Welcome back to the third event of the Oxford Mathematics Public Lectures. Home Edition. My
name is Alan Guerrieri and I'm in charge of external relations for the Mathematical Institute,
as usual. Special thanks to the sponsors, execs, market execs market are leading
quantitative driven electronic market maker with offices in London, Singapore and New York.
The ongoing support is crucial in creating quality content. There has been
a lot of talk in the last two months about modelling related to the ongoing pandemic crisis.
I found that there is a lot of misunderstanding of what modelling is all about, and we thought it
would be important to take a couple of steps back and give you a very broad overview of mathematical
modelling in the context of biological system. What does it mean to model in biology where so
much is known and so little is understood? Is the function of a model does to make predictions
or to understand processes that we cannot observe directly or both to discuss
these ideas? I've invited one of the most distinguished experts on the subject
in the world, my colleague and friend, Professor Philip Me, Philip Oulds,
the Professorship of Mathematical Biology at Oxford, and he's a fellow at St John's College.
He has made countless contribution to both biology and medicine within the UK.
He has also played a very significant role in fostering a strong school of mathematical biology,
which has been a wonderful leader and a mentor to many young student and researcher.
The breath of Professor Menees contribution is truly remarkable. Spanning developmental
biology system biology or pathogen interaction, cancer biology, biology,
multi scale modelling, morphogenesis and pattern formation, amongst many others
for his contribution. He has received too many prises to list tonight, but maybe the best indicator
of his Murty disciplinary work is his election as a fellow to both the Academy
of Medical Sciences and the Royal Society. Finally, I want to add that Philip
is also a wonderful lecturer and I am very keen to hear him tonight. Please start. No, Philip.
I'd like to begin by thanking Aladdin for inviting me to give this talk. I'm going to talk about
mathematical modelling in biology, ecology and medicine.
So that's your first question to ask is what is a model and the best definition
I've seen is that by Alan Turing. Who,
talking about his model in 1952, said this model will
be a simplification and an idealisation and consequently falsification.
It is to be hoped that the features retained for discussion are those of greatest
importance in the present state of knowledge. So let's see exactly what he
said in here. First sentence model would be simplification.
And therefore, falsification, that means the model is wrong. Because
you're trying to understand some complex system. You
simplify it, which means you throw. Things on it,
and therefore, by definition, it's wrong.
It is to be hoped smart means that modelling
is not a science, it's an art. That the features
are 10 for discussion are those of greatest importance in the present state of knowledge.
So if you're working in some medical area
or some logical area. The present state of knowledge, you probably
don't know that she's probably going to have to work as a collaborator.
The rule of modulated is to advance the present state of knowledge
in the application domain to advance the state of knowledge,
then you might have to re-evaluate your model and start
again. Change the model, refine it and continue
to advance the present state of knowledge. But I'm going to do this talk
is focus really on the modelling side of things
and the modelling side of things is where you do the substraction. And
what I'm going to try to show is that by doing this abstraction, one can
draw analogies between many different areas in science and transfer
knowledge from one area of science to another area of science.
So I'm going to begin by talking about ecology and medicine.
So concer so roughly speaking, in our body
every day, lots of sounds die.
And Charles prefer it defined to compensate
for that death. And you have a balance,
the birth balances the death and you have something called homeostasis
or steady state. It happens in cancer.
Is it a group of cells through mutations?
Their properties change so that this balance of birth and death
is upset. And the net result of that is you get a growth.
So the question counsellor is trying to recruit, troll this population of
cells that are growing, that are actually invading our body.
In ecology, there's a long history of analysing invasion.
So if we think of cancer as a disease, invasion of our body.
By sounds, can't we draw lessons? From
ecology. So what are the invasions that the
U.K. experienced and is continued to experience? It's a squirrel
invasion. So this is the red squirrel. The indigenous squirrel.
The host population. And last century, this character
appeared, the grey squirrel imported from the US
and the grey squirrel has invaded the red squirrel squirrels
territory. So I apologise for the
very bad quality of this very old drawing. So
here we see the south east part of England.
There is the coast there. And in 1964, these
black. Spots are where people
have seen red squirrels during that year. And then
this is 64, 65, 66, 67, and
then seventy five up to seventy nine. And you can see going from here to
here, the red squirrels have begun to disappear. And that's because
they've been replaced by the grey swans. So this
is something that I worked on, actually, when I was a postdoc with Jim Marry a cute couple.
And Mark Williams and we developed actually Lockable Care, a well known
petition mogel to try and understand this invasion. So
these equations here, G, is a population of grey squirrels.
Red is a pop, art is a population of red swarms and years time.
So what are these equations mean? Well, let's look at the red squirrel, because that's the home team.
OK. So every author. Trumped every other
thing. Our one K one or two K to a one the day two. These are all
positive fixed values. OK, so suppose
we just had this term. So our two times K. To give you something.
No positive number multiplied by Ah. Let's consider that first of all
we'd have Diyab ADT something positive. Times are.
That means you have exponential growth. And in fact, exponential growth
is a pretty good description of how a lot of population start to grow.
But of course, populations do not grow to infinity because
as they start to grow, there's competition for resource and you'd expect
the growth rate to decrease. The simplest way to model that is
to put this term here. So what this term is saying is that as
the population increases, the growth rate instead of being
are two times K2 becomes are two times
K to minus R, it decreases because of the presence
of threat. Now, here we have a competition,
and so with the greys also decrease control
threat. So you can think of this as competition for resources
that decrease the growth rate. The same thing for the grace.
And if we think of the greys as being battered competitors, then the Reds
we'd have this is because this. OK, so then
on top of this, we would have to fusion, which I haven't put in here, but we would have
movement of the praise movement of the rich.
So let's see how this system with movement in it behaves.
So this is a video that was, in fact, generated by former graduate
should mine a long time ago, Anthony Pollner. So what we've done
here, we've scaled the populations to lie between zero
and one. So here we have. The red squirrels,
they're all a population. Grey squirrels.
Population zero, except for some pockets here of population
one. And then we're going to ask what happens if we run the small.
You run the model and what you're seeing happening is.
Waves of grey squirrels by competing threats
to invade land, the grey squirrels. No one
understands answer.
So let's try to understand how it was that the GREs, a computer
threat and the way to do this, you go back to these equations
here and we look at something called the no fly
and no clines or where these derivatives are zero.
OK, so if we look at the R equation, the R by the cheese zero,
if R zero. Which is this line here.
Or if this is the. That's the straight line here.
Likewise, Jean. We have either G a zero
or this. OK. So if we look at these two lines, these are two straight lines.
And they can. Look differently, depending on the values of Kay.
One, a one, K two and a two. Take a look like this, or they could look like this
or they could have other combinations. Okay, so let us look
at this one to be less disposable sitting here. So we've got
some grais small number of reds. The Greys have already come into the
domain fixed on the domain and see what's going to happen now. Ah,
not here is shorthand for Driftin. So at this point,
we are below this line where they are guided to zero.
So these are small, which means this is positive.
So they are digestives positive. So are will grow.
Also d.g by the team's roster, because we've be below
this line. So Julie will drill. So the net result will be
if G is growing and R is growing. We'll move like this
now. Crucially, they crossed the red line below
with a great line by the black line here. We cross the red
line first when we crossed the red line. Now
the sign off to your back. He switches NTR banditti becomes
negative. Which means power starts to decrease.
But we're still below the black line, so the graves are still actressy.
So we go off like this. And we reach this point here
where the red squirrels are zero scray squirrel. This is supposed to
be very happy. Pretty squirrel. And we have had competitive education.
So suppose we now think, well, let's say if the Reds and there'd
been lots of campaigns. Throughout the century, last century
and this century to save the Red Square. Part of those campaigns are
let us comb the grey square. So we call the grey squirrels,
which means we decrease the population of the grey swans.
We've killed a lot of grey squirrels. What's going to happen? Well below
both lines. No claims, so the populations whisstock to pro.
And we crossed the red knoll first. And so
we go up and we finish up. It's own race,
so it hasn't worked. On the other hand, suppose
these no zones look like this. Here we are
again. Start off here with below both lines,
both g you know, our increase. We cross the red line first.
Which means are starts to decrease. And we go to here
happy grey squirrels, red squirrels are all dead. Now, let's do the
same experiment we did on the right hand side. Let's call
some Grayskull. We've talked to here once again, we're
below. Both know Clynes. So both genau will
start to increase. Now there's a difference. We crossed
the black line first. Which means that the grey start to decrease.
Then we move on to here and we finish shop. Excluding the greys
and happy Red Square. So two points to note from here.
Same experiment. Gives different results here.
Drays Wollman. Here, Reg., for.
Second thing to notice is that if we are in this situation
here, Colleen will never work. We have to do something
else. We have to change the slope. Sloops of these
lines so that they intersect like this. Now, the sloops
of these mines and their intersections are determined
by. These values here. So what we would have to
do is we have to somehow make the red squirrels
better competitors in the grey squirrels. Or make
the environment such that the red squirrels are better suited
to the environment than the grey squirrels. So that's
the insight we get out of this.
These insights have been used by.
My friend and collaborator Bob Gayton B, who
began life as a physicist and now as a cancer biologist and two radiologist
who's at the Moffett Cancer Centre in Tampa.
It's well known that cancer cells sometimes
use a metabolism that is less efficient
than normal cells. It's. This metabolism
is called. Anaerobic metabolism, and this is called the Warburg.
So basically, if we have lots of oxygen for ourselves, lots of oxygen,
we we use a real big metabolism and that produces lots of energy.
Oxygen decreases the cells, then use.
Anaerobic metabolism, and that involves costs.
And that is. Lower
production of energy gives lower production of energy. And what Warburton
noted was that in some. Situations
where there is enough oxygen. Tumour cells use
this anaerobic tablets. That doesn't seem to make sense.
Why would you use a lower. Energy
generating system in the competition.
So Hope's theory was the following are his hypothesis was the following.
He said that, OK, you've got these invaders.
By using the clitic pathway, one of the by-products of the black Lytic
pathway in the Fotopoulos and lactic acid. And that makes
the environment sediq. And the civic environment.
Favours. Chuba Cell Division found it's more toxic
to normal cells than to the tumour cells. So what
the cancer cell is doing is it's changing the parameter values
to make the environment more suitable for it.
And for the normal cells and to help turn fit. So he
wrote down a system of equations. Bit different to the ones we just
looked at. But the essence is the same. A competition
model. And he solved this competition model. And so what we see
here is this is in space.
This is his model where this. This black line
are the tumour cells that are invading. So they're forming travelling wave. It's moving
along like this. This is the acid that they're producing
and these are the normal cells. So the normal cells
initially were at one everywhere. And they're still at one here
because there's no not no tumour cells, but the tumour cells have started
to invade. They're producing the acid. The acid
is changing the environment, making it more suitable for the tumour
cells and for the normal cells. And so this will grow. You see what's
happened here? The normal cells. It worked. One contact point
for. In other PROMETA regimes, it was
even more dramatic. What you find here, you say started off
again with normal cells everywhere. Then he introduced
some tumour cells. And now these tumour cells are invading.
And they have pushed the normal cells to extinction.
So this is competitive exclusion. Now, what you find, which
was a novel thing, was this strange business here. Here,
there are no. So the prediction of this
model was. If
tumours are using acid to mediate the invasion,
there should be a gap with no cells between the invading front
of tumour cells and the regress in front of
the normal cells. So then Bob went
and looked at some human head and neck cancers
and had several of these head and neck cancers. He found his.
So. This leads to various
obvious next experiment. If Schumer
is changing the environment, making it sitting so that it can invade.
Then what would happen if we neutralise the. This
is an experiment by a real B and various people.
This is done on mice. Two mice
were injected with tumours. The control makes.
We're just left to do their own thing. The treated
mice were given water that had
carbonate. And then what they looked
at was how many metastases
reformed, in other words, from this initial tumour? How many groups of tumours
broke off and invaded? And what we see here
in the old treated here is the average.
I didn't. Treated. Here's the average. So the treatment
didn't stop. Totally, totally stop the invasion.
Could have significantly reduced the invasion.
So the mathematical model. Which tested and
validated the hypotheses led to these experiments.
That, again, would be more. Now,
this is a nice Melhuish is a model. We're
humans, we're not mice. And then the question is, just because
it works in the mice cannot work in the human. We've got
a completely different metabolism, completely different immune system. Lots
of things are different between us and the mice. And so this but this
work is actually instigated. A whole series of experiments
don't work that. No, I'm modelling that this group at the Moffitt
Cancer Centre are doing and they continue to work on the rule of metabolism
in cancer progression and can be altered metabolism
in order to stop cancer invasion. They work
the cancer at the Moffitt Cancer Centre and in fact, there are
a group of seven or eight full time mathematicians
working with clinicians and the experimentalists headed
by Sandy Anderson. So
I'm not going to move on to something to do with.
Slime mould, which is developmental biology and fact tax,
which is physiology. So
but before we do, that is just the chemistry.
In the 1950s, the loose of.
And then later dropped, Pensky worked on this
chemical problem. And this was a chemical reaction
where the chemical here. This is time.
And it is.
And then it goes black. It was back to the head.
And this child happily. So what's going on
here? These this chemical that they're looking at, this chemical
reaction is a redox reaction where the chemical, when it
is oxidised, it's in certain colour.
And when it is reduced to deoxygenated, it's a different colour.
So what we're seeing here is oxidation reduction, oxidation reduction,
etc cetera. So why is this happening? And in fact, if you look
at this in. A dish.
You get all you get waves like you did for the squirrels. But
these are much more interesting. You get these what are called target patterns
because they look like targets. You can get here's target pattern. And here
are. Cheryl Pratchett's. So what's happening here?
These patterns propagate and they break up and they form spines.
So go to the Internet and you can see videos of these.
OK, so what can we understand? So lots of people have developed models for these.
And this is the sort of the field noise model.
And a very, very simplified version of it. So we I think
in this setup, X and Y to the chemical players
and we told the no fly. No Clines.
For X. And here's one adult by.
It's a Kubic. And the no fine
for why. I've just drawn it by simply
here as a straight line. The important thing is that the no plan
only it intersects along this branch. So there's no climate
could be like this. Doesn't really matter. The key thing is it intersects
along here. OK. And it turns out in
this system, when you are above the black,
no climb. Not only is DXP by positive,
it's very possible the number is low. Crime is where the derivative is here.
So one part of the no crime, the derivative would be positive on the other part
would be negative diff. One of the differences between this and the previous
model is that this x2 of a diff is very big.
On this side, positive, very big on this side, Nick.
So suppose we start at this point. P. So at this point, p p
y by two zero. The X by DTL zero. This is a steady
state. So the system would stay there. Nothing's changing. But in
biological system, there will be small fluctuations.
So suppose there's a small fluctuation that pushes us to clear. So
if we look at to why by detainees, negative, so why will
decrease? To expire, duty is positive,
so X will increase. But it's very big. So it'll increase rapidly.
So the net result will be we should to hear. And we crossed
the no fly zone and nobody explained t is negative.
So we start to go down. And the thing is that we stick
to the no climb, because if we try to fall off. No. To explain to T.
is very large negative. It'll push us back. So we go time
and then we reach the point B. So ideally, we would like to
stick with no climb. We'd like to move up.
But that would mean why increasing? But look, why
does it increase? Why decreases so we fall off the loped fine
and we should approach us to this point. And not at this point,
do what I buy duty is my positive. So we increase.
Go up to here. Reach this point and then at this point.
We fall off the Noel Klein again, shoot across to a cheap on an
ongoing. What does the X
look like as a function of T? Let's look down here. Say,
Andy X increases very quickly to a it
off to a. Slowly decreases to be
slowly goes dark, to be quickly decreases time to see and slowly
moves up to tea and keeps on going. And that's the oscillation.
So there'll be other chemicals that will be oscillating either fingers
with this. So with this one is high. And you've got,
say, an oxidised version and the reduced version is low to one colour
and then they swap. Then the. Reduced one is high.
The oxidised one is low. It's a different colour. And this keeps on happening. And this is
called the vexation relaxation. Also.
Now, what would happen instead of this no fine intersecting
here. Suppose it intersected here.
On this branch near this maximum.
Now, I suppose we do the perturbation again from P
and we move to Q so a Q does X by detail is
negative because we're below the X, no client. So X
will deepfreeze. Why is negative? Why will decrease
and will should die? But suppose instead we
moved to here in a perturbation. Now to explain, to tease
positive. And we should in the opposite direction.
Right across to here come time to hear. Right
across to hear again. Up to here. And then we stop at this
steady state. So crucially, here we have what's called the
threshold for Nordgren. If we make a small perturbation.
We go back to where we started. But if we breach the threshold,
we then have a big excursion before we go back.
And that is this type of behaviour here.
That's called an excitable system. Now, one
of the amazing examples where we see an excitable system is something called the slime mould.
And again, I would urge you to go look at movies of this on the Internet to
slime mould our single celled creatures who
live around you enjoy life if he'd be producing, but then times
become bad and they run out of food. So what did he do?
Well, some of them start to secrete a chemical called Cycler GameSpy.
This chemical diffuses and the
chemical and cell combination is
an excitable system, so that when this chemical.
Close to the next cell. It perturbs at.
And so it then releases a chemical.
And this isn't in time, but in space, as we saw here,
moving back here, we get travelling waves. So this would be a travelling
cross. So in this way, the signal
gets relayed. And these. Cells
move up the chemical gradient and form the streams,
which eventually aggregate and form.
A slok, a mind and then a stroke. So basically this idea,
you sound the chemical and the reason to send up the chemical is to signal to
all your friends. We've got to get together and form a asla.
The importance of this slug formation is that when the cells come
close to each other, they can exchange certain key
proteins that allow some of them to be common.
Stock sales and so on become spoors.
So what you get here is a fruiting body where.
These things grew up and then you have spores at the top and stop
at the bottom and the spore cells are the ones that survive. So
an animal comes. Pushes them away or the wind comes
blows the stock cells of the sport out away and then hopefully
blows them away to a better environment where this has to.
Now, in the 1980s and 90s, this was studied a lot
because. It had processes that are
vital in higher organisms, signal transduction, which
is how you really signal. Table taxes. The movement
up, a chemical grading, which is very important in higher blood vessels form
in higher immune system, responds to to disease and differentiation
of how these cells change from one cell type to another. These are all
key processes in higher organisms. But the
slime mould is a really good study. A really good model
system to study.
And in fact, if you look at this in two days, here are cyclic
empey waves and you can see they form spirals and target
patterns. They much like the.
The list of top 10 scary action you can see here we've got.
A biological system. There we had a chemical system.
They both exhibit similar patterns from a mathematical point
of view. We see that there are several provinces between.
And then here are the cells responding and forming streams.
So we see the equivalence there between a.
Chemical system and a biological system. So excitable systems
are very important. Here's another
example of an excitable system. Flushing toilet.
The one with a handle, not the one that you wave. OK.
Think about a flushing toilet. If you go along and you push the handle
just a little bit, what happens? Little bit of water dribbles out.
You perturb a little bit from the steady state and you go by. Pull
the handle a little bit more when you breach the threshold, then wish.
A whole pile of water comes down and then you relax back to the stand.
So that is an excitable system.
So moving on quickly. In 1952,
Hodgkin and Huxley wrote a series of papers for which they won the Nobel Prise.
Some years later, they were looking at the giant
squid giant axon. So I nearly said it wrong. Their giant squid oxen. That's quite scary.
This is the squid giant Daxam. So what does that mean? Well, basically
act so he's looking at the nerve cells. And
the nerve cells. Communicate with each other via long protrusions
that are called accents, and the communication is the electrical communication.
And basically what happens is you have an electrical pulse
that bleeps in your cell. And then that propagates
it kickstarts. The next show.
The policies and pick stocks, the next step it touches.
And in that way, you propagate the signal. And
this work, they're there on paper for this mathematical
model, for this is a very complicated model of four equations. And it was
later reduced to the equations by fortune. You know, this showed
that this behaviour was identical to the behaviour we've just discussed in terms
of relaxation, oscillator and the acceptable.
And in fact, there's been a lot of work on marketing of the heart,
and in particular people like Dennis Gilbert and Peter Hunter have done a huge amount
of work on developing models of the heart because these excite so into the heart.
You basically have an electrical pulse that sweeps time the heart.
That triggers a mechanical response that causes the heart to beat.
Now, as we've seen these excitable systems propagate
this. Signal. But if there is
a bit of the heart that maybe has died, maybe there wasn't
enough. There was a blood clot was not enough
uterine getting to that vet. Then also six hours this wave comes to that
point. That point is going to have different properties to the rest.
Therefore, the flow of this wave through thought part of the heart
is going to be a different speed and the wave of Break-Up.
And as we saw with the slime mould and with the
Robert Pinsky reaction, once these things break up, they form spa,
which. And that means your heart would then try to beat
not as a coherent beat, mechanical signal, coherently
beating the heart, but the signal would be a spiral signal
causing the heart crew to act. And so that can't be properly. That's called
fibrillation. And in fact, these researchers
and many other researchers have actually developed very, very sophisticated models
of four heart models where they take into account a lot of them
the detail, anatomical detail of the heart
and the lettre physiology and mechanical aspects of the heart
to see how you get these beats and how it is then that
you can get fibrillation, etc., and then you can use
these if you're going to give a drug. Some drop
and you want to make sure that that drug doesn't lead to heart problems.
Then you can say, well, if that drug affects the ion channels
and therefore the electrical system and excitable system, can
we make sure that it doesn't lead to these instabilities
that lead to different relation to fibrillation? So what
we're seeing here, then, is Phenomenon Romila, a phenomenon
that occurs in. Chemistry,
developmental biology, physiology and plumbing.
The last thing I'm going to look at is animal poop markets.
And football shirts. OK, so here's an animal.
It is a cheater and it's sleeping. OK? So I didn't do
anything back to it is just having a nap. I'm just talking and this is
what I mean by pattern. She only spots. It was a package.
Stripey tale. Here's another powter digits,
five digits pattern of Gore's.
Question is, how do these patterns arise? So I mentioned the beginning to talk
Alan Turing. He starts to look at this problem in the early 1950s
before he tried to. So
he had this notion here, think of this tree. And the tree is growing.
Suppose you were to look at this tree at this point here. What do you would say
if you look down on it? It be a circle? And the tree is growing.
And then at some point here. That's circular symmetry is broken
by this branch cheering had the following idea. There must be some
growth hormone that's causing this to grow.
And then it's. And so if the growth is symmetric, that growth hormone
must be symmetrically.
Then at a certain point, that cemetery must break, must become
destabilised. And then you have a.
Asymmetric.
Growth through asymmetric concentration profile. Of
the growth hormone. And that leads to an asymmetry in the drugs.
So you can think about taking this even further and you could say that
instead of that asymmetry in the chemical resulting in a symmetry
and growth, it could result in an asymmetry in Fed determination.
So, for example, where the chemical concentration is high, that could result
in cells adopting a certain type of fat where the chemical concentration is
low. The cells do something different. So to form
bone with the chemical concentrations high do form ball. It's chemical castration.
And he called these chemicals morphogenesis.
That's what he said. It is suggested that system of chemicals called morphogenesis reacting
together and diffusing. Through tissue is adequate
to account for the main phenomena with morphogenesis, meaning after information.
And in fact, many, many years later, people actually find more fidgets.
These were predicted by Alan Turing. Thirty years before it was.
So the question is, how do you get this?
Symmetry breaking or instability tertrais. And what Turing
said was. Diffusion costs the instable.
Now, this seems really counterintuitive because think schools, you start
off here, here's some water and you put a globe think into the water.
You have a pattern here, ink here. Nothing else here.
Come back a little while later, the patterns disappeared. There's no spatial
structure here. It's all the same. That's because the diffusion, the random
movement of the molecules around has ended up meaning they distribute uniformly.
And yet Turing was saying. That diffusion causes Patrick.
That seems really counterintuitive.
So he will turn to a system of two equations. Two
chemicals, you and V. So and there's a diffusion term here.
And here's the. We action terms and what he said
was for this to happen. You need a number of key
things. You need one chemical to produce.
No other chemical. Outproduced itself.
This chemical inhibits this chemical. So this is what he
called the activator, because it activates inhibitor
and inhibitor inhibits the activity. So let's think about this
spoon's. You've got a pile of a activator. It's catalysed
in itself, producing more of itself, but it's also producing the inhibitor.
Then the inhibitor inhibits the activate. So there's more activator.
There's more inhibitor and that pulls activator die, so you can imagine getting
to the equilibrium. Well, you've got a level of activity
and a level of inhibitor, and that's stable because if you added
more and more activator. That would activate the inhibitor,
which would make more inhibitor, which would pull the activator packed on
less activator Shlash inhibitor inhibitor. We'll come back and you get back to the
steady state. So this is still. Now, I suppose we have
to fusion. And suppose that the inhibitor
diffuses faster than the activator.
Then what will happen if you do that prohibition again? The
activator Stotch produce itself starts to grow, produces
the inhibitor. Now, the inhibitor, some of it
inhibits the activator, but a lot of it gets out of the way. So
there's not enough inhibitor to inhibit the activity you the activator
grows even more. Produces more inhibitor. It goes away.
So what you'll end up getting will be lots of activator, surrounded by
lots of inhibitor, a spatial pattern. So our
intuition, that diffusion. Gets rid of
pattern is based on one thing diffusing.
And that's the singular case. As soon as you have more than one
species, chemical fusing diffusion can drive
Patrick. So this is a nice example of mathematics, pointing out
where our intuition is wrong and building our intuition.
And Cherry used mathematical analysis to prove
this. And this idea
of reaction to fusion forming a pre pattern.
For animal, quote, markings, patterns and various other
things has been used a lot. And so Jim Murray, who is my supervisor, nice
graduate student here in Oxford. Very famous paper of his here, where he
shows how you can get patterns that look like the spots
on a leopard. And you can get those reactions. Fusion system.
And here's another idea. I here hear this often looks like an animal
put marking. And what's happening here so you can think of right here. There's no pattern.
The chemical doesn't form any pattern. And then as you move here,
the demand is got bigger. But he scaled the domain box, which fits
on the graph, on another piece paper. Otherwise the pictures would get too big. But this
is. For larger to mean now you start getting some pattern
as you increase the domain further, you get more pattern. Further, you could more
you get more and more. So we can sort of think
about these. Is it basically these patterns, it turns out if you look at them,
they are what are called Igen functions of the class in. And on these sorts of
diagrams here, they're essentially mixtures of signs and cosine.
And what you can think of is also demand gets larger. You can fit more
signs and put signs in and get more complicated patterns.
So therefore, the prediction is that there should be nominal dips like this one, the analytics like
this, you might have seen an animal that looks like this. Maybe not seen an animal
like this. But here we are. These are not photo shopped. This is Volly
Goat. This is about the goat.
So as I mentioned, these Igen functions, Pacien.
Form the pattern. So are there other places
in mathematics or in science where we see Igen functions of the class? Well,
if you vibrate a plate or a drum or strong a guitar.
The vibrating modes are Igen function plasty.
So these authors here. Had the idea that if
you vibrated a plate, it looked like an animal put marking.
And you could visualise the vibration then, since the vibration.
Or Igen functions of the reclass in. And they're
proposing that animal coop markings or Igen functions as a claassen, then
you should see animal markings. So this is the vibrating plate.
Look. These are very like, um, local blockage.
Look at here. You got that sort of thing here. Here's a separate fossas.
And hands my heart. And this is from taken from his book, The Algorithmic
Beauty of Seashells. A beautiful book where this is
a real seashell and this is a pattern of sea shell generated
by system of reaction fusion equations. And the book is full of
beautiful pictures like this.
The Igen functions of the PACIEN have certain properties. One of their properties
is that. Ask them to man the geometry
changes, the pattern changes in particular
as the domain becomes Nardwuar and more one dimensional.
The pattern changes from being intrinsically two dimensional exports
to becoming one dimensional like. So here is the model.
And here is an. Michael and.
So if we take this a bit further, what this town, us? Is
that if you have a domain and you see a pattern on
this. The only thing you change. Is the demand
geometry. So, for example, you go from a fact being
to in to a small to man. Then the pattern.
Cannot get more complicated because the complication of the pattern
increases with domain size. So
if you have an animal that's got spots
on its body. And then you look at the tail, which has got
a narrower, more one dimensional geometry. You will either
see spots or stripes. More likely
strikes, on the other hand, if the animal
body you see stripes, then I should knock with the to man.
You will either see stripes or no pattern. You will not see spots
because that would mean the pattern getting more complicated.
This is their idea of developmental constraints. This is a paper where two theoreticians,
OSTREM Murray, came up with this idea from the mathematics.
And these are two biologists who came up with the idea for actually looking
at skeletons, looking at put markings, etc. So
these were data driven. These were mechanistically driven
results. So the idea is
that a spotted animal can have a stripy tail, a striped animal
Congleton spotted. OK, so now
let's move to. Premier
League football shirts or football shirts? The Premier League
here, stripey body.
Similar pattern on the art spots. Simple pattern,
stripes, stripes, stripes, no pattern.
So these will be Turing's model.
Pattern on the sleeves is not more complicated than the pattern
on the body. Here, season 2016, 2017,
the Premier League and virtually all of these satisfy cheerleaders
theory, a pattern on the sleeves is not more
complicated than the pattern on the body. Rugby shirts.
Same thing, pattern on the body.
As complicated or more complicated and part on the streets.
But there are contradictions. This is the daughter of a former graduate
student of mine. And he sent me this photograph. And instead of said, look.
How beautiful is my daughter? He said, look, contradiction
to cheering small. Stripey body spotted arms.
Here are some cuddly examples, too. You see these animals, you should do them because they don't
satisfy tributes, theory, stripey body, spotted tail,
no pattern pattern. These her
country examples. So what do you think is going on here? Is that not only
is the domain geometry changing, but something else is changing.
Now, the question is, do these occur in nature,
in the actual electoral system? We know that morphogenesis prayer
is still a huge controversy has to do with these patterns to occur
because of. The way to insert the action diffusion.
They've been shown to a crowd chemistry. So in chemistry, these were cheering
patterns. They've found in chemistry. And here is the simulation.
This is actually just taken from this paper showing that the model
predicts you can get these types of patterns like this. You can get Spaull hexagonal
patterns like this. In fact, I haven't got the picture here, but you can also get these lab
redesigned. So hopefully
what I've shown here is that by understanding the mathematics in one
area of biology. On chemistry.
Mechanics, you can. Understand? Other areas
of science. And that is the power of mathematics
because by abstracting by the art of modelling, which means you abstract.
Processes, many different processes. How fundamentally
the same principles. So you could argue that mathematics.
Is what unifies science. A drinking
night. These.
Processes that are common across different areas so they can transfer
knowledge from one area to site to another.
Shows that mathematics, really, Prof. But remember,
with great power comes great responsibility. And you have to
be careful not to push your luck, not to look at some thing,
some scientific phenomenon and say, oh, that patriot or that structure
looks like this other area of science. So since I understand that area of science,
I understand that that is not science.
He's got to know the signs. You've got to know the biology. Know
the medicine, the physiology. Work with the key
people in the field. The experts in the field to make sure that your
mathematics is relevant and realistic in that area.
In other words, you've got Cropper's. So thank you for
your attention.