0:07

Hello, we're now at our third lecture on mathematical models of the cell cycle.

Previously, we discussed the biological background as necessary

to understand these models of the cell cycle.

And then we discussed it in some detail the

model developed by Novak and Tyson and published in 1993.

Now we're going to take a step back in some

ways and, and discuss in a more historical or

even a more philosophical level, how these types of

mathematical models develop and, and how they evolve over time.

Which I think it's a useful exercise because it helps you to

understand the, the assumptions that go

into developing these dynamical mathematical models.

0:47

And our, our theme in this lecture is going to

be that of phenomen, phenomenology versus mechanism in, in mathematical models.

And the general idea, what I, what I mean by this

is that sometimes models describe

the actual biochemical or molecular mechanisms.

But sometimes they just describe a phenomenon that's observed.

And both both types of approach's can, can be useful in

some cases and that's what were going to argue in this lecture.

And to illustrate what I, what I mean by this, what

I mean by phenomenology versus mechanism, we're going to provide some examples.

And the examples we're going to show are going to be to contrast the

1993 Novak and Tyson model that we discussed in, in lecture two.

We're going to contrast that with an earlier

model that was published by, by one of

those same authors, by Tyson a couple of

years previous to the Novak and Tyson model.

And then we're going to also contrast the 1993

Novak and Tyson model versus more contemporary models.

1:46

The general sort of conclusion we're going to reach from this, is that

models generally become more mechanistic and, and more complex as they evolve.

And that part's not really so surprising,

because we learn more about biology every day.

And we learn more about the complexity of biological systems every day.

So, the fact that models generally become more complex isn't so surprising.

But, then we're going to give at least one example

for, for when models can become simpler over time.

[SOUND].

To review, we previously discussed this model, the

1993 Novak and Tyson model of the cell cycle.

This shows the what the, it looks like on the

title page and this gives the the full reference down here.

This was discussed in the second lecture on cell cycle models.

2:34

But what I didn't tell you during that

lecture, is that two years previous to that one

of the same authors John Tyson published another

paper with a mathematical model of the cell cycle.

2:48

And the reference for this paper is, is given down here.

What we're going to do in the next few slides is we're

going to, we're going to contrast the 1991 model with the 1993 model.

And, the reason I think it's useful is that when

we compare these two models, we look at particular processes

and how they may have been represented in one model

versus how they were represented in, in the other model.

This can illustrate this, this theme that I want to emphasize,

of phenomenology versus mechanism in the development of mathematical models.

[SOUND].

When I use this phrase, phenomenology versus mechanism, what do I mean?

3:30

We can see what I mean by this,

by comparing the 1991 model with the 1993 model.

The scheme for the 1991 model is given over here.

And then the scheme for the 1993 model that we

discussed in, in lecture two is, is given over here.

And the first thing you notice is that the 1993 model

has, has more in it, it contains more species, it contains processes.

And we, you know, conclude from that that between

1991 and 1993, new processes were added to the model.

[COUGH]

This occurred because this was a very active time

in, in cell cycle research and between, in, in

those intervening two years, our people learned more about

exactly what was going on in the cell cycle.

So you can see a Cdc25 up here in the 1993

model and you don't see a Cdc25 in the 1991 model.

Similarly, Wee, Wee1 is included in 1993 and is not included in 1991.

So, we're going to discuss of couple of these processes in a little more detail to

illustrate what I mean when I talk

about a phenomenological representation versus a mechanistic representation.

4:42

For instance, let's contrast the 1999 model versus the 1993 model.

And look at how these two

models represented autocatalytic activation of, of MPF.

What I mean by autocatalytic activation of MPF is what we've

discussed in the in the very first lecture on the cell cycle.

Is that when MPF activity goes up, that leads to even greater MPF activity.

5:07

In 1991 Tyson represented it this way, where the top species here, we have cyclin

bound to Cdc2 and Cdc2 which, again, is a synonym for CDK, cyclin-dependent kinase.

The Cdc2 doesn't have a phosphate group on it.

And then down here we have the inactive MPF.

Where you have cyclin bound to Cdc2, but

Cdc2 does have the inactivating phosphate, all right.

In 1991 Tyson did draw, Tyson did include autocatalytic activation of MPF.

But you see that you, the MPF goes directly

the arrow goes directly to this rate constant right here.

So, he said this was a direct affect of MPF.

And he wrote down an equation that looked like this.

Where the rate, it was pre-MPF, or inactive MPF, gets converted to MPF,

is pre-MPF concentration, so this is a substrate, times this rate constant here.

And you can see this rate constant here depends on MPF itself.

6:09

By the time Novak and Tyson published a more detailed model

in, in 1993, they didn't represent the process this way any more.

Here you have inactive MPF on the left-hand

side and active MPF on the right-hand side.

And, you can see what we talked about in, in lecture two on the, on the cell cycle.

6:44

Where you have the rate at which pre-MPF gets

converted into MPF depends on your concentration of phosphorylated Cdc25.

And then the rate at which Cdc25 gets converted from

the unphosphorylated form to the phosphorylated form depends on MPF.

So here we have a process where MPF catalyzes phosphorylation of Cdc25.

And then Cdc25, which is a phosphotase,

7:12

catalyzes dephosphorylation of the inactive MPF process.

So in 1991, this was represented in a phenomenological way, where MPF

concentration goes up and that increases the rate at which MPF gets formed.

In 1993, more info, more information had been

gained by that point about the action of Cdc25.

And so when Novak and Tyson published their model a couple years later, they

actually included Cdc25 explicitly, and they represented the process this way.

8:02

Again, this is the scheme in Tyson's 1991 model.

And you can see that you get active MPF

goes back to inactive MPF through this reaction five here.

And in the 1991 model, the rate constant k5, which is the perimeter, which is the

constant number, there's no inclusion of of the species we want in this 1991 model.

But as we discussed in the previous lecture, when Novak and Tyson

published their model two years later in 1993 they explicitly included Wee1.

8:39

So, in 1991, this conversion of active MPF back to the

inactive MPF form just occurred with a constant rate rate constant.

But in 1993, this phosphorylation reaction occurred through the

protein that was known to immediate this phosphorylation reaction.

Which at that point was known to be Wee1.

So again, this, this changed from being a

very sort of general phenomenological representation in 1991.

To a more mechanistic representation in 1993 where

the actual kinase that, that added the phosphate

group to the CDK, that kinase being Wee1,

was explicitly included in the later mathematical model.

9:25

As a third example, let's consider what

happens after MPF get's activated, after you

have a large increase in MPF that

initiates the mitosis phase and initiates cell division.

9:37

As we've discussed, cyclin gets degraded at that point,

and then MPF concentration goes down, because MPF needs cyclin

in order for the, for the kinase activity to be

active, and then the cycle is able to begin again.

9:54

So, what what happens to terminate this is, you have a, this reaction here, from

active MPF, degradation of cyclin, and out here you have Cdc2, also

known as CDK all by itself, no longer bound to cyclin.

In 1991, Tyson represented this with a constant rate.

He said degradation occurs at a constant rate, so this rate constant k6 that

mediates this, is again, just a, a normal parameter, it's just a constant.

But in 1993, additional processes were included

in the model, where active MPF phosphorylates this

protein here called IE for intermediate enzyme

and then phosphorylated IE activates the anaphase-promoting complex.

And it's the anaphase-promoting complex that explicitly degrades cyclin.

10:43

So, this was yet another process in the

model that went, moved from a phenomenological representation here.

To a more mechanistic representation over here, where MPF indirectly activates the

APC and the APC, the anaphase-promoting

complex is what explicitly degrades the cyclin.

11:07

consider the species in the model that we

just talked about called IE, or intermediate enzyme.

Most of the other variables in the model that we've, that we've discussed have

more descriptive names than this, and some

of these proteins you maybe even heard about.

You may have heard about the anaphase-promoting

complex you may have heard about cyclin.

IE, or intermediate enzyme is a, is a much more general term.

11:31

So what does IE represent?

Well, when Novak and Tyson actually published this

model back in 1993 they didn't know, in

fact they they just hypothesized that, that there

must be an intermediate enzyme in this case.

So this is actually something they included in the model, only to account

for the delay between the increase in

MPF and the activation of anaphase-promoting complex.

The idea was that if they went directly

from active MPF to activation of the anaphase-promoting

complex they thought that the APC turned on

too quickly, and then cyclin got degraded too quickly.

And they said, well, there, the APC turns on more slowly

than the way it's being simulated right now in the model.

And so maybe there's something in the middle, maybe there's something

in between and they just called it IE for intermediate enzyme.

And they said this is something we put into the model that was hypothetical.

We, we think it might exist but but we're not sure what, what it

is and so therefore we're going to give it a very, a very general term.

12:32

And so, again, it was put into the model

only to account for the delay that they knew

needed to be present, but when they put it

in the model, they didn't know what it was.

12:42

Later people did experiments and they were able to

identify what this was, and it's now known as Fizzy.

That's a term from Drosiphila or Cdc20 which is the name of it in the yeast.

So, these are the two corresponding analogous genes.

But it's known that these are the two these

are what are code this this intermediate enzyme here.

13:03

In the previous lecture we talked about how some of the

predictions of the Novak Tyson model were later proven to be correct.

I think that intermediate enzyme can be can

represent yet another experimentally confirmed prediction of this model.

Another one of the successes of the Novak and Tyson 1993 cell cycle model.

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Now as we go, go through this process, and we compare

the 1991 model to the 1993 model the over theme we

come up with is that several processes were modeled in a phenomenological way

in 1991, and then were described more mechanistically in 1993.

And the reason we go through this process is to,

to illustrate that this is how dynamical models typically evolve.

It's very common that when you, when you're first developing

a model, you might not know what all the intermediates are.

You might not know what all the all the players are.

And you might not be, have enough knowledge to be able

to represent all the mechanisms that are part of this model.

But it's still very useful to represent things in a,

in a, in a very general way, and then later

when you get when more data are obtained, then the

process can be represented in a much more mechanistic way.

So, what happened in the development of these cell cycle models by Tyson and

coworkers is very typical of what happens in a, in a lot of fields.

Sometimes things start very phenomenological and then, as

time goes on the representations become more mechanistic.

14:53

So, I just want to note that, you know, since the development of Novak and

Tyson model in 1993, it's not that they, these investigators quit at that point.

They have continued to develop these models

and more components have, have been included.

We're not going to go through these more recent models in, in-depth.

But I provide these schemes here just for reference.

There's a generic model of cell cycle regulation and one

of the, the somewhat recent studies here was in 19 2006 in the Biophysical Journal.

And this shows the the overall scheme here.

You can see the Cdc25 as we're talking about.

We have one here as we're talking about.

But you can see many more components that have been discovered subsequently that

are also included in this, in this generic model of, of cell cycle regulation.

15:43

In addition to the generic model from 2006, there

was another more more detailed model that has been published.

This is specific to budding yeast, there are a

lot of data from yeast because in yeast just

about any knockout or any combination of knockouts you

can imagine can be produced and has been produced.

So, the cell, cell cycle model from Novack and Tyson in 1993 has now diverged into

two models, one specific for yeast and another one generic to a, to vertebrates.

16:15

And this shows the, the overall scheme for the 2004 model of yeast cell cycle.

And you can see from these two schemes that many more

components have been included in the model as as biological knowledge increases.

As we learn more about what's important in the cell cycle.

These elements get included into these mathematical representations.

[BLANK_AUDIO]

From that discussion of how the cell cycle

models have evolved since 1993, you might conclude that

models, dynamical models and biological processes invariably get

more and more complicated as we gain biological knowledge.

And it's true that that's how these things usually evolve.

Usually we learn something new about how a process works.

And then, that that novel mechanism of

regulation gets included into the mathematical model.

But I think it's worth pointing out that

occasionally things get simpler rather than more complicated.

17:16

In discussing the Novak and Tyson model, I've

been using this diagram over here, because I think

this is a nice diagram that comes from a

review article published by Sible and Tyson in 2007.

But, one of the original diagrams from the

Novak and Tyson model looks like this over here.

This part is just showing the process of convergent of pre-MPF into MPF.

And if we look at the 1993 diagram, we can see that it actually includes more species

and is, it is more complicated than the 2007 version.

In 1993, when Novak and Tyson actually published

this, they included the effects of, of CAKs.

These are the kinases that put on the,

the activating kinase that occurs at 3NE and 161.

And this phosphorylation at 3NE and 161 by CAK is no longer included in the model.

So, this is something that actually got simpler rather than more complicated.

18:13

They, they included these steps here, putting on the activating phosphate.

And taking off the activating phosphate which can occur either when you have the

inhibitory phosphate tyrosine 15 on it, or, or not.

But this phosphorylation of 3NE and 161 by

the CAK, is no longer included in the model.

And this is something that actually came

out of the simulations with the model itself.

The simulations showed that 3NE and 161 was almost always phosphorylated.

So, you can put a process like this where you, you put on you can

include a process like this where you put

on this activating phosphate or take it off.

And over here, you put it on by moving up or take it off by moving down.

But if your simulations show that you're always

going to be in these upper states up here, then

these lower states down here that don't include the

activating phosphate, aren't really necessary any more, are they?

And, that's what these investigators concluded by working with this model.

Since 3NE and 161 was almost always in the phosphorylated

form, it was okay to exclude the unphosphorylated form the model.

And so, this is an example of where things got simpler rather

than more complex and by you were able to make it simpler.

You were able to justify that, in part

through the simulations with the dynamical mathematical model.

19:37

To summarize this third lecture on cell cycle models.

What we've seen is that

dynamical mathema-, mathematical models frequently evolve.

And one of the way, the primary ways in which they evolve is that they start with

a phenomenological representation or phenomenological description, and then

that can be modified into a more mechanistic description.

And, again, to redefine what I mean

by that, phenomenology means that you might measure

some species B and you can observe that

species B increases when some species A increases.

But, you might not know if there's something in

between A and B or if that's a direct effect.

You might only know that B is going to go up when A goes

up and you might not know the mechanism by which this this occurs.

You can still put that in a model but

in that case that would be a phenomenological representation.

20:37

So, when this is put in explicitly as a

phosphorylation reaction, you can consider this a mechanistic representation.

But, if you're just saying that B is going to

increase when A increases, that would be a more phenomenological representation.

[SOUND].

However, I don't want to make it sound like I'm being extremely

critical of phenomenology, I think phenomenological

representations can still be extremely useful.

And they can be especially useful when, when mechanistic detail is lacking.

21:10

And what we seen by looking at some of

these examples is that cell cycle models by, developed

by Tyson and co-workers, they do provide excellent examples

of, of how models can evolve in this way.

And how they can change from

phenomenological representations to mechanistic representations over time.

And as additional biological knowledge gets

incorporated into the dynamical mathematical models.

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