0:28

We're going to discuss the structure of the

Â Novak and Tyson model, in particular the biochemical

Â reactions that are a part of this model,

Â and the differential equations that constitute this model.

Â And, then we're going to talk about the relevance of the Novak-Tyson model.

Â Why do we think this is something that's

Â important to discuss, 20 years after it was published?

Â So we're going to discuss some of the insight that

Â was gained from the simulations that Novak and Tyson performed.

Â And then, we're also going to discuss model

Â predictions that were confirmed in subsequent experiments.

Â And we really want to emphasize this last point, because this is

Â going to illustrate one of the

Â great strengths of dynamical mathematical models.

Â Is that they can generate predictions, sometimes these

Â are nobel predictions, and these predictions can inspire

Â experiments, and occasionally there's a success story, where

Â these experiments confirm their predictions of the mathematical model.

Â 1:23

As management, we're going to discuss the 1993 Novak Tyson cell

Â cycle model, this was described in a very influential paper that

Â these two authors published in the Journal of Cell science in

Â 1993, and the complete reference for this work is given down here.

Â Now

Â 1:54

There's one module here that's about regulation of cyclin and Cdk dimers.

Â Let's review what these species are here.

Â These are, these should sound familiar to you.

Â These are things that we defined and talked

Â about in the first lecture on the cell cycle.

Â 2:21

When it has both phosphate groups on it, it's inactive form.

Â So this species over here is pre NPF, and then

Â the transition from left to right involves removal of this phosphate.

Â This is removal of the inhibitory phosphate,

Â so this species over here is NPF.

Â Now let's recall what we, what we discussed about

Â how phosphorelation reaction and dephosphorelation

Â occur and how they're regulated.

Â 2:47

Dephosphorelation occurs through this phosphorylated form of cdc25

Â and this phosphorylation of cdc25 remember its regulated by itself.

Â And this part up here is how you have the regulation of the

Â de-phosphorylation reaction of, of MPF, and then down here

Â you have the regulation of the phosphorylation reaction of MPF.

Â 3:16

MPF will catalyze the phosphorylation of Wee1.

Â Phosphorylize phosphorylating Wee1 will put it into a, a less active form.

Â And it's the active form of Wee1 over here is the unphosphorylated form.

Â And Wee1, when it's active, will put the phosphate group

Â back on Cdk, converting the active MPF into pre-MPF over here.

Â And then of course we also have synthesis of cyclin from amino acids.

Â And cyclin and Cdk have to find one another,

Â and they have to bind to form a diagram.

Â 3:52

So these are the reactions in the Novak-Tyson cell cycle

Â model that have to do with regulation of the cyclin/Cdk dimers.

Â The second module in the Novak-Tyson, cell

Â cycle model involves regulation of cyclin degradation.

Â This is another, these are reactions we talked

Â about in in the previous lectures as well.

Â Remember we said that when MPF gets active, it

Â triggers its own degradation, and that's what's expressed here.

Â Active MPF will lead to phosphorylation

Â protein, are called IE for intermediate enzyme.

Â When IE gets phophorolated it's going to lead to

Â activation of this complex here, called the anaphase-promoting complex.

Â And when the anaphase-promoting complex gets

Â turned on it's going to degrade cyclin.

Â So that's how you're going to go from having

Â MPF over here which is Cdk bound to cyclin.

Â The Cdk all by itself over here, and the reason

Â it's by itself is that the cyclin has been degraded.

Â So this part of this module, model here involves regulation of cyclin degradation.

Â 4:59

And the overall, or what you might call

Â the wiring diagram of the model combines these two.

Â And this, this diagram comes from a very nice

Â review article by Sible and Tyson published in 2007.

Â But this describes the scheme of 1993 model by Novak and Tyson.

Â 5:21

Now let's consider the ODEs that are in the Novac-Tyson model.

Â We're not going to go through all the

Â equations, cause there's quite a few of them.

Â But I think it's helpful to, to point out that you can

Â divide the equations that are in

Â the Novak-Tyson model into two main categories.

Â One are, are those equations that involve synthesis or degradation of cyclin.

Â 6:05

Well this first term represents synth, synthesis of

Â cyclin, from amino acids, this step in the scheme.

Â The second term represents dimer formation.

Â When cyclin and Cdk come together to form either pre-MPF or active MPF.

Â That's going to make the free cyclin concentration go down.

Â And another thing that can make the free

Â cyclin concentration go down is, is degradation of cyclin.

Â So, that's represented here.

Â 6:31

So, as with other ODEs that we've seen

Â in biochemical reactions, the equation can look quite complicated.

Â But if you break it down term by term

Â usually the individual terms within the equation make sense.

Â The other thing I want to point out is that this rate constant k2 in this

Â case, this is typical of a lot of

Â the equations in the, in the Novak-Tyson model.

Â 6:51

This represents, this k2 will have a separate equation that represents it.

Â And why does it have this separate equation here?

Â Well k2 will have some rate, times the amount of anaphase promoting

Â complexes off, plus some other rate times, amount of anaphase promoting complexes on.

Â Remember that when, when cyclin gets degraded it occurs through

Â the anaphase-promoting complex, which we've abbreviated as APC in this case.

Â 7:21

The way that Novak and Tyson set up this model is, they're saying that if, when the

Â APC is off, it can still degrade cyclin, it can just do it at a much lower rate.

Â But then when it gets turned on, it

Â degrades cyclin at a, at a much higher rate.

Â And the way that you calculate the overall rate

Â at which cyclin gets degraded, that's k2, is you have

Â some weighted average of one rate times the amount that's

Â off, plus some other rate times the amount that's on.

Â And given those considerations, we can, we can deduce that

Â this term here, this v2_2 is much greater than v2_1.

Â And there are quite a few terms in the,

Â in the Novak-Tyson model that look, that look like that.

Â 8:09

The other main category is those

Â that involve only deep phosphorylation and dephosphorylation.

Â And those equations will look like this.

Â If we just look at Weee1 for instance, and if we

Â want to write down a differential equation for the amount of phosphrylated we

Â want, we can say that there's it's some rate, remember that

Â MPF is the, is the kinase that actually catalyzes phosphorylation of E1.

Â So there's going to be some [UNKNOWN] times

Â enzyme concentration, times MPF, times substrate concentration, which

Â in this case is the amount of unphosphorylated

Â B1, over unphosphorylated B1 plus some [UNKNOWN] constant.

Â So this is substrate, substrate plus

Â [UNKNOWN] constant, in the, in the denominator.

Â This is for the phosphorylation reaction.

Â And, of course, what opposes

Â the phosphorylation reaction is the dephosphorylation

Â reaction, which is also expressed with a Michaleis-Menten equation over here, and

Â PPase, in this case, just represents

Â generic phosphatase activity, where you, you

Â know exactly what the cynase is that's putting the phosphate group on.

Â You don't necessarily know what the phosphate, the

Â phosphatases that's going to take off this phosphate group.

Â And so to represent it mathematically, you just lump all the cellular phosphatases

Â together in a, in a parameter and it turns out we, they call PPase.

Â 9:33

But in general the you know, almost every differential equation in

Â a Novak-Tyson model can be understood by you know by putting into

Â one of these categories and breaking down the individual term whether it's

Â the phosphorylation term, is it a synthesis term or degradation term etc.

Â 9:52

At this point, we have established the structure of the Novak-Tyson

Â model and talk a little bit about the, the ODEs that earn

Â the Novak-Tyson model, now let's move on to think about, talk about

Â what are some of the results that they've obtained with Novak-Tyson model?

Â 10:06

Well, one important result they saw is that with the

Â controlled parameters, they get spontaneous

Â oscillations of MPF and cyclin.

Â And they said these were analogous to the

Â rapid divisions that occur in newly fertilized frog oocyte.

Â And so it was known from experiments that as soon

Â as the frog oocyte gets fertilized, it will undergo several stages

Â of of cell division, these are rapid cell divisions, and

Â those provided experimental data to which Novak-Tyson could compare their results.

Â And so this is an important result just as the validation

Â of the model that [UNKNOWN] shown out cycling will increase and decrease

Â as a function of time it's the black line here, and

Â then MPF will also increase and decrease as a function of time.

Â And this is what happens when you have the

Â control parameters, and you can plot these as a time

Â course, cyclin and MPF versus time, or you can

Â plot cyclin and MPF in the phase plane like this.

Â And you can see that with

Â control parameters you get these spontaneous oscillations.

Â So, that's one one good validation of the cell cyclin model.

Â If it wasn't cyclin then it might not be a such a good model.

Â They might have to consider something else.

Â But when they did put in the the actions they

Â thought were important, sure enough they saw they saw cyclin.

Â They saw these oscillations.

Â 11:27

We just saw that Novak and Tyson saw a spontaneous

Â oscillations of cyclin and MPF in their cell cycle model.

Â Is that enough to conclude that it's a, a good model,

Â and a, and a, and a worthwhile study, and a worthwhile result?

Â Usually not by itself.

Â But Novak and Tyson pushed their model a lot

Â further, and, and gained important insights by analyzing their model.

Â And one of the insights they gained is that

Â they observed bistability between total cyclin and an MPF activity.

Â 11:57

And in order to analyze this in a simplified

Â way, they used non-degradable cyclin in their in their,

Â their model, so that way you could control the

Â total amount of cyclin, that you added to the cells.

Â And they did this in part because there were

Â experimental observations that they, that they could compare this to.

Â 12:17

And, so this was a there were experimental results that they were, that

Â they could go by in the literature, that use this non-degradable form of cyclin.

Â And what do these experimental results show?

Â What we're finding here is time on the x-axis, and active MPF on the Y axis.

Â And what's interesting about this is that so you go from the,

Â from the bottom curve to the top curve as you add more cyclin.

Â 0.2 you get barely any activation of MPF.

Â 0.3 you get tiny bit more activation of MPF.

Â And then suddenly when you go to 0.4, from 0.3

Â to 0.4, you get a much bigger increase in MPF activity.

Â And then after that, you see a, a, you also see small small increases in the

Â amount of MPF activity, but the biggest jump in this case, is between 0.3 and 0.4.

Â And Novak and Tyson looked at this, and they said, this looks like a threshold

Â behavior, and a threshold behavior is something

Â that you often see when you have bi-stability.

Â We talked about bi-stability in the context of an all

Â or none response, and this is that's one of the

Â things that you see when you have an all or

Â none response, is that once you exceed some threshold level.

Â You get a large change in your output even for a small change in the stimulus.

Â 13:37

And so Novak and Tyson analyzed this in the model and

Â they said if we plot cyclin level on the X axis.

Â Again, this is non-degradable cyclin and the total amount of

Â cyclin you add experimentally and MPF activity on the y axis.

Â You see this bi-stable behavior, where cyclin goes

Â up and MPF goes up a little bit, a

Â little bit, a little bit and suddenly you get

Â this big jump here on this right hand arrow.

Â 14:19

So the threshold behavior that they saw here on the

Â left, was similar to experimental

Â observations that had already been obtained.

Â But this analysis they did here on the right showing that

Â the relationship between cycling and, and MPF is in fact bi-stable.

Â That was a nobel prediction of, of the model.

Â 14:37

And there's a general theme that, that I want

Â to emphasize in, in talking about this, is that

Â quantitative data obtained in a simplified preparation can be

Â very valuable when you're constructing a, a systems level model.

Â 14:54

understand this and analyze this is the

Â fact that it was done with non-degradable cyclin.

Â Right?

Â Because, if you add normal cyclin, what will happen?

Â MPF will go up and then cyclin will, and the MPF will trigger degradation of

Â cyclin and so then, it will go down

Â and you'll, you'll see a transient time course.

Â The fact that they were able to see this sustained rise

Â in MPF only occurred because they had non-degradable cyclin they were using.

Â 15:17

And this is something that's encountered, repeatedly

Â when you're talking about interactions between, ex,

Â between, between quantitative dynamical models and experimental

Â data is that when you can obtain

Â the data in a quantitative way and you can do it in a simplified

Â preparation, that can be very, very valuable

Â for constructing your systems of a model.

Â 15:38

And just to emphasize this point, of, how you

Â can obtain data, constrain your model of a complicated system.

Â The value that you get, when you have experiments

Â that remove one or more variables, experiments that simplify the

Â system, in other words, just provide a little bit

Â of foreshadowing of what we're going to discuss, in subsequent lectures.

Â 15:58

Which is a mathematical model of the of

Â the action potential in neurons, where you're looking

Â at how voltage changes with respect to time

Â in an excitable cell such as a neuron.

Â When everything is changing all at once, you can see

Â that your voltage wave form in this case is very complicated.

Â One thing you see, this is analogous to what we were just

Â talking about is threshold behavior, where these small changes in voltage will decay.

Â But larger changes will undergo much, much

Â great excursions in voltage from the resting level.

Â But one a, one of the things that we're

Â going to discuss when we talk about the actual potential model

Â is that, when voltage is changing as a function

Â of time, you see that the waveform is very complex.

Â The experiments that allowed the construction of a,

Â of a robust and very valuable model of the

Â neuronal action potential, were obtained by experiments that

Â fixed the voltage and measure the individual ionic currents.

Â So this is what we're going to talk about with the exponential models.

Â That when everything is changing at all once it's complicated.

Â 17:00

We're going to discuss experiments where the voltage is fixed.

Â And the currents were measured individually.

Â And we're going to argue that voltage clamp, which is

Â what this technique is called, where you're able fix the voltage.

Â Voltage clamp was the key advance that made the mathematical model

Â of the action potential developed by two investigators named Hodgkin and Huxley.

Â Voltage clamp is what made this model possible.

Â 17:44

It was already known from experiments, that if you edit unreplicated DNA into

Â your preparation that was undergoing cell cycle, unreplicated DNA would change the

Â way, the way that MPF was able to regulate Wee1 and Cdc25.

Â And we can look at these curves to see how,

Â how the unreplicated DNA changes the regulation of Wee1 and Cdc25.

Â These are plots of Wee1 activity vs MPF activity for no unreplicated DNA.

Â So this one on the left is a control

Â curve, and then if you add some unreplicated DNA, it

Â shifts it to the right, and then if you

Â add more it shifts it to the right even more.

Â So what you can conclude from this, if you just look at Wee1 versus active MPF, is

Â that unreplicated DNA makes it harder for MPF to inhibit E1.

Â And, and a way to look at that is, you just pick one level of MPF, for instance,

Â 0.1 down here, you would have, say, three percent E1 activity or 70% inhibition.

Â But no unreplicated DNA, but then you would

Â have to move up to this curve here to

Â have only 20% inhibition or 80% active Wee

Â 1, when you have a lot of unreplicated DNA.

Â So, unreplicated DNA makes it more difficult for MPF to inhibit Wee1.

Â 19:06

You would see the same thing when you look

Â at how unreplicated DNA changes the activation of CDC 25.

Â Again, this is for, this is the curve for CDC 25 activity versus MPF.

Â Remember, MPF turns CDC 25 on.

Â So zero here is the control curve for no replicated DNA.

Â And then we can compare that to what happens

Â when you have a lot of unreplicated DNA, 300.

Â This would be the amount of almost complete

Â activation you would get with no unreplicated DNA, and

Â then you would move down to this curve

Â here when you have a lot of unreplicated DNA.

Â So the unreplicated DNA makes it more difficult for MPF to turn on

Â CDC 25 and it makes it more difficult for MPF to inhibit Wee1.

Â 19:55

So more wee1 would make it, would mean that you'd

Â have more pre-mpf and less mpf and then less cdc25 would have the same effect.

Â It would make it harder to move from the left, preMDF, to the right over here.

Â So both of these effects would give you more MPF that's in the pre-stage with

Â the inhibitory phosphate on, and less MPF

Â that's in the activated state stage over here.

Â So, what would that do to your cell cycle oscillations?

Â [SOUND] It would slow, it would slow it down.

Â This is what you would see in control.

Â Rapid oscillations of cyclin and MPF.

Â And then this is what you would see when you added unreplicated DNA.

Â Your oscillations are, are larger in amplitude, in terms of

Â the total amount of cyclin, but you can see that

Â they are much more spread out you get, you get

Â fewer oscillations in a, in a given amount of time.

Â So that oscillation will become slower but unreplicated

Â DNA then they were in under the control conditions.

Â And this is what Novak-Tyson observe with their

Â model and this is consistent with experiments that

Â had already been done to add the unreplicated

Â DNA and observe the effects on cell cycle oscillation.

Â So this was an, a further validation of

Â their model, showing that yes, it does rep,

Â it does reproduce the effects that you see

Â when you add the unreplicated DNA to these preparations.

Â 21:24

Now let's move on to some results that Novak and

Â Tyson obtained with their model that people hadn't seen before.

Â We've already discussed how they validated their model by

Â comparing it to data that already existed in the literature.

Â 21:36

But one of the, the useful aspects of these

Â dynamical models is you can make novel predictions, and

Â then sometimes these novel prediction can inspire experiments that

Â will later determine what of the model prediction was right.

Â 21:50

And one of the predictions that Novak and Tyson made in their

Â model is they predicted there was hysteresis in the cyclin MPF relationship.

Â What we mean by that is what we saw before.

Â If you plot cyclin versus MPF, and these

Â were experiments remember that we're doing with non-degradable

Â cyclin, so you don't have to worry about

Â the fact that MPF will trigger cyclin degradation.

Â If you plot MPF on a y-axis versus cyclin on

Â the x-axis, you see that it jumps up over here on

Â the right, at Ta, what they're calling T activation, but then

Â when it jumps down, it jumps down at a different level.

Â It jumps down to this level on the left called T inhibition.

Â This was a novel model prediction.

Â 22:30

People didn't know whether it was right at the time they ran these simulations.

Â [BLANK_AUDIO]

Â The simulations that were run by Novak and

Â Tyson and the predictions that were made in

Â these simulations, inspired new experiments to test whether

Â this hysteresis in the [cyclin]-[MPF] relationship actually existed.

Â And in 2003, two, two independent studies, confirmed

Â that these investigators were right about this prediction.

Â I'm not going to go into details of, of these studies.

Â But the idea is that this is checking the the entry into mitosis as a

Â function of cyclin concentration, and then exit

Â from mitosis as a function of cyclin concentration.

Â So the key thing this, that they're looking for here, is that as

Â you add more non-degradable cyclin, when you see these spindles start to form.

Â 23:21

And when they're adding more and more cyclin they see that it the threshold

Â where you get this entry into mitosis that occurs somewhere between 32 and 40.

Â So this would be the level at which you

Â have an increase in MPF activity when you add cyclin.

Â And then they did some other experiments where they

Â tested to see how you'd get this to exit.

Â And they saw this exiting somewhere between 16 and, and 24.

Â So if you look down here, you can see here you have

Â these spindles, and then here, you know longer have these spindles forming.

Â 23:53

So, this is how Sha et al concluded

Â that Novak and Tyson were correct about their prediction.

Â That the threshold for an, an increase in MPF activity,

Â or entry into mitosis, is different from the threshold to

Â induce exit from mitosis, and this threshold is, has to

Â be higher than this threshold for this to have hysteresis.

Â 24:14

There was another similar study performed the same year by Pomerening

Â et al, that they showed the same thing and this was even

Â expressed in the, in the form of a biofication diagram similar

Â to those what we have seen when we talked about bistable system.

Â This is plotting MPF activity versus non-degradable cyclin concentration,

Â and sure enough they saw that the amount, where it

Â jumps up on the way up is different and it's

Â higher than where it jumps down on the way down.

Â So if you look at the solid symbols here this is MPF versus cyclin on

Â the way up and then the open symbols here are MPF versus cyclin on the way down.

Â Confirming this prediction of hysteresis in the relationship.

Â 25:05

A second prediction that was made by Novak and Tyson that was later

Â confirmed experimentally, was that unreplicated DNA

Â will change the location of the bifurcation.

Â In other words, unreplicated DNA will change where you get

Â a sudden jump in MPF activity, as you add non degradable

Â 25:24

cyclin.

Â And I'm not going to go into the details of

Â this experiment to explain CH acts, and APH ecetra.

Â The reference is given down here for those

Â of you who might be very interested in this.

Â But the overall message here is that, your looking for where

Â you have this transition between no

Â spindles forming, and spindles forming here.

Â And that gives you some sense of what you're, you're bifurcation is.

Â This is an indicative of a jump in

Â MPF activity as you end at non degradable Cyclin.

Â And then when you add a drug, like, that increases the amount of

Â unreplicated DNA in this replication, you see

Â that this threshold occurs at higher level.

Â So here the threshold is somewhere between 80 and 100.

Â Here the threshold is somewhere between zero and 40,

Â and that's due to the presence of the unreplicated DNA.

Â One other thing I should mention is that we showed some simulations

Â that unreplicated DNA will will change, will slow down the speed of oscillations.

Â Novak and Tyson took that analysis further and showed that unreplicated DNA does do

Â that by changing the location at which you get the sudden increase in MPF activity.

Â That is what they did in 1993, when the published their model and then 10

Â years later, Sha et al confirmed that,

Â that novel prediction of the dynamical mathematical model.

Â 26:40

So to summarize, 1993 Novak and Tyson cell cycle model has truly become a classic.

Â It was a very important study, in the

Â field of understanding of mechanism in the cell cycle.

Â But I think it's also a very interesting paper to discuss in this sort of context

Â because this model illustrates several of the steps

Â that are involved in the dynamical modeling study.

Â 27:03

One is that when you build the model, it very much helps if

Â you can match you're simulations to data that are obtained in a simplified system.

Â And we saw an example of that.

Â How experiments were done using non-degradable cyclin,

Â 27:17

and those experiments were really critical for

Â them to be able to pick the parameter values in their, in their model

Â by matching the data that they obtained by matching the results they obtained in

Â the simulations to the data that were,

Â that were obtained in the simplified system.

Â 27:30

It also illustrates a second step that's important in a dynamic modeling studies.

Â You have to validate the model against, norm results.

Â So we saw how, one of the things that give

Â them confidence in their model was the idea that when you

Â add unreplicated DNA it will slow down, the speed of

Â cell cycle, cell cycling and sure enough, their model showed that.

Â That gave them confidence in their, in their model.

Â And then probably the most important step is down here, number three is, you

Â don't want to just use your mathematical model

Â to confirm things that you already know.

Â You want to use it to generate nobel predictions and hopefully,

Â you can present those in a way that these are predictions

Â that can subsequently be tested experimentally, and that was one of

Â the ways in which the Novak and Tyson model was extremely successful.

Â [BLANK_AUDIO]

Â