The Mathematical Code Hidden In Nature
Summary
TLDRThis script explores the mathematical underpinnings of nature's patterns, focusing on Alan Turing's 'reaction-diffusion' theory. It reveals how simple equations can explain complex biological phenomena like zebra stripes and leopard spots. Despite initial skepticism, Turing's work has been validated by recent biological discoveries of actual 'morphogens'. The narrative also poignantly touches on Turing's tragic life and his profound contributions to both computing and biology, emphasizing the beauty and power of mathematics in understanding our world.
Takeaways
- 𧏠The diversity of life's patterns can be explained by simple mathematical equations, challenging the notion that biology is too complex and unpredictable for such simplicity.
- đŠ The mystery of zebra stripes is not just about camouflage but also serves to confuse biting flies, highlighting the multifaceted purposes of biological patterns.
- đą Alan Turing's mathematical model, known as 'Turing patterns', provides a framework for understanding how complex biological patterns emerge from simple chemical interactions.
- đ Turing's work in mathematical biology was initially overlooked, possibly due to the overshadowing of DNA's discovery and the skepticism towards a mathematician's contribution to the field.
- đ§Ș The concept of 'morphogens', as introduced by Turing, refers to chemicals that spread and react differently to create various biological patterns, a key to understanding morphogenesis.
- đ The reaction-diffusion equations describe how two chemicals interact and spread, leading to the formation of patterns like those seen on animals' skins.
- đ By adjusting variables in Turing's equations, such as the rate of chemical production and diffusion, different patterns like spots or stripes can be generated.
- đ± The rediscovery of Turing patterns in the 1970s by Gierer and Meinhardt spurred further interest in applying mathematics to understand biological development.
- đŹ Recent biological findings have identified actual morphogens, supporting Turing's theories and demonstrating the practical application of his mathematical models in nature.
- đĄ Turing's legacy extends beyond his mathematical and computational contributions, as his work in biology has inspired new avenues of research and deepened our understanding of life's complexity.
Q & A
What is the mystery underlying the variety of patterns in the living world?
-The mystery is how such a variety of patterns, like spots and stripes, can arise from the same simple biological building blocks: cells and their chemical instructions.
What is a Turing pattern and who discovered it?
-A Turing pattern is a set of patterns in nature, such as spots, stripes, and waves, that can be explained by a set of mathematical equations. They were discovered by mathematician Alan Turing in 1952.
Why were Turing's ideas initially ignored when he published his work on biological patterns?
-Turing's ideas were largely ignored at the time of publication possibly because they were overshadowed by other significant discoveries in biology, such as the double helix structure of DNA, or because the scientific community was not yet ready to accept mathematical explanations for biological phenomena.
What is 'morphogenesis' and how does it relate to Turing's work?
-Morphogenesis is the biological process that causes an organism to develop its shape. Turing's work is related to this process as he published a paper called 'The Chemical Basis of Morphogenesis' which included equations describing how complex shapes can arise spontaneously from simple initial conditions.
What is a reaction-diffusion system and how does it create patterns?
-A reaction-diffusion system is a model that involves two or more chemicals moving around and reacting with each other. It creates patterns by having one chemical (the activator) that promotes its own production and the production of an inhibitor, while the second chemical (the inhibitor) suppresses the activator. The interaction and diffusion of these chemicals can lead to the formation of various patterns.
How do the concepts of activator and inhibitor chemicals relate to the formation of biological patterns?
-In the context of biological pattern formation, the activator chemical promotes its own production and that of an inhibitor, while the inhibitor chemical suppresses the activator. This interplay, along with the differential rates of diffusion of these chemicals, can lead to the formation of various patterns such as spots or stripes.
How did Turing's mathematical models predict the patterns on a cheetah's fur?
-Turing's models predicted patterns by simulating a reaction-diffusion system where an activator (like a fire) and an inhibitor (like firefighters) interact. The activator promotes its own production, while the inhibitor suppresses it. The key to getting spots, as opposed to a uniform color, is that the inhibitor diffuses faster than the activator.
What role did Alan Turing play in World War II and the development of computer science?
-Alan Turing was a famous wartime codebreaker who played a significant role in decoding the Enigma machine, which is estimated to have shortened World War II in Europe by more than two years. After the war, he was instrumental in developing the core logical programming that forms the basis of modern computers.
What challenges did Turing face in his lifetime, and how were they related to his work?
-Alan Turing faced significant personal challenges, including criminal charges for being in a homosexual relationship at a time when it was illegal in the UK. He chose chemical castration treatment over prison, but tragically died from cyanide poisoning two years after his treatment began, likely a suicide. These challenges were unrelated to his scientific work but tragically cut short a brilliant career.
How have biologists verified Turing's mathematical models in the context of actual biological systems?
-Biologists have verified Turing's models by finding actual morphogensâchemicals or proteins inside cellsâthat behave as his model predicts. Examples include patterns on a mouse's mouth, bird feathers, human arm hair, and shark denticle scales, all of which are sculpted by the diffusion and reaction of molecular morphogens.
What is the significance of Turing's work in the field of biology and mathematics?
-Turing's work is significant because it demonstrated that mathematics could be used to explain complex biological patterns, leading to a new field of study called mathematical biology. His work has inspired new questions and approaches in understanding the underlying processes of biological development and pattern formation.
Outlines
𧏠The Mystery of Biological Patterns
This paragraph introduces the complexity and beauty of biological patterns, questioning how such diversity arises from basic cellular components. It highlights the role of mathematics in explaining these patterns, referencing Alan Turing's work on 'Turing patterns' that can account for various natural designs. The paragraph also touches on Turing's lesser-known contributions to biology, emphasizing the interdisciplinary nature of his pursuits.
đ The Power of Reaction-Diffusion Systems
The second paragraph delves into the concept of 'reaction-diffusion systems' as explained by Turing. It describes how the interaction between two chemicals, an activator and an inhibitor, can lead to the formation of complex patterns, such as those seen on a cheetah's fur. The paragraph uses an analogy of a forest fire and firefighters to illustrate this process, emphasizing how different rates of diffusion can result in various biological patterns. It also discusses how these mathematical models can be adapted to irregular shapes, reflecting the natural world's diversity.
đ”ïžââïž Turing's Legacy and the Future of Mathematical Biology
The final paragraph reflects on Turing's legacy, noting the initial lack of recognition for his work in biological patterns due to overshadowing scientific discoveries and societal attitudes. It discusses the rediscovery of Turing patterns and the subsequent validation of his theories through the identification of actual morphogens in nature. The paragraph also touches on the complexity of biological systems, such as the patterning in mammalian limbs, and Turing's pioneering role in computer science and codebreaking. It concludes with a tribute to Turing's multifaceted genius and the enduring impact of his work across disciplines.
Mindmap
Keywords
đĄMorphogenesis
đĄTuring Patterns
đĄReaction-Diffusion
đĄActivator and Inhibitor
đĄDiffusion
đĄMorphogens
đĄBinary Pattern
đĄHomosexual Relationship
đĄChemical Castration
đĄPardon
đĄBinary Code
Highlights
The mystery of biological diversity arises from simple ingredients like cells and chemical instructions.
Mathematics, not DNA, can explain many of biology's varied patterns.
Alan Turing's mathematical rules, known as 'Turing patterns', explain patterns in nature like stripes and spots.
Mathematical biology helps describe and predict complex biological systems.
Turing's work on 'morphogenesis', the generation of form, introduced a series of equations for complex shapes.
Reaction-diffusion equations describe how chemicals move and react, leading to pattern formation.
Diffusion and reaction combined can explain biological patterns, contrary to previous beliefs.
Biological patterns like cheetah spots are formed by the interaction of activator and inhibitor chemicals.
Turing's equations can create a variety of patterns by adjusting variables like chemical diffusion rates.
The shape of the surface on which patterns form can influence the final biological pattern.
Turing's 1952 article on biological patterns was initially ignored, later gaining recognition.
Biologists have recently found molecular morphogens that fit Turing's mathematical model.
Turing's work has inspired new questions in biology and the use of mathematics to understand nature's beauty.
Alan Turing's legacy includes contributions to codebreaking, computer science, and mathematical biology.
Turing's notes contained a unique binary code, showcasing his advanced thinking even without modern computers.
The world may have missed out on further groundbreaking work had Alan Turing not passed away prematurely.
Turing's multifaceted genius is celebrated for his work in codebreaking, computer logic, and biological pattern formation.
Transcripts
The living world is a universe of shapes and patterns. Beautiful, complex, Â
and sometimes strange. And beneath all of them is a mystery: How does so much variety Â
arise from the same simple ingredients:Â cells and their chemical instructions?
There is one elegant idea that describes many of biologyâs varied patterns, Â
from spots to stripes and in between. Itâs 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? Â
How accurately can mathematics truly predict reality? Â
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? Â
This is not a trick question. The answer? Is black with white Â
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 Â
that question like this: the stripes aid in camouflage from predators. And that would Â
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. Â
Not where the stripes come from, or why patterns like this are even possible.
Our best answer to those questions doesnât come from a biologist at all.
In 1952, mathematician Alan Turing published a set of surprisingly simple mathematical rules Â
that can explain many of the patterns that we see in nature, ranging from stripes to spots Â
to labyrinth-like waves and even geometric mosaics. All now known as âTuring patternsâ
Most people know Alan Turing as a famous wartime codebreaker, and the father of modern computing. Â
You might not know that many of the problems that most fascinated him throughout his life were, Â
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 Â
so much we don't know about it. If you think about a living system, like a human being, Â
there's just so many different things going on. And really, we don't know everything.
The movements of animals, population trends, evolutionary relationships, Â
interactions between genes, or how diseases spread. All of these are Â
biological problems where mathematical models can help describe and predict what we see in Â
real life. But mathematical biology is also useful for describing things we canât see.
Joe (05:44) What do you say when people ask, Â
why should we care about math in biology?
Natasha (05:54): Why should we care Â
about what mathematics describes in biology? Â
The reason is because there's things about biology that we can't observe.
We canât follow every animal all the time in the wild, or observe their every moment. Â
Itâs impossible to measure every gene and chemical in a living thing at every instant. Â
Mathematical models can help make sense of these unobservable things. And one of the Â
most difficult things to observe in biology is the delicate process of how living things grow Â
and get their shape. Alan Turing called this âmorphogenesisâ, the âgeneration of formâ.
In 1952, Turing published a paper called âThe Chemical Basis of Morphogenesisâ. Â
In it was a series of equations describing how complex shapes like these Â
can arise spontaneously from simple initial conditions.
According to Turingâs model, all it takes to form these patterns is two chemicals, spreading out the Â
same way atoms of a gas will fill a box, and reacting with one another. Turing called these Â
chemicals âmorphogens.â But there was one crucial difference: Instead of spreading out evenly, Â
these chemicals spread out at different rates. Natasha (15:49):Â
So the way that we create a Turing pattern is with some equations called reaction-diffusion Â
equations. And usually they describe how two or possibly more chemicals are moving around Â
and reacting with each other. So diffusion is the process of sort of spreading out. Â
So if you can imagine, I don't know, if you had a dish with two chemicals Â
in (GFX). They're both spreading out across the dish, they're both reacting with each other. Â
This is what reaction-diffusion equations are describing.
This was Turingâs first bit of genius. To combine these two Â
ideasâdiffusion and reactionâto explain patterns.
Because diffusion on its own doesnât create patterns. Just think of ink in water.Â
Simple reactions donât create patterns either. Reactants become products and⊠thatâs that.
Natasha (20:48): Everybody thought back Â
then that if you introduce diffusion into systems, it would stabilize it. Â
And that would basically make it boring. What I mean by that is you wouldn't see a lovely pattern. Â
You'd have an animal, just one color, but actually Turing showed that when you introduce diffusion Â
into these reacting chemical systems, it can destabilize and form these amazing patterns.
A âreaction-diffusion systemâ may sound intimidating, but itâs actually pretty simple: Â
There are two chemicals. An activator & an inhibitor. The activator makes more of itself Â
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 Â
can think of its fur as a dry forest. In this really dry forest, little fires break out. Â
But firefighters are also stationed throughout our forest, and they can travel faster than the Â
fire. The fires canât be put out from the middle, so they outrun the fire and spray it back from the Â
edges. Weâre 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 Â
are the inhibitor chemical, reacting to the fire and extinguishing it. Fire Â
and firefighters both spread, or diffuse, throughout the forest. The key to getting Â
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, Â
Turingâs simple set of mathematical rules can create a staggering variety of patterns.
Natasha (34:18): These equations that Â
produce spotted patterns like cheetahs, the exact same equations can also produce stripy patterns Â
or even a combination of the two. And that depends on different numbers inside the equations. For Â
example, there's a number that describes how fast the fire chemical will produce itself. Â
There's a number that describes how fast the fire chemical would diffuse and how fast Â
the water chemical would diffuse as well. And all of these different numbers inside the equations Â
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 youâre creating the pattern on. Â
A circle or a square is one thing, but animalsâ skins arenât simple geometric shapes. When Â
Turingâs mathematical rules play out on irregular surfaces, different patterns can form on different Â
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 Â
two versions of the same thing, identical rules playing out on different surfaces.
Turingâs 1952 article wasâŠÂ largely ignored at the time. Â
Perhaps because it was overshadowed by other groundbreaking discoveries in biology, Â
like Watson & Crickâs 1953 paper describing the double helix structure of DNA. Or perhaps Â
because the world simply wasnât ready to hear the ideas of a mathematician when it came to biology.
But after the 1970s, when scientists Alfred Gierer and Hans Meinhardt Â
rediscovered Turing patterns in a paper of their own, biologists began to take notice. Â
And they started to wonder: Creating biological patterns using mathematics may work on paper, Â
or inside of computers. But how are these patterns *actually* created in nature?
Itâs been a surprisingly sticky question to untangle. Turingâs mathematics simply Â
and elegantly model reality, but to truly prove Turing right, biologists needed to Â
find actual morphogens: chemicals or proteins inside cells that do what Turingâs model predicts.
And just recently, after decades of searching, biologists have finally begun to find molecules Â
that fit the math. The ridges on the roof of a mouseâs mouth, the spacing of bird feathers Â
or the hair on your arms, even the toothlike denticle scales of sharks: Â
All of these patterns are sculpted in developing organisms by the diffusion Â
and reaction of molecular morphogens, just as Turingâs math predicted.
But as simple and elegant as Turingâs math is, some living systems have proven to be Â
a bit more complex. In the developing limbs of mammals, for example, three Â
different activator/inhibitor signals interact in elaborate ways to create the pattern of fingers: Â
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 Â
published his paper on biological patterns, he admitted to being in a homosexual relationship, Â
which at the time was a criminal offense in the United Kingdom. Rather than go to prison, Â
he submitted to chemical castration treatment with synthetic hormones. Two years later, in Â
June of 1954, at the age of 41, he was found dead from cyanide poisoning, likely a suicide. In 2013, Â
Turing was finally pardoned by Queen Elizabeth, nearly 60 years after his tragic death.
Now I donât like to make scientists sound like mythical heroes. Even the greatest discoveries Â
are the result of failure after failure and are almost always built on the work of many others, Â
theyâre never plucked out of the aether and put in someoneâs head by some angel of genius.Â
But that being said, Alan Turingâs work decoding Â
zebra stripes and leopard spots leaves no doubt that he truly was a singular mind
Natasha (37:55): The equations that produce these patterns, Â
we can't easily solve them with pen and paper. And in most cases we can't at all, Â
and we need computers to help us. So what's really amazing is that when Alan Turing was writing Â
these theories and studying these equations, he didn't have the computers that we have today.
Natasha (39:01):Â
So this here is some of Alan's Turing's notes that were found in his house when he died. Â
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 Â
code. Itâs his secret code. It's in binary actually, but instead of writing binary out, Â
because you've got the five digits, he had this other code that kind of coded out the binary. So Â
Alan Turing could describe the equations in this way that required millions of Â
calculations by a computer, but you didn't really have, you know, Â
really didn't have a fast computer to do it. So it would have taken him absolutely ages.
Joe (40:15) What has Â
the world missed out on by the fact that we lost Alan Turing?
Natasha (40:25): Itâs extremely hard Â
to describe what the world's missed out on with losing Alan Turing. Because so often he Â
couldn't communicate his thoughts to other people because they were so far ahead of other people Â
and they were so complicated. They seemed to come out of nowhere sometimes.Â
Natasha (25:52) When you read accounts Â
of people who knew him, they were saying the same thing. We don't know where we got this idea from,Â
Natasha (40:42) So what, what he could have achieved. Â
I don't think anyone could possibly say. Natasha (42:14)Â
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 Â
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 Â
heart of every computer on Earth today, including the one youâre watching this video on.
And decades later, his lifelong fascination with the mathematics underlying natureâs beauty Â
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 Â
of a rare and special mind. One that could see that the true beauty of mathematics is Â
not just its ability to describe reality, it is to deepen our understanding of it.
Stay curious.
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