Hi. It's Paul Andersen and I was cruising around the internet earlier this
year when I happened upon a journal article by Professor Joel Cohen. It's entitled, "Mathematics
is Biology's Next Microscope Only Better." It's a fascinating article, I'll include a
link in the video description down below. I reached out to Professor Cohen and asked
him if I could put together this video to summarize the major points. He thought that
was great. Thanks for being a teacher he said. And he included a picture of himself. And
so what does that mean? Well let me give you an example. I was recently listening to a
mathematician at Montana State University. He works with biologists to do mathematical
modeling. And he's had over 60 peer reviewed articles. So he really knows what he's talking
about. And in this research they were looking at e.coli. E.coli takes about 20 minutes to
reproduce but what they were wondering is, you know, why doesn't it reproduce every one
minute or every five minutes? And what they found is that the limiting factor was the
ribosomes. The ribosomes have to drive along the RNA. And they make the proteins. Because
the proteins are going to make the bacteria. But they go one after another. It's not just
one ribosome. That would take years. But they're going to go one after another after another.
And what ends up happening is that they get traffic. It's almost like all of the cars
driving on a road. As one slows down, the one behind it slows down. It slows down the
whole thing. And so what he had done is created a mathematical model using high level mathematics
to explain like what's the limit. How fast can they actually go? And he was excited because
he got the wrong answer. He got a round 60 minutes instead of 20 minutes. And what that
told him is that there had to be some other mechanism, something that the bacteria were
using to communicate so they wouldn't have these big traffic jams. And so what really
mathematics is doing is opening up these new fields in biology. But let's first talk about
the first microscopes. And the first microscopes opened up a whole new world of biology. And
so imagine looking at a leaf under the first microscope. And all of a sudden you don't
see a flat piece of green. You actually see cells popping out. And you can see organelles,
in this case you can see the chloroplasts inside those cells. It opens up new realms
and the whole idea of cell theory wouldn't have been built until we eventually had the
microscopes. But let's say you were the first biologist looking at pond water. And no one
had every seen this life, like an amoeba or these little algae. It must have opened a
brand new world and just to look at that, it must have been feeling like you were exploring
space for the first time. And even Charles Darwin, one of the most famous biologists
said that "Mathematics seems to endow one with something like a new sense." And so mathematics
acts like a microscope. It opens up new worlds that we haven't seen before. And mathematics
has been associated with other sciences in the past. And so the word geometry comes from
earth measure. And so they were first figuring out how we can quantify and measure the earth.
But they had to build a mathematics to explain that. Or calculus didn't really show up until
Newton and Leibniz actually had a reason to do that.
\b \b0 And for me I didn't really get calculus
until I took physics and understand that calculating the area under a curve is really important.
But what's interesting is that biology is like physics but better. In other words it's
way more complex than physics is. And to give you an example of that just think of our planet.
So our planet is going to have simple atoms that organize into molecules. And if we were
to just talk about the inorganic things on our planet, so minerals and rocks, it's fairly
simple. But start thinking about the complexity of life and you understand how complex it
is. And it's not really understood. And it's not clearly understood with the mathematics
that we have today. And we even see that through history. And so one of the big turning points
in biology is we had this understanding of natural selection that was put forward by
Darwin. We had this understanding of genetics that was put forward by Mendel. They lived
at the same time, but nobody really put their two ideas together until a couple of mathematicians
came up with the Hardy-Weinberg equation and showed how genes were the units that were
being selected in a population. And as we go forward let's get to some more complex
mathematics. This is the Radon transform. This is an Austrian mathematician who what
just doing math to start with. What he was looking at is imagine a line that's moving
through material. And the material is not homogenous. It's going to have different densities.
And as a result it's going to bend the object as it moves forward through it. And so we
could think of that as like an x-ray. Imagine we're making x-rays and we have a source on
this side. And we have a sensor on this side. And then we put some kind of an object in
the middle. Well as we rotate that object in the middle, the x-rays that are moving
through it are going to be refracted and they're going to be bent. And so what we get is something
called a sinogram. And so what Radon said is let's start with the sinogram itself. Let's
start with this pattern. Could we work backwards and figure out what that shape inside that
sensor is actually going to be? And so the mathematics gets incredibly complex. These
are things that I have no idea what this means. Euclidean space. Hough space. Large vectors.
And so basically what we can find is if we work backwards through all of this we can
figure out what that structure is. Mathematics reveals this to us. But he wasn't really looking
at an application. But we have a huge application of this. If we put an object in the middle,
shoot x-rays through it and then rotate that, we can figure out what it is using mathematics.
And that's how a CT scan works. You are in the middle of x-rays and as those go through
you it's exposing things that we couldn't see. And so you can see how this is like a
new microscope. It's allowing us to see things that we'd never seen before. And so there's
huge implications here. And what we're getting now is an explosion in this new field called
mathematical biology. Where is it coming from? Well we have all of this data, genomic data,
data that's locked in the DNA. It's being unlocked. We have Moore's Law where were getting
faster computers that get cheaper. We have fractals. This right here is a rule of thirty.
It was developed by the person who started WolframAlpha. And basically what you get is
patterns that are predicting what life looks like. This cone snail actually has a fractal
growing on its shell. Or we're having now insilico experiments. What's that? Experiments
that we can do in a computer rather than doing experiments in the lab or experiments using
living material. And so let me throw up this quote from Dr. Cohen. "The calculus of Newton
and Leibniz and probability theory is the line between ancient thought and modern thought.
Without an understanding of the concepts of analysis, especially the concept of a limit,
it is not possible to grasp much of modern science, technology or economic theory. Those
who understand the calculus, ordinary and partial differential equations and probability
theory have a way of seeing and understanding the world, including the biological world,
that is unavailable to those who do not." And so what he's saying there is that if you
really want to be a biologist, it's not an excuse now to say that I don't understand
the mathematics. And when I looked at this quote I saw that the limit ends up being the
limit between really understanding biology and not understanding it. And so I wanted
to understand like what's the importance of a limit. I mean I wanted to consult somebody
who really knew what they were talking about. And if you're talking about the internet and
you're talking about mathematics there's really only one person who knows that. And that's
going to be PatrickJMT. And so what I've done is I've written my question, "Why are limits
important in mathematics and calculus?" And what I'm going to do is put this in a manilla
envelope. I don't know if this will work for you, but it works for me. Put it in an envelope.
I'm going to put my return address here, bozemanbiology. And I'm going to send this to PatrickJMT.
And I'm going to see what he can do. If he can come up with a good answer to why limits
are important? And what do they do? Okay. So I recently got this package from Paul Andersen
of bozemanbiology bozemancscience, teacher extraordinare. And he asked me two, two simple
questions that you can spend a lot of time on and gave me the Herculean task of doing
it in about five minutes or less. So he asked what are limits and why are they important?
And I wish we had ten hours because I think that's what it would take to do it justice.
But let me give kind of a quick intuitive idea. So let's start off by saying what is
a limit? So intuitively let's start with a sequence of numbers, say 1, 1.5, 1.9, 1.99,
1.999, etc. It look like those numbers are getting closer and closer to the number 2.
So if these numbers get arbitrarily close to one single number, in this case the number
2, we say that number is the limit. So take a sequence of numbers until they get close
to some number. If so, we say yes. That's the limit. That's to me intuitively a limit.
Now why do mathematicians like limits? Well they give us a way to make a lot of intuitive
and I should say non-intuitive ideas much more formal. Mathematicians need this formal,
these technical definitions. That gives us something to test our theories and our ideas
against. Okay? So that's the basic idea. Now, so I kind of gave you the hand wavy definition.
If a mathematician said what's the limit to a number? There is the technical definition.
It's actually interesting because it took historically a couple of hundred years after
calculus was developed for people to agree upon this technical definition of a limit.
And it's been giving calculus students nightmares ever since. So Newton and Leibniz actually
used what were called infinitesimals and there were some philosophical objections to that.
Although it's now been shown that they, there in fact are no issues. You can use these,
what are called infinitesimals. But we actually use this definition, I believe it's typically
attributed to Koshi is who came up with this. So interesting, it actually took a couple
hundred years to come up with that. Well in calculus where do we need, you know, where
do we need these definitions? Or where do we need these limits I should say to produce
these definitions? Well two huge places is to find the derivative of a function. That
tells us about instantaneous rates of change. And also in definite integrals. That calculates
the net change of a function. So just again, two specific applications here of limits.
And these, finding the derivative and also being able to find definite integral, just
has tons and tons and tons and tons and tons of applications. Just, the idea is that you
can deal with sort of functions that are constantly changing. And that was an issue in mathematics.
It was easy to, you know, to deal with sort of static situations. It was this problem
where things keep changing that provided a real, a real mechanical nightmare for mathematicians.
How do you compute things when things are constantly changing? And calculus solved that
problem. So there's the technical definition of a derivative of a function. Again, it involves
a limit. Well I would actually do want to. We don't have time to. And also to calculate,
well I should say to define a definite integral. Again we use limits. And this time we actually
have a limit as n goes to infinity. Okay. So the real magic was it also is how do you
actually compute these things, right? You're letting something get arbitrarily large. How
do you mechanically deal with that? And that was one of the magical things about calculus
was it got over that hurdle. So maybe something a little closer to home for those of you that
haven't had calculus. There's these really neat things called Markov Chains. And in this
case it's called a transition matrix. So this has to do with a little bit of genetics. So
suppose we've got a plant species and it has red, white or pink flowers. And each one of
those genotypes is going to be crossed with a red flowering plant to produce the next
generation. We want to know what happens in the long run. So according to this transition
matrix, it says if you start with a red plant, you cross it with a red plant, it says there's
a 100% chance that the next generation will produce red plants or will produce red flowers.
No way that it's going to make pink or white flowers. If it starts pink you cross it with
a red flowering plant, it says there a 50% chance the next generation will give red flowers.
50% chance it will make pink flowers. If you start off with a white plant cross it with
a red plant, it says there's no way it gives you red flowers, but it does always give you
pink flowers, the next generation. Well if we keep crossing these plants over and over
and over. If we keep, you know, keep this process up where we keep the next generation
we again cross it with a red flowering plant, we can ask ourselves, what happens in the
the long run? And it turns out, you know, in the long run, suppose we call this our
matrix T. It turns out in the long run to compute this what we do is we start taking
our matrix T and we multiply it by itself. We raise it to the power of n. And we can
actually show that as n goes to infinity, what happens, if you keep multiplying this
matrix by itself, you'll get a new one. Again multiply it by this one, you'll get a new
one. Again multiply it by this one, you'll get a new one. You can actually show that
you produce the following matrix. So red, pink, white, red, pink, white. And what this
tells us is in the long run it's actually all the plants will eventually be red flowering
plants. So limits help you determine what happens in the long run. So you know certainly
if you are, you could easily collect this data in the lab, right? You know you've got
some plants. You see what happens to the offspring. You collect this data. You obviously don't
have to do this for 100 years and that's where mathematics comes in. It says, hey, you know
we can analyze what happens using powers of the matrix and taking a limit and then we
don't have to wait a million years to see what happens. And, yeah, okay, we're talking
about pretty flowers, but certainly this is genetics. There's gobs of applications here.
So, alright, I hope I this does justice to limits. I know it's super quick. Again, you
know, so I would say it's been instrumental in defining ideas in calculus. And also it
gives us a way to deal with things mechanically. When things get really large, and also really
small. Thanks Patrick. That's amazing. I think mathematics offers us a new way to look at
biology. It's going to open up new possibilities that we didn't understand before. And I think
biology is going to stimulate new mathematics that didn't exist before. It's going to allow
us to understand big concepts like the biosphere, molecules, macroevolution, photosynthesis,
epidemiology. How disease flows through humans. And could maybe even unlock the brain. But
when I was listening to Professor Gedeon talk about the brain and talk about all these complexities,
what he said was interesting to me. What he said was, let's imagine we want to take a
brain and build a mathematical model of that. So how do you do that? Well there's a hundred
billion neurons in the brain. We're going to have to create a model that has a hundred
billion parts. And when we've done that, we've now created something as complex as the things
that we hope to understand. So that didn't get us anywhere. And so I think were just
starting down this pathway into understanding how biology really works. Mathematics is going
to allow us to look at that. But we're going to need people like you. Bright students who
understand biology and also understand mathematics. And so work hard and I hope that was helpful.