What is Life - Mathematics of Life

Mathematics of Life (2011)

Chapter 17. What is Life?

Biology is the study of life, in all its forms – on this planet.

Since we currently know of no other place in the universe where life definitely exists, or used to exist, ‘on this planet’ may seem superfluous. However, it points to a gap in today’s biological knowledge, one that would be present even if there were no life anywhere else in the universe.

In its strongest form, the gap is the general question ‘What is life?’ Must all life in principle be similar to the living creatures of this planet – built from carbon chemistry, controlled by DNA, composed of cells ... in short, just like us? Is there no alternative, not even hypothetically? Or could entities that reproduce, and are sufficiently complex and organised to qualify as ‘living’, be made from different materials, be organised in different ways? More strongly, do such entities exist, somewhere in our galaxy or another galaxy?

More bluntly: could aliens exist – and do they?

The first part of this question is a lot easier than the second. We can investigate the potential for exotic life forms without exploring the planets that circle distant stars. But even then, we run into deep problems. We’ve already seen that biologists disagree about whether viruses are alive, so ‘life’ is to some extent a matter of definition. To answer the second part, we have to make contact with extraterrestrial life – whatever we decide that means. We might do this by visiting another world, by observing chemical signatures of living processes through powerful telescopes, by receiving messages from an alien civilisation, or by waiting for aliens to visit us.

In the next chapter I will argue that UFO reports and claims of alien abduction are not sufficiently convincing to conclude that the fourth of those options has already occurred. The SETI project has been pursuing the third option since 1961,1 so far without success, but it could pay off at any moment if advanced aliens actually do exist. We are just beginning to attempt the second option. The first is now being done using robotic explorers, and is currently confined to our own Solar System. Humans last landed on the Moon in 1970, and the proposed project to land a human on Mars has been cancelled.

Some biologists define life to be just like Earthly life: based on carbon, water, organic chemistry, DNA, proteins, the whole shebang. It works here, but this surely begs the point, by making a gigantic assumption for which no evidence exists. Worse, such a concept of life has to be continually adjusted as we discover more about our own planet’s more exotic inhabitants. A lot of Earthly life is distinctly different from what we believed was ‘normal’ fifty years ago.

I can’t help imagining two cavemen discussing the definition of ‘tool’. They quickly agree on two fundamental points: a tool has to be made of flint and it has to fit in your hand. Otherwise there would be no way to make it, and people would not be able to use it. Now imagine their faces if some time-traveller turns up with a bulldozer.

If we are to discuss potential forms of life – like ours or not – the first step is to agree on a working definition of ‘life’, and I’ll spend the rest of this chapter on that question, returning to aliens in the next one. Life is one of those annoying concepts that can usually be recognised when you see it, but turns out to be hard to pin down precisely. I don’t personally find that to be either a surprise or an obstacle: in my experience, the only scientific concepts that can be pinned down with absolute precision are in areas that were mined out long ago. Think of all the fuss about whether Pluto counts as a ‘planet’. Even in mathematics, where precise definitions are de rigeur, it is common for them to evolve as new research reveals new aspects. We’ve already seen this for such basic terms as ‘space’ and ‘dimension’.

Biologists don’t have a universally accepted definition of ‘life’; instead, they have several competing definitions, none totally satisfactory. At one extreme, you can build carbon chemistry and DNA explicitly into the definition. If you do, then anything acceptable as a life form will use carbon chemistry and DNA – end of story. However, this begs most of the interesting questions. From the viewpoint of mathematics and physics, terrestrial life looks like an example (rather, a gigantic number of closely related examples) of what ought to be a far more general process. Many biologists feel the same way: they dislike defining life in terms of what it is made of: what it does and how it works seem more appropriate and less limiting. It’s rather like having a mathematics that is limited to numbers between 1 and 100, and wondering whether a more general concept of number could preserve most of the interesting properties observed in that range.

The upshot is that the current working definitions of life concentrate on what it does, rather than what it is. The main features of life are:

• possessing an organised structure;

• regulating internal behaviour in response to short-term changes in the environment;

• maintaining both the above by extracting energy from the environment;

• responding to external stimuli, say by moving towards a food source;

• growing – in a way that does not merely accumulate more and more stuff while doing nothing with it;

• reproducing;

• adapting to long-term changes in the environment.

These are not the only things that living creatures do, they are not mutually exclusive, some are less important than others, and some might even be dispensed with altogether. But in broad-brush terms, if some system in nature exhibits most of the features on the list, then it may qualify as a form of life.

To appreciate the difficulties, think about a flame. Flames have a definite physical structure. They change their dynamics in response to their surroundings, growing in the presence of fuel and oxygen, dying down if these are absent. They extract chemical energy from the reaction between fuel and oxygen. They invade adjacent sources of fuel. They grow. They reproduce: a forest fire starts as a single flame. But the chemistry of flames today is the same as it was a billion years ago, so they fall at the final hurdle.

With a vivid imagination, you could invent plausible aliens that were complex systems of flames. If their chemistry could change over long periods of time, depending on what the environment can provide, they might evolve. In a way, we are like that: the energy cycle that keeps our cells working – and us – is an internalised flame. It is an exothermic reaction: it gives off heat. But we are more than just an exothermic reaction.

Many alternatives to this list of properties of life have been proposed. Nearly all of them feature the most obvious of life’s characteristics: reproduction. They distinguish this from a closely related ability: replication. The distinction is important – indeed, vital.

An object or system replicates if it makes exact copies of itself, or copies that are so similar that it is hard to tell the difference. It reproduces if the copies have some degree of variability. A photocopier replicates black-and-white text documents: aside from differences in paper, and possible enlargement or reduction, the copy is essentially the same as the original. In particular, the text, which is what usually matters, is essentially identical, though even here there are smudges and gaps: true replication in the strongest sense is rare. Even copies of computer files may contain errors. In contrast, a cat reproduces: its kittens, even when fully grown, do not resemble it in detail, and they often have totally different markings, sizes and gender. But they grow into cats and can, on the whole, father or give birth to their own kittens. So the system ‘cat’ reproduces, but does not replicate.

One of the pithiest and potentially broadest definitions of life was devised by Stuart Kauffman, and it is based on thinking of a living organism as a complex system. This phrase has a specific technical meaning: something composed of a large number of relatively simple agents, or entities, which interact according to relatively simple rules. The mathematics of complex systems shows that despite the simplicity of the ingredients, the combined system often (indeed usually) displays complicated ‘emergent’ behaviour, not evident in the entities and rules. In Kauffman’s view, life is a complex system that can reproduce, and can carry out at least one thermodynamic work cycle (which converts heat into work and can be repeated over and over again, given a reliable heat source). All other properties of life can be seen as possible – sometimes inevitable – consequences of these basic features. This definition emphasises general mathematical features of the system, not the ingredients it is made from. However, it has yet to find favour among the majority of biologists.

It is of course fascinating to realise that life on Earth is heavily dependent on the information-bearing properties of DNA, and only a scientist who is brain-dead could fail to want to know how it all works. But this does not imply that when the question is ‘Life?’, the answer is ‘DNA’. If the question were ‘Cake?’, few scientists would settle for the answer ‘Baking soda’. They’d want to know just what the baking soda contributed, and whether anything else might have a similar effect. So it can’t be the details of terrestrial biochemistry that really explain why life is possible: it must be the abstract process that the biochemistry realises.

Life on Earth is one example, the only one we know. The question is: an example of what?

Must life be based on DNA? If you think that DNA is the key to life, you have to answer ‘yes’. Numerous interesting questions crop up, even then. Would the ‘genetic code’ that turns DNA triplets into proteins have to be the same as the code used on Earth? The code is implemented by transfer RNA, and experiments show that transfer RNA could implement a different code with the same facility, using the same chemical processes. You can synthesise nonstandard transfer RNA and the whole system still functions perfectly well. So let us accept as a working hypothesis that in principle Earthly life does not employ the only system that would work.

Upping the ante slightly, it is also clear that any molecule sufficiently similar to DNA could play a similar role. There could be minor variants in which different bases occurred: even on Earth some viruses employ RNA in place of DNA, and RNA uses uracil instead of thymine; RNA also plays several key roles in the reproduction of nearly all organisms. Synthetic ‘exotic bases’ have been created in the laboratory and inserted into DNA double helices. The code has been extended to a four-base code, in the laboratory, so the range of amino acids encoded could be in the hundreds. So it seems unlikely that DNA is the only possible linear ‘information-bearing’ molecule (more properly, class of molecules, since the whole point is that different organisms have different DNA).

Nor need an information-bearing molecule be linear – tree-like structures, or two- or three-dimensional polymer arrays, could also be used. Because of the unique properties of carbon, it seems probable that any large information-bearing molecule would have to be organic (carbon-based) – but silicon could be a possible alternative in the presence of the occasional metal atom, which stabilises complex silicon-based molecules. In fact, the kind of organised complexity that is required to make a living creature could in principle be found in, say, magnetohydrodynamic vortices in the photosphere of a star, crystalline monolayers on the surface of a neutron star, wave-packets of electromagnetic radiation crossing the wastes of interstellar space, quantum wave-functions, or even non-material creatures inhabiting pocket universes whose physics is very different from ours.

Whether or not such creatures actually exist, the thought experiment ‘What if they did?’ suggests that there might be a meaningful definition of ‘life’ that is valid in much greater generality than ‘organised chemical systems that reproduce using DNA’. This would be a definition that emphasised the abstract processes of life, not special material constituents. Such a definition would presumably involve the apparent ability of living creatures to self-organise and self-complicate.

Until the middle of the twentieth century, the list of properties used to define life could have been shortened to just one: reproduction. (Some minor tweaks in what that word means would rule out flames.) No known non-living object, or system, could replicate, let alone reproduce. So living systems had a unique, mysterious ability which non-living systems could not emulate. Early proposals for the cause of this ability included possession of a soul or the presence of some vital force (élan vital) that animated non-living matter. But for scientists, explanations like these are unsatisfactory unless the soul can be located somewhere in the organism, or the vital force can be identified. Otherwise we are back to the medieval habit of explaining why objects fall by appealing to some tendency to ‘move to their natural place’ – on the ground. They fall because they move towards the ground ... brilliant!

By the middle of the twentieth century it had become clear that the ability to replicate is a system property: an ability arising from the way the system is organised. Replication is a consequence of the system’s structure; it does not require some magic extra ingredient that no one can locate or identify. The system that proved this definitely had no such ingredient, because it was a mathematical abstraction. Its inventor was John Von Neumann, whom we have already encountered as the creator of game theory.

Von Neumann’s first step in this direction came in lectures he gave in 1948, and it was a thought experiment. Imagine a programmable robot living in a warehouse filled with spare parts, which it can manipulate. It also has a tape containing instructions. The instructions tell it to wander round the warehouse and build a copy of itself by picking up all the parts required – except for the tape. Finally, it makes a duplicate of the tape and inserts this into the copy.

There are problems with this description: for example, the robot has to be able to copy the tape, so isn’t replication built in from the start? Von Neumann’s point was that in this scenario, neither the robot alone, nor its program, can replicate. Only the combined system does that. The program replicates the robot; the robot replicates the program. This division of roles was a key insight, because it got round what had previously seemed to be an insurmountable logical obstacle to a self-replicating device.

Suppose that a self-replicating entity exists. Then it must contain within itself a complete specification of its own structure, in order to know how to construct the copy. However, the copy must also be able to replicate (or else it’s not a true copy), so this internal specification of structure must contain something that specifies the part of the copy that provides a complete specification of its own structure ...

More carefully: the device must contain a representation of itself. Inside that representation must be a representation of the secondgeneration device. Inside that must be a representation of the thirdgeneration device, and so on. So it looks as though any self-replicating device has to resemble a set of Russian dolls, each nested inside the previous one, and this set of dolls must go on for ever. If it didn’t, the smallest doll would not contain a representation of itself, and so could not replicate.

No physical entity can do this: at some point, the doll has to become smaller than the smallest fundamental particle. Of course no living entity can do it either, but that’s not a problem if you believe that life rests on some supernatural ‘essence’. The supernatural doesn’t have to make sense.

Von Neumann’s proposal for the architecture of his device avoided the Russian-doll objection. It did so by interpreting the same physical object – the program on the tape – in two conceptually distinct ways. In one, the program consisted of instructions, to be obeyed. In the other, it consisted of symbols, to be copied. In one, a piece of paper with ‘put the kettle on’ causes a kettle to be boiled. In the other, it leads to a second piece of paper bearing the message ‘put the kettle on’.

Now the program can replicate the robot when the robot obeys the instructions, and the robot can replicate the program by copying it but not obeying the instructions.

Von Neumann wasn’t satisfied with this set-up, because he couldn’t see a good way to analyse it mathematically, or to build a real machine that could carry it out. At the time, he was working at the Los Alamos National Laboratory in New Mexico, and one of his colleagues was the mathematician Stanislaw Ulam. Ulam, renowned for his original turn of mind, had been modelling the growth of crystals using a lattice: a square grid like a large chessboard, without the chequered pattern. He suggested that Von Neumann might be able to implement his self-replicating machine by employing a similar trick. In detail, Ulam’s idea was to define a self-replicating cellular automaton (see Figure 73).

In this context, an automaton is a mathematical system that can obey simple rules – in effect, perform elementary computations. A cellular automaton is a grid with rules, like a simple video game. It is a special kind of complex system, with cells as entities and – well, rules as rules. Each square on the grid – each cell – can exist in a variety of states. One way to visualise the states is to colour the cells, so that possible states correspond to a list of colours. Each cell now obeys a specific system of rules, in which its own colour and those of its neighbours determines the next colour for that cell. For instance, with two colours ‘red’ and ‘blue’ the rules might be a list of statements like this:

• If you are red and your four immediate neighbours are all blue, turn blue.

• If you are red and exactly three of your neighbours are blue, turn blue.

• If you are red and exactly two of your neighbours are blue, remain red.

096

Fig 73 Schematic of Von Neumann’s replicating automaton.

The full list would cover all possible patterns of states.

With the colours and rules in place, you start the automaton in some pattern of colours (initial state), apply the rules (simultaneously on all cells) to get the next pattern, repeat to get the pattern after that, and so on. It sounds simple, but the consequences can be complex. A suitable cellular automaton can mimic any calculation that a real computer can carry out.

Inspired by Ulam’s suggestion, Von Neumann worked out a set of rules for an automaton with 29 cell colours.2 The replicator device occupied about 200,000 squares; the rest were left blank – in effect, another colour, which changed only if a neighbouring cell ceased to be blank. Von Neumann proved that by following the simple list of rules, the automaton would build a copy of itself. Which would then build another copy, which would build another ...

He never published his results – he may have seen them as a diversion from his main research, or he may have lacked the time or the inclination. Whatever his reasons were, it’s a pity that he didn’t put his ideas into print, because they would have constituted an important mathematical prediction about real organisms: namely, when an organism reproduces, it must employ some list of data (the tape) that has two distinct functions – to control the replication procedure, and to be copied. The discovery of the structure of DNA, and its role in the reproduction of organisms, would have verified that prediction. As it happens, Von Neumann’s work came to public attention only in 1955, just after Crick and Watson’s epic paper. And it was not until 1960 that the American mathematician and computer pioneer Arthur Burks gave the first complete proof that Von Neumann’s mathematical machine could replicate.3 So the chance to predict a basic mechanism of biological reproduction from general mathematical principles went begging.

Several people took up Von Neumann’s ideas. Conway (whom we last came across in knot theory) was among them: he invented a cellular automaton with dynamics so flexible and ‘unpredictable’ that he named it the Game of Life.

‘Life’, as it is usually known, is played with counters on a square grid. The game begins by setting up some finite configuration of counters, the initial state of the automaton. Then a short set of simple rules, involving the number of immediate neighbours of each counter, is applied to get the next configuration. These rules govern the survival, birth or death of counters. Dead counters are removed from the grid, newborn ones are added and the rest stay where they are.

The precise rules are:

• A counter with 0 or 1 neighbours dies.

• A counter with more than 3 neighbours dies.

• A counter with 2 or 3 neighbours remains alive.

• An empty space with exactly 3 neighbouring counters gives birth to a new counter.

There is a host of information about Life on the Internet, together with free software to run the game.4 Life runs on rigid rules, so the future of any given initial configuration is completely determined: if the game is run again starting from the same shape, the subsequent history is the same as before. Nevertheless, the outcome is unpredictable in the sense that there is no short cut that can predict what happens – all you can do is run the game and see what evolves. This is one of several ways in which ‘deterministic’ and ‘predictable’ differ in practice, despite being essentially the same in principle.

Despite the simplicity of the rules of Life, its behaviour can be astonishingly rich. So rich, in fact, that it is sometimes unpredictable in a very strong sense, even though the initial state completely determines everything that happens later. In 1936, Alan Turing provided a solution to the halting problem, proving that in general it is not possible to forecast ahead of time whether a computer program will terminate with an answer, or go on for ever – for example, getting stuck in a loop and repeating it indefinitely. Conway and others proved that there exists a configuration in Life that forms a universal Turing machine, a mathematical representation of a programmable computer.5 So there is no way to predict whether a given Life configuration will live for ever or die out.

In 2000, Matthew Cook found a simpler universal Turing machine, by proving a conjecture that the English polymath Stephen Wolfram, had made in 1985: a cellular automaton whose states form a line of cells, rather than square grid, can also mimic a universal Turing machine.6 This automaton is known as ‘Rule 110’. It has two states, say 0 and 1, and its rules are very simple. To find the next state of a cell, look at that cell and its two neighbours to the left and the right. If the pattern is 111, 100 or 000, that state becomes 0; otherwise, it becomes 1. It is remarkable that such a simple system of rules can in principle do anything that a computer can do – for example, calculate π to a billion decimal places. This reinforces the main message of artificial life: never underestimate the complexity of the behaviour that can result from simple rules.

When Conway invented the Game of Life, Chris Langton, who at the time was working in a hospital programming mainframe computers, found the game so interesting that he began to experiment with computer simulations of features of living creatures. Burks was running a postgraduate programme at the University of Michigan, and in 1982 Langton joined it. The outcome was a new sub-branch of science: artificial life. Some people object that this name is an exaggeration, but it should be obvious that the name is not intended to indicate the creation, by artificial means, of real life. Instead, it refers to non-biological systems that mimic, or emulate, some of the key features of living organisms, such as replication. Those features are puzzling in their own right, independently of their physical realisation, so it makes sense to study them in mathematical systems that separate the features from what they are made from.

Langton described the new field at the first conference on the topic, saying

Artificial life is the study of artificial systems that exhibit behavior characteristic of natural living systems. It is the quest to explain life in any of its possible manifestations, without restriction to the particular examples that have evolved on earth ... The ultimate goal is to extract the logical form of living systems.7

As a demonstration of what was possible, Langton had already invented the first self-replicating ‘organism’ to be implemented in a real computer. But replication is only one of the puzzling features of living organisms. Reproduction – replication with occasional errors – opens up the possibility of evolution; all that is required is a selection principle, to decide which changes to keep and which to discard.

Over the past thirty years, a seemingly endless stream of artificial life systems, defined in many different ways, has made three things abundantly clear – all of them contrary to most previous intuition:

1. Almost any rule-based system, capable of any kind of behaviour more complex than steady states or periodic cycles, is capable of very complex behaviour indeed. In rule-based systems, complex behaviour is the norm.

2. There is no significant connection between the complexity or simplicity of the rules, and the complexity or simplicity of the resulting behaviour. Complex rules can lead to behaviour that is simple, or complex. Simple rules can lead to behaviour that is simple, or complex. There is no ‘conservation of complexity’ between rules and behaviour.

3. Evolution is a remarkably powerful way to create highly complex structures and processes, without designing the desired features into the evolving entities in any explicit way.

Langton’s basic idea has been implemented in many different forms, in systems with names like Tierra, Avida and Evolve. Darwinbots, introduced in 2003 by Carlo Cormis, is typical.8 Individual organisms, the aforementioned bots, are represented on the computer screen as circles. Each bot is equipped with simulated genes, which affect its behaviour. It acquires energy by feeding, and its energy runs down as it carries out its activities. If the energy level gets too low, it dies. The bots display a rich variety of behaviour. Unlike the cells in Life and Rule 110, they can wander around all over a plane, and are not confined to specific, discrete cells.

The philosophical stance known as ‘weak alife’ holds that the only way to create a living process is through chemistry. (‘Alife’ is jargon for artificial life.) Since we already know that the standard DNA-based system that occurs naturally on Earth can be changed and will still work, it is not possible to retreat further and insist that the form of life that we know is the only kind that is possible. However, we have no solid evidence of non-chemical life, so it is reasonable to argue that all life must be chemical.

The main message of artificial life is more imaginative and more speculative. It goes back to Von Neumann, and is called ‘strong alife’. This position maintains that life is not a specific chemical process, but a general type of process that does not depend on the medium used to implement it.

If strong alife is right, what matters is not what life is made from, but what it does.

Synthetic life may sound rather similar to artificial life, but the term refers to organisms that run on conventional biochemistry but are synthesised in the laboratory from inorganic ingredients.

In 2010 a team at the J. Craig Venter Institute in Rockville, Maryland, announced the creation of an organism nicknamed Synthia. First, the team made a copy of the genome of the bacterium Mycoplasma mycoides, 1.2 million base pairs, by purely chemical techniques – no living organisms involved. (They also added some coded messages to distinguish the copy from the original, challenging other scientists to break the code. They did, fast.) Then they removed the DNA from a related bacterium and replaced it with this synthetic genome. The resulting bacterium was then able to replicate, proving that the replacement genome worked.

The achievement gained worldwide publicity as the creation of the first synthetic life form – but that is an exaggeration. It’s like overwriting part of your computer’s memory with an exact copy of the same code, typed in by hand, and claiming to have built a new computer. The manufacture of Synthia was also condemned as ‘playing God’, an equally exaggerated criticism.

Synthia is important, though not to the extent that the hype suggested. It shows that long DNA sequences can be assembled from scratch in the laboratory. It adds weight to the belief that the activity of DNA in an organism follows from the laws of chemistry, rather than some mysterious aspect of life. And it is a useful step towards the Minimal Genome Project, whose objective is to make a synthetic bacterium with the smallest genome that allows it to replicate.9 This hoped-for bacterium has been dubbed Mycoplasma laboratorium, and unlike Synthia, its genome will not be a copy of an existing natural one. But the rest of the cell’s biochemical machinery will still be taken from a pre-existing organism.

That would be like writing a new operating system and loading it into an existing computer: closer to making a brand new computer from scratch, but not there yet. So genuine synthetic life is still some way off.