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Hello.

We are now onto the second lecture

on mathematical modelling with partial differential equations.

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In the second lecture on PDEs, we're

going to continue discussing the Reaction-Diffusion equation.

We've seen two examples of this already.

And we'll just sort of reiterate why is it

that this equation occurs so frequently in biological systems.

And then we're going to the rest of the lecture is

going to be to introduce some PDE examples from the literature.

How have how have PDE simulations,

and in particular, simulations of Reaction-Diffusion

systems, how have they been used to gain new insight, into biological processes?

We're going to show an example of simulations of intracellular calcium.

We're going to show the predicted effect of how

cell shape can affect signaling and then we're going to

show the study that generated predictions and also

experimental validation of activity of some kinases within cells.

Where we left off before was looking at the

one dimensional cable equation versus the epithelial reaction-diffusion equation.

We showed the cable equation that looks like this and we discussed how

this term is the diffusion term and this term is the reaction term.

This is a reaction that, that increases

or, or decreases membrane potential and membrane voltage.

And then we also showed how that was analogous

to this equation here for diffusion of bicarbonate across

the epithelium, where this is the diffusion term and

this was, this term described a reaction that consumes bicarbonate.

Now we're going to show some other biological

processes where reaction-diffusion equations also come into play.

One example of where reaction-diffusion equations come into play is when

we're thinking of changes in intracellular calcium during release.

Now if we wanted to derive equations for intracellular calcium we would use the

same sort of rules, same procedure that

we used for describing bicarbonate across the cells.

So if you could just replace bicarbonate with, with calcium

in the derivation that we had in the last lecture.

So we're not going to go through that whole derivation again,

but how did how did reaction-diffusion equations come into play here?

Well, the important thing to note is that in,

in heart cells, which is one cell type where

release of intracellular calcium is extremely important, in heart

cells, calcium is released from discrete clusters of channels.

And we can, that can be illustrated here.

We're not going to go through all of the, all

of the biological details, and all of the terminology.

But basically, this represents the cell membrane up here.

You have these discrete membrane channels within the cell membrane.

These membrane channels can let in calcium.

When calcium comes in through an open membrane channel,

so the way that we've represented this here is

that this red one is open and it's letting

calcium in, all the other membrane channels are closed.

This is, this calcium can go up and this calcium can bind

to these release channels that are part of the sarcoplasmic reticulum membrane.

So the sarcoplasmic reticulum is the endoplasmic particulum in other cells,

but in muscle we refer to this as the sarcoplasmic reticulum.

And these channels, these release channels in

the sarcoplasmic reticulum membrane are sensitive to calcium.

So when calcium comes in through this

membrane channel, those calcium ions can bind to

these release channels and these release channels can

then open and they can have calcium flow.

They can release calcium.

Calcium will flow from the inside of the sarcoplasmic reticulum into the cytosol.

And the key point here is that we're

talking about discrete clusters of, of these channels.

So if this is the only membrane channel that's open,

then you're only going to get release from these release channels here.

Whereas these other ones over here and these over

here and these over here are not going to release calcium.

So what does that mean?

Well, what that means is that when calcium comes out, it's going to diffuse.

It's going to move from left to right here or from right to left over here.

And again, we're going to have a reaction-diffusion equation, where

the reaction in this case is the release process.

And then after the calcium gets released,

it diffuses from, from one location to another.

So again, the released calcium diffuses within the cytoplasm and

the reaction, in this case, is the release of calcium.

And once the calcium gets into the,

the cytoplasm, then there will buffering reactions.

Calcium combined to, to various instances of other proteins and so those buffering

reactions are also going to be part of reaction term, in this case.

So again, release of calcium from discrete clusters of channels in heart

cells will also yield a reaction-diffusion equation describing the process.

What's another example of a reaction-diffusion

equation that's that's encountered in biological processes?

Well, other second messengers another very common one besides calcium is cyclic AMP.

Other intracellular second messengers are

also going to obey reaction-diffusion equations.

Cyclic AMP is produced by an enzyme that's illustrated

up here called adenylyl cyclase, or AC for short.

And adenylyl cyclase is located in a cell membrane.

If adenylyl cyclase were located everywhere within the cytoplasm,

then the cyclic AMP might not be able to diffuse.

But when cyclic AMP is produced at the adenylyl cyclase, it can diffuse away.

As it diffuses away, it combines to different effector molecules.

Protein kinase A is one, Epac is another effector molecule, and it

can also be degraded and the degradation process is shown here, with PDE.

For.

For, this is not for partial differential equation in this case, this

is for phosphor-diesterase so the cyclic AMP is produced at a discreet location.

It's produced at the cell membrane by adenylyl cyclase.

It can diffuse a way and then it can bind to the different proteins or be degraded.

And so these binding and degradation processes are a reaction term.

The diffusion results from the fact that it is produced at a discrete location.

So these examples show that these sorts of processes are very common.

It's very very frequently within biology, you will encounter something

where a species is produced or generated at a particular location.

It can move away from that and then as it moves away, other things can happen to it.

Often degradation or, or binding to other, other species.

So because these processes are common,

reaction-diffusion equations are also very common.

And this, these are the reasons why reaction-diffusion

equations are encountered so frequently in describing biological processes.

Now, for the remainder of this lecture we're going to

show some examples of partial differential equation-based modeling space.

I feel like we don't have enough time in this class to really

go into PDEs with the same depth with which we went into ODEs.

And it's also true that there have been a lot more modeling studies

in Biology that have used ordinary

differential equations, compared to partial differential equations.

So we did sort of bias it in a way that was related to, to how frequently

these things are used so we're not going to have

the same amount of depth that we had previously.

But I do want to show some examples of how PDEs have been used.

We're going to go through these in a little bit of a cursory manner, however.

One example is calcium in the cytosol and

calcium in sarcoplasmic reticulum during release in heart cells.

This is actually one of the studies from my lab

that was published a few years ago in the Biophysical Journal.

And this is showing a simulation

of cytosolic calcium and sarcoplasmic reticulum calcium.

And what you can see is that these are

plotted as a function of time, which goes along

this axis here, and location, which goes along this

axis here and the concentrations are, are color coded.

The details aren't so important here.

But this and the bottom panels here, which, which

zoom in a little bit on the images that

are shown in the top, you can see simulations

from one and these are the locations that release calcium.

And we can see simulations that as the calcium

moves from one location to to the next location.

And we looked at here at slow diffusion

within sarcoplasmic reticulum and then a medium speed

of diffusion within the sarcoplasmic reticulum and fast

speed of, speed of diffusion within the sarcoplasmic reticulum.

Take all these numbers down here, which are in

units of length squared per time, are our diffusion constants.

And again, without getting into all the details the main message here is that the

calcium diffusion within the sarcoplasmic reticulum matters just

as much as calcium diffusion within the cytosol.

So in this particular example, we actually

looked at two coupled partial differential equations,

one for calcium within the cytosol, another

one for calcium within the sarcoplasmic reticulum.

And in terms of the, the patterns that, that emerged,

in terms of what we saw during the release process,

we, we found that the calcium diffusion within the sarcoplasmic

reticulum is as important as calcium diffusion within the cytosol.

A second example that I want to discuss was a study that looked at

how does the cell shape influence the signaling that is observed within a cell.

And these are some images that are shown from this paper

here by Meyers Craig, and Odde from 2006 in Current Biology.

This was a great paper where they looked at different

shapes of cells and they looked at how different shapes

of cells could could affect the the patterns that were

observed In terms of the signaling molecules that were produced.

And these images here, the, the, heat maps that are at these discrete locations

within cell, are showing relative activity of,

of the various kinases or various signalling molecules.

And the message of this cell, sorry, the message

of this particular study, this paper, was that geometry matters.

The degree of phosphorylation will be different in

thick regions, compared to thin regions of the cell.

For instance, in the thin region here you

see you get very little activation in the center.

But in the thin region here, at the very leading edge of this

polarized cell, you get very, very strong

activation of the, whatever your kinase is.

And I think that you know, biologists and biochemists are very

used to thinking that the amount of signal that you get

depends a lot on the on, on the activity of your

enzymes, depends on, on the whatever species are, are producing that signal.

So if you're talking about phosphorylation or

a particular molecule, biologists are very comfortable

with the idea that the amount of

phosphorylation you get depends on the kinase

activity, but this was a, this was an influential study and this was a

very cool study, because it illustrated that

while it's not just the, this biochemistry.

It's not just the kinase activity that matters.

It's also the geometry that matters very much.

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And finally, a third example that I want to discuss is a study

that predicted and then also measured

spacial gradients in, in signalling molecules.

And this was a study that actually came

out from some colleagues of mine at Mount Sinai.

The first author, Dr. Susanna Neves and the senior author on this study was

Dr. Rob A Engar, who also teaches a course on Systems Biology in Coursera.

You can see that these are some really cool

simulations they did of protein kinase-A activity and MAP-kinase activity.

We encountered MAP-kinase in some of the some of the earlier lectures.

This is mitagen-activated protein kinase activity in realistic Geometry.

So they took the neurons, they traced the geometries and then they simulated how the

activity of these would vary at different

locations within these neurons that had realistic shapes.

And in addition to doing simulations with these realistic

geometries, they also did some simulations in idealized geometries.

And these are some space-time images similar to

what we saw two slides ago in the simulations

of intercellular calcium, where you have distance along

this axis here and time along this axis here.

And these are idealized neurons with different diameters.

You can see you go from one micrometer diameter, to

1.5 micrometers, down to two and down to three micrometer diameters.

And they did simulations of how protein kinase-A activity or MAP-kinase activity

varied as a function of distance and as a function of time.

And the message in this in this paper was,

one message was, that spacial signals depend on geometry.

We saw that in the previous study, however, by Myers et al.

What was unique about this, this study by Neves, et al, was he said that spacial

signals don't only depend on Geometry, but they

depend on whatever processes are turning the signals off.

So protein kinase-A gets activated when you have when you have an increase

in cyclic AMP due to, you know, due to activation of adenylyl cyclase.

This gets shut off in part because of the phosphodiesterases.

Those are the proteins those are the enzymes that are degrading cyclic AMP.

And what was what they looked at in this study was not just the was not just how

the geometry affected these processes, but also how the,

the, regulatory networks that these processes were embedded within.

In particular, whatever the, the signals that shut off the signals were.

Whatever those, processes were, how those affected the spatial gradients.

So this was one that, that looked at both the geometry and the biochemistry.

And one other thing that was really cool about this particular study

by, by Neves et al was that they didn't just make predictions using

the mathematical model, but then they

also made measurements of these spacial gradients

that were observed in in signaling

molecules, such as protein kinase-A or MAP-kinase.

So this is showing the the prediction and

then over here, you're seeing the, the experimental data.

So the two images on the left are, are

control images and then on the right you see how

the how activation of a particular a signalling cascade will

change your, your signal as a function of, of location.

And then here you have the the prediction when you're inhibiting whatever

the, the negative feedback is, when

you're inhibiting whatever, whatever shuts it off.

But the main message here is that what was cool about this paper was that they didn't

just make the predictions using the mathematical model, then

they also went in and tested those predictions experimentally.

And we some examples of that in our, in the earlier lectures.

In particular, on, on some of the

cell cycling, the cell cycle mathematical models.

You know, the purpose of doing the mathematical

modeling is not just to build the model.

One of the main purposes is to be able to generate predictions,

that then you can, that you can then go in and test experimentally.

To summarize this lecture, what we've seen is that

reaction-diffusion equations occur repeatedly in biology, and these occur

whenever the processes that produce some species and the

processes that degrade some species are, are spatially separated.

If the, if the production and degradation processes are,

are everywhere, then you won't necessarily get a reaction-diffusion equation.

In that case, ordinary differential equations will suffice.

But if the species is being produced at

one location and then it's being degraded at

some other locations, what you're going to end

up with is a reaction-diffusion equation describing that.

So that's why these sorts of PDEs are

encountered so often when we're looking at biological processes.

Another thing we see is that modeling studies based on PDEs

can provide insight into the relative

importance of biological versus geometrical factors.

So when you have something that is described by a PDE and when it's

in particular when it's described by a

reaction-diffusion equation, that reaction part can be important.

The diffusion part can also be important and that's

why you need a mathematical model in order to

be able to to tell you, well, you know,

in which circumstance is one more important than the other.

And that's how the, the model can really help.

It can give you insight into the relevance

importance of, of the biochemistry versus the geometry.

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Lecture 27 - Modeling with Partial Differential Equations - Part 2

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