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 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.
