Alan Turing: Pioneer of Mathematical Biology

Alan Turing is best known today for his work, his contribution to cracking the Enigma code in World War II and v's theoretical work underpinning computer science. But what's often overlooked, and I think unfairly, is his pioneering work on one of the great challenges of biology, the mystery of what's known as morphogenesis. How is it that complex living organisms can develop, in particular, how do the complex shapes and forms and patterns, uh, that we see in the leopard with his spots and the zebra with stripes? How do those develop from, you know, the initial collection of tiny cells that look identical to each other, um, uniform? How does that happen physically? What's the process? So in today's lecture, I'm going to show you, uh, about the work that Alan Sherin did in this area in particular, he wrote a paper that came out in 1952 called The Chemical Basis of Morphogenesis, the Emergence of Form. That's what morphogenesis means, and I'm gonna talk you through what he did and how the ideas from that paper can explain how these patterns in animals conform, uh, in nature. I, so to begin with this paper that Alan Shering wrote, uh, he begins by saying that the mathematical model he's suggesting is a simplification and idealization, and consequently a falsification, but that he hopes the ideas underlying it. The essential concepts are going to be useful to describe, uh, the important salient points of what's going on. So in that spirit, I'm gonna begin with a very brief biographical sketch of Alan. She, um, who he was, and just a tiny recap of some of his other work, um, that he did, his contributions that he's really very well known for. And then we'll move on to talk about, uh, the contributions in biology. But we'll start a little bit with, uh, with plants anyway, because there's this beautiful, beautiful, uh, little drawing that Alan Turing's mother did, uh, of Turing. And this kind of is a, is a key perhaps to his character as a child. There's a hockey match going on. He's supposed to be playing hockey, but he isn't. He's over here just away from the action, just looking at the flowers and thinking about how they grow. So, you know, this is an early indication and he's interested in this kind of thing. How do things fit together? How do they work? Where do these beautiful patterns come from? And as a boy, yeah, he was interested in so many things. He had a wonderful imagination he made of imaginary languages. He was interested in patterns and, and in biology and all sorts of things like this. He was a bit of a dreamer. He did not enjoy school. Uh, he didn't get on particularly well at school. He, he didn't really know how to people <laugh>, you know, to, to, to play the games, to get along with, with the, with the other kids at school, because his mind was always just wandering off in different directions. Incredibly bright, of course. But even academically, he didn't get great marks. His teachers always going, oh, he's, he's disorganized. His writing is messy. Even in mathematics, you know, he makes silly mistakes cause he is not really paying attention, because of course he's thinking about some, you know, hyperdimensional object. Uh, he's thinking about some abstract thing. So even in maths lessons, you know, the maths he was thinking about while staring outta the window was not necessarily what he was supposed to be thinking about. So he did got sort of very, uh, mediocre reports, really, from the teachers at school. But it did become clear that he was exceptionally able in mathematics, and that this would be what he would study. So, uh, he, he was born in June, 1912. So this year, you know, we would celebrate his 11 first birthday. That's the nearest to a round number I'm gonna come up with. Um, but by 1931, he was off to Cambridge King's College to study mathematics. And here's the amazing thing, uh, by 1935, he already had a fellowship at Kings College. So that's kind of quite a stratospheric rise. Um, and, and it was, he was awarded this on the basis of a paper he wrote in Probability Theory, which is not anything that he's well known for doing. He just produced this amazing thing, got a fellowship. And then in 1936, uh, he produced a paper, and it's kind of the first thing I think is major contribution. And it was in what I guess we now call theoretical computer science. The question he was trying to answer was, what kind of things are computable? What could a machine do? It was a, it wasn't a machine that existed. It was a theoretical, sort of a thought experiment of a very, very simple, the simplest possible really idea for what a computing machine could be like. So the idea was you had this, I won't say an infinite amount of tape, but as much as you need, um, as much tape as you need. And the, the computer, the machine, we now call them Turing machines. They're these theoretical things. I think people have actually built physical examples now, but that's not the purpose of what he was talking about. The machine was very, very simple on this tape. Were just, um, zeros and ones possibly blank spaces. And as I said, um, what I do is gonna be a simplification, an idealization and a falsification of what's going on. But roughly speaking, you have zeros and ones on this tape. And all the machine can do, depending on kind of what state it's in, where it's at, at the program, at any point, it can change the symbol from a zero to a one or a one to a zero. It can leave it the same, and it can, if it wants, move one step to the left or one step to the right. On this tape, you can put, initially, you can have an input. So you can put your ones and nos, however you like on the tape. And then there will be a program that it follows, and then it'll give you an output. Now that is an extremely simple setup, just a couple of symbols. A very small finite collection of possible instructions. And yet Turing managed to show that a machine like this so, so simple, if any machine at all could do a particular calculation, then a Turing machine could do it. So this means it kind of reduced the question of what might be computable by machines to what is computable by this extremely simple machine. And this was a very important, uh, contribution to, to these kinds of questions. What could machines do in a totally different area? And this is much later, but in about 1950 after the war, he made another brilliant contribution. And this is in the area, what we might now call, well, we would call AI artificial intelligence. And people are asking, you know, these newfangled computer things, could they ever be as intelligent as a human? And he didn't like that question because it's, it's sort of why, why was, why is that the criterion for intelligence? They, they're gonna be as intelligent as machines. You know, it's a different kind of thinking, whatever that is. It's a different kind of intelligence potentially. Um, but if you really wanted to understand, could a machine at least mimic human intelligence enough to convince us that it is a human, then, you know, that might be some criteria for, for, for a test you could pass. And he called, he, he talked about this test. Um, here it is, uh, he called it the imitation Game. We now call it the Turing test. It's another thing named after Turing. So the idea is, um, you test your, your machine to see if it can pass as human by having a machine and a human somewhere. And they are sending messages and they are, those messages are received and you're having a sort of conversation with the tester who is a human being, who is trying to decide which of these two, uh, things that I'm interacting with, which one's the machine and which one's the human. And if the machine can fool at human and, and pass itself off as human, then it has passed the Turing test. So that's has become, you know, again, a very popular idea, um, when we're talking about ai. So these, this is kind of work loosely, you might think, call roughly in theoretical computer science. The thing that Turing is now most famous for is, of course, his work on Enigma in the Second World War at Bletchley Park. But he wasn't famous for that at the time. It's all official secrets act nobody knew about it for, for decades after, after this, you know, life-saving work that that team were doing. Nobody knew, um, which is, you know, real shame, um, the enigma machine. So this is what the Germans were using to send encrypted messages. This is a picture of one. It, as you can see, there's a very long number up there, which I'm not going to even try to say, but it's a big number. Something, you know, 159 septillion, I think, um, settings on this machine. So you can't just test them all. You can't crack this by brute force. Um, it takes a message, it gobbles it up in one of up to 159 sillion, uh, ways. And you don't know which one, if you were to try and test each, each setting to see if you got, you know, could decipher the message. Um, if you were managing to check a million settings every hour, it would still take you 300 million years to work through all the settings. So that is impossible. I mean, I think it would still be impossible, but certainly it was impossible at that time. So I have a hope of doing it by brute force, even with a machine. But what Turing did, the breakthrough that he made, which made this thing attractable problem, was you see at the front there, you've got this kind of, uh, board with little, little holes on and um, kind of wires that are going in between them. This plug board was part of the setup of the machine. And it, you swapped over, um, letters, 10 pairs of letters you swapped over as part of the encryption process. And the number of ways of doing that, even if setting the plug board is, is huge. And ensuring managed to find a way to take out that bit of the, of the process mathematically, sort of mathematical way to make the effect of that plug board go away. And that reduced the number of settings that would have to be checked, um, to a maximum of about a million. Now, that's still obviously a lot, but they had several of the machines, the bomb machines going, they would run them in parallel and they got to a point where they could crack the code within a few hours. Now that's, you might think, well, why? You know, that's great, but once you've cracked it, you've cracked it. The thing was, they were always up against a deadline. Cause this code, it was changed the settings every single day. So you have 24 hours to crack it, otherwise, you know, it's all pointless. So this was a huge, huge victory in kind of, uh, decrypting. This, this machine. Um, they got it down to a few hours each day, and then they could read all the messages and, and the Germans were none the wiser. And it's believed that this work of Turing and his team, uh, actually shortened the war by, by two years. So really saved hundreds of thousands, perhaps millions of lives, and nobody knew for 30 years. Um, so after the war then what do you do? You, Alan Turing, you've made groundbreaking, uh, contributions to like two different areas of mathematics. Well, of course, you then make a groundbreaking contribution to a third area. So this is the first page of his, his paper. We're not going to read this. Um, but this is, this is the first page of this paper, the chemical basis of Morphogenesis. I'm gonna tell you a bit about today. Um, he submitted it in 1951. It came out in 1952. And it, he was interested in, yes, how can mathematics help us to understand how patterns and shapes and, and forms of animals and plants develop. And this was something he had been interested in. I mean, we saw him watching the daisies grow <laugh>. Um, you know, when he was a young child, it was a lifelong interest. Uh, and even so, you probably know that he, you know, sadly tragically died young. He died in 1954. Um, he had had the bad experience after the war of, unfortunately, he was convicted of what was then gross indecency because he'd had a relationship with another man and he was forced to, um, undergo this horrific regime of, of taking large amounts of hormones to, you know, treat this and this. It was just had a terrible side effect and it was a really awful time for him. He died in, uh, in June, 1954, probably by suicide, will never absolutely know for sure. But, so his, his life is tragically cut short cause of this. So he, he died, you know, within two years, just after two years after this paper came out. We don't know what he would've gone on to achieve. But in his unpublished papers, uh, after he died, there was one called outline of the development of the daisy. So, you know, he was still thinking about daisies even all those years later. Now, why is it that a mathematician would be interested in biology? Well, this is not, you know, it's not new to find mathematics in biology. What he did was very new and interesting and will, will look at that. But it's often been remarked that forms in nature, the shapes of animals and plants. There's a lot of mathematics to them. For example, symmetry. So it's been known for a long time or observed that animals like humans have a kind of bilateral symmetry, a mirror symmetry left, right? There's vitruvian man, uh, to, to show us this symmetry is prevalent in nature. And one reason might be that it's, it's an efficient way of solving a problem. You know, if your solution works in one direction, then it's gonna work in the other direction, unless there are some other constraints that prevent it. So why not be left, right symmetric, that's fine. Now, with a big animal like a human, um, we, we are affected by gravity. We are moving around. So we can't be symmetrical kind of on the top, top to bottom on the vertical axis. Um, that, that wouldn't make sense. And also, if we are walking around, we've got eyes, we're looking forward, we are hunting. Um, we, we are, we do have a front and a back kind of necessarily. So we are not symmetrical in that, in that axis, but we can be symmetrical left, right? And that sort of halves the number of instructions somehow that it takes to make huge saying, well, do whatever you did over here, do the mirror image over here, kind of thing. I, I, I simplify, I idealize, I falsify. Um, but that's the basic idea that symmetry. Cause it's, it's an efficient way to make something. Just make it the same in all directions. If you have smaller, less sophisticated animals or plants, then you can have more symmetry cuz they have fewer constraints. So for instance, uh, another thing like this, but less sophisticated a starfish. Now these do do not have to be then they're less sophisticated animal. Um, but you can see that it's got this lovely five pointed star radial symmetry that it has. So that's kind of, uh, the next step, kind of two kinds of symmetry. But the tiniest animals, and these are things we could only start to see when microscopes got good enough, you can have even more symmetry. And this started to be seen for real, uh, in the late 19th century with the work of, for instance, Ernst Tekel, who was a German naturalist, uh, and artist. And he went on an expedition on a ship called the Challenger in, uh, 1875, I think. And he captured these, he drew these pictures using his microscope of tiny little sea creatures called radial aria. And these are sort of a 10th of a millimeter across. They're tiny. They are not affected by gravity, for instance. They're just sort of bobbing up and down in a lovely time in, in the, in the ocean. So they're not sort of affected by those kinds of constraints. Um, they, they are simple organisms. They don't, you know, need to see or anything like that. They're just, they're proto are they're single celled organisms. And as a result, they can be very symmetrical. And if you look at, for example, the one on the top left there, I mean, that looks like one of the platonic solids, the most symmetrical of, of all polyhedron. It looks like an icosahedron, a 20 sided, uh, shape made of equilateral triangles. Um, and it's, it's amazing that we see these shapes that are so, so symmetrical, look like mathematically perfect designs. It's wonderful to see this symmetry coin. So this, you know, his, his books, Kels books, and these beautiful, uh, drawings and paintings were very popular. You know, they caused kind of a sensation of, of, of amazement to see these tiny creatures for the first time. So we see a lot of symmetry in nature and, and there's the mathematics behind that. Another thing you see is in the, in the forms of plants and in particular in, in what's called otax, the way leaves are arranged on stems, the way the way seeds are arranged. So that's on the left, that's a sunflower. You see things like the Fibonacci sequence occurring. So that's the one that goes 1, 2, 3, 5, 8. And so on each term is the sum of the previous two. The reason you see it, um, is linked to the fact that the Fibonacci sequence is related to the, the famous golden ratio. I'm not gonna say anything more about that now, but I did give a talk last year on harmony and proportion where I talk a bit more about the golden ratio. So you can, you can go back and look at that if you like, but that's another way that mathematical patterns arise. And Turing certainly was interested in these and, and talked about fur cones and things, having these nice spiral designs. Um, there's another way that mathematics can impinge, and this was mentioned, uh, in among many other things in a huge and an influential book that came out in 1917 by Darcy Thompson. Uh, Darcy went with Thompson. So he wrote about, uh, this book was called On Growth and Form, and it was about how animals and plants grow and the forms of them. And his case was that these things are not just sort of randomly chosen things, arrangements, forms, designs, they depend on physical laws. So the size you are is relates to physical laws. The shape that you are for a given size relates to physical laws. And then, so there's mathematics behind those. And I'll just give you one example of the, of some of the things he was saying that actually you can see mathematically that different species of a particular animal. So here these are fish, different species, which superficially look different in shape and size. You can relate the shapes by very simple mathematical transformations. So in fact, they're much more similar to each other and could have almost the same, let's say, recipe to make them, um, they're more similar than, than they initially look. And so he gives examples of these transformations you can do. So on the top left there, you've got a kind of a sheer, a shearing kind of process. Uh, and then you've got other thing where you sort of are magnifying from a particular point. And over here on the right, you've got this sunfish, which is being produced from the fish on the left. Um, it's, it's shape can be obtained by taking these kind of, uh, rectilinear, uh, grid and turning some of the parallel lines into concentric circles. And all of these things can be precisely mathematically described and show us that these species that look quite different are actually really mathematically very relatable to each other. So again, it's a sort of simplification of, of what's the underlying structure going on. Um, couple other things, uh, spirals. We've seen this, I've mentioned spirals already, but Christopher Ren, uh, architect but also mathematician, he studied this kind of spiral. It's called a logarithmic spiral. The reason it's called that is, so when you make a spiral, what have you got? You've got a curve that, uh, the distance away from the center is increasing with the angle that you've gone through. And in the case of a logarithmic spiral, the relationship is a logarithmic one. The logarithm of the distance is proportional to the angle. Okay? So that's why it's called it. We see a lot of these type of spirals in nature, the shape of shells. But this is just, you know, this is a, this is a curve, a flat curve. Christopher Ren, um, came up with a way to create a kind of three dimensional version of this. And he postulated that the, um, the shapes that you could produce with his, with his equation is, is logarithm. Spiraling in three dimensions could, um, describe the shape of, of shells seashells or snail shells. Now, he didn't have a machine like I do a computer that I could just plug in the equations, tweak with the parameters and obtain these kind of pictures that look, I mean, just depending on what parameters you put into his, to his, uh, to his work, you can produce loads of different kinds of shell like shapes that really do look very like seashells that you might find. Um, so he didn't have a computer to try doing that. And this is a thing that Turing did have a computer, but you know, was not, uh, it couldn't do the, the amazing things that computers now could do. Uh, cuz we have lots more memory and and com print out images and things. Um, so that we'll see later that that perhaps slowed the progress on, on the kind of ideas that Juing had. Um, the final one, so this is, um, not something that during would've been aware of. He certainly was aware of Kels work and, uh, Darcy Thompson's work. I dunno if you knew about Christopher's work on shells. He probably did, um, the next one though, but, uh, anachronistic, but I wanted to mention it. Here is another interface between mathematics and biology, and that is, um, fractals. So on the left here, this is not a beautifully drawn picture of a plant. It's about the best I could do without computer help, right? But it's, it, that's, that's, there's a plant, isn't it? Lovely. But what we can do with the, with fractals, so fractals are produced using an iterative process, and what you do is you have an initial design and then you, you add into it or you, you replace components of that design with smaller copies of the hole that you've got. And you keep doing that infinitely many times. And if you do this with a design like this, you can get something that looks more like a plant structure and you can see how it might work in, in nature if you have a particular blueprint and then you reproduce that, um, over and over and over. And of course as these things grow, you reproduce and reproduce, you can see that it can kind of makes sense. It might be something that could happen in nature. So let's see how this very badly drawn plant, um, becomes a marginally more realistic. So this is step two. And you can see for example, that top left little stalk there has now been replaced where copy of the, of the whole design, and we do it again, step three, step four, step five, step six. And by now, this isn't looking so terrible. It's looking like a reasonable approximation of a, a sort of fern like plant. So fractal kind of designs where you see a lot in nature. And you know, I could not have drawn that by hand.<laugh> would've taken a month, but a machine can just do that very quickly. And so it can test out these sorts of, um, designs. And the, this has become possible because of computers being a lot better now. And, and we'll see this is, this has also been an effect that kinda the legacy of Turing's work has, has become more available to us when computers have improved. Okay? So we've seen the kind of background, there are links between maths and biology, what the things I've talked about here though. They give us the what, they may even give us the why, but they do not give us the how, how is this actually happening, you know, what physical processes can make these sorts of things happen. And the other ones I mentioned the leopard spots and the zebra stripes, and this is what Alan Turing was talking about in his paper. How do you get something like a tiny, you know, the, the, the initial cell that's going to turn into a adorable little tiger cub. How does that, which looks the same all over know to create at some point stripes? What's, what's going on? Of course, it doesn't know anything. There must be some physical process, a chemical process that is doing this, but nobody had any idea what that might be. This is what Turing is suggesting. And you know, there's this sentence about it's being, it's a simplification. So what he does is he says, look, here's proof of concept. He sets up, um, some examples for us in the paper. So it's a very simple setup, which is, you know, not necessarily biologically real realistic, but it's realistic enough. It gives us kind of the, the bare bones of the idea that could lead to some of the effects that we see. So what I want to do in maybe the next five, 10 minutes, I want to go through, uh, and show you this very first example he gives in his paper because it's not very often that, um, I have the joy of being able to say, look, here is an actual thing from, you know, a a paper written, you know, in the 20th century. It's not super, super long ago. And I can give you the actual things that are involved in that example. So I thought I would do that, but I've drawn some pretty pictures as well just to keep this. So if, if at any point I do, I I have to say there is, there is a differential equation coming up. So if that's worrying <laugh> just should let your eyes glaze over for that bit and don't worry, and we'll be back with, it'll only last five minutes, but for, but I thought it'd be really lovely to be able to show you this. Um, and especially if you, you know, have, uh, have done an A level in mathematics, this, this, you can go and actually write these things down. You can play with them, you can solve these equations yourself and see what happens. So it's, I really wanted to show you this. So, um, let's give you the example. And this is the kind of process that Alan chewing is suggesting. And you can get more complicated. You can, you can, you know, add more uh, chemicals to this, but this is the basic idea. So the, he doesn't know what the chemicals are going to be that are doing this, this morphogenesis this uh, helping form be created. So he calls them morphogen, so we dunno what they are, but let's call 'em morphos. And those are the two chemicals that are going to be interacting and causing effects to happen. So I've got one called x and one called y X is kinda the pink one, and the y is the green one. And there are two ways in which they interact. Um, one's diffusion and one's reaction, I'll talk about those. And in this very simple example that he's, it's sort of a toy example that has a nice mathematical solution to it, we can get more complicated, but then we can't solve exactly and we have to use numerical approximations and that's where computers come in. But if this one, we can solve it exactly. If you've got two cells next to each other and there are some interactions, so there's some of each morphogen in each cell and what's, what will happen in this system? So the first thing that happens is diffusion. And that's, that's the movement of these chemicals, these morphs between cells. And so diffusion happens when, for example, if you've got, you know, a big room and there's um, some gas released at somewhere in one end of the room, it will spread out. You know, if you spray air freshen at one end of the room, the smell from that will spread out through the room as the gas diffuses through the room, through the area of high concentration to low concentration. And the rate at which it happens, that diffusion will depend on the difference in concentration. So if you've got loads here and none here, then then that will, the, the spread will, the rate of that spread will be faster, um, than if they're more or less the same. So as the amounts gradually become closer and closer to equal, the diffusion slows down and eventually it would go to nothing when, when everything's equaled out. So you've got diffusion happening, um, I've got to put some symbols. So our, our morphogen X I'll say that the concentration in cell one is X one, the concentration in cell two is X two. And so in, in this, uh, in this picture I've got, I've got clearly got more of, of, of X in cell one than I have in cell two. So X one is greater than X two. So the diffusion will be that it will gradually be spreading from cell one to cell two. And Turing says, okay, I'm gonna put some numbers on this. The rate of this diffusion, so the rate of change of the the chemical, um, x will be half the difference of concentrations. I'm not giving any units or anything you can put your own units in as long as you are consistent with them. So that's what during tells us. So there's the sum spread. So here, for example, where I've got, um, five amount of, uh, of X in cell one, and I've got three in cell two. So five minus three is two, not 0.5 of that is one. So in this little tiny instant example, the rate of change, how quickly the concentration is changing is it would be one half the difference between, okay, that's diffusion and that is happening and, but it's the rate that it is the rate of diffusion, the rate of change of these chemicals is constantly changing as well because the amounts are changing. So it's this constantly evolving scenario. The other thing that's happening is that there are chemical reactions. So diffusion is happening and reactions are happening and you can, I mean there there are many kinds of chemical reaction and the rates that reactions happen at can be affected by the presence or absence of different chemicals. Some chemicals are catalysts, they speed up the rate of the reaction, some inhibit the rate of the reaction, and in some cases the chemical itself that's involved its presence can accelerate or inhibit the rate of reaction. So some things are self catalyzing, they, the more of of chemical X there is in in in the cell, the faster it will be produced. And this is what happens here. So the reaction's happening inside the cells. So once you are inside the cell, there's chemical reactions going on that produce or destroy these X and y morphogen. So in this particular example, Turing tells us that the production of rate of morphogen X in the first cell is this given by this, uh, formula five X one minus six Y one plus one. So in our case, so X one is the concentration in cell, one of morphogen x Y one is the concentration of, uh, of chemical Y in cell one. So in our example, I've got five x one is five, uh, uh, y one is three. So five five is 25 minus six times three 18. That's seven plus one eight. Okay? So in, in our little diagram we've got the rate at which X one is being produced. Uh, so that's the, the chemical X in cell one that is eight. So it's being produced, we're getting a net increase from this chemical reaction. So you need to ensuring does tell us what's happening with, with, uh, morphogen y and what's happening in cell two. Um, and you get a system of equations that govern the rates of change. So this is not the amount of the chemicals, but how quickly those are changing. So it's sort of like the, you know, distance traveled compared to speed, the rate of the rate of production of these things, things and the notation we use in mathematics for rates of change. Um, we've put a little dot above the, the variable. So I'm gonna, I'm gonna show you what the equations are. We are not gonna solve them in real time, but they can be solved <laugh>, they can be solved, um, um, using not too horrible mathematical techniques. There are some equations, they're just, don't worry about them, but they're there. And these are four things and they tell you how to calculate the changes in the concentrations of these chemicals. In the two cells you have four equations, four variables. And that means we can solve it in this case, um, and we can work out what's gonna happen in this system. Now the reason this example is a good one, you don't know this yet, I'm about to tell you, it's not obviously amazing, but it is amazing because this is set up by Turing as this kind of, uh, toy example to show that a tiny deviation from equilibrium, so this cell, you know, to begin with, it looks like it's just completely uniform all the way around. And then suddenly differences appear discrepancies. You get spots and stripes, you get things cut, you know, pigments are not evenly distributed around the skin, whereas to begin with they are. So how is this possible when things start off the same? And what he says is okay, nothing can ever be, you know, a cell is not a perfect sphere. There are always gonna be tiny little random fluctuations that are gonna deviate just a tiny weeny bit from total equality uniformness uniformity. And so this system actually gives us a way in which tiny teeny little discrepancies from an equilibrium point can balloon and mushroom and grow exponentially. So in this particular case, um, if you just happen to have everything be the same, completely the same. So all of the concentrations are just one, one amount per per cell. I'm not specifying what amount that is. If all of these things are equal to one, then if you put, if you plug those into the right hand side, if everything equals one, let me look at the top one. So X one and X two are equal to one, they cancel each other out. Uh, X one is one. So you've got plus five minus six, but plus one. So on the right hand side that becomes zero. That's true for all of these equations. What that means is the rates of change are zero, there isn't any change happening. So if everything is, is equal to one, it stays where it is, it's equilibrium. Nothing changes, there's no diffusion cause all the concentrations are equal, the any reaction and, uh, production and destroying of things cancels out. So you have equilibrium, but, but, um, this is what the example you use in the paper if you just deviate a tiny little bit, um, from perfect equality. So his examples has the concentrations in cell one are just slightly above one. Next one, Y one, there's slightly bigger than one. And in cell two, there's slightly less than one, not very much, just a little bit. Then something interesting happens, you can plug those numbers into these equations again. And if you do this, you find that the change in X one how quickly it's changing, that's a positive number. That means X one is growing, X two is shrinking by the same amount, Y one is growing, Y two is shrinking by the same amount. What this means is we started off with tiny, teeny bit more than more in cell one than in cell two. But the direction of travel is to increase that discrepancy. And if you actually solve these equations, you find that the increase in the discrepancy actually goes up. It's an exponential rise. So at this point, let's dispense of the equation and just solve them and show you the graphs of what happens at the initial starting point. All of these numbers, these are the concentrations, X one, Y one, that's what's in cell one X two, Y two, that's what's in cell two. The concentrations initially are all almost identical, almost identical, but then they start to diverge exponentially. And so you reach a point where cell one has loads of these and cell two has almost none. And so you've started off from an almost uniform position, just a tiny random fluctuation has created, um, within, you know, not much time, completely different situations. So, you know, maybe this, the cell one is gonna be a spot <laugh> and cell two is gonna be the gap between the spots. So this is kind of how it works. It's a proof of concept that chewing, uh, sets out. He goes on in the rest of his paper to do, uh, lots of, uh, examples, perhaps slightly more biologically realistical realistic examples. He goes up to kind of two dimensions and you know, he, at some point there's a diagram in the paper which gives using these kind of ideas, the reaction diffusion model. He shows that you could get, uh, a dappling sort of pattern like you might see on a, a frisian cow or something. This can come out of a reaction diffusion system as he's set out mathematically. So this is a really interesting idea and you know, it was, it was very interesting for people. The paper dealt with kind of one and two-dimensional setups, so, you know, skin pigmentation. But after the paper came out, so in 19 52, 3 start of 1954, he started thinking about what about the three-dimensional kind of analog of the equations he'd created. And the very simplest kind of three-dimensional creatures he could think of were those radial aria that he had read about in Hale's work, these strikingly symmetrical, single celled organisms which start out as essentially ace sphere, but then the, the kind of adult if you were, if you will, version of them. Um, different kinds of them have different numbers of spikes that are coming out. So we saw some examples of that in the picture I showed you. Um, could, could he use these reaction diffusion, uh, models to kind of replicate these different species of radial laia? Some of them have two spikes, some of 'em have six, some of them have 20. Could you, could you produce this? And this is, you know, essentially if you think of a spot or spot or something, instead of spots on your sphere, you have these spikes, but somewhere, you know, in particular places, can we produce this? And he was working with a, a student of his, a research student called Bernard Richards. And they did not have, you know, modern computers. What they had was an amazingly, you know, brilliant new thing. The fer mark one machine, which it, on the left, it's shown now there's the sort of console there on the desk there, that's not the whole machine. It filled all of these cupboards in a great big room. Um, and it, you know, it was very advanced for its time. Not one of the first computers really, but the output was not, uh, it could not produce images. Not like any graphics calculator now can draw your graph. This machine could not do that yet. All it could do is give you numbers. So Turing, uh, uh, set be Richards' student. The question of can we produce a three dimensional version of the reaction diffusion model that will allow us to kind of predict to model the shapes of these very simple, simple kind of spherical things, but with spikes in various places. And so, um, with those generalized equations, with the three dimensional version we wanted, or they wanted to know what, what's the output, what kind of shapes are possible? So they couldn't get the ferranti to actually print out a picture of what it had got, draw a graph. So what Bernard Richards had to do, this is not the actual output by the way. I've just, I've just done a grid of figures, but this is the sort of thing it looked like he produced. If you want to sort of see what's happening on the surface of a sphere, you basically look at the latitude and longitude. And if you want to see a spike, you need to know how far away from the center of the sphere is, is the point on the surface of your organism. So the spikes are gonna be suddenly very far away, surrounded by uh, uh, kind of roughly elsewhere, uniform, constant distance, the radius of the sphere. So he, he got the machine, the ferranti, to print out an array. So latitude and longitude of every degree, kind of zero to 90. And he did two pages of this. So you could see the front and the back. So zero to 90, zero to 90 in this array. And the number for a given latitude and longitude was how far from the center of the sphere is the, the point on the animal. So if it could be here for, if it's just part of the sphere or it could be further away if it was one of these spikes. And so then what you do, he had this array of, uh, figures and he drew contour lines on it. So he found the places of equal distance and he joined them together and drew contour lines. And that enabled him to create this, you know, rudimentary picture of what was predicted by the model with different parameters. And so he told Shering, um, that he was going to sort of gonna have a look at this on the machine. Um, it sort of towards the end of May, 1954, he told Shering about this and Shering said, right, he hadn't mentioned radio aria to Bernard Richards yet cuz he didn't want to, perhaps he didn't wanna jeopardize, you know, make him deliberately produce things that he knew what he was looking for at that point, Shering says, right, go to the library, get Kels book, look at the pictures of Radiolaria and see if any of them match up with what your model produces. And so Bevi Richards did this and he was delighted. He found that he was getting things like this. He was able to produce using the reaction diffusion model, uh, outputs of, you know, spheres with these spikes in the right places with 20 spikes with six spikes just like the Radiolaria. And here's the tragic thing. He had an appointment with Turing to tell him about his progress. The appointment was for June the eighth, 1954. Turing died on June the seventh. So he never got to see that his work ha had successfully predicted the shape of these radio aria. But it did, and this is kind of an a lovely early thing, even with these perimeter computers, they managed to do it. So what happened after this? What's the legacy of, of Turing's work? Well, it's interesting, the theory itself kind of, it feels like to me it sort of mirrors what happens in the theory, the reaction diffusion, uh, cause basically nothing seemed to happen for about 40 years. And then suddenly, apparently out of nowhere things happened, just like in the reaction diffusion model itself, everything looks uniform and then suddenly structure, but it's not really sudden. It's an exponential growth. And there, there are a couple of reasons for this. The first thing is that, um, at almost the same time as this exciting new paper on Morphogenesis comes out, 1952 and it was published and the biologists were interested and thought it was fantastic, but then something that drew their attention away from Morphogenesis and that was DNA and the structure of DNA being discovered. And quite understandably, uh, everyone's like, oh, let's, we've gotta find out about dna, we've gotta investigate this. And so that was became instantly a focus and perhaps thinking about the, the Turing's work on Morphogenesis got slightly left for a while. Another problem, as I've mentioned, computers were not, um, you know, powerful enough to give, to show us what the numerical, uh, approximations to, to solutions to these kind of systems could give you. So that kind of held things back a little bit. Um, but then there what, you know, when computers got good enough, kind of a similar time as was when everyone was thinking about fractals can, we just started to be able to actually do these kind of calculations. And it was realized that the crucial thing about the reaction diffusion, uh, models was that you have, um, your, your chemicals involved. You have to have what are called activators and what are called inhibitors, and that's what's kind of making it tick. So the activator, these are still these morpheus. So activators like our morphogen X in our example, they are self catalyzing. So what does that mean? It means the more of X there is the faster the rate at which it's produced. And we saw in one of the equations that we had, we had kind of five x minus six Y, so that's five x, that's a positive coefficient. So that's helping the production of x. And the other chemical is called an inhibitor. It slows down the production. So in our equation we had minus six Y. That minus sign is telling you that the presence of Y is slowing down the production of X. So you've got one thing that's self catalyzing producing more and more of itself. It's being slowed down by the other morphogen, the inhibitor. And the crucial thing is that the inhibitor should diffuse faster, it should spread through the substance faster than the activator. What that means is there's a tendency to kind of get islands of activator activity surrounded by a sea of inhibitor. So you could sort of clumps of particular different things surrounded by the rest of what's of, of the fluid or the substance, the tissue. So that's that kind of system is what is what is making all this tick. Um, other developments. So computers are getting better. We understand the under what's going, you know, what kind of processes chemicals will lead to a successful, uh, production of these Turing type patterns. And we can also look at having more than two morphogen. So the example I've got here is it's recently been shown that they kind of shape the, the, the walls, the patterns, um, of fingerprints can be produced by a system of three morphogen or interacting with, uh, reaction and diffusion and producing these kind of, they are kind of striped, but they're curvy. And this can be produced with three morphogen. Um, so let's look then at, you know, now we've got the technology to be able to model these sorts of behaviors. Let's look at some examples. And the first thing to say is that size is important when you are laying down these pigmentation patterns, like on the, on the skins of animals and those pigments, the size of the animal matters. And I don't mean the eventual size, I mean the size it is, you know, in, in, uh, in the growth pros, in the, in the development process, the size of the embryo of that animal or the fetus. Um, when the patterns are laid down and the reason is that diffusion happens, um, it's sort of, it's a local process, things spread to their nearby areas. The bigger the animal gets, the, the, the, the less the impact of diffusion because, you know, things can't diffuse to the furthest extremes. And so some solutions become then not stable and can't sustain. Uh, so this is, I'm such an artist <laugh> here, here is, here is a very well depicted animal. Um, it's small, right? It's small and I, it basically mated it with, uh, two cylinders, one cylinder for the body, one for the tail, and it's for small sizes. Stripes are, um, a stable solution. You can get stripes, big sizes, stripes don't work anymore. It's kind of like collapsing a wave form. But you can get spots and there's a intriguing gap in the middle here where we could postulate that, okay, maybe there's a point in between where the body is too big for stripes, but the tail is still strippable. I've just invented a new word. You could this happen. So here's the question. So then we go and look at nature and we say, okay, we know we can get animals that are all stripes. That's lovely tiger. We know we can get animals that are all spots the leopard, but have a look at this cheetah stripy body. We get stripes on the tail button, see the last sort of third or half of the tail, uh, we get spots on the body, on the tail starts off stripes, and then look at it starts off spots and then look, there's stripes at the bottom because the tail was just small enough still at the end of the tail to still have stripes be be a solution to the the equations. So you can get these kind of intermediate steps. But you know, the theorem of today is you can have stripy body and uh, you can have spotty body and stripy tail, but you cannot have, there cannot be, you know, if this is all correct, you cannot have an animal with a stripy body and a spotty tail and you will never see such an animal. She said confidently <laugh>. So yeah, you know this, this comes out of the theory and it, and it's shown, you know, in nature you can get all stripes, you can get all spots, you can get spotty body at stripy tape. Okay, so next exciting thing that's coming now I've said that, um, machines can, can model these kind of, uh, reaction diffusion systems and there are lots of online, uh, places where you can go, where you can try these things for yourself. So if you look on the transcript, I've put a link to one of these, um, and I'll show you a couple of things that I've, I've made using it, um, by tweaking the parameters. You can, you can affect just slightly what happens by mimicking, you know, what's the size of the animal, how fast are things, react, reacting, and so on. Um, but there were several of these things online. So you'd go and investigate and see what you can manage to produce. So I thought I would try to produce, uh, spots and I've started with, well you can see here's Gresham written, but now the reaction is gonna start. And uh, you can see there are still some bits of lines, but they are getting sort of chopped up as the inhibitor is having its effect. We are isolating more and more these little islands. And this one might go, I don't know, it's one or two just, yeah, it's gone. Okay, <laugh>, so this is spots, right? So you can produce these with these two chemicals. Um, next I thought, let's try for stripes. So you have to change the parameters a little bit. And here, here are, I'm trying to make sort of lots of lines being produced and you can see it's gradually diffusing through the, the, the, the tissue I guess is what we're mimicking. And the smaller, shorter lines, some of them are joining up with each other. And you can see that this is a developing pattern, but little tiny dots are kind of being gathered in by longer lines. And the effect is that you get these kind of parallel curves developing. And if you look, so it's not kind of straight like the, the tiger, but if you look at, for example, this is a close up of a zebra, you can see that you get exactly this kind of pattern, um, developing. And so, and you can also imagine, you can think of other, uh, animal plant forms where you might see this, I just made me think perhaps of coral or something like that, patterns on leaves. Um, so you can produce these, it's quite fun to play with them. Also a bit mesmerizing. You may not get any work done or the rest of the afternoon while he just watched these things developing. Um, so these are the kind of patterns that we can produce that can mimic with these systems. And crucially, it's not just saying we can make things look like nature, but it's giving a mechanism for how it might actually be done. In, in, in biological systems you need these morphin, these chemicals, activators inhibitors. Um, but this tells you that this is kind of a recipe for how it can happen and that's what makes it really different and important. And the legacy of Turing's work is it's profound more and more, you know, every day almost there's a new paper which says, here's a Turing pattern, here's another Turing pattern, you know, whether it's fingerprints or animals, pigmentation or, um, they, I saw one that was like the hair follicles on, on particular animals' bodies are arranged in these kind of Turing patterns. Um, it even goes beyond biology. So I'm gonna give you just a another couple of examples to finish up. One is, look at this. This is kind of the ripples that created in, in, in, in the desert in the sand when wind is blowing across. Obviously there's not a chemical process there, it's a physical thing. But if you could think of the activator inhibitor, um, set up, if you imagine that the sand's essentially flat to begin with, maybe a tiny little fluctuation that gives a mini ridge. Once you've got a mini ridge that's happening, that's just a little bit more likely to capture sand grains than the surrounding area. And so it's inclined to get a little bit bigger. So that's kind of like the activator, but then the surrounding area is gonna get less sand, grains of sand because they've been captured by the ridge. So not only is the ridge making itself get bigger, but it's inhibiting the presence of ridge other ridges nearby. So you don't get one big, uh, Hillel with these, with the wind blowing cross, you get these ripley effects that look exactly like those Turing patterns that we were generating before. So that's kind of sand dunes. The other example is, well, it's to do with animals, but it's homo sapiens and that's, that's crime rates not committed by this cow. That's just to show you <laugh>. This is like the dappling effect that Turing predicted even in 1952. But this is a map of crime, crime hotspots, and you can produce, you can model these effectively using, um, the reaction diffusion kind of models. And in this case, the activator is, um, the fact that once there are crimes committed in an area, it seems to be more likely that there'll be more crime. You know, if you've a burglar has successfully, um, got away with robbing a house, uh, in a particular street, then they're more likely to go back. They think they have success and this has been shown to be the case. So the kind of the mor crime breeds crime, right? Um, and what's inhibiting it, well, hopefully law enforcement is the inhibitor here. And you can, using these things, you, you can produce these, I mean, this is a real crime hotspot map, but you can model them mathematically using the, the same kinds of equations as, as Turing used for, for biological situations. Um, so I hope what I've shown you today is the immense reach of this one brilliant idea of Turings and the way he carried it through, set up the mathematical model that have all these far reaching applications, both inside biology and beyond biology. Um, and with discovering more every day. Um, so this was the last academic lecture of the academic year from me, but I'm coming back in the autumn with the, with my fourth and final year as Gresham Pressive geometry, starting with the massive board games on the 10th of October. But if you cannot wait that long, which I'm sure you can't, cause maths is fantastic. Never fear.

00 PM Robin Wilson, our friend, former Gretchen, professor of geometry himself will be giving a talk called Connecting the dots, milestones in Graph theory. So that will keep you going. Thank you. Super. Well, um, there are plenty of questions online. I'm sure there are some in the, uh, audience too. Um, let's start with an online one. Um, so someone in the audience was icing violets. She was making a cake. I said she could have been, he could have been anyone, right? They're making a cake. And uh, what they said was that they noticed that every violet on the planet was different and they wondered, uh, how that happened. And then I think what they might have meant there was, when you look at variation in nature, how can you tell what caused the variation? Because what seems to be happening a lot here is you sort of hold up, you say, oh, well I've forgot this equation looks a bit like a cow q e d <laugh>. Could you talk a bit more Mr. Farra about how you can really know it was caused by reaction diffusion or, and many of the other interesting techniques, rather than just sort of, Hey look, it looks Cool. It looks cool. Yeah. And, and this is, this is kind of, it almost, it stands or falls on this exact question, right? You can, you can make things look like other things <laugh>, but it's how can we be sure. And I think this is one of the great things that's happened in the last few years really with, with this kind of idea, the morphogenesis idea, Turing patterns, is that they've actually found, you know, the, the chemicals, the reagents that are doing this. So with the fingerprints paper, I'm pretty sure that they, it wasn't just, oh, look, we can make something that looks like a fingerprint. They really did find that there are three chemicals that are reacting and diffusing in this way. So it's not just, it looks like this, which might be useful if you are a computer animator or something, but it, these things really do seem to be actually present and happening, um, in, in at least, you know, several of the biological systems that we've been able to study recently. Yeah. So that's a mark of a great scientist, isn't it? Not only that, did it look right? Yes. But it also happened to match the physical reality. Exactly. And you're not Only you, you a good scientist. You're lucky, You're lucky. Lucky. And I mean, this is sort thing, so in 1952, I mean, it's just perhaps a bad year to be making suggestions about how things are happened genetically because you know, then DNA comes and potentially blasts everything outta the water. But it's the fact that yeah, that this idea does, does have relevance and does work, um, perhaps using different terminology. And of course he was carefully said morphogen, he wasn't specifying what he thought they were. But still it has been found that there are, there really are places where exactly this model is happening. Yeah. Alright. Could that model predict a zebra with both polka dots and stripes? Well, I think you would have to. I mean, the size thing really matters. So I think you would have to think about how that would be laid. Like if you've got the same basic recipe, I don't think you could have both of those things at the same time. Unless it was one part of the body, it was smaller than that. So theoretically you could imagine animal that perhaps like its legs had stripes and its body had spots. But I don't, I I think it'd be difficult to have both of those on parts of the animal that are the same size because of what I've said about the stability of the, you seen Zebra Canada, You've seen that zebra in Canada. I spoke, Well if you, but if you look at what's going on. So he's showing me a picture with some spots in some str, but the stripes kind of, I mean, look, they're kind of on the, on the leg, on the thigh and the leg. So this is a kind of a smaller part of the body at least. So, you know, I'd have to, I'd have to look at, uh, the exact thing, <laugh>, but I was full the other day by a picture. It was supposed to be a giraffe with dwarfism, and it looked very, very cute. But then, uh, someone told me it's photoshopped. I'm not, I'm not saying that is <laugh>. I'm not saying that is for real. That's for real. Yeah, yeah, yeah. It's not a, it's not a generative AI is, it's a it's a real one. Yeah. Sorry. But I'm really not agreeing with this size and spot. It's cheaper than it's j and the tigers actually the biggest, so I don't think the size is the Thing. Not, well, as I said, <laugh>, it's not the size of the eventual animal, it's the size that, that it is, you know, in utero when the pattern is laid down. So what we, if it's stripy, that means that that stripy pattern is laid down when the, the, the creature is small enough to be, to be able to have stripes. And then once it's set, you know, once it's the paint's dried, then, then it stays like that. So it's not the eventual size of the animal, it's when the pattern develops. I mean, what I interpreted due to, perhaps I misunderstood this, was when you look at a pattern, you can tell whether the domain of influence or its genesis was big or large. Exactly. Exactly. Yeah. Yeah. So that's where the size comes in. Yeah. That was a marvelous talk. Thank you. I really enjoyed it. Uh, members of the audience and those online, let me draw your attention to this fantastic book, once Upon A Prime. It was written by Professor Hart is really good read. Thank, thank you. Um, if you're looking for early Christmas presents or birthday presents or so on, it's thoroughly recommended A good beach read. And You're, I think you're going to be on radio four to 32 couple weeks. Yeah. So end of the season, end of a lecture, and a fantastic one. Thank you very much. Thank.