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  • The living world is a universe of  shapes and patterns. Beautiful, complex,  

  • and sometimes strange. And beneath all of  them is a mystery: How does so much variety  

  • arise from the same simple ingredientscells and their chemical instructions?

  • There is one elegant idea that describes  many of biology's varied patterns,  

  • from spots to stripes and in between. It's a code  written not in the language of DNA, but in math.

  • Can simple equations really explain something  as messy and un-predictable as the living world?  

  • How accurately can mathematics  truly predict reality?  

  • Could there really be one universal  code that explains all of this?

  • [OPEN]

  • Hey smart people, Joe here.

  • What color is a zebra? Black with white  stripes? Orwhite with black stripes?  

  • This is not a trick questionThe answer? Is black with white  

  • stripes. And we know that because some  zebras are born without their stripes.

  • It might make you wonder, why do zebras have  stripes to begin with? A biologist might answer  

  • that question like this: the stripes aid in  camouflage from predators. And that would  

  • be wrong. The stripes actual purpose? Is most  likely to confuse bloodthirsty biting flies. Yep.

  • But that answer really just  tells us what the stripes do.  

  • Not where the stripes come from, or why  patterns like this are even possible.

  • Our best answer to those questions  doesn't come from a biologist at all.

  • In 1952, mathematician Alan Turing publishedset of surprisingly simple mathematical rules  

  • that can explain many of the patterns that we  see in nature, ranging from stripes to spots  

  • to labyrinth-like waves and even geometric  mosaics. All now known asTuring patterns

  • Most people know Alan Turing as a famous wartime  codebreaker, and the father of modern computing.  

  • You might not know that many of the problems that  most fascinated him throughout his life were,  

  • well, about life: About biology.

  • But why would a mathematician be  interested in biology in the first place?

  • That's a really good question!

  • I'm Dr. Natasha Ellison, and I'm from the  University of Sheffield, which is in the UK.

  • I think so many mathematicians are interested in  biology because it's so complicated and there's  

  • so much we don't know about it. If you think  about a living system, like a human being,  

  • there's just so many different things going  on. And really, we don't know everything.

  • The movements of animals, population  trends, evolutionary relationships,  

  • interactions between genes, or how  diseases spread. All of these are  

  • biological problems where mathematical models  can help describe and predict what we see in  

  • real life. But mathematical biology is also  useful for describing things we can't see.

  • Joe (05:44) What do you say when people ask,  

  • why should we care about math in biology?

  • Natasha (05:54): Why should we care  

  • about what mathematics describes in biology?  

  • The reason is because there's things  about biology that we can't observe.

  • We can't follow every animal all the time  in the wild, or observe their every moment.  

  • It's impossible to measure every gene and  chemical in a living thing at every instant.  

  • Mathematical models can help make sense of  these unobservable things. And one of the  

  • most difficult things to observe in biology is  the delicate process of how living things grow  

  • and get their shape. Alan Turing called this  “morphogenesis”, thegeneration of form”.

  • In 1952, Turing published a paper called  “The Chemical Basis of Morphogenesis”.  

  • In it was a series of equations  describing how complex shapes like these  

  • can arise spontaneously from  simple initial conditions.

  • According to Turing's model, all it takes to form  these patterns is two chemicals, spreading out the  

  • same way atoms of a gas will fill a box, and  reacting with one another. Turing called these  

  • chemicalsmorphogens.” But there was one crucial  difference: Instead of spreading out evenly,  

  • these chemicals spread out at different rates. Natasha (15:49): 

  • So the way that we create a Turing pattern is  with some equations called reaction-diffusion  

  • equations. And usually they describe how two  or possibly more chemicals are moving around  

  • and reacting with each other. So diffusion  is the process of sort of spreading out.  

  • So if you can imagine, I don't knowif you had a dish with two chemicals  

  • in (GFX). They're both spreading out across the  dish, they're both reacting with each other.  

  • This is what reaction-diffusion  equations are describing.

  • This was Turing's first bit of  genius. To combine these two  

  • ideasdiffusion and reactionto explain patterns.

  • Because diffusion on its own doesn't create  patterns. Just think of ink in water

  • Simple reactions don't create patterns eitherReactants become products andthat's that.

  • Natasha (20:48): Everybody thought back  

  • then that if you introduce diffusion  into systems, it would stabilize it.  

  • And that would basically make it boring. Whatmean by that is you wouldn't see a lovely pattern.  

  • You'd have an animal, just one color, but actually  Turing showed that when you introduce diffusion  

  • into these reacting chemical systems, it can  destabilize and form these amazing patterns.

  • A “reaction-diffusion systemmay sound  intimidating, but it's actually pretty simple:  

  • There are two chemicals. An activator & an  inhibitor. The activator makes more of itself  

  • and makes inhibitor, while the  inhibitor turns off the activator.

  • How can this be translated to actual biological  patterns? Imagine a cheetah with no spots. We  

  • can think of its fur as a dry forest. In this  really dry forest, little fires break out.  

  • But firefighters are also stationed throughout  our forest, and they can travel faster than the  

  • fire. The fires can't be put out from the middleso they outrun the fire and spray it back from the  

  • edges. We're left with blackened spots surrounded  by unburned trees in our cheetah forest.

  • Fire is like the activator chemical: It  makes more of itself. The firefighters  

  • are the inhibitor chemical, reacting  to the fire and extinguishing it. Fire  

  • and firefighters both spread, or diffusethroughout the forest. The key to getting  

  • spots (and not an all-black cheetah) is that  the firefighters spread faster than the fire.

  • And by adjusting a few simple variables like that,  

  • Turing's simple set of mathematical rules  can create a staggering variety of patterns.

  • Natasha (34:18): These equations that  

  • produce spotted patterns like cheetahs, the exact  same equations can also produce stripy patterns  

  • or even a combination of the two. And that depends  on different numbers inside the equations. For  

  • example, there's a number that describes how  fast the fire chemical will produce itself.  

  • There's a number that describes how fast  the fire chemical would diffuse and how fast  

  • the water chemical would diffuse as well. And all  of these different numbers inside the equations  

  • can be altered very slightly. And then we'd see  instead of a spotted pattern, a stripy pattern.

  • And one other thing that affects the pattern  is the shape you're creating the pattern on.  

  • A circle or a square is one thing, but animalsskins aren't simple geometric shapes. When  

  • Turing's mathematical rules play out on irregular  surfaces, different patterns can form on different  

  • parts. And often, when we look at naturethis predicted mix of patterns is what we see.

  • We think of stripes and spots as very  different shapes, but they might be  

  • two versions of the same thing, identical  rules playing out on different surfaces.

  • Turing's 1952 article was…  largely ignored at the time.  

  • Perhaps because it was overshadowed by  other groundbreaking discoveries in biology,  

  • like Watson & Crick's 1953 paper describing  the double helix structure of DNA. Or perhaps  

  • because the world simply wasn't ready to hear the  ideas of a mathematician when it came to biology.

  • But after the 1970s, when scientists  Alfred Gierer and Hans Meinhardt  

  • rediscovered Turing patterns in a paper of  their own, biologists began to take notice.  

  • And they started to wonder: Creating biological  patterns using mathematics may work on paper,  

  • or inside of computers. But how are these  patterns *actually* created in nature?

  • It's been a surprisingly sticky question  to untangle. Turing's mathematics simply  

  • and elegantly model reality, but to truly  prove Turing right, biologists needed to  

  • find actual morphogens: chemicals or proteins  inside cells that do what Turing's model predicts.

  • And just recently, after decades of searchingbiologists have finally begun to find molecules  

  • that fit the math. The ridges on the roof of  a mouse's mouth, the spacing of bird feathers  

  • or the hair on your arms, even the  toothlike denticle scales of sharks:  

  • All of these patterns are sculpted in  developing organisms by the diffusion  

  • and reaction of molecular morphogensjust as Turing's math predicted.

  • But as simple and elegant as Turing's math  is, some living systems have proven to be  

  • a bit more complex. In the developing  limbs of mammals, for example, three  

  • different activator/inhibitor signals interact in  elaborate ways to create the pattern of fingers:  

  • Stripe-like signals, alternating on and offLike 1s and 0s. A binary pattern ofdigits.

  • Sadly, Alan Turing never lived to see  his genius recognized. The same year he  

  • published his paper on biological patterns, he  admitted to being in a homosexual relationship,  

  • which at the time was a criminal offense in  the United Kingdom. Rather than go to prison,  

  • he submitted to chemical castration treatment  with synthetic hormones. Two years later, in  

  • June of 1954, at the age of 41, he was found dead  from cyanide poisoning, likely a suicide. In 2013,  

  • Turing was finally pardoned by Queen Elizabethnearly 60 years after his tragic death.

  • Now I don't like to make scientists sound like  mythical heroes. Even the greatest discoveries  

  • are the result of failure after failure and are  almost always built on the work of many others,  

  • they're never plucked out of the aether and  put in someone's head by some angel of genius

  • But that being said, Alan Turing's work decoding  

  • zebra stripes and leopard spots leaves no  doubt that he truly was a singular mind

  • Natasha (37:55): The equations that produce these patterns,  

  • we can't easily solve them with pen and  paper. And in most cases we can't at all,  

  • and we need computers to help us. So what's really  amazing is that when Alan Turing was writing  

  • these theories and studying these equations, he  didn't have the computers that we have today.

  • Natasha (39:01): 

  • So this here is some of Alan's Turing's notes  that were found in his house when he died.  

  • If you can see that, you'll notice  that they're not actually numbers.

  • Joe (39:17): It's like a secret code!

  • Natasha (39:20): Yeah. It's like a secret  

  • code. It's his secret code. It's in binary  actually, but instead of writing binary out,  

  • because you've got the five digits, he had this  other code that kind of coded out the binary. So  

  • Alan Turing could describe the equations  in this way that required millions of  

  • calculations by a computer, but  you didn't really have, you know,  

  • really didn't have a fast computer to do itSo it would have taken him absolutely ages.

  • Joe (40:15) What has  

  • the world missed out on by the  fact that we lost Alan Turing?

  • Natasha (40:25): It's extremely hard  

  • to describe what the world's missed out on  with losing Alan Turing. Because so often he  

  • couldn't communicate his thoughts to other people  because they were so far ahead of other people  

  • and they were so complicated. They  seemed to come out of nowhere sometimes

  • Natasha (25:52) When you read accounts  

  • of people who knew him, they were saying the same  thing. We don't know where we got this idea from

  • Natasha (40:42) So what, what he could have achieved.  

  • I don't think anyone could possibly say. Natasha (42:14) 

  • I have no idea where we would have got  to, but it would have been brilliant.

  • One war historian estimated that the work of  Turing and his fellow codebreakers shortened  

  • World War II in Europe by more than two yearssaving perhaps 14 million lives in the process.

  • And after the war, Turing was instrumental in  developing the core logical programming at the  

  • heart of every computer on Earth todayincluding the one you're watching this video on.

  • And decades later, his lifelong fascination  with the mathematics underlying nature's beauty  

  • has inspired completely new questions in biology.

  • Doing any one of these things would be worth  celebrating. To do all of them is the mark  

  • of a rare and special mind. One that could  see that the true beauty of mathematics is  

  • not just its ability to describe realityit is to deepen our understanding of it.

  • Stay curious.

The living world is a universe of  shapes and patterns. Beautiful, complex,  

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The Mathematical Code Hidden In Nature

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    Ψ( ̄▽ ̄)Ψ に公開 2022 年 06 月 02 日
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