492 | Jack Copeland
we do not yet understand.9 Minsky likens intelligence to the concept ‘unexplored
regions of Africa’: it disappears as soon as we discover it.
Analogy and Creativity
Turing sketches an interesting mechanical explanation of how analogy works in
the human brain, and suggests that a digital computer can be made to do the
same (p. 499). Another proposal which does not appear elsewhere in Turing’s
writings concerns the vexed issue of creativity. A machine that combines words
more or less at random and then scores the combinations ‘for various merits’
would, he says, be able to Wnd useful new concepts. His example is ‘lumping
together rain, hail, snow and sleet, under the word ‘‘precipitation’’ ’ (p. 499). He
agrees with Newman that this process would be ‘shockingly slow’. Nevertheless,
this brief suggestion of Turing’s is an economical illustration of the possibility of
a machine’s acquiring new concepts for itself.
Machine Learning
‘Can Automatic Calculating Machines Be Said to Think’ contains a lengthy
discussion of machine learning, in which Turing alludes to the P-type machines
of Chapter 10 (pp. 425–9). He says in response to JeVerson’s point that real
learning involves intervention by teachers, ‘I have made some experiments in
teaching a machine to do some simple operation, and a very great deal of such
intervention was needed’ (p. 497). Turing emphasizes several times that a
learning machine, if it is to be eVective, must not only learn Wrst-order facts
but must also be able to learn to improve its learning methods (compare the
discussion of the Mathematical Objection in Chapter 12 and elsewhere).
Newman gives a simple illustration (on p. 496) of how a computer can learn to
do better with practice. His example makes use of the idea of the computer
modifying its own programme, mentioned by Turing in Chapter 9 (p. 393).
Faced with a two-move chess problem, the machine initially follows an instruc-
tion to choose a move at random. If the move, say B-Q5, is found to lead to
forced mate in two moves, then the machine changes the instruction to ‘Try B-
Q5’. When presented with the same problem again, the machine immediately
gives the right answer.
Newman’s Test
Towards the end of the discussion Newman suggested a test—and the term
‘Newman’s test’ seems appropriate—for when machines have ‘begun to think’
9 M. Minsky, The Society of Mind (London: Pan Books, 1988), 71.
Can Automatic Calculating Machines Be Said To Think? | 493
(pp. 504–5). That stage of development has been reached, he said, when a
machine can solve a mathematical problem for which no eVective method exists.
Provenance
The discussion is taken from a BBC script (which is marked ‘Not checked in
Talks Department with ‘‘as broadcast’’ script’).10
10 The script is in the Modern Archive Centre, King’s College, Cambridge (catalogue reference B 6). It
was Wrst published in B. J. Copeland (ed.), ‘A Lecture and Two Radio Broadcasts on Machine Intelligence by
Alan Turing’, in K. Furukawa, D. Michie, and S. Muggleton (eds.), Machine Intelligence 15 (Oxford: Oxford
University Press, 1999).
Can Automatic Calculating Machines
Be Said To Think?
Braithwaite : We’re here today to discuss whether calculating machines can be
said to think in any proper sense of the word. Thinking is ordinarily regarded
as so much a speciality of man, and perhaps of other higher animals, that the
question may seem too absurd to be discussed. But, of course, it all depends
on what is to be included in thinking. The word is used to cover a multitude of
diVerent activities. What would you, JeVerson, as a physiologist, say were the
most important elements involved in thinking?
JeVerson: I don’t think that we need waste too much time on [a] deWnition of
thinking since it will be hard to get beyond phrases in common usage, such as
having ideas in the mind, cogitating, meditating, deliberating, solving prob-
lems or imagining. Philologists say that the word ‘Man’ is derived from a
Sanskrit word that means ‘to think’, probably in the sense of judging between
one idea and another. I agree that we could no longer use the word ‘thinking’
in a sense that restricted it to man. No one would deny that many animals
think, though in a very limited way. They lack insight. For example, a dog
learns that it is wrong to get on cushions or chairs with muddy paws, but he
only learns it as a venture that doesn’t pay. He has no conception of the real
reason, that he damages fabrics by doing that.
The average person would perhaps be content to deWne thinking in very
general terms such as revolving ideas in the mind, of having notions in one’s
head, of having one’s mind occupied by a problem, and so on. But it is only
right to add that our minds are occupied much of the time with trivialities.
One might say in the end that thinking was the general result of having a
suYciently complex nervous system. Very simple ones do not provide the
creature with any problems that are not answered by simple reXex mechan-
isms. Thinking then becomes all the things that go on in one’s brain, things
that often end in an action but don’t necessarily do so. I should say that it was
the sum total of what the brain of man or animal does. Turing, what do you
think about it? Have you a mechanical deWnition?
Turing : I don’t want to give a deWnition of thinking, but if I had to I should
probably be unable to say anything more about it than that it was a sort of
buzzing that went on inside my head. But I don’t really see that we need to
agree on a deWnition at all. The important thing is to try to draw a line
between the properties of a brain, or of a man, that we want to discuss, and
those that we don’t. To take an extreme case, we are not interested in the fact
Printed with the permission of the BBC, Lewis C. Braithwaite, Antony A. JeVerson, Edward Newman, and
the Estate of Alan Turing.
Can Automatic Calculating Machines Be Said To Think? | 495
that the brain has the consistency of cold porridge. We don’t want to say
‘This machine’s quite hard, so it isn’t a brain, and so it can’t think.’ I would like
to suggest a particular kind of test that one might apply to a machine. You
might call it a test to see whether the machine thinks, but it would be better to
avoid begging the question, and say that the machines that pass are (let’s say)
‘Grade A’ machines. The idea of the test is that the machine has to try and
pretend to be a man, by answering questions put to it, and it will only pass if
the pretence is reasonably convincing. A considerable proportion of a jury,
who should not be expert about machines, must be taken in by the pretence.
They aren’t allowed to see the machine itself—that would make it too easy. So
the machine is kept in a far away room and the jury are allowed to ask it
questions, which are transmitted through to it: it sends back a typewritten
answer.
Braithwaite : Would the questions have to be sums, or could I ask it what it had
had for breakfast?
Turing : Oh yes, anything. And the questions don’t really have to be questions,
any more than questions in a law court are really questions. You know the sort
of thing. ‘I put it to you that you are only pretending to be a man’ would be
quite in order. Likewise the machine would be permitted all sorts of tricks so
as to appear more man-like, such as waiting a bit before giving the answer, or
making spelling mistakes, but it can’t make smudges on the paper, any more
than one can send smudges by telegraph. We had better suppose that each jury
has to judge quite a number of times, and that sometimes they really are
dealing with a man and not a machine. That will prevent them saying ‘It must
be a machine’ every time without proper consideration.
Well, that’s my test. Of course I am not saying at present either that
machines really could pass the test, or that they couldn’t. My suggestion is
just that this is the question we should discuss. It’s not the same as ‘Do
machines think,’ but it seems near enough for our present purpose, and raises
much the same diYculties.
Newman : I should like to be there when your match between a man and a
machine takes place, and perhaps to try my hand at making up some of the
questions. But that will be a long time from now, if the machine is to stand any
chance with no questions barred?
Turing : Oh yes, at least 100 years, I should say.
JeVerson : Newman, how well would existing machines stand up to this test? What
kind of things can they do now?
Newman : Of course, their strongest line is mathematical computing, which they
were designed to do, but they would also do well at some questions that don’t
look numerical, but can easily be made so, like solving a chess problem or
looking you up a train in the time-table.
Braithwaite : Could they do that?
496 | Turing, Braithwaite, Jefferson, Newman
Newman: Yes. Both these jobs can be done by trying all the possibilities, one after
another. The whole of the information in an ordinary time-table would have
to be written in as part of the programme, and the simplest possible routine
would be one that found the trains from London to Manchester by testing
every train in the time-table to see if it calls at both places, and printing out
those that do. Of course, this is a dull, plodding method, and you could
improve on it by using a more complicated routine, but if I have understood
Turing’s test properly, you are not allowed to go behind the scenes and criticise
the method, but must abide by the scoring on correct answers, found reason-
ably quickly.
JeVerson: Yes, but all the same a man who has to look up trains frequently gets
better at it, as he learns his way about the time-table. Suppose I give a machine
the same problem again, can it learn to do better without going through the
whole rigmarole of trying everything over every time? I’d like to have your
answer to that because it’s such an important point. Can machines learn to do
better with practice?
Newman: Yes, it could. Perhaps the chess problem provides a better illustration of
this. First I should mention that all the information required in any job—the
numbers, times of trains, positions of pieces, or whatever it is, and also the
instructions saying what is to be done with them—all this material is stored in
the same way. (In the Manchester machine it is stored as a pattern on something
resembling a television screen.) As the work goes on the pattern is changed.
Usually it is the part of the pattern that contains the data that changes, while the
instructions stay Wxed. But it is just as simple to arrange that the instructions
themselves shall be changed now and then. Well, now a programme could be
composed that would cause the machine to do this: a 2-move chess problem is
recorded into the machine in some suitable coding, and whenever the machine
is started, a white move is chosen at random (there is a device for making
random choices in our machine). All the consequences of this move are now
analysed, and if it does not lead to forced mate in two moves, the machine
prints, say, ‘P-Q3, wrong move’, and stops. But the analysis shows that when the
right move is chosen the machine not only prints, say, ‘B-Q5, solution’, but it
changes the instruction calling for a random choice to one that says ‘Try B-Q5.’
The result is that whenever the machine is started again it will immediately print
out the right solution—and this without the man who made up the routine
knowing beforehand what it was. Such a routine could certainly be made now,
and I think this can fairly be called learning.
JeVerson: Yes, I suppose it is. Human beings learn by repeating the same exercises
until they have perfected them. Of course it goes further, and at the same time
—we learn generally to shift the knowledge gained about one thing to another
set of problems, seeing relevances and relationships. Learning means remem-
bering. How long can a machine store information for?
Can Automatic Calculating Machines Be Said To Think? | 497
Newman: Oh, at least as long as a man’s lifetime, if it is refreshed occasionally.
JeVerson: Another diVerence would be that in the learning process there is much
more frequent intervention by teachers, parental or otherwise, guiding the arts
of learning. You mathematicians put the programme once into the machine
and leave it to it. You wouldn’t get any distance at all with human beings if that
is what you did. In fact, the only time you do that in the learning period is at
examinations.
Turing : It’s quite true that when a child is being taught, his parents and teachers
are repeatedly intervening to stop him doing this or encourage him to do that.
But this will not be any the less so when one is trying to teach a machine. I have
made some experiments in teaching a machine to do some simple operation,
and a very great deal of such intervention was needed before I could get any
results at all. In other words the machine learnt so slowly that it needed a great
deal of teaching.
JeVerson: But who was learning, you or the machine?
Turing : Well, I suppose we both were. One will have to Wnd out how to make
machines that will learn more quickly if there is to be any real success. One
hopes too that there will be a sort of snowball eVect. The more things the
machine has learnt the easier it ought to be for it to learn others. In learning to
do any particular thing it will probably also be learning to learn more
eYciently. I am inclined to believe that when one has taught it to do certain
things one will Wnd that some other things which one had planned to teach it
are happening without any special teaching being required. This certainly
happens with an intelligent human mind, and if it doesn’t happen when one
is teaching a machine there is something lacking in the machine. What do you
think about learning possibilities, Braithwaite?
Braithwaite : No-one has mentioned what seems to me the great diYculty about
learning, since we’ve only discussed learning to solve a particular problem. But
the most important part of human learning is learning from experience—not
learning from one particular kind of experience, but being able to learn from
experience in general. A machine can easily be constructed with a feed-back
device so that the programming of the machine is controlled by the relation of
its output to some feature in its external environment—so that the working of
the machine in relation to the environment is self-corrective. But this requires
that it should be some particular feature of the environment to which the
machine has to adjust itself. The peculiarity of men and animals is that they
have the power of adjusting themselves to almost all the features. The feature
to which adjustment is made on a particular occasion is the one the man is
attending to and he attends to what he is interested in. His interests are
determined, by and large, by his appetites, desires, drives, instincts—all the
things that together make up his ‘springs of action’. If we want to construct a
machine which will vary its attention to things in its environment so that it
498 | Turing, Braithwaite, Jefferson, Newman
will sometimes adjust itself to one and sometimes to another, it would seem to
be necessary to equip the machine with something corresponding to a set of
appetites. If the machine is built to be treated only as a domestic pet, and is
spoon-fed with particular problems, it will not be able to learn in the varying
way in which human beings learn. This arises from the necessity of adapting
behaviour suitably to environment if human appetites are to be satisWed.
JeVerson: Turing, you spoke with great conWdence about what you are going to be
able to do. You make it sound as if it would be fairly easy to modify
construction so that the machine reacted more like a man. But I recollect
that from the time of Descartes and Borelli on people have said that it would
be only a matter of a few years, perhaps 3 or 4 or maybe 50, and a replica of
man would have been artiWcially created. We shall be wrong, I am sure, if we
give the impression that these things would be easy to do.
Newman: I agree that we are getting rather far away from computing machines as
they exist at present. These machines have rather restricted appetites, and they
can’t blush when they’re embarrassed, but it’s quite hard enough, and I think a
very interesting problem, to discover how near these actually existing ma-
chines can get to thinking. Even if we stick to the reasoning side of thinking, it
is a long way from solving chess problems to the invention of new mathemat-
ical concepts or making a generalisation that takes in ideas that were current
before, but had never been brought together as instances of a single general
notion.
Braithwaite : For example?
Newman: The diVerent kinds of number. There are the integers, 0, 1, À2, and so
on; there are the real numbers used in comparing lengths, for example the
pcircumference of a circle and its diameter; and the complex numbers involving
À1; and so on. It is not at all obvious that these are instances of one thing,
‘number’. The Greek mathematicians used entirely diVerent words for the
integers and the real numbers, and had no single idea to cover both. It is
really only recently that the general notion of kinds of number has been
abstracted from these instances and accurately deWned. To make this sort of
generalisation you need to have the power of recognising similarities, seeing
analogies between things that had not been put together before. It is not just a
matter of testing things for a speciWed property and classifying them accord-
ingly. The concept itself has to be framed, something has to be created, say the
idea of a number-Weld. Can we even guess at the way a machine could make
such an invention from a programme composed by a man who had not the
concept in his own mind?
Turing : It seems to me, Newman, that what you said about ‘trying out possibil-
ities’ as a method applies to quite an extent, even when a machine is required
to do something as advanced as Wnding a useful new concept. I wouldn’t like
to have to deWne the meaning of the word ‘concept’, nor to give rules for rating
Can Automatic Calculating Machines Be Said To Think? | 499
their usefulness, but whatever they are they’ve got outward and visible forms,
which are words and combinations of words. A machine could make up such
combinations of words more or less at random, and then give them marks for
various merits.
Newman: Wouldn’t that take a prohibitively long time?
Turing : It would certainly be shockingly slow, but it could start on easy things,
such as lumping together rain, hail, snow and sleet, under the word ‘precipi-
tation.’ Perhaps it might do more diYcult things later on if it was learning all
the time how to improve its methods.
Braithwaite : I don’t think there’s much diYculty about seeing analogies that can
be formally analysed and explicitly stated. It is then only a question of
designing the machine so that it can recognise similarities of mathematical
structure. The diYculty arises if the analogy is a vague one about which little
more can be said than that one has a feeling that there is some sort of
similarity between two cases but one hasn’t any idea as to the respect in
which the two cases are similar. A machine can’t recognise similarities when
there is nothing in its programme to say what are the similarities it is expected
to recognise.
Turing : I think you could make a machine spot an analogy, in fact it’s quite a
good instance of how a machine could be made to do some of those things
that one usually regards as essentially a human monopoly. Suppose that
someone was trying to explain the double negative to me, for instance, that
when something isn’t not green it must be green, and he couldn’t quite get it
across. He might say ‘Well, it’s like crossing the road. You cross it, and then
you cross it again, and you’re back where you started.’ This remark might just
clinch it. This is one of the things one would like to work with machines, and I
think it would be likely to happen with them. I imagine that the way analogy
works in our brains is something like this. When two or more sets of ideas
have the same pattern of logical connections, the brain may very likely
economise parts by using some of them twice over, to remember the logical
connections both in the one case and in the other. One must suppose that
some part of my brain was used twice over in this way, once for the idea of
double negation and once for crossing the road, there and back. I am really
supposed to know about both these things but can’t get what it is the man is
driving at, so long as he is talking about all those dreary nots and not-nots.
Somehow it doesn’t get through to the right part of the brain. But as soon as
he says his piece about crossing the road it gets through to the right part, but
by a diVerent route. If there is some such purely mechanical explanation of
how this argument by analogy goes on in the brain, one could make a digital
computer do the same.
JeVerson: Well, there isn’t a mechanical explanation in terms of cells and con-
necting Wbres in the brain.
500 | Turing, Braithwaite, Jefferson, Newman
Braithwaite : But could a machine really do this? How would it do it?
Turing : I’ve certainly left a great deal to the imagination. If I had given a longer
explanation I might have made it seem more certain that what I was describing
was feasible, but you would probably feel rather uneasy about it all, and you’d
probably exclaim impatiently, ‘Well, yes, I see that a machine could do all that,
but I wouldn’t call it thinking.’ As soon as one can see the cause and eVect
working themselves out in the brain, one regards it as not being thinking, but a
sort of unimaginative donkey-work. From this point of view one might be
tempted to deWne thinking as consisting of ‘those mental processes that we
don’t understand’. If this is right then to make a thinking machine is to make
one which does interesting things without our really understanding quite how
it is done.
JeVerson: If you mean that we don’t know the wiring in men, as it were, that is
quite true.
Turing : No, that isn’t at all what I mean. We know the wiring of our machine, but
it already happens there in a limited sort of way. Sometimes a computing
machine does do something rather weird that we hadn’t expected. In principle
one could have predicted it, but in practice it’s usually too much trouble.
Obviously if one were to predict everything a computer was going to do one
might just as well do without it.
Newman: It is quite true that people are disappointed when they discover what the
big computing machines actually do, which is just to add and multiply, and use
the results to decide what further additions and multiplications to do. ‘That’s
not thinking’, is the natural comment, but this is rather begging the question. If
you go into one of the ancient churches in Ravenna you see some most beautiful
pictures round the walls, but if you peer at them through binoculars you might
say, ‘Why, they aren’t really pictures at all, but just a lot of little coloured stones
with cement in between.’ The machine’s processes are mosaics of very simple
standard parts, but the designs can be of great complexity, and it is not obvious
where the limit is to the patterns of thought they could imitate.
Braithwaite: But how many stones are there in your mosaic? JeVerson, is there a
suYcient multiplicity of the cells in the brain for them to behave like a
computing machine?
JeVerson: Yes, there are thousands, tens of thousands more cells in the brain than
there are in a computing machine, because the present machine contains—
how many did you say?
Turing: Half a million digits. I think we can assume that is the equivalent of half a
million nerve cells.
Braithwaite: If the brain works like a computing machine then the present
computing machine cannot do all the things the brain does. Agreed; but if a
computing machine were made that could do all the things the brain does,
wouldn’t it require more digits than there is room for in the brain?
Can Automatic Calculating Machines Be Said To Think? | 501
JeVerson: Well, I don’t know. Suppose that it is right to equate digits in a machine
with nerve cells in a brain. There are various estimates, somewhere between
ten thousand million and Wfteen thousand million cells are supposed to be
there. Nobody knows for certain, you see. It is a colossal number. You would
need 20,000 or more of your machines to equate digits with nerve cells. But it
is not, surely, just a question of size. There would be too much logic in your
huge machine. It wouldn’t be really like a human output of thought. To make
it more like, a lot of the machine parts would have to be designed quite
diVerently to give greater Xexibility and more diverse possibilities of use. It’s a
very tall order indeed.
Turing : It really is the size that matters in this case. It is the amount of infor-
mation that can be stored up. If you think of something very complicated that
you want one of these machines to do, you may Wnd the particular machine
you have got won’t do, but if any machine can do it at all, then it can be done
by your Wrst computer, simply increased in its storage capacity.
JeVerson: If we are really to get near to anything that can be truly called ‘thinking’
the eVects of external stimuli cannot be missed out; the intervention of all
sorts of extraneous factors, like the worries of having to make one’s living, or
pay one’s taxes, or get food that one likes. These are not in any sense minor
factors, they are very important indeed, and worries concerned with them may
greatly interfere with good thinking, especially with creative thinking. You see
a machine has no environment, and man is in constant relation to his
environment, which, as it were punches him whilst he punches back. There
is a vast background of memories in a man’s brain that each new idea or
experience has to Wt in with. I wonder if you could tell me how far a calculating
machine meets that situation. Most people agree that man’s Wrst reaction to a
new idea (such as the one we are discussing today) is one of rejection, often
immediate and horriWed denial of it. I don’t see how a machine could as it
were say ‘Now Professor Newman or Mr. Turing, I don’t like this programme
at all that you’ve just put into me, in fact I’m not going to have anything to do
with it.’
Newman: One diYculty about answering that is one that Turing has already
mentioned. If someone says, ‘Could a machine do this, e.g. could it say ‘‘I
don’t like the programme you have just put into me’’ ’, and a programme for
doing that very thing is duly produced, it is apt to have an artiWcial and ad hoc
air, and appear to be more of a trick than a serious answer to the question. It is
like those passages in the Bible, which worried me as a small boy, that say that
such and such was done ‘that the prophecy might be fulWlled which says’ so
and so. This always seemed to me a most unfair way of making sure that the
prophecy came true. If I answer your question, JeVerson, by making a routine
which simply caused the machine to say just the words ‘Newman and Turing, I
don’t like your programme’, you would certainly feel this was a rather childish
502 | Turing, Braithwaite, Jefferson, Newman
trick, and not the answer to what you really wanted to know. But yet it’s hard
to pin down what you want.
JeVerson: I want the machine to reject the problem because it oVends it in some
way. That leads me to enquire what the ingredients are of ideas that we reject
because we instinctively don’t care for them. I don’t know why I like some
pictures and some music and am bored by other sorts. But I’m not going to
carry that line on because we are all diVerent, our dislikes are based on our
personal histories and probably too on small diVerences of construction in all
of us, I mean by heredity. Your machines have no genes, no pedigrees.
Mendelian inheritance means nothing to wireless valves. But I don’t want to
score debating points! We ought to make it clear that not even Turing thinks
that all that he has to do is to put a skin on the machine and that it is alive!
We’ve been trying for a more limited objective whether the sort of thing that
machines do can be considered as thinking. But is not your machine more
certain than any human being of getting its problem right at once, and
infallibly?
Newman: Oh!
Turing : Computing machines aren’t really infallible at all. Making up checks on
their accuracy is quite an important part of the art of using them. Besides
making mistakes they sometimes haven’t done quite the calculation one had
expected, and one gets something that might be called a ‘misunderstanding’.
JeVerson: At any rate, they are not inXuenced by the emotions. You have only to
upset a person enough and he becomes confused, he can’t think of the answers
and may make a fool of himself. It is high emotional content of mental
processes in the human being that makes him quite diVerent from a machine.
It seems to me to come from the great complexity of his nervous system with
its 1010 cells and also from his endocrine system which imports all sorts of
emotions and instincts, such as those to do with sex. Man is essentially a
chemical machine, he is much aVected by hunger and fatigue, by being ‘out of
sorts’ as we say, also by innate judgements, and by sexual urges. This chemical
side is tremendously important, not the least so because the brain does
exercise a remote control over the most important chemical processes that
go on in our bodies. Your machines don’t have to bother with that, with being
tired or cold or happy or satisWed. They show no delight at having done
something never done before. No, they are ‘mentally’ simple things. I mean
that however complicated their structure is (and I know it is very compli-
cated), compared with man they are very simple and perform their tasks with
an absence of distracting thoughts which is quite inhuman.
Braithwaite: I’m not sure that I agree. I believe that it will be necessary to provide
the machine with something corresponding to appetites, or other ‘springs of
action’, in order that it will pay enough attention to relevant features in its
environment to be able to learn from experience. Many psychologists have
Can Automatic Calculating Machines Be Said To Think? | 503
held that the emotions in men are by-products of their appetites and that they
serve a biological function in calling higher levels of mental activity into play
when the lower levels are incapable of coping with an external situation. For
example, one does not feel afraid when there is no danger, or a danger which
can be avoided more or less automatically: fear is a symptom showing that the
danger has to be met by conscious thought. Perhaps it will be impossible to
build a machine capable of learning in general from experience without
incorporating in it an emotional apparatus, the function of which will be to
switch over to a diVerent part of the machine when the external environment
diVers too much from what would satisfy the machine’s appetites by more
than a certain amount. I don’t want to suggest that it will be necessary for the
machine to be able to throw a Wt of tantrums. But in humans tantrums
frequently fulWl a deWnite function—that of escaping from responsibility;
and to protect a machine against a too hostile environment it may be essential
to allow it, as it were, to go to bed with a neurosis, or psychogenic illness—just
as, in a simpler way, it is provided with a fuse to blow, if the electric power
working it threatens its continued existence.
Turing : Well, I don’t envisage teaching the machine to throw temperamental
scenes. I think some such eVects are likely to occur as a sort of by-product of
genuine teaching, and that one will be more interested in curbing such
displays than in encouraging them. Such eVects would probably be distinctly
diVerent from the corresponding human ones, but recognisable as variations
on them. This means that if the machine was being put through one of my
imitation tests, it would have to do quite a bit of acting, but if one was
comparing it with a man in a less strict sort of way the resemblance might
be quite impressive.
Newman: I still feel that too much of our argument is about what hypothetical
future machines will do. It is all very well to say that a machine could easily be
made to do this or that, but, to take only one practical point, what about the
time it would take to do it? It would only take an hour or two to make up a
routine to make our Manchester machine analyse all possible variations of the
game of chess right out, and Wnd the best move that way—if you didn’t mind
its taking thousands of millions of years to run through the routine. Solving a
problem on the machine doesn’t mean Wnding a way to do it between now and
eternity, but within a reasonable time. This is not just a technical detail that
will be taken care of by future improvements. It’s most unlikely that the
engineers can ever give us a factor of more than a thousand or two times
our present speeds. To assume that runs that would take thousands of millions
of years on our present machines will be done in a Xash on machines of the
future, is to move into the realms of science Wction.
Turing : To my mind this time factor is the one question which will involve all the
real technical diYculty. If one didn’t know already that these things can be
504 | Turing, Braithwaite, Jefferson, Newman
done by brains within a reasonable time one might think it hopeless to try
with a machine. The fact that a brain can do it seems to suggest that the
diYculties may not really be so bad as they now seem.
Braithwaite: I agree that we ought not to extend our discussion to cover whether
calculating machines could be made which would do everything that a man
can do. The point is, surely, whether they can do all that it is proper to call
thinking. Appreciation of a picture contains elements of thinking, but it also
contains elements of feeling; and we’re not concerned with whether a machine
can be made that will feel. Similarly with moral questions: we’re only con-
cerned with them so far as they are also intellectual ones. We haven’t got to
give the machine a sense of duty or anything corresponding to a will: still less
need it be given temptations which it would then have to have an apparatus
for resisting. All that it has got to do in order to think is to be able to solve,
or to make a good attempt at solving, all the intellectual problems with which
it might be confronted by the environment in which it Wnds itself. This
environment, of course, must include Turing asking it awkward questions as
well as natural events such as being rained upon, or being shaken up by an
earthquake.
Newman: But I thought it was you who said that a machine wouldn’t be able to
learn to adjust to its environment if it hadn’t been provided with a set of
appetites and all that went with them?
Braithwaite : Yes, certainly. But the problems raised by a machine having appetites
are not properly our concern today. It may be the case that it wouldn’t be able to
learn from experience without them; but we’re only required to consider
whether it would be able to learn at all—since I agree that being able to learn
is an essential part of thinking. So oughtn’t we to get back to something centred
on thinking? Can a machine make up new concepts, for example?
Newman: There are really two questions that can be asked about machines and
thinking, Wrst, what do we require before we agree that the machine does
everything that we call thinking? This is really what we have been talking about
for most of the time; but there is also another interesting and important
question: Where does the doubtful territory begin? What is the nearest thing
to straight computing that the present machines perhaps can’t do?
Braithwaite: And what would your own answer be?
Newman: I think perhaps to solve mathematical problems for which no method
is known, in the way that men do; to Wnd new methods. This is a much more
modest aim than inventing new mathematical concepts. What happens when
you try to solve a new problem in the ordinary way is that you think about it
for a few seconds, or a few years, trying out all the analogies you can think of
with problems that have been solved, and then you have an idea. You try it out
in detail. If it is no good you must wait for another idea. This is a little like the
chess-problem routine, where one move after another is tried, but with one
Can Automatic Calculating Machines Be Said To Think? | 505
very important diVerence, that if I am even a moderately good mathematician
the ideas that I get are not just random ones, but are pre-selected so that there
is an appreciable chance that after a few trials one of them will be successful.
Henry Moore says about the studies he does for his sculpture, ‘When the work
is more than an exercise, inexplicable jumps occur. This is where the imagin-
ation comes in.’ If a machine could really be got to imitate this sudden pounce
on an idea, I believe that everyone would agree that it had begun to think, even
though it didn’t have appetites or worry about the income tax. And suppose
that we also stuck to what we know about the physiology of human thinking,
how much would that amount to, JeVerson?
JeVerson: We know a great deal about the end-product, thinking itself. Are not
the contents of our libraries and museums the total up to date? Experimental
psychology has taught us a lot about the way that we use memory and
association of ideas, how we Wll in gaps in knowledge and improvise from a
few given facts. But exactly how we do it in terms of nerve cell actions we don’t
know. We are particularly ignorant of the very point that you mentioned just
now, Newman, the actual physiology of the pounce on an idea, of the sudden
inspiration. Thinking is clearly a motor activity of the brain’s cells, a sugges-
tion supported by the common experience that so many people think better
with a pen in their hand than viva voce or by reverie and reXection. But you
can’t so far produce ideas in a man’s mind by stimulating his exposed brain
here or there electrically. It would have been really exciting if one could have
done that—if one could have perhaps excited original thoughts by local
stimulation. It can’t be done. Nor does the electro-encephalograph show us
how the process of thinking is carried out. It can’t tell you what a man is
thinking about. We can trace the course, say, of a page of print or of a stream
of words into the brain, but we eventually lose them. If we could follow them
to their storage places we still couldn’t see how they are reassembled later as
ideas. You have the great advantage of knowing how your machine was made.
We only know that we have in the human nervous system a concern compact
in size and in its way perfect for its job. We know a great deal about its
microscopical structure and its connections. If fact, we know everything except
how these myriads of cells allow us to think. But, Newman, before we say ‘not
only does this machine think but also here in this machine we have an exact
counterpart of the wiring and circuits of human nervous systems,’ I ought to
ask whether machines have been built or could be built which are as it were
anatomically diVerent, and yet produce the same work.
Newman: The logical plan of all of them is rather similar, but certainly their
anatomy, and I suppose you could say their physiology, varies a lot.
JeVerson: Yes, that’s what I imagined—we cannot then assume that any one of
these electronic machines is a replica of part of a man’s brain even though the
result of its actions has to be conceded as thought. The real value of the
506 | Turing, Braithwaite, Jefferson, Newman
machine to you is its end results, its performance, rather than that its plan
reveals to us a model of our brains and nerves. Its usefulness lies in the fact
that electricity travels along wires 2 or 3 million times faster than nerve
impulses pass along nerves. You can set it to do things that man would need
thousands of lives to complete. But that old slow coach, man, is the one with
the ideas—or so I think. It would be fun some day, Turing, to listen to a
discussion, say on the Fourth Programme, between two machines on why
human beings think that they think!
Artificial Life
Jack Copeland
1. What is ‘Artificial Life’?
The highly interdisciplinary Weld of ArtiWcial Life was so named by Christopher
Langton, a physicist working at the Los Alamos National Laboratory.1 In 1987
Langton organized at Los Alamos what he described as an ‘interdisciplinary
workshop on the synthesis and simulation of living systems’.2 The workshop
was a rallying point, bringing together researchers with shared interests and
diverse backgrounds.
ArtiWcial Life (‘A-Life’) aims to achieve theoretical understanding of naturally
occurring biological life, in particular of the most conspicuous feature of living
matter, its ability to self-organize, i.e. to develop form and structure spontan-
eously. Langton deWnes ArtiWcial Life as ‘the study of man-made systems that
exhibit behaviors characteristic of natural living systems’.3 A-Life, he says, ‘com-
plements the traditional biological sciences concerned with the analysis of living
organisms by attempting to synthesise life-like behaviors within computers and
other artiWcial media’.4
The use of computers to simulate living and life-like systems is central to
A-Life. Langton says:
Computers should be thought of as an important laboratory tool for the study of life,
substituting for the array of incubators, culture dishes, microscopes, electrophoretic gels,
pipettes, centrifuges, and other assorted wet-lab paraphernalia, one simple-to-master
piece of experimental equipment.5
Langton even suggests that the ‘ultimate goal of the study of artiWcial life
would be to create ‘‘life’’ in some other medium, ideally a virtual medium where
the essence of life has been abstracted from the details of its implementation in
any particular hardware’.6
1 C. G. Langton, ‘Studying ArtiWcial Life with Cellular Automata’, Physica D, 22 (1986), 120–49.
2 C. G. Langton (ed.), ArtiWcial Life: The Proceedings of an Interdisciplinary Workshop on the Synthesis and
Simulation of Living Systems (Redwood City, Calif.: Addison-Wesley, 1989).
3 C. G. Langton, ‘ArtiWcial Life’, in Langton, ArtiWcial Life, 1.
4 Ibid.
5 Ibid. 32.
6 Langton, ‘Studying ArtiWcial Life with Cellular Automata’, 147.
508 | Jack Copeland
2. Morphogenesis
Turing was the earliest pioneer of computer-based A-Life: he was the Wrst to use
computer simulation to investigate a theory of the development of organization
and pattern in living things.7 Early in 1951, the world’s Wrst commercially
manufactured general-purpose electronic digital computer, the Manchester-
built Ferranti Mark I, was installed in the Computing Machine Laboratory at
Manchester University.8 Turing immediately set about using the machine to
model biological growth. In February 1951 he wrote to a colleague at the
National Physical Laboratory:
Our new machine [the Ferranti Mark I] is to start arriving on Monday. I am hoping as one
of the Wrst jobs to do something about ‘chemical embryology’. In particular I think one
can account for the appearance of Fibonacci numbers in connection with Wr-cones.9
In a Fibonacci number-series, each number, except for the Wrst two, is the sum of
the two previous numbers: for example 1, 3, 4, 7, 11 . . .
In brief, Turing’s ‘chemical embryology’ is the hypothesis that the develop-
ment of anatomical structure in the animal or plant embryo is a result of the fact
that diVusing chemicals reacting with one another can form spatial patterns. This
is a thoroughly reductionist view: forms found in living matter are accounted for
by the fact that, under appropriate conditions, sheer chemical reaction produces
pattern and form.
Turing took his cue from the zoologist D’Arcy Thompson (to whose work
Turing refers at the end of Chapter 15). D’Arcy Thompson held that the forms of
living things, no less than naturally occurring forms in inorganic matter, are to
be explained in terms of ‘the operation of physical forces or mathematical laws’.10
Cell and tissue, shell and bone, leaf and Xower, are so many portions of matter, and it is in
obedience to the laws of physics that their particles have been moved, moulded and
conformed. . . . The form . . . of any portion of matter, whether it be living or dead, and
7 Another early pioneer of A-Life was the neurophysiologist W. Grey Walter (a founder of the Ratio
Club, of which Turing was also a member). His famous ‘tortoises’—built from about 1949 onwards—were
mobile, battery-powered devices exhibiting life-like properties. He constructed them in order to show that
seemingly complex behaviour can result from simple mechanisms (W. Grey Walter, The Living Brain
(London: Gerald Duckworth, 1953), 82–7 and appendix B). Although digital computers were not involved,
Grey Walter’s work certainly conforms to Langton’s description given above: ‘ArtiWcial Life . . . attempt[s] to
synthesise life-like behaviors within computers and other artiWcial media’ (italics added).
8 A digital facsimile of Turing’s Programmers’ Handbook for Manchester Electronic Computer (University
of Manchester Computing Machine Laboratory, 1950) is in The Turing Archive for the History of
Computing <www.AlanTuring.net/programmers_handbook>.
9 Letter from Turing to Michael Woodger, undated, received 12 Feb. 1951 (in the Woodger Papers,
National Museum of Science and Industry, Kensington, London; a digital facsimile is in the Turing Archive
for the History of Computing <www.AlanTuring.net/turing_woodger_feb51>).
10 W. D’Arcy Thompson, On Growth and Form (2nd edn. Cambridge: Cambridge University Press,
1942), 3.
Artificial Life | 509
the changes of form which are apparent in its . . . growth, may in all cases alike be described
as due to the action of force.11
Concerning the puzzling fact that the scales of a Wr-cone, or the Xorets of a
sunXower, are grouped together in numbers that form a Fibonacci series, D’Arcy
Thompson boldly declared:
while the Fibonacci series stares us in the face in the Wr-cone, it does so for mathematical
reasons; and its supposed usefulness, and the hypothesis of its introduction into plant-
structure through natural selection, are matters which deserve no place in the plain study
of botanical phenomena.12
Turing summarized his own theory as the suggestion that ‘certain well-known
physical laws are suYcient to account for many of the facts’ of morphogenesis
(p. 519). (‘Morphogenesis’ means ‘generation of form’.) Turing described an
idealized chemical mechanism, now called the reaction-diVusion model. He
showed that this mechanism could lead to a number of simple but life-like
patterns and forms. The reaction-diVusion model is the topic of Chapter 15,
‘The Chemical Basis of Morphogenesis’. ‘The Chemical Basis of Morphogenesis’
has been widely cited in the biological literature and today reaction-diVusion
remains a possible, although still unconWrmed, explanation of aspects of the
generation of biological pattern and form.
The geneticist C. H. Waddington commented in a letter to Turing in 1952 that
the most clear-cut application of Turing’s theory appeared to be ‘in the arising of
spots, streaks, and Xecks of various kinds in apparently uniform areas such as the
wings of butterXies, the shells of molluscs, the skin of tigers, leopards, etc’.13
Modern computer simulations of Turing’s reaction-diVusion mechanism have
indeed produced leopard-like spots, cheetah-like spots, and giraVe-like stripes, as
well as textures reminiscent of reptile skin, corals, and the surfaces of some
fungi.14 Turing himself, however, envisaged diverse applications of his theory
(as he indicated in the letter to biologist J. Z. Young quoted below), includ-
ing leaf arrangements and the appearance of Fibonacci sequences, and
phenomena such as gastrulation. (Gastrulation is a process of rearrange-
ment of cells in the spherical embryo, involving the folding inwards of part
of the surface, producing a depression akin to the dent formed by poking a
balloon with a Wnger.) Furthermore, Turing described his research into
11 Ibid. 10, 16.
12 Ibid. 933.
13 Letter from Waddington to Turing (11 Sept. 1952). The letter is among the Turing Papers in the
Modern Archive Centre, King’s College, Cambridge (catalogue reference D 5).
14 J. D. Murray, ‘A Pre-pattern Formation Mechanism for Animal Coat Markings’, Journal of Theoretical
Biology, 88 (1981), 161–99; G. Turk, ‘Generating Textures on Arbitrary Surfaces Using Reaction-DiVusion’,
Computer Graphics, 25 (1991), 289–98; A. Witkin and M. Kass, ‘Reaction-DiVusion Textures’, Computer
Graphics, 25 (1991), 299–308.
510 | Jack Copeland
morphogenesis as ‘not altogether unconnected’ to his work on neural net-
works (see Chapter 10).
Turing simulated the reaction-diVusion mechanism using the Ferranti com-
puter. A mathematician using paper and pencil to analyse the behaviour of a
reaction-diVusion system risks becoming overwhelmed by formidable math-
ematical complexity, and must make (what Turing calls) ‘simplifying assump-
tions’. Turing explains that use of the computer enables such assumptions to be
dispensed with to some extent. This freedom enables Turing to employ non-
linear diVerential equations to describe the chemical interactions hypothesized
by his theory.15 Non-linear diVerential equations are mathematically intractable.
Turing used the computer to explore in detail particular cases of interactions
governed by equations of this type. He may well have been the Wrst researcher to
engage in the computer-assisted exploration of non-linear systems.16 (It was not
until Benoit Mandelbrot’s discovery of the ‘Mandelbrot set’ in 1979 that the
computer-assisted investigation of non-linear systems gained widespead
attention.17)
Turing’s work on morphogenesis was in every respect ahead of its time. He
died while in the midst of this groundbreaking work, leaving a large pile of
handwritten notes and various programmes.18 This material is still not fully
understood.
The computer programme shown in Figure 1, which is in Turing’s own
hand, formed part of his study of the development of the Wr-cone. He also
investigated the development of the sunXower. The photograph and diagram
shown in Figure 2 are from his notes.19
3. The Reaction-Diffusion Model
In reaction-diVusion, two or more chemicals diVuse through the embryo reacting
with each other. Turing showed that under certain conditions a stable pattern
of chemical concentrations will be reached. For example, in the case of an artiW-
cially simple embryo consisting of nothing but a ring of twenty cells, reaction-
diVusion will produce a stable, regular pattern of concentration and rarefaction
around the circumference of the ring. The points of highest concentration
15 See p. 561 of Chapter 15; also section 2 of Turing’s ‘Morphogen Theory of Phyllotaxis’, in Morphogen-
esis: Collected Works of A. M. Turing, ed. P. T. Saunders (Amsterdam: North-Holland, 1992).
16 Unless perhaps the computer-assisted modelling of non-linear systems was Wrst undertaken a little
earlier, in secret, by members of the Los Alamos group in connection with nuclear explosions.
17 B. B. Mandelbrot, The Fractal Geometry of Nature (New York: Freeman, 1977; revised and expanded
1982, 1983). For a popular exposition see J. Gleick, Chaos: Making a New Science (London: Cardinal, 1988).
18 Some—but by no means all—of this material appears in Morphogenesis, ed. Saunders.
19 The notes and programme sheets are among the Turing Papers in the Modern Archive Centre, King’s
College, Cambridge (catalogue references C 25, C 27).
Artificial Life | 511
Figure 1.
occur equidistantly from one another. Turing picturesquely describes this pattern
as a stationary ‘chemical wave’. His suggestion is that at the points of high
concentration around the ring, the chemical acts as a trigger to stimulate growth.
For example, a ring of cells might sprout leaves at these points, or tentacles,
producing a structure reminiscent of Hydra (‘something like a sea-anenome but
liv[ing] in fresh water and hav[ing] from about Wve to ten tentacles’ (p. 556)).
Regular but non-stationary patterns of concentration—travelling waves—are
also possible. Turing suggested that the movements of the tail of a spermatozoon
may provide an example of these travelling waves.
512 | Jack Copeland
It might be wondered how genes Wt into this picture. Given our current
knowledge of the role played by genes in the determination of anatomical
structure, is Turing’s theory archaic? Not at all. On Turing’s account, reaction-
diVusion is the mechanism by which genes determine the anatomical structure of
the resulting organism. The function of genes, he suggested, is catalytic. The
genes are presumed to catalyse the production of appropriate chemicals, so
setting reaction-diVusion in train.
Turing calls the chemicals that diVuse and react ‘morphogens’, suggesting
hormones as an example. However, his aim is not to give a taxonomy of
morphogens or to describe speciWc morphogens, but rather to demonstrate in
the abstract that, given certain realistic assumptions about unspeciWed morpho-
gens, reaction-diVusion will produce pattern. These assumptions include the
rates at which the morphogens diVuse between the cells, the rates at which the
various chemical reactions between the morphogens take place, and the ways in
which these rates change (due, for example, to increases in temperature in the
tissue, or increases in concentration of morphogens that act as catalysts). The
initial concentrations of the morphogens must be speciWed, and the number,
dimensions, and positions of the cells making up the mass of tissue through
which the morphogens diVuse. This mass—the embryo—is assumed to be
initially homogeneous. (Other features of the situation, such as the motions
and elasticities of the cells, are ignored in order to simplify the model.)
Turing represents the reactions between morphogens purely schematically,
making no assumptions concerning the actual chemical compositions of the
substances involved. For example, two reactions might be speciWed like this:
morphogens X and Y react to produce Z; Z and morphogen A react to produce
Figure 2. Turing’s numbering of the individual florets of a sunflower.
20 Editor’s note. A blastula is a hollow sphere of cells, one cell in thickness.
Artificial Life | 513
2Y. The Wrst reaction therefore depletes the supply of morphogen Y, while the
second tends to build up the concentration of Y.
Uniform diVusion and uniform reaction in a uniform mass of tissue can
produce only uniformity, not diVerentiation and pattern. Some sort of sym-
metry-breaker must be thrown into the mix. As Turing put it (p. 525):
There appears superWcially to be a diYculty confronting this theory of morphogenesis,
or, indeed, almost any other theory of it. An embryo in its spherical blastula20 stage
has spherical symmetry, or if there are any deviations from perfect symmetry, they cannot
be regarded as of any particular importance, for the deviations vary greatly from embryo
to embryo within a species, though the organisms developed from them are barely
distinguishable. One may take it therefore that there is perfect spherical symmetry. But
a system which has spherical symmetry, and whose state is changing because of chemical
reactions and diVusion, will remain spherically symmetrical forever. . . . It certainly
cannot result in an organism such as a horse, which is not spherically symmetrical.
Turing suggested various possible symmetry-breakers, including small disturb-
ances caused by the presence of anatomical structures neighbouring the embryo,
and purely statistical Xuctuations at the molecular level. An example of the latter is
statistical Xuctuation in the number of molecules of a given morphogen passing
through the wall of a cell, so producing small Xuctuations in the concentration of
that morphogen within the cell. Turing’s point is that in certain circumstances,
small Xuctuations such as this can bring about large eVects, just as a small nudge
that under normal circumstances would have no eVect on a person’s balance could
be enough to topple someone balancing on one foot.
Turing demonstrated that under appropriate conditions small departures
from uniformity can indeed lead to the formation of chemical waves.
4. Genetic Algorithms
An important concept both in ArtiWcial Life and in ArtiWcial Intelligence is that
of a genetic algorithm (GA). GAs employ methods analogous to the processes of
natural evolution in order to produce successive generations of software entities
that are increasingly Wt for their intended purpose. Turing anticipated the
concept of a genetic algorithm in a brief passage of his ‘Intelligent Machinery’,
where he described what he called a ‘genetical or evolutionary search’ (Chapter
10, p. 431; see also Chapter 11, p. 463, and the introduction to Chapter 16,
p. 565). The actual term ‘genetic algorithm’ was introduced circa 1975 by John
Holland and his research group at the University of Michigan.21 Holland’s work
is responsible for the current intense interest in GAs. (Holland, a student of
Arthur Burks, was inXuenced by von Neumann’s ideas—see below.)
21 J. H. Holland, Adaptation in Natural and ArtiWcial Systems (Cambridge, Mass.: MIT Press, 1992), p. x.
514 | Jack Copeland
Turing described an early example of a GA in connection with his chess-player
in Chapter 16.
One of the Wrst GAs to be implemented (in the 1950s) formed part of the
learning mechanism of Samuel’s checkers or draughts programme mentioned
above in ‘ArtiWcial Intelligence’.22 Samuel’s programme used heuristics to rank
moves and board positions (the programme ‘looked ahead’ as many as ten turns
of play). To speed up learning, Samuel would set up two copies of the pro-
gramme, Alpha and Beta, on the same computer and leave them to play game
after game with each other. The learning procedure consisted in the computer
making small numerical changes to Alpha’s ranking procedure, leaving Beta’s
unchanged, and then comparing Alpha’s and Beta’s performance over a few
games. If Alpha played worse than Beta, these changes to the ranking procedure
were discarded, but if Alpha played better than Beta then Beta’s ranking proced-
ure was replaced with Alpha’s. As in biological evolution, the Wtter survived. Over
many such cycles of mutation and selection, the programme’s quality of play
increased markedly.
The use of GAs is burgeoning, in AI and elsewhere. In one application a GA-
based system and a witness to a crime cooperate to generate on-screen faces that
become closer and closer to the recollected face of the criminal.23 In A-Life,
researchers study GAs as a means of studying the process of evolution itself.
5. John von Neumann and A-Life
John von Neumann was another important early pioneer of ArtiWcial Life. In his
1948 Hixon Symposium, entitled ‘The General and Logical Theory of Automata’,
he said:
Natural organisms are, as a rule, much more complicated and subtle, and therefore much
less well understood in detail, than are artiWcial automata. Nevertheless . . . a good deal of
our experiences and diYculties with our artiWcial automata can be to some extent
projected on our interpretations of natural organisms.24
Arthur Burks (who edited and completed von Neumann’s posthumously
published volume Theory of Self-Reproducing Automata, listed in the section of
Further Reading) wrote this concerning von Neumann’s research on the problem
of self-reproduction:
22 A. L. Samuel, ‘Some Studies in Machine Learning Using the Game of Checkers’, IBM Journal of
Research and Development, 3 (1959), 211–29; reprinted in E. A. Feigenbaum and J. Feldman (eds.),
Computers and Thought (New York: McGraw-Hill, 1963).
23 D. E. Goldberg, ‘Genetic and Evolutionary Algorithms Come of Age’, Communications of the Associ-
ation for Computing Machinery, 37 (1994), 113–19.
24 J. von Neumann, ‘The General and Logical Theory of Automata’, in vol. v of von Neumann’s Collected
Works, ed. A. H. Taub (Oxford: Pergamon Press, 1963), 288–9.
Artificial Life | 515
Von Neumann had the familiar natural phenomenon of self-reproduction in mind . . . but
he was not trying to simulate the self-reproduction of a natural system at the levels of
genetics and biochemistry. He wished to abstract from the natural self-reproduction
problem its logical form.25
This passage is quoted approvingly by Langton, who italicizes the Wnal sentence
and comments:
This approach is the Wrst to capture the essence of ArtiWcial Life. To understand the Weld
of ArtiWcial Life, one need only replace references to ‘self-reproduction’ in the above with
references to any other biological phenomenon.26
Von Neumann was thinking about issues relevant to A-Life at least as early as
1946. In a letter to the cyberneticist Norbert Wiener (dated 29 November 1946),
von Neumann wrote: ‘I did think a good deal about self-reproductive mechan-
isms. I can formulate the problem rigorously, in about the style in which Turing
did it for his mechanisms.’27 There is no doubt that von Neumann’s theorizing
about self-reproduction was strongly inXuenced by Turing’s discovery of the
universal computing machine (or ‘universal automaton’, as von Neumann called
it). Von Neumann’s colleague Herman Goldstine wrote:
von Neumann had a profound concern for automata. In particular, he always had a deep
interest in Turing’s work. . . . Turing proved a most remarkable and unexpected result. . . .
In essence what he showed is that any particular automaton can be described by a Wnite set
of instructions, and that when this is fed to his universal automaton it in turn imitates the
special one. . . . Von Neumann was enormously intrigued with these ideas, and he started
in 1947 working on . . . how complex a device or construct needed to be in order to be self-
reproductive.28
In his Hixon Symposium, von Neumann said:
For the question which concerns me here, that of ‘self-reproduction’ of automata,
Turing’s procedure is too narrow in one respect only. His automata are purely computing
machines. Their output is a piece of tape with zeros and ones on it. What is needed . . . is
an automaton whose output is other automata. There is, however, no diYculty in
principle in dealing with this broader concept and in deriving from it the equivalent of
Turing’s result. . . . The problem of self-reproduction can . . . be stated like this: Can one
build an aggregate out of . . . elements in such a manner that if it is put into a reservoir, in
which there Xoat all these elements in large numbers, it will then begin to construct other
aggregates, each of which will at the end turn out to be another automaton exactly like the
25 A. W. Burks (ed.), Essays on Cellular Automata (Urbana: University of Illinois Press, 1970), p. xv.
26 C. G. Langton, ‘ArtiWcial Life’, in M. A. Boden (ed.), The Philosophy of ArtiWcial Life (Oxford: Oxford
University Press, 1996), 47.
27 Letter from von Neumann to Wiener, 29 Nov. 1946 (in the von Neumann Archive at the Library of
Congress, Washington, DC).
28 H. H. Goldstine, The Computer from Pascal to von Neumann (Princeton: Princeton University Press,
1972), 271, 274–5.
516 | Jack Copeland
original one? This is feasible, and the principle on which it can be based is closely related
to Turing’s [universal automaton] outlined earlier.29
In lectures von Neumann described a ‘universal constructor’ (remarking, ‘You
see, I’m coming quite close to Turing’s trick with universal automata’).30 Just as
complete descriptions of Turing machines can be fed into the universal automa-
ton in the form of programmes, complete descriptions of automata can be
inserted into the universal constructor. The universal constructor Xoats in a
medium—or sea—in which also Xoat, in practically unlimited supply, the
components from which the universal constructor is made. Given a complete
description of an automaton, the universal constructor will assemble that
automaton.
If what is inserted into the universal constructor is a description I of the
universal constructor itself, the constructor assembles a duplicate of itself—self-
reproduction. Since the duplicate is to be an exact copy, it must contain the
description I. This is done by a copying mechanism in the universal constructor
and the copy of I is inserted into the oVspring.31 Thus the oVspring, too, is capable
of self-reproduction. Von Neumann likened the description I to the gene, saying
that the copying mechanism ‘performs the fundamental act of reproduction, the
duplication of the genetic material, which is clearly the fundamental operation in
the multiplication of living cells’.32 Allowing the copying mechanism to make
occasional random errors aVords the possibility of random mutation in the
genes of the oVspring, thus opening the door to Darwinian evolution.33
In his letter to Wiener (29 November 1946), von Neumann voiced some
concerns that might equally well be raised concerning the focus of modern
research in A-Life and ArtiWcial Intelligence. Von Neumann pointed out that
when automata theorists choose the human nervous system as their model,
they are unrealistically selecting ‘the most complicated object under the sun—
literally’. Moreover, he said, there is little advantage in choosing instead simpler
organisms with fewer neurons, for example, the ant: any nervous system exhibits
‘exceptional complexity’. Von Neumann suggested that automata theorists ‘turn
to simpler systems’, and he recommended attention to ‘organisms of the virus or
bacteriophage type’. These, he pointed out, are ‘self-reproductive and . . . are able
to orient themselves in an unorganized milieu, to move towards food, to
appropriate it and to use it’. He estimated that a typical bacteriophage might
consist of 6 million atoms grouped into a few hundred thousand ‘mechanical
elements’, saying that this represents ‘a degree of complexity which is not
29 von Neumann, ‘The General and Logical Theory of Automata’, 315.
30 J. von Neumann, Theory of Self-Reproducing Automata, ed. and completed by A. W. Burks (Urbana:
University of Illinois Press, 1966), 83.
31 von Neumann ‘The General and Logical Theory of Automata’, 316–17.
32 Ibid., 317.
33 Ibid. 317–18.
Artificial Life | 517
necessarily beyond human endurance’. By following this path, he said, the
‘decisive break’ might be achieved.
6. Letter from Turing to Young
Shortly before the Ferranti computer arrived in 1951, Turing wrote about his
work on morphogenesis in a letter to the biologist J. Z. Young. The letter
connects Turing’s work on morphogenesis with his interest in neural networks
(Chapter 10), and moreover to some extent explains why he did not follow up
his earlier suggestion (Chapter 10, p. 428) and use the Ferranti computer to
simulate his ‘unorganised machines’.
I am afraid I am very far from the stage where I feel inclined to start asking any
anatomical questions [about the brain]. According to my notions of how to set about it
that will not occur until quite a late stage when I have a fairly deWnite theory about how
things are done.
At present I am not working on the problem at all, but on my mathematical theory of
embryology . . . This is yielding to treatment, and it will so far as I can see, give satisfactory
explanations of
i) Gastrulation.
ii) Polyogonally symmetrical structures, e.g., starWsh, Xowers.
iii) Leaf arrangement, in particular the way the Fibonacci series (0, 1, 1, 2, 3, 5, 8,
13, . . . ) comes to be involved.
iv) Colour patterns on animals, e.g., stripes, spots and dappling.
v) Patterns on nearly spherical structures such as some Radiolaria, but this is more
diYcult and doubtful.
I am really doing this now because it is yielding more easily to treatment. I think it is
not altogether unconnected with the other problem. The brain structure has to be one
which can be achieved by the genetical embryological mechanism, and I hope that this
theory that I am now working on may make clearer what restrictions this really implies.
What you tell me about growth of neurons under stimulation is very interesting in this
connection. It suggests means by which the neurons might be made to grow so as to form
a particular circuit, rather than to reach a particular place.34
Further reading
Boden, M. A., Mind as Machine: A History of Cognitive Science (Oxford: Oxford University
Press, 2005).
—— (ed.), The Philosophy of ArtiWcial Life (Oxford: Oxford University Press, 1996).
34 Letter from Turing to Young, 8 Feb. 1951 (a copy of the letter is in the Modern Archive Centre, King’s
College, Cambridge (catalogue reference K 78)).
518 | Jack Copeland
Holland, J. H., ‘Genetic Algorithms’, ScientiWc American, 267 (July 1992), 44–50.
Swinton, J., ‘Watching the Daisies Grow: Turing and Fibonacci Phyllotaxis’, in C. Teuscher
(ed.), Alan Turing: Life and Legacy of a Great Thinker (Berlin: Springer-Verlag, 2004).
Turing, A. M., Morphogenesis: Collected Works of A. M. Turing, ed. P. T. Saunders
(Amsterdam: North-Holland, 1992).
Turk, G., ‘Generating Textures on Arbitrary Surfaces Using Reaction-DiVusion’, Computer
Graphics, 25 (1991), 289–98.
von Neumann, J., Theory of Self-Reproducing Automata, ed. and completed by A. W. Burks
(Urbana: University of Illinois Press, 1966).
CHAPTER 15
The Chemical Basis of
Morphogenesis (1952)
Alan Turing
It is suggested that a system of chemical substances, called morphogens, reacting together
and diVusing through a tissue, is adequate to account for the main phenomena of
morphogenesis. Such a system, although it may originally be quite homogenous, may
later develop a pattern or structure due to an instability of the homogeneous equilibrium,
which is triggered oV by random disturbances. Such reaction-diVusion systems are
considered in some detail in the case of an isolated ring of cells, a mathematically
convenient, though biologically unusual system. The investigation is chieXy concerned
with the onset of instability. It is found that there are six essentially diVerent forms which
this may take. In the most interesting form stationary waves appear on the ring. It is
suggested that this might account, for instance, for the tentacle patterns on Hydra and for
whorled leaves. A system of reactions and diVusion on a sphere is also considered. Such a
system appears to account for gastrulation. Another reaction system in two dimensions
gives rise to patterns reminiscent of dappling. It is also suggested that stationary waves in
two dimensions could account for the phenomena of phyllotaxis.
The purpose of this paper is to discuss a possible mechanism by which the genes of a
zygote may determine the anatomical structure of the resulting organism. The theory does
not make any new hypotheses; it merely suggests that certain well-known physical laws are
suYcient to account for many of the facts. The full understanding of the paper requires a
good knowledge of mathematics, some biology, and some elementary chemistry. Since
readers cannot be expected to be experts in all of these subjects, a number of elementary
facts are explained, which can be found in text-books, but whose omission would make
the paper diYcult reading.
1. A model of the embryo. Morphogens
In this section a mathematical model of the growing embryo will be described.
This model will be a simpliWcation and an idealization, and consequently a
falsiWcation. It is to be hoped that the features retained for discussion are those
of greatest importance in the present state of knowledge.
This article Wrst appeared in Philosophical Transactions of the Royal Society of London, Series B, 237
(1952–54), 37–72. It is reprinted with the permission of the Royal Society and the Estate of Alan Turing.
520 | Alan Turing
The model takes two slightly diVerent forms. In one of them the cell theory is
recognized but the cells are idealized into geometrical points. In the other the
matter of the organism is imagined as continuously distributed. The cells are not,
however, completely ignored, for various physical and physico-chemical charac-
teristics of the matter as a whole are assumed to have values appropriate to the
cellular matter.
With either of the models one proceeds as with a physical theory and deWnes
an entity called ‘the state of the system’. One then describes how that state is to be
determined from the state at a moment very shortly before. With either model
the description of the state consists of two parts, the mechanical and the
chemical. The mechanical part of the state describes the positions, masses,
velocities and elastic properties of the cells, and the forces between them. In
the continuous form of the theory essentially the same information is given in
the form of the stress, velocity, density and elasticity of the matter. The chemical
part of the state is given (in the cell form of theory) as the chemical composition
of each separate cell; the diVusibility of each substance between each two adja-
cent cells must also be given. In the continuous form of the theory the concen-
trations and diVusibilities of each substance have to be given at each point. In
determining the changes of state one should take into account
(i) The changes of position and velocity as given by Newton’s laws of motion.
(ii) The stresses as given by the elasticities and motions, also taking into
account the osmotic pressures as given from the chemical data.
(iii) The chemical reactions.
(iv) The diVusion of the chemical substances. The region in which this
diVusion is possible is given from the mechanical data.
This account of the problem omits many features, e.g. electrical properties and
the internal structure of the cell. But even so it is a problem of formidable
mathematical complexity. One cannot at present hope to make any progress with
the understanding of such systems except in very simpliWed cases. The interde-
pendence of the chemical and mechanical data adds enormously to the diYculty,
and attention will therefore be conWned, so far as is possible, to cases where these
can be separated. The mathematics of elastic solids is a well-developed subject,
and has often been applied to biological systems. In this paper it is proposed to
give attention rather to cases where the mechanical aspect can be ignored and the
chemical aspect is the most signiWcant. These cases promise greater interest, for
the characteristic action of the genes themselves is presumably chemical. The
systems actually to be considered consist therefore of masses of tissues which are
not growing, but within which certain substances are reacting chemically, and
through which they are diVusing. These substances will be called morphogens,
the word being intended to convey the idea of a form producer. It is not intended
to have any very exact meaning, but is simply the kind of substance concerned in
The Chemical Basis of Morphogenesis | 521
this theory. The evocators of Waddington provide a good example of morpho-
gens (Waddington 1940). These evocators diVusing into a tissue somehow
persuade it to develop along diVerent lines from those which would have been
followed in its absence. The genes themselves may also be considered to be
morphogens. But they certainly form rather a special class. They are quite
indiVusible. Moreover, it is only by courtesy that genes can be regarded as
separate molecules. It would be more accurate (at any rate at mitosis) to regard
them as radicals of the giant molecules known as chromosomes. But presumably
these radicals act almost independently, so that it is unlikely that serious errors
will arise through regarding the genes as molecules. Hormones may also be
regarded as quite typical morphogens. Skin pigments may be regarded as mor-
phogens if desired. But those whose action is to be considered here do not come
squarely within any of these categories.
The function of genes is presumed to be purely catalytic. They catalyze the
production of other morphogens, which in turn may only be catalysts. Eventu-
ally, presumably, the chain leads to some morphogens whose duties are not
purely catalytic. For instance, a substance might break down into a number of
smaller molecules, thereby increasing the osmotic pressure in a cell and promot-
ing its growth. The genes might thus be said to inXuence the anatomical form of
the organism by determining the rates of those reactions which they catalyze. If
the rates are assumed to be those determined by the genes, and if a comparison
of organisms is not in question, the genes themselves may be eliminated from the
discussion. Likewise any other catalysts obtained secondarily through the agency
of the genes may equally be ignored, if there is no question of their concen-
trations varying. There may, however, be some other morphogens, of the nature
of evocators, which cannot be altogether forgotten, but whose role may neverthe-
less be subsidiary, from the point of view of the formation of a particular organ.
Suppose, for instance, that a ‘leg-evocator’ morphogen were being produced in a
certain region of an embryo, or perhaps diVusing into it, and that an attempt was
being made to explain the mechanism by which the leg was formed in the
presence of the evocator. It would then be reasonable to take the distribution
of the evocator in space and time as given in advance and to consider the
chemical reactions set in train by it. That at any rate is the procedure adopted
in the few examples considered here.
2. Mathematical background required
The greater part of this present paper requires only a very moderate knowledge of
mathematics. What is chieXy required is an understanding of the solution of linear
diVerential equations with constant coeYcients. (This is also what is chieXy
required for an understanding of mechanical and PeleActerbitc,awl hoesrceiltlhateioqnusa.n)tiTtihees
solution of such an equation takes the form of a sum
522 | Alan Turing
A, b may be complex, ip.e. of the form a þ ib, where a and b are ordinary
(real) numbers and i ¼ À1. It is of great importance that the physical sign-
iWcance of the various possible solutions of this kind should be appreciated, for
instance, that
(a) Since RthPe sAoleubttioonr sPwRillAneobtrm(Ralmlyebaensre‘raelaol npearctaonf also write them in the
form ’).
(b) That if A ¼ A0eif and b ¼ a þ ib, where A0, a, b, f are real, then
RAebt ¼ A0eat cos (bt þ f):
Thus each such term represents a sinusoidal oscillation if a ¼ 0, a damped
oscillation if a < 0, and an oscillation of ever-increasing amplitude if
a > 0.
(c) If any one of the numbers b has a positive real part the system in question
is unstable.
(d) After a suYciently great lapse of time all the terms Aebt will be negligible
in comparison with those for which b has the greatest real part, but unless
this greatest real part is itself zero these dominant terms will eventually
either tend to zero or to inWnite values.
(e) That the indeWnite growth mentioned in (b) and (d) will in any physical or
biological situation eventually be arrested due to a breakdown of the
assumptions under which the solution was valid. Thus, for example, the
growth of a colony of bacteria will normally be taken to satisfy the equa-
tion dy=dt ¼ ay(a > 0), y being the number of organisms at time t, and
this has the solution y ¼ Aeat . When, however, the factor eat has reached
some billions the food supply can no longer be regarded as unlimited and
the equation dy=dt ¼ ay will no longer apply.
The following relatively elementary result will be needed, but may not be
known to all readers:
XN ! if 0 < s < N,
exp 2pirs ¼ 0
r¼1 N
but ¼ N if s ¼ 0 or s ¼ N :
The Wrst case can easily be proved when it is noticed that the left-hand side is a
geometric progression. In the second case all the terms are equal to 1.
The relative degrees of diYculty of the various sections are believed to be as
follows. Those who are unable to follow the points made in this section should only
attempt §§3, 4, 11, 12, 14 and part of §13. Those who can just understand this
section should proWt also from §§7, 8, 9. The remainder, §§5, 10, 13, will probably
only be understood by those deWnitely trained as mathematicians.
The Chemical Basis of Morphogenesis | 523
3. Chemical reactions
It has been explained in a preceding section that the system to be considered
consists of a number of chemical substances (morphogens) diVusing through a
mass of tissue of given geometrical form and reacting together within it. What
laws are to control the development of this situation? They are quite simple. The
diVusion follows the ordinary laws of diVusion, i.e. each morphogen moves from
regions of greater to regions of less concentration, at a rate proportional to the
gradient of the concentration, and also proportional to the ‘diVusibility’ of the
substance. This is very like the conduction of heat, diVusibility taking the place of
conductivity. If it were not for the walls of the cells the diVusibilities would be
inversely proportional to the square roots of the molecular weights. The pores of
the cell walls put a further handicap on the movement of the larger molecules in
addition to that imposed by their inertia, and most of them are not able to pass
through the walls at all.
The reaction rates will be assumed to obey the ‘law of mass action’. This states
that the rate at which a reaction takes place is proportional to the concentrations
of the reacting substances. Thus, for instance, the rate at which silver chloride
will be formed and precipitated from a solution of silver nitrate and sodium
chloride by the reaction
Agþ þ ClÀ ! AgCl
will be proportional to the product of the concentrations of the silver ion Agþ
and the chloride ion ClÀ. It should be noticed that the equation
AgNO3 þ NaCl ! AgCl þ NaNO3
is not used because it does not correspond to an actual reaction but to the Wnal
outcome of a number of reactions. The law of mass action must only be applied to
the actual reactions. Very often certain substances appear in the individual reac-
tions of a group, but not in the Wnal outcome. For instance, a reaction A ! B may
really take the form of two steps A þ G ! C and C ! B þ G. In such a case the
substance G is described as a catalyst, and as catalyzing the reaction A ! B.
(Catalysis according to this plan has been considered in detail by Michaelis &
Menten (1913).) The eVect of the genes is presumably achieved almost entirely by
catalysis. They are certainly not permanently used up in the reactions.
Sometimes one can regard the eVect of a catalyst as merely altering a reaction
rate. Consider, for example, the case mentioned above, but suppose also that A
can become detached from G, i.e. that the reaction C ! A þ G is taken into
account. Also suppose that the reactions A þ G ! C both proceed much faster
than C ! B þ G. Then the concentrations of A, G, C will be related by
the condition that there is equilibrium between the reactions A þ G ! C
and C ! A þ G, so that (denoting concentrations by square brackets)
524 | Alan Turing
[A][G] ¼ k[C] for some constant k. The reaction C ! B þ G will of course
proceed at a rate proportional to [C], i.e. to [A][G]. If the amount of C is always
small compared with the amount of G one can say that the presence of the
catalyst and its amount merely alter the mass action constant for the reaction
A ! B, for the whole proceeds at a rate proportional to [A]. This situation does
not, however, hold invariably. It may well happen that nearly all of G takes the
combined form C so long as any of A is left. In this case the reaction proceeds at a
rate independent of the concentration of A until A is entirely consumed. In either
of these cases the rate of the complete group of reactions depends only on the
concentrations of the reagents, although usually not according to the law of mass
action applied crudely to the chemical equation for the whole group. The same
applies in any case where all reactions of the group with one exception proceed at
speeds much greater than that of the exceptional one. In these cases the rate of
the reaction is a function of the concentrations of the reagents. More generally
again, no such approximation is applicable. One simply has to take all the actual
reactions into account.
According to the cell model then, the number and positions of the cells are
given in advance, and so are the rates at which the various morphogens diVuse
between the cells. Suppose that there are N cells and M morphogens. The state of
the whole system is then given by MN numbers, the quantities of the M
morphogens in each of N cells. These numbers change with time, partly because
of the reactions, partly because of the diVusion. To determine the part of the rate
of change of one of these numbers due to diVusion, at any one moment, one only
needs to know the amounts of the same morphogen in the cell and its neigh-
bours, and the diVusion coeYcient for that morphogen. To Wnd the rate of
change due to chemical reaction one only needs to know the concentrations of all
morphogens at that moment in the one cell concerned.
This description of the system in terms of the concentrations in the various
cells is, of course, only an approximation. It would be justiWed if, for instance, the
contents were perfectly stirred. Alternatively, it may often be justiWed on the
understanding that the ‘concentration in the cell’ is the concentration at a certain
representative point, although the idea of ‘concentration at a point’ clearly itself
raises diYculties. The author believes that the approximation is a good one,
whatever argument is used to justify it, and it is certainly a convenient one.
It would be possible to extend much of the theory to the case of organisms
immersed in a Xuid, considering the diVusion within the Xuid as well as from cell
to cell. Such problems are not, however, considered here.
4. The breakdown of symmetry and homogeneity
There appears superWcially to be a diYculty confronting this theory of morpho-
genesis, or, indeed, almost any other theory of it. An embryo in its spherical
The Chemical Basis of Morphogenesis | 525
blastula stage has spherical symmetry, or if there are any deviations from perfect
symmetry, they cannot be regarded as of any particular importance, for the
deviations vary greatly from embryo to embryo within a species, though the
organisms developed from them are barely distinguishable. One may take it
therefore that there is perfect spherical symmetry. But a system which has
spherical symmetry, and whose state is changing because of chemical reactions
and diVusion, will remain spherically symmetrical for ever. (The same would
hold true if the state were changing according to the laws of electricity and
magnetism, or of quantum mechanics.) It certainly cannot result in an organism
such as a horse, which is not spherically symmetrical.
There is a fallacy in this argument. It was assumed that the deviations from
spherical symmetry in the blastula could be ignored because it makes no
particular diVerence what form of asymmetry there is. It is, however, important
that there are some deviations, for the system may reach a state of instability in
which these irregularities, or certain components of them, tend to grow. If this
happens a new and stable equilibrium is usually reached, with the symmetry
entirely gone. The variety of such new equilibria will normally not be so great as
the variety of irregularities giving rise to them. In the case, for instance, of the
gastrulating sphere, discussed at the end of this paper, the direction of the axis of
the gastrula can vary, but nothing else.
The situation is very similar to that which arises in connexion with electrical
oscillators. It is usually easy to understand how an oscillator keeps going when
once it has started, but on a Wrst acquaintance it is not obvious how the
oscillation begins. The explanation is that there are random disturbances always
present in the circuit. Any disturbance whose frequency is the natural frequency
of the oscillator will tend to set it going. The ultimate fate of the system will be a
state of oscillation at its appropriate frequency, and with an amplitude (and a
wave form) which are also determined by the circuit. The phase of the oscillation
alone is determined by the disturbance.
If chemical reactions and diVusion are the only forms of physical change
which are taken into account the argument above can take a slightly diVerent
form. For if the system originally has no sort of geometrical symmetry but is a
perfectly homogeneous and possibly irregularly shaped mass of tissue, it will
continue indeWnitely to be homogeneous. In practice, however, the presence of
irregularities, including statistical Xuctuations in the numbers of molecules
undergoing the various reactions, will, if the system has an appropriate kind of
instability, result in this homogeneity disappearing.
This breakdown of symmetry or homogeneity may be illustrated by the case of
a pair of cells originally having the same, or very nearly the same, contents. The
system is homogeneous: it is also symmetrical with respect to the operation of
interchanging the cells. The contents of either cell will be supposed describable
by giving the concentrations X and Y of two morphogens. The chemical reactions
526 | Alan Turing
will be supposed such that, on balance, the Wrst morphogen (X) is produced at
the rate 5X À 6Y þ 1 and the second (Y ) at the rate 6X À 7Y þ 1. When,
however, the strict application of these formulae would involve the concentration
of a morphogen in a cell becoming negative, it is understood that it is instead
destroyed only at the rate at which it is reaching that cell by diVusion. The Wrst
morphogen will be supposed to diVuse at the rate 0Á5 for unit diVerence of
concentration between the cells, the second, for the same diVerence, at the rate
4Á5. Now if both morphogens have unit concentration in both cells there is
equilibrium. There is no resultant passage of either morphogen across the cell
walls, since there is no concentration diVerence, and there is no resultant
production (or destruction) of either morphogen in either cell since
5X À 6Y þ 1 and 6X À 7Y þ 1 both have the value zero for X ¼ 1, Y ¼ 1.
But suppose the values are X1 ¼ 1Á06, Y1 ¼ 1Á02 for the Wrst cell and
X2 ¼ 0Á94, Y2 ¼ 0Á98 for the second. Then the two morphogens will be being
produced by chemical action at the rates 0Á18, 0Á22 respectively in the Wrst cell
and destroyed at the same rates in the second. At the same time there is a Xow
due to diVusion from the Wrst cell to the second at the rate 0Á06 for the Wrst
morphogen and 0Á18 for the second. In sum the eVect is a Xow from the second
cell to the Wrst at the rates 0Á12, 0Á04 for the two morphogens respectively. This
Xow tends to accentuate the already existing diVerences between the two cells.
More generally, if
X1 ¼ 1 þ 3x, X2 ¼ 1 À 3x, Y1 ¼ 1 þ x, Y2 ¼ 1 À x,
at some moment the four concentrations continue afterwards to be expressible in
this form, and x increases at the rate 2x. Thus there is an exponential drift away
from the equilibrium condition. It will be appreciated that a drift away from the
equilibrium occurs with almost any small displacement from the equilibrium
condition, though not normally according to an exact exponential curve.
A particular choice was made in the above argument in order to exhibit the
drift with only very simple mathematics.
Before it can be said to follow that a two-cell system can be unstable, with
inhomogeneity succeeding homogeneity, it is necessary to show that the reaction
rate functions postulated really can occur. To specify actual substances, concen-
trations and temperatures giving rise to these functions would settle the matter
Wnally, but would be diYcult and somewhat out of the spirit of the present
inquiry. Instead, it is proposed merely to mention imaginary reactions which
give rise to the required functions by the law of mass action, if suitable reaction
constants are assumed. It will be suYcient to describe
(i) A set of reactions producing the Wrst morphogen at the constant rate 1,
and a similar set forming the second morphogen at the same rate.
(ii) A set destroying the second morphogen (Y ) at the rate 7Y.
The Chemical Basis of Morphogenesis | 527
(iii) A set converting the Wrst morphogen (X) into the second (Y ) at the rate
6X.
(iv) A set producing the Wrst morphogen (X ) at the rate 11X.
(v) A set destroying the Wrst morphogen (X) at the rate 6Y, so long as any of
it is present.
The conditions of (i) can be fulWlled by reactions of the type A ! X, B ! Y ,
where A and B are substances continually present in large and invariable concen-
trations. The conditions of (ii) are satisWed by a reaction of the form Y ! D, D
being an inert substance, and (iii) by the reaction X ! Y or X ! Y þ E. The
remaining two sets are rather more diYcult. To satisfy the conditions of (iv) one
may suppose that X is a catalyst for its own formation from A. The actual
reactions could be the formation of an unstable compound U by the reaction
A þ X ! U , and the subsequent almost instantaneous breakdown U ! 2X. To
destroy X at a rate proportional to Y as required in (v) one may suppose that a
catalyst C is present in small but constant concentration and immediately
combines with X, X þ C ! V . The modiWed catalyst reacting with Y, at a rate
proportional to Y, restores the catalyst but not the morphogen X, by the reactions
V þ Y ! W , W ! C þ H, of which the latter is assumed instantaneous.
It should be emphasized that the reactions here described are by no means
those which are most likely to give rise to instability in nature. The choice of the
reactions to be discussed was dictated entirely by the fact that it was desirable
that the argument be easy to follow. More plausible reaction systems are de-
scribed in §10.
Unstable equilibrium is not, of course, a condition which occurs very natur-
ally. It usually requires some rather artiWcial interference, such as placing a
marble on the top of a dome. Since systems tend to leave unstable equilibria
they cannot often be in them. Such equilibria can, however, occur naturally
through a stable equilibrium changing into an unstable one. For example, if a rod
is hanging from a point a little above its centre of gravity it will be in stable
equilibrium. If, however, a mouse climbs up the rod the equilibrium eventually
becomes unstable and the rod starts to swing. A chemical analogue of this
mouse-and-pendulum system would be that described above with the same
diVusibilities but with the two morphogens produced at the rates
(3 þ I)X À 6Y þ I À 1 and 6X À (9 þ I)Y À I þ 1:
This system is stable if I < 0 but unstable if I > 0. If I is allowed to increase,
corresponding to the mouse running up the pendulum, it will eventually become
positive and the equilibrium will collapse. The system which was originally
discussed was the case I ¼ 2, and might be supposed to correspond to the
mouse somehow reaching the top of the pendulum without disaster, perhaps
by falling vertically on to it.
528 | Alan Turing
5. Left-handed and right-handed organisms
The object of this section is to discuss a certain diYculty which might be thought
to show that the morphogen theory of morphogenesis cannot be right. The
diYculty is mainly concerned with organisms which have not got bilateral
symmetry. The argument, although carried through here without the use of
mathematical formulae, may be found diYcult by non-mathematicians, and
these are therefore recommended to ignore it unless they are already troubled
by such a diYculty.
An organism is said to have ‘bilateral symmetry’ if it is identical with its own
reXexion in some plane. This plane of course always has to pass through some
part of the organism, in particular through its centre of gravity. For the purpose
of this argument it is more general to consider what may be called ‘left–right
symmetry’. An organism has left-right symmetry if its description in any right-
handed set of rectangular Cartesian co-ordinates is identical with its description
in some set of left-handed axes. An example of a body with left–right symmetry,
but not bilateral symmetry, is a cylinder with the letter P printed on one end, and
with the mirror image of a P on the other end, but with the two upright strokes
of the two letters not parallel. The distinction may possibly be without a
diVerence so far as the biological world is concerned, but mathematically it
should not be ignored.
If the organisms of a species are suYciently alike, and the absence of left–right
symmetry suYciently pronounced, it is possible to describe each individual as
either right-handed or left-handed without there being diYculty in classifying
any particular specimen. In man, for instance, one could take the X-axis in the
forward direction, the Y-axis at right angles to it in the direction towards the side
on which the heart is felt, and the Z-axis upwards. The specimen is classed as left-
handed or right-handed according as the axes so chosen are left-handed or right-
handed. A new classiWcation has of course to be deWned for each species.
The fact that there exist organisms which do not have left–right symmetry
does not in itself cause any diYculty. It has already been explained how various
kinds of symmetry can be lost in the development of the embryo, due to the
particular disturbances (or ‘noise’) inXuencing the particular specimen not
having that kind of symmetry, taken in conjunction with appropriate kinds of
instability. The diYculty lies in the fact that there are species in which the
proportions of left-handed and right-handed types are very unequal. It will be
as well to describe Wrst an argument which appears to show that this should not
happen. The argument is very general, and might be applied to a very wide class
of theories of morphogenesis.
An entity may be described as ‘P-symmetrical’ if its description in terms of one
set of right-handed axes is identical with its description in terms of any other set
of right-handed axes with the same origin. Thus, for instance, the totality of
The Chemical Basis of Morphogenesis | 529
positions that a corkscrew would take up when rotated in all possible ways about
the origin has P-symmetry. The entity will be said to be ‘F-symmetrical’ when
changes from right-handed axes to left-handed may also be made. This would
apply if the corkscrew were replaced by a bilaterally symmetrical object such as a
coal scuttle, or a left–right symmetrical object. In these terms one may say that
there are species such that the totality of specimens from that species, together
with the rotated specimens, is P-symmetrical, but very far from F-symmetrical.
On the other hand, it is reasonable to suppose that
(i) The laws of physics are F-symmetrical.
(ii) The initial totality of zygotes for the species is F-symmetrical.
(iii) The statistical distribution of disturbances is F-symmetrical. The indi-
vidual disturbances of course will in general have neither F-symmetry
nor P-symmetry.
It should be noticed that the ideas of P-symmetry and F-symmetry as deWned
above apply even to so elaborate an entity as ‘the laws of physics’. It should also
be understood that the laws are to be the laws taken into account in the theory in
question rather than some ideal as yet undiscovered laws.
Now it follows from these assumptions that the statistical distribution of
resulting organisms will have F-symmetry, or more strictly that the distribution
deduced as the result of working out such a theory will have such symmetry. The
distribution of observed mature organisms, however, has no such symmetry. In
the Wrst place, for instance, men are more often found standing on their feet than
their heads. This may be corrected by taking gravity into account in the laws,
together with an appropriate change of deWnition of the two kinds of symmetry.
But it will be more convenient if, for the sake of argument, it is imagined that
some species has been reared in the absence of gravity, and that the resulting
distribution of mature organisms is found to be P-symmetrical but to yield more
right-handed specimens than left-handed and so not to have F-symmetry. It
remains therefore to explain this absence of F-symmetry.
Evidently one or other of the assumptions (i) to (iii) must be wrong, i.e. in a
correct theory one of them would not apply. In the morphogen theory already
described these three assumptions do all apply, and it must therefore be regarded
as defective to some extent. The theory may be corrected by taking into account
the fact that the morphogens do not always have an equal number of left-and
right-handed molecules. According to one’s point of view one may regard this as
invalidating either (i), (ii) or even (iii). Simplest perhaps is to say that the totality
of zygotes just is not F-symmetrical, and that this could be seen if one looked at
the molecules. This is, however, not very satisfactory from the point of view of
this paper, as it would not be consistent with describing states in terms of
concentrations only. It would be preferable if it was found possible to Wnd
more accurate laws concerning reactions and diVusion. For the purpose of
530 | Alan Turing
accounting for unequal numbers of left- and right-handed organisms it is
unnecessary to do more than show that there are corrections which would not
be F-symmetrical when there are laevo- or dextrorotatory morphogens, and
which would be large enough to account for the eVects observed. It is not very
diYcult to think of such eVects. They do not have to be very large, but must, of
course, be larger than the purely statistical eVects, such as thermal noise or
Brownian movement.
There may also be other reasons why the totality of zygotes is not
F-symmetrical, e.g. an asymmetry of the chromosomes themselves. If these also
produce a suYciently large eVect, so much the better.
Though these eVects may be large compared with the statistical disturbances
they are almost certainly small compared with the ordinary diVusion and reac-
tion eVects. This will mean that they only have an appreciable eVect during a
short period in which the breakdown of left–right symmetry is occurring. Once
their existence is admitted, whether on a theoretical or experimental basis, it is
probably most convenient to give them mathematical expression by regarding
them as P-symmetrically (but not F-symmetrically) distributed disturbances.
However, they will not be considered further in this paper.
6. Reactions and diffusion in a ring of cells
The original reason for considering the breakdown of homogeneity was an
apparent diYculty in the diVusion-reaction theory of morphogenesis. Now
that the diYculty is resolved it might be supposed that there is no reason for
pursuing this aspect of the problem further, and that it would be best to proceed
to consider what occurs when the system is very far from homogeneous. A great
deal more attention will nevertheless be given to the breakdown of homogeneity.
This is largely because the assumption that the system is still nearly homoge-
neous brings the problem within the range of what is capable of being treated
mathematically. Even so many further simplifying assumptions have to be made.
Another reason for giving this phase such attention is that it is in a sense the
most critical period. That is to say, that if there is any doubt as to how the
organism is going to develop it is conceivable that a minute examination of it
just after instability has set in might settle the matter, but an examination of it at
any earlier time could never do so.
There is a great variety of geometrical arrangement of cells which might be
considered, but one particular type of conWguration stands out as being particu-
larly simple in its theory, and also illustrates the general principles very well. This
conWguration is a ring of similar cells. One may suppose that there are N such
cells. It must be admitted that there is no biological example to which the theory
of the ring can be immediately applied, though it is not diYcult to Wnd ones in
which the principles illustrated by the ring apply.
The Chemical Basis of Morphogenesis | 531
It will be assumed at Wrst that there are only two morphogens, or rather only
two interesting morphogens. There may be others whose concentration does not
vary either in space or time, or which can be eliminated from the discussion for
one reason or another. These other morphogens may, for instance, be catalysts
involved in the reactions between the interesting morphogens. An example of a
complete system of reactions is given in §10. Some consideration will also be
given in §§8, 9 to the case of three morphogens. The reader should have no
diYculty in extending the results to any number of morphogens, but no essen-
tially new features appear when the number is increased beyond three.
The two morphogens will be called X and Y. These letters will also be used to
denote their concentrations. This need not lead to any real confusion. The
concentration of X in cell r may be written Xr , and Yr has a similar meaning.
It is convenient to regard ‘cell N’ and ‘cell O’ as synonymous, and likewise ‘cell 1’
and cell ‘N þ 1’. One can then say that for each r satisfying 1 < r < N cell r
exchanges material by diVusion with cells r À 1 and r þ 1. The cell-to-cell
diVusion constant for X will be called m, and that for Y will be called n. This
means that for unit concentration diVerence of X, this morphogen passes at the
rate m from the cell with the higher concentration to the (neighbouring) cell with
the lower concentration. It is also necessary to make assumptions about the rates
of chemical reaction. The most general assumption that can be made is that for
concentrations X and Y chemical reactions are tending to increase X at the rate
f(X, Y) and Y at the rate g(X, Y). When the changes in X and Y due to diVu-
sion are also taken into account the behaviour of the system may be described by
the 2N diVerential equations
dXr ¼ f (Xr , 9 (r ¼ 1, ..., N ): (6:1)
¼ g (Xr , Yr ) þ m(Xrþ1 À 2Xr þ XrÀ1) >=
dt Yr ) þ n(Yrþ1 À 2Yr þ YrÀ1) >;
dYr
dt
If f (h, k): g(h, k) ¼ 0, then an isolated cell has an equilibrium with concen-
trations X ¼ h, Y ¼ k. The ring system also has an equilibrium, stable or
unstable, with each Xr equal to h and each Yr equal to k. Assuming that
the system is not very far from this equilibrium it is convenient to put
Xr ¼ h þ xr , Yr ¼ k þ yr . One may also write ax þ by for f (h þ x, y þ k) and
cx þ dy for g(h þ x, y þ k). Since f (h, k) ¼ g(h, k) ¼ 0 no constant terms are
required, and since x and y are supposed small the terms in higher powers of x
and y will have relatively little eVect and one is justiWed in ignoring them. The
four quantities a, b, c, d may be called the ‘marginal reaction rates’. Collectively
they may be described as the ‘marginal reaction rate matrix’. When there are M
morphogens this matrix consists of M2 numbers. A marginal reaction rate has
the dimensions of the reciprocal of a time, like a radioactive decay rate, which is
in fact an example of a marginal (nuclear) reaction rate.
532 | Alan Turing
With these assumptions the equations can be rewritten as
dxr ¼ axr þ byr þ m(xrþ1 À 2xr þ xrÀ1), 9
dt =>
dyr ;> (6:2)
dt
¼ cxr þ dyr þ n(yrþ1 À 2yr þ yrÀ1):
To solve the equations one introduces new co-ordinates x0, . . . , xNÀ1 and
Z0, . . . , ZNÀ1 by putting
NXÀ1 2pirs ! 9
¼ exp N =>>>>
xr xs ,
s¼0
! (6:3)
NXÀ1 2pirs >>>>;
yr ¼ exp N Zs :
s¼0
These relations can also be written as
XN !9
1 exp À 2pirs xs, >>>=>
xr ¼ N
N s¼1
! (6:4)
1 XN 2pirs ;>>>> (6:5)
Zr ¼ exp À N ys ,
N s¼1
as may be shown by using the equations
XN ! if 0<r < N,
exp 2pirs ¼ 0
s¼1 N
¼ N if r ¼ 0 or r ¼ N ,
(referred to in §2). Making this substitution one obtains
dxs 1 XN ! À 2pis ! 2pis ! !
dt exp À 2pirs axr þ byr þ m exp N N
¼ N s¼1 À 2 þ exp xs
N !
2pis !
Àþ4bmZssiþn2mpNs exxps þÀb2ZNps:is N
¼ axs À 2 þ exp xs
¼
(6:6)
a
Likewise dZs ¼ cxs þ À 4n sin2 pNs Zs : (6:7)
dt d
The equations have now been converted into a quite manageable form, with the
variables separated. There are now two equations concerned with x1 and Z1, two
concerned with x2 and Z2, etc. The equations themselves are also of a well-known
The Chemical Basis of Morphogenesis | 533
standard form, being linear with constant coeYcients. Let ps and ps0 be the roots
of the equation ps p ps
p
À a þ 4m sin2 À d þ 4n sin2 ¼ bc (6:8)
NN
(with Rps > Rps0 for deWniteness), then the solution of the equations is of the
form xs ¼ Asepst þ Bse ps0 t , )
Zs ¼ Csepst þ Dse p0st ,
(6:9)
where, however, the coeYcients As, Bs, Cs, Ds are not independent but are
restricted to satisfy 9
=>
À a þ 4m sin2 ps ¼ bCs , ;> (6:10)
À a þ 4m sin2 pNs ¼ bDs :
As ps N
Bs ps0
If it should happen that ps ¼ ps0 the equations (6.9) have to be replaced by
xs ¼ (As þ Bst )e pst , )
Zs ¼ (Cs þ Dst )e pst :
(6:9)0
and (6.10) remains true. Substituting back into (6.3) and replacing the variables
xr , yr by Xr , Yr (the actual concentrations) the solution can be written
Xr XN þ Bse ps0t ) exp 2pirs! 9
¼ h þ (Ase pst N, >>=>>
s¼1 2pirs!: (6:11)
N
Yr XN þ Dse ps0t ) exp >>>;>
¼ k þ (Cse pst
s¼1
Here As, Bs, Cs, Ds are still related by (6.10), but otherwise are arbitrary
complex numbers; ps and ps0 are the roots of (6.8).
The expression (6.11) gives the general solution of the equations (6.1) when one
assumes that departures from homogeneity are suYciently small that the func-
tions f(X, Y) and g(X, Y) can safely be taken as linear. The form (6.11) given is not
very informative. It will be considerably simpliWed in §8. Another implicit as-
sumption concerns random disturbing inXuences. Strictly speaking one should
consider such inXuences to be continuously at work. This would make the
mathematical treatment considerably more diYcult without substantially altering
the conclusions. The assumption which is implicit in the analysis, here and in §8, is
that the state of the system at t ¼ 0 is not one of homogeneity, since it has been
displaced from such a state by the disturbances; but after t ¼ 0 further disturb-
ances are ignored. In §9 the theory is reconsidered without this latter assumption.
534 | Alan Turing
7. Continuous ring of tissue
As an alternative to a ring of separate cells one might prefer to consider a
continuous ring of tissue. In this case one can describe the position of a point
of the ring by the angle y which a radius to the point makes with a Wxed reference
radius. Let the diVusibilities of the two substances be m0 and n0. These are not
quite the same as m and n of the last section, since m and n are in eVect referred to
a cell diameter as unit of length, whereas m0 and n0 are referred to a conventional
unit, the same unit in which the radius r of the ring is measured.
Then
m ¼ m0 N 2 , n ¼ n0 N 2 :
2pr 2pr
The equations are
@X ¼ a(X À h) þ b(Y À k) þ m0 @2X , 9
@t r2 @y2 >=>
@Y n0 @2Y >>; (7:1)
@t r2 @y2
¼ c(X À h) þ d(Y À k) þ ,
which will be seen to be the limiting case of (6.2). The marginal reaction rates a,
b, c, d are, as before, the values at the equilibrium position of
@f =@X, @f =@Y , @g=@X, @g=@Y . The general solution of the equations is
X ¼ h þ X1 (Ase pst þ Bse ps0t )eisy, 9
>>>=>
s¼À1
(7:2)
X1 (Cse pst þ Dse ps0t )eisy, ;>>>>
Y ¼kþ
s¼À1
where ps, ps0 are now roots of
À a þ m0s 2 À d þ n0s 2 ¼ bc (7:3)
p r2 p r2
and m0s 2 9
As ps r2 >>=>
À a þ ¼ bCs ,
m0s 2 >>;> (7:4)
Bs ps0 r2
À a þ ¼ bDs :
This solution may be justiWed by considering the limiting case of the solution
(6.11). Alternatively, one may observe that the formula proposed is a solution, so
that it only remains to prove that it is the most general one. This will follow if
values of As, Bs, Cs, Ds can be found to Wt any given initial conditions. It is well
The Chemical Basis of Morphogenesis | 535
known that any function of an angle (such as X) can be expanded as a ‘Fourier
series’
X(y) ¼ X1 Gseisy (X(y) being values of X at t ¼ 0),
s¼À1
provided, for instance, that its Wrst derivative is continuous. If also
Y (y) ¼ X1 Hseisy (Y (y) being values of Y at t ¼ 0),
s¼À1
then the required initial conditions are satisWed provided As þ Bs ¼ Gs
and Cs þ Ds ¼ Hs. Values As, Bs, Cs, Ds to satisfy these conditions can be
found unless ps ¼ ps0. This is an exceptional case and its solution if required
may be found as the limit of the normal case.
8. Types of asymptotic behaviour in the ring after
a lapse of time
As the reader was reminded in § 2, after a lapse of time the behaviour of an
expression of the form of (6.11) is eventually dominated by the terms for which
the corresponding ps has the largest real part. There may, however, be several
terms for which this real part has the same value, and these terms will together
dominate the situation, the other terms being ignored by comparison. There will,
in fact, normally be either two or four such ‘leading’ terms. For if ps0 is one of
them then pNÀs0 ¼ ps0 , since
sin2 p(N À s0) ¼ sin2 ps0 ,
NN
so that ps0 and pNÀs0 are roots of the same equation (6.8). If also ps0 is complex
then Rps0 ¼ Rps00 , and so in all
Rps0 ¼ Rps00 ¼ RpNÀs0 ¼ RpN0 Às0 :
One need not, however, normally anticipate that any further terms will have to be
included. If ps0 and ps1 are to have the same real part, then, unless s1 ¼ s0
or s0 þ s1 ¼ N the quantities a, b, c, d, m, n will be restricted to satisfy some
special condition, which they would be unlikely to satisfy by chance. It is possible
to Wnd circumstances in which as many as ten terms have to be included if such
special conditions are satisWed, but these have no particular physical or biological
importance. It is assumed below that none of these chance relations hold.
It has already been seen that it is necessary to distinguish the cases where the
value of ps0 for one of the dominant terms is real from those where it is complex.
These may be called respectively the stationary and the oscillatory cases.
536 | Alan Turing
Stationary case. After a suYcient lapse of time Xr À h and Yr À k approach
asymptotically to the forms !9
Xr À h ¼ 2RAs0 exp 2pis0r þ It , >=>> (8:1)
Yr À k ¼ 2RCs0 exp N! : >>>;
2pis0r þ It
N
Oscillatory case. After a suYcient lapse of time Xr À h and Yr À k approach the
forms !' 9
&! À 2pis0r À ivt
2pis0r þ ivt N !' >>>=
Xr À h ¼ 2eIt R As0 exp N! þ ANÀs0 exp , >>;> (8:2)
& þ CNÀs0 exp À 2pis0r À ivt :
2pis0r þ ivt N
Yr À k ¼ 2eIt R Cs0 exp N
The real part of ps0 has been represented by I, standing for ‘instability’, and in the
oscillatory case its imaginary part is v. By the use of the R operation (real part
of), two terms have in each case been combined in one.
The meaning of these formulae may be conveniently described in terms of
waves. In the stationary case there are stationary waves on the ring having s0
lobes or crests. The coeYcients As0 and Cs0 are in a deWnite ratio given by (6Á10),
so that the pattern for one morphogen determines that for the other. With the
lapse of time the waves become more pronounced provided there is genuine
instability, i.e. if I is positive. The wave-length of the waves may be obtained by
dividing the number of lobes into the circumference of the ring. In the oscillatory
case the interpretation is similar, but the waves are now not stationary but
travelling. As well as having a wave-length they have a velocity and a frequency.
The frequency is v=2p, and the velocity is obtained by multiplying the wave-
length by the frequency. There are two wave trains moving round the ring in
opposite directions.
The wave-lengths of the patterns on the ring do not depend only on the
chemical data a, b, c, d, m0, n0 but on the circumference of the ring, since they
must be submultiples of the latter. There is a sense, however, in which there is a
‘chemical wave-length’ which does not depend on the dimensions of the ring.
This may be described as the limit to which the wave-lengths tend when the rings
are made successively larger. Alternatively (at any rate in the case of continuous
tissue), it may be described as the wave-length when the radius is chosen to give
the largest possible instability I. One may picture the situation by supposing that
the chemical wave-length is true wave-length which is achieved whenever pos-
sible, but that on a ring it is necessary to ‘make do’ with an approximation which
divides exactly into the circumference.
Although all the possibilities are covered by the stationary and oscillatory
alternatives there are special cases of them which deserve to be treated separately.
The Chemical Basis of Morphogenesis | 537
One of these occurs when s0 ¼ 0, and may be described as the ‘case of extreme
long wave-length’, though this term may perhaps preferably be reserved to
describe the chemical data when they are such that s0 is zero whatever the
dimensions of the ring. There is also the case of ‘extreme short wave-length’.
This means that sin2 (ps0=N ) is as large as possible, which is achieved by s0 being
either 1 N , or 1 (N À 1). If the remaining possibilities are regarded as forming
2 2
the ‘case of Wnite wave-length’, there are six subcases altogether. It will be shown
that each of these really can occur, although two of them require three or more
morphogens for their realization.
(a) Stationary case with extreme long wave-length. This occurs for instance if
m ¼ n ¼ 1 , b ¼ c ¼ 1, a ¼ d. Then ps ¼ a À sin2 ps þ 1. This is certainly real
4 N
and is greatest when s ¼ 0. In this case the contents of all the cells are the same;
there is no resultant Xow from cell to cell due to diVusion, so that each is
behaving as if it were isolated. Each is in unstable equilibrium, and slips out of
it in synchronism with the others.
(b) Oscillatory case with extreme long wave-length. This occurs, for instance, if
m¼ n ¼ 1 , b ¼ Àc ¼ 1, a ¼ d. Then ps ¼ a À sin2 ps ± i. This is complex and its
4 N
real part is greatest when s ¼ 0. As in case (a) each cell behaves as if it were isolated.
The diVerence from case (a) is that the departure from the equilibrium is oscillatory.
(c) Stationary waves of extreme short wave-length. This occurs, for instance, if
n ¼ 0, m ¼ 1, d ¼ I, a ¼ I À 1, b ¼ Àc ¼ 1 : ps is
s( 1 2À1),
I À 1 À 2 sin2 ps þ 2 sin2 ps þ 2
2 N N
and is greatest when sin2(ps=N ) is greatest. If N is even the contents of each cell
are similar to those of the next but one, but distinctly diVerent from those of its
immediate neighbours. If, however, the number of cells is odd this arrangement
is impossible, and the magnitude of the diVerence between neighbouring
cells varies round the ring, from zero at one point to a maximum at a point
diametrically opposite.
(d) Stationary waves of Wnite wave-length. This is the case which is of greatest
interest, and has most biological application. It occurs, for instance,
if a ¼ 2I À 2, b ¼ 2Á5, c ¼ À1Á25, d ¼ I þ 1Á5, m0 ¼ 1, n0 ¼ 1 , and m ¼ n ¼
2 m0 n0
N . As Ubeffoorre rpNr is2tshine2 radius of the ring, and N the number of cells in it. If
2pr writes equation (6.8) can, with these special values,
ps , then
one N
be written
(p À I )2 þ ( 1 þ 3 U )(p À I) þ 1 (U À 1 )2 ¼ 0 : (8:3)
2 2 2 2
This has a solution p ¼ I if U ¼ 21. On the other hand, it will be shown that if U
has any other (positive) value then both roots for p À I have negative real parts.
Their product is positive being 1 (U À 1 )2, so that if they are real they both have
2 2
538 | Alan Turing
the same sign. Their sum in this case is À 1 À 3 U which is negative. Their
2 2
common sign is therefore negative. If, however, the roots are complex their
real parts are both equal to À 1 À 3 U, which is negative.
4 4
If the rad ius 2 r of the ring be chosen so that for some integer
¼U there will be stationary waves with s0 lobes and a
s0, 1 ¼ N sin2 ps0 ,
2 pr N
wave-length which is also equal to the chemical wave-length, for ps0 will be equal
to I, whereas every tohtahter ppNrs w2sililnh2 apNvs e¼a21rceaanl npoartthsomldalwleirththaanninI.teIfg,rhalosw, etvheern, the
radius is chosen so (in
this example) the actual number of lobes will be one of the two integers nearest
to the (non-integral) solutions of this equation, and usually the nearest.
Examples can, however, be constructed where this simple rule does not apply.
Figure 1 shows the relation (8.3) in graphical form. The curved portions of the
graphs are hyperbolae.
The two remaining possibilities can only occur with three or more morpho-
gens. With one morphogen the only possibility is (a).
(e) Oscillatory case with a Wnite wave-length. This means that there are genuine
travelling waves. Since the example to be given involves three morphogens it is
not possible to use the formulae of § 6. Instead, one must use the corresponding
three morphogen formulae. That which corresponds to (6Á8) or (7Á3) is most
conveniently written as
U
0 0.2 0.4 0.6 0.8 1.0
−0.1
−0.2
p
−0.3
−0.4
Figure 1. Values of Rp (instability or growth rate), and jIpj (radian frequency of
oscillation), related to wave-length 2pUÀ12 as in the relation (8.3) with I ¼ 0. This is a
case of stationary waves with Wnite wave-length. Full line, Rp; broken line, ÀjIpj (zero
for U > 0 Á 071); dotted line, Rp0. The full circles on the curve for Rp indicate the values
of U, p actually achievable on the finite ring considered in §10, with s ¼ 0 on the extreme
left, s ¼ 5 on the right.
The Chemical Basis of Morphogenesis | 539
a11 À pÀ m1U a12 a13
a21 a22 À p À m2U a23
a31 pÀ ¼ 0, (8:4)
a32
a33 À m3U
where again U has been written for pNr 2sin2 ps . (This means essentially that
N reactivities are
À2pÁ2
U ¼ , where l is the wave-length.) The four marginal
l
superseded by nine a11, . . . , a33, and the three diVusibilities are m1, m2, m3.
Special values leading to travelling waves are
m1 ¼ 2 , m2 ¼ 1 , m3 ¼ 0, 9
3 3 a13 ¼ À1, =>>>
a11 ¼ À 10 , a12 ¼ 3, (8:5)
3
a21 ¼ À2, a22 ¼ 7 , a23 ¼ 0, >;>>
3
a31 ¼ 3, a32 ¼ À4, a33 ¼ 0,
and with them (8.4) reduces to
p3 þ p2(U þ 1) þ p(1 þ 29(U À 1)2) þ U þ 1 ¼ 0 : (8:6)
If U ¼ 1 the roots are ± i and À2. If U is near to I they are approximately
(U À1)2
À1 À U and ± i þ 18 (± i À 1), and all have negative real parts. If the greatest
real part is not the value zero, achieved with U ¼ 1, then the value zero must be
reached again at some intermediate value of U. Since P is then pure imaginary
the even terms of (8.6) must vanish, i.e. (p2 þ 1)(U þ 1) ¼ 0. But this can only
happen if p ¼ ± i, and the vanishing of the odd terms then shows that U ¼ 1.
Hence zero is the largest real part for any root p of (8.6). The corresponding p is
± i and U is 1. This means that there are travelling waves with unit (chemical)
radian frequency and unit (chemical) velocity. If I is added to a11, a22 and a33,
the instability will become I in place of zero.
(f) Oscillatory case with extreme short wave-length. This means that there is
metabolic oscillation with neighbouring cells nearly 1808 out of phase. It can be
achieved with three morphogens and the following chemical data:
m ¼ 1, m2 ¼ m3 ¼ 0, 9
a11 ¼ À1, a12 ¼ À1, a13 ¼ 0, =>>>
a21 ¼ 1,
a31 ¼ 0, a22 ¼ 0, a23 ¼ À1, >>;> (8:7)
a32 ¼ 1, a33 ¼ 0 :
With these values (8.4) reduces to
p3 þ p2(U þ 1) þ 2p þ U þ 1 ¼ 0 : (8:8)
This may be shown to have all the real parts of its roots negative if U 50, for if
U ¼ 0 the roots are near to À0Á6, À 0Á2 ± 1Á3i, and if U be continuously
increased the values of p will alter continuously. If they ever attain values with
540 | Alan Turing
a positive real part they must pass through pure imaginary values (or zero). But
if p is pure imaginary p3 þ 2p and (p2 þ 1)(U þ 1) must both vanish, which is
impossible if U 50. As U approaches inWnity, however, one of the roots app-
roaches i. Thus Rp ¼ 0 can be approached as closely as desired by large values of
U, but not attained.
9. Further consideration of the mathematics of the ring
In this section some of the Wner points concerning the development of wave
patterns are considered. These will be of interest mainly to those who wish to do
further research on the subject, and can well be omitted on a Wrst reading.
(1) General formulae a foW2rllpat hme2etniwnt)o(,7mo.n3o)erpamhnoadgyeownbrctiaatiesne . TpNark in2gsinth2epNlSim¼itUing¼ca se2lpo f a2
ring of large radius (or
in (6.11) or s2 ¼ U ¼
r2
(p À a þ m0U )(p À d þ n0U ) ¼ bc, (9:1)
which has the solution s( m0 )
p ¼ aþd À Á
m0 þ n0 ± À n0 þ d À a 2 þbc (9:2)
U U
22 22
One may put I(U) for the real part of this, representing the instability for waves
of wave-length l ¼ 2pU À21. The dominant waves correspond to the maximum of
I(U). This maximum may either be at U ¼ 0 or U ¼ 1 or at a stationary point
on the part of the curve which is hyperbolic (rather than straight). When this last
case occurs the values of p (or I) and U at the maximum are
p ¼ I ¼ (dm0 À an0 À 2p(m0n0)p( Àbc)(m0 À n0)À1, 9
U ¼ =>
a À d þ pm0(þm0nn00) p Àbc) (m0 À n0)À1Á >; (9:3)
(
The conditions which lead to the four cases (a), (b), (c), (d ) described in the last
section are
(a) (Stationary waves of extreme long wave-length.) This occurs if either
(i) bc > 0, (ii) bc < 0 and pd À a ) > pm0(þm0nn00), (iii) bc < 0 and pd Àa < À2.
( À bc ( Àbc)
The condition for instability in either case is that either bc > ad or a þ d > 0.
(b) (Oscillating case with extreme long wave-length, i.e. synchronized oscilla-
tions.)
This occurs if 4p(m0n0)
m0 þ n0
bc < 0 and À 2 < pd Àa < :
( Àbc)
The Chemical Basis of Morphogenesis | 541
There is instability if in addition a þ d > 0.
(c) (Stationary waves of extreme short wave-length.) This occurs if
bc < 0, m0 > n0 ¼ 0.
There is instability if, in addition, a þ d > 0.
(d) (Stationary waves of Wnite wave-length.) This occurs if
bc <0 and 4p(m0n0) < pd Àa < m0 þ n0 , (9:4a)
m0 þ n0 ( Àbc) (pm0n0)
and there is instability if also
p d r À p a rn0 > 2 : (9:4b)
( Àbc) m0 ( Àbc) m0
n0
It has been assumed that n04m0 > 0. The case where m04n0 > 0 can be obtained
by interchanging the two morphogens. In the case m0 ¼ n0 ¼ 0 there is no co-
operation between the cells whatever.
Some additional formulae will be given for the case of stationary waves of
Wnite wave-length. The marginal reaction rates may be expressed parametrically
in terms of the diVusibilities, the wave-length, the instability, and two other
parameters a and w. Of these a may be described as the ratio of X À h to Y À k
in the waves. The expressions for the marginal reaction rates in terms of these
parameters are
a ¼ m0(n0 À m0)À1(2n0U0 þ w) þ I , 9
b ¼ m0(n0 À m0)À1((m0 þ n0)U0 þ w)a, >=>>>
c ¼ n0(m0 À n0)À1((m0 þ n0)U0 þ w)aÀ1, >;>>> (9:5)
d ¼ n0(m0 À n0)À1(2m0U0 þ w) þ I,
and when these are substituted into (9.2) it becomes
p ¼ I À 21w À m0 þ n0 U þ s( m0 þ n0 U þ 2 À ) (9:6)
2 2 12w Àm0n0(U U0)2 Á
Here 2pU0À12 is the chemical wave-length and 2pU À21 the wave-length of
the Fourier component under consideration. w must be positive for case (d) to
apply.
If s be regarded as a continuous variable one can consider (9.2) or (9.6)
as relating s to p, and dp=ds and d2p=ds2 have meaning. The value of d2p=ds2
at the maximum is of some interest, and will be used below in this section. Its
value is
d2p ¼ À p(m0n0) Á 8p(m0n0) cos2 ps (1 þ wU0À1(m0 þ n0)À1)À1Á (9:7)
ds2 r2 m0 þ n0 N
The words you are searching are inside this book. To get more targeted content, please make full-text search by clicking here.
Home
Explore
Alan M. Turing, B. Jack Copeland - The essential Turing_ seminal writings in computing, logic, philosophy, artificial intelligence, and artificial life, plus the secrets of Enigma (2004, Clarendon Press_ Oxford Universit
View in Fullscreen
Alan M. Turing, B. Jack Copeland - The essential Turing_ seminal writings in computing, logic, philosophy, artificial intelligence, and artificial life, plus the secrets of Enigma (2004, Clarendon Press_ Oxford Universit
Discover the best professional documents and content resources in AnyFlip Document Base.
Search
Alan M. Turing, B. Jack Copeland - The essential Turing_ seminal writings in computing, logic, philosophy, artificial intelligence, and artificial life, plus the secrets of Enigma (2004, Clarendon Press_ Oxford Universit
- 1 - 50
- 51 - 100
- 101 - 150
- 151 - 200
- 201 - 250
- 251 - 300
- 301 - 350
- 351 - 400
- 401 - 450
- 451 - 500
- 501 - 550
- 551 - 600
- 601 - 622
Pages: