have to be the root of the number of genes (the number of attractors is the root of the number of elements). If we suppose that the number of genes is proportional to the quantity of DNA we can predict the number of cell types. Humans have about 30 000 genes and should have 250 – 300 types. We have counted 254 therefore this is in the exact order of magnitude.

Let us examine this in the different phyla: bacteria have one or two cell types, sponges 12 to 15 and ringworms 60. Because not all DNA encodes for proteins the values are lower than those calculated, but well within the range.

The number of types of cells in organisms seems to have a mathematical relationship to the number of genes in an organism. In this diagram we suppose that the number of genes is related to the quantity of DNA in a cell. If the gene regulatory systems are K=2 networks, then the number of attractors in a system is the square root of the number of genes. The actual number of cell types in various organisms appears to rise accordingly as the amount of DNA increases. Source: Scientific American, August 1991

The diagram is based on the idea that the number of genes is directly related to the amount of DNA. We now know that this is not true. What has been called ‘junk DNA’ now is shown to have a function too. But as long as the true role of the so-called non-coding DNA is not clear it is not possible to make conclusions about this part. In fact the clear relation between amount of DNA and number of cell types seems to be another proof of the activity of the non-gene-DNA.