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

arise from the same simple ingredients:  cells 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? Or… white with black stripes?  

This is not a trick question.  The 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 published a  set 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 as “Turing 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”, the “generation 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  

chemicals “morphogens.” 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 know,  if 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  

ideas–diffusion and reaction–to explain patterns.

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

Simple reactions don't create patterns either.  Reactants become products and… that'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. What I  mean 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 system” may 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 middle,  so 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 diffuse,  throughout 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 animals'  skins 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 nature,  this 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 searching,  biologists 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 morphogens,  just 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 off.  Like 1s and 0s. A binary pattern of… digits.

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 Elizabeth,  nearly 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 it.  So 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 years,  saving 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 today,  including 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 reality,  it is to deepen our understanding of it.

Stay curious.