[BLANK_AUDIO] Hello, we are now on the second lecture on mathematical modeling of the cell cycle. What we are going to discuss in this lecture part two of our series of lectures on cell cycle. This is a particular mathematical model which was published in 1993 by Novak and Tyson. We're going to discuss the structure of the Novak and Tyson model, in particular the biochemical reactions that are a part of this model, and the differential equations that constitute this model. And, then we're going to talk about the relevance of the Novak-Tyson model. Why do we think this is something that's important to discuss, 20 years after it was published? So we're going to discuss some of the insight that was gained from the simulations that Novak and Tyson performed. And then, we're also going to discuss model predictions that were confirmed in subsequent experiments. And we really want to emphasize this last point, because this is going to illustrate one of the great strengths of dynamical mathematical models. Is that they can generate predictions, sometimes these are nobel predictions, and these predictions can inspire experiments, and occasionally there's a success story, where these experiments confirm their predictions of the mathematical model. As management, we're going to discuss the 1993 Novak Tyson cell cycle model, this was described in a very influential paper that these two authors published in the Journal of Cell science in 1993, and the complete reference for this work is given down here. Now let's discuss the structure for the 1993 Novak Tyson cell cycle model Overall, this model can be divided in, in two general modules. There's one module here that's about regulation of cyclin and Cdk dimers. Let's review what these species are here. These are, these should sound familiar to you. These are things that we defined and talked about in the first lecture on the cell cycle. Over here, you have cyclin bound to Cdk, and both cyc, and, Cdk has two phosphate groups on it. When it has both phosphate groups on it, it's inactive form. So this species over here is pre NPF, and then the transition from left to right involves removal of this phosphate. This is removal of the inhibitory phosphate, so this species over here is NPF. Now let's recall what we, what we discussed about how phosphorelation reaction and dephosphorelation occur and how they're regulated. Dephosphorelation occurs through this phosphorylated form of cdc25 and this phosphorylation of cdc25 remember its regulated by itself. And this part up here is how you have the regulation of the de-phosphorylation reaction of, of MPF, and then down here you have the regulation of the phosphorylation reaction of MPF. MPF will catalyze the phosphorylation of Wee1. Phosphorylize phosphorylating Wee1 will put it into a, a less active form. And it's the active form of Wee1 over here is the unphosphorylated form. And Wee1, when it's active, will put the phosphate group back on Cdk, converting the active MPF into pre-MPF over here. And then of course we also have synthesis of cyclin from amino acids. And cyclin and Cdk have to find one another, and they have to bind to form a diagram. So these are the reactions in the Novak-Tyson cell cycle model that have to do with regulation of the cyclin/Cdk dimers. The second module in the Novak-Tyson, cell cycle model involves regulation of cyclin degradation. This is another, these are reactions we talked about in in the previous lectures as well. Remember we said that when MPF gets active, it triggers its own degradation, and that's what's expressed here. Active MPF will lead to phosphorylation protein, are called IE for intermediate enzyme. When IE gets phophorolated it's going to lead to activation of this complex here, called the anaphase-promoting complex. And when the anaphase-promoting complex gets turned on it's going to degrade cyclin. So that's how you're going to go from having MPF over here which is Cdk bound to cyclin. The Cdk all by itself over here, and the reason it's by itself is that the cyclin has been degraded. So this part of this module, model here involves regulation of cyclin degradation. And the overall, or what you might call the wiring diagram of the model combines these two. And this, this diagram comes from a very nice review article by Sible and Tyson published in 2007. But this describes the scheme of 1993 model by Novak and Tyson. Now let's consider the ODEs that are in the Novac-Tyson model. We're not going to go through all the equations, cause there's quite a few of them. But I think it's helpful to, to point out that you can divide the equations that are in the Novak-Tyson model into two main categories. One are, are those equations that involve synthesis or degradation of cyclin. For instance, if we just look at if we just look at free cyclin, in this case, we can see where this differential equation comes from. The derivative of cyclin with respect to time, is equal to some constant k1, minus k3 times cyclin times Cdk, minus k2 times cyclin. What do these represent? Well this first term represents synth, synthesis of cyclin, from amino acids, this step in the scheme. The second term represents dimer formation. When cyclin and Cdk come together to form either pre-MPF or active MPF. That's going to make the free cyclin concentration go down. And another thing that can make the free cyclin concentration go down is, is degradation of cyclin. So, that's represented here. So, as with other ODEs that we've seen in biochemical reactions, the equation can look quite complicated. But if you break it down term by term usually the individual terms within the equation make sense. The other thing I want to point out is that this rate constant k2 in this case, this is typical of a lot of the equations in the, in the Novak-Tyson model. This represents, this k2 will have a separate equation that represents it. And why does it have this separate equation here? Well k2 will have some rate, times the amount of anaphase promoting complexes off, plus some other rate times, amount of anaphase promoting complexes on. Remember that when, when cyclin gets degraded it occurs through the anaphase-promoting complex, which we've abbreviated as APC in this case. The way that Novak and Tyson set up this model is, they're saying that if, when the APC is off, it can still degrade cyclin, it can just do it at a much lower rate. But then when it gets turned on, it degrades cyclin at a, at a much higher rate. And the way that you calculate the overall rate at which cyclin gets degraded, that's k2, is you have some weighted average of one rate times the amount that's off, plus some other rate times the amount that's on. And given those considerations, we can, we can deduce that this term here, this v2_2 is much greater than v2_1. And there are quite a few terms in the, in the Novak-Tyson model that look, that look like that. So that's one main category of equations in the Novak-Tyson model: synthesis and degradation of cyclin. The other main category is those that involve only deep phosphorylation and dephosphorylation. And those equations will look like this. If we just look at Weee1 for instance, and if we want to write down a differential equation for the amount of phosphrylated we want, we can say that there's it's some rate, remember that MPF is the, is the kinase that actually catalyzes phosphorylation of E1. So there's going to be some [UNKNOWN] times enzyme concentration, times MPF, times substrate concentration, which in this case is the amount of unphosphorylated B1, over unphosphorylated B1 plus some [UNKNOWN] constant. So this is substrate, substrate plus [UNKNOWN] constant, in the, in the denominator. This is for the phosphorylation reaction. And, of course, what opposes the phosphorylation reaction is the dephosphorylation reaction, which is also expressed with a Michaleis-Menten equation over here, and PPase, in this case, just represents generic phosphatase activity, where you, you know exactly what the cynase is that's putting the phosphate group on. You don't necessarily know what the phosphate, the phosphatases that's going to take off this phosphate group. And so to represent it mathematically, you just lump all the cellular phosphatases together in a, in a parameter and it turns out we, they call PPase. But in general the you know, almost every differential equation in a Novak-Tyson model can be understood by you know by putting into one of these categories and breaking down the individual term whether it's the phosphorylation term, is it a synthesis term or degradation term etc. At this point, we have established the structure of the Novak-Tyson model and talk a little bit about the, the ODEs that earn the Novak-Tyson model, now let's move on to think about, talk about what are some of the results that they've obtained with Novak-Tyson model? Well, one important result they saw is that with the controlled parameters, they get spontaneous oscillations of MPF and cyclin. And they said these were analogous to the rapid divisions that occur in newly fertilized frog oocyte. And so it was known from experiments that as soon as the frog oocyte gets fertilized, it will undergo several stages of of cell division, these are rapid cell divisions, and those provided experimental data to which Novak-Tyson could compare their results. And so this is an important result just as the validation of the model that [UNKNOWN] shown out cycling will increase and decrease as a function of time it's the black line here, and then MPF will also increase and decrease as a function of time. And this is what happens when you have the control parameters, and you can plot these as a time course, cyclin and MPF versus time, or you can plot cyclin and MPF in the phase plane like this. And you can see that with control parameters you get these spontaneous oscillations. So, that's one one good validation of the cell cyclin model. If it wasn't cyclin then it might not be a such a good model. They might have to consider something else. But when they did put in the the actions they thought were important, sure enough they saw they saw cyclin. They saw these oscillations. We just saw that Novak and Tyson saw a spontaneous oscillations of cyclin and MPF in their cell cycle model. Is that enough to conclude that it's a, a good model, and a, and a, and a worthwhile study, and a worthwhile result? Usually not by itself. But Novak and Tyson pushed their model a lot further, and, and gained important insights by analyzing their model. And one of the insights they gained is that they observed bistability between total cyclin and an MPF activity. And in order to analyze this in a simplified way, they used non-degradable cyclin in their in their, their model, so that way you could control the total amount of cyclin, that you added to the cells. And they did this in part because there were experimental observations that they, that they could compare this to. And, so this was a there were experimental results that they were, that they could go by in the literature, that use this non-degradable form of cyclin. And what do these experimental results show? What we're finding here is time on the x-axis, and active MPF on the Y axis. And what's interesting about this is that so you go from the, from the bottom curve to the top curve as you add more cyclin. 0.2 you get barely any activation of MPF. 0.3 you get tiny bit more activation of MPF. And then suddenly when you go to 0.4, from 0.3 to 0.4, you get a much bigger increase in MPF activity. And then after that, you see a, a, you also see small small increases in the amount of MPF activity, but the biggest jump in this case, is between 0.3 and 0.4. And Novak and Tyson looked at this, and they said, this looks like a threshold behavior, and a threshold behavior is something that you often see when you have bi-stability. We talked about bi-stability in the context of an all or none response, and this is that's one of the things that you see when you have an all or none response, is that once you exceed some threshold level. You get a large change in your output even for a small change in the stimulus. And so Novak and Tyson analyzed this in the model and they said if we plot cyclin level on the X axis. Again, this is non-degradable cyclin and the total amount of cyclin you add experimentally and MPF activity on the y axis. You see this bi-stable behavior, where cyclin goes up and MPF goes up a little bit, a little bit, a little bit and suddenly you get this big jump here on this right hand arrow. And they continue to analyze this even more and said well if we decrease cyclin from this level, does MPF jump down at the same level? No it doesn't jump down to the same level. It only jumps down when you go to a much lower level of, of cyclin. So the threshold behavior that they saw here on the left, was similar to experimental observations that had already been obtained. But this analysis they did here on the right showing that the relationship between cycling and, and MPF is in fact bi-stable. That was a nobel prediction of, of the model. And there's a general theme that, that I want to emphasize in, in talking about this, is that quantitative data obtained in a simplified preparation can be very valuable when you're constructing a, a systems level model. What made this, pos, what made it possible for them tuh, tuh, understand this and analyze this is the fact that it was done with non-degradable cyclin. Right? Because, if you add normal cyclin, what will happen? MPF will go up and then cyclin will, and the MPF will trigger degradation of cyclin and so then, it will go down and you'll, you'll see a transient time course. The fact that they were able to see this sustained rise in MPF only occurred because they had non-degradable cyclin they were using. And this is something that's encountered, repeatedly when you're talking about interactions between, ex, between, between quantitative dynamical models and experimental data is that when you can obtain the data in a quantitative way and you can do it in a simplified preparation, that can be very, very valuable for constructing your systems of a model. And just to emphasize this point, of, how you can obtain data, constrain your model of a complicated system. The value that you get, when you have experiments that remove one or more variables, experiments that simplify the system, in other words, just provide a little bit of foreshadowing of what we're going to discuss, in subsequent lectures. Which is a mathematical model of the of the action potential in neurons, where you're looking at how voltage changes with respect to time in an excitable cell such as a neuron. When everything is changing all at once, you can see that your voltage wave form in this case is very complicated. One thing you see, this is analogous to what we were just talking about is threshold behavior, where these small changes in voltage will decay. But larger changes will undergo much, much great excursions in voltage from the resting level. But one a, one of the things that we're going to discuss when we talk about the actual potential model is that, when voltage is changing as a function of time, you see that the waveform is very complex. The experiments that allowed the construction of a, of a robust and very valuable model of the neuronal action potential, were obtained by experiments that fixed the voltage and measure the individual ionic currents. So this is what we're going to talk about with the exponential models. That when everything is changing at all once it's complicated. We're going to discuss experiments where the voltage is fixed. And the currents were measured individually. And we're going to argue that voltage clamp, which is what this technique is called, where you're able fix the voltage. Voltage clamp was the key advance that made the mathematical model of the action potential developed by two investigators named Hodgkin and Huxley. Voltage clamp is what made this model possible. This is analogous to what we were just discussing with the nondegradable cyclin experiments, and how valuable they were for Novak and Tyson when they built their model. Back to the Novak and Tyson model now, what else did they see? Well one interesting simulation they performed was of the effects of unreplicated DNA on cell cycle oscillations. It was already known from experiments, that if you edit unreplicated DNA into your preparation that was undergoing cell cycle, unreplicated DNA would change the way, the way that MPF was able to regulate Wee1 and Cdc25. And we can look at these curves to see how, how the unreplicated DNA changes the regulation of Wee1 and Cdc25. These are plots of Wee1 activity vs MPF activity for no unreplicated DNA. So this one on the left is a control curve, and then if you add some unreplicated DNA, it shifts it to the right, and then if you add more it shifts it to the right even more. So what you can conclude from this, if you just look at Wee1 versus active MPF, is that unreplicated DNA makes it harder for MPF to inhibit E1. And, and a way to look at that is, you just pick one level of MPF, for instance, 0.1 down here, you would have, say, three percent E1 activity or 70% inhibition. But no unreplicated DNA, but then you would have to move up to this curve here to have only 20% inhibition or 80% active Wee 1, when you have a lot of unreplicated DNA. So, unreplicated DNA makes it more difficult for MPF to inhibit Wee1. You would see the same thing when you look at how unreplicated DNA changes the activation of CDC 25. Again, this is for, this is the curve for CDC 25 activity versus MPF. Remember, MPF turns CDC 25 on. So zero here is the control curve for no replicated DNA. And then we can compare that to what happens when you have a lot of unreplicated DNA, 300. This would be the amount of almost complete activation you would get with no unreplicated DNA, and then you would move down to this curve here when you have a lot of unreplicated DNA. So the unreplicated DNA makes it more difficult for MPF to turn on CDC 25 and it makes it more difficult for MPF to inhibit Wee1. So we can con, conclude that when you add DNA you would have more wee1 and you would see less cdc25. So more wee1 would make it, would mean that you'd have more pre-mpf and less mpf and then less cdc25 would have the same effect. It would make it harder to move from the left, preMDF, to the right over here. So both of these effects would give you more MPF that's in the pre-stage with the inhibitory phosphate on, and less MPF that's in the activated state stage over here. So, what would that do to your cell cycle oscillations? [SOUND] It would slow, it would slow it down. This is what you would see in control. Rapid oscillations of cyclin and MPF. And then this is what you would see when you added unreplicated DNA. Your oscillations are, are larger in amplitude, in terms of the total amount of cyclin, but you can see that they are much more spread out you get, you get fewer oscillations in a, in a given amount of time. So that oscillation will become slower but unreplicated DNA then they were in under the control conditions. And this is what Novak-Tyson observe with their model and this is consistent with experiments that had already been done to add the unreplicated DNA and observe the effects on cell cycle oscillation. So this was an, a further validation of their model, showing that yes, it does rep, it does reproduce the effects that you see when you add the unreplicated DNA to these preparations. Now let's move on to some results that Novak and Tyson obtained with their model that people hadn't seen before. We've already discussed how they validated their model by comparing it to data that already existed in the literature. But one of the, the useful aspects of these dynamical models is you can make novel predictions, and then sometimes these novel prediction can inspire experiments that will later determine what of the model prediction was right. And one of the predictions that Novak and Tyson made in their model is they predicted there was hysteresis in the cyclin MPF relationship. What we mean by that is what we saw before. If you plot cyclin versus MPF, and these were experiments remember that we're doing with non-degradable cyclin, so you don't have to worry about the fact that MPF will trigger cyclin degradation. If you plot MPF on a y-axis versus cyclin on the x-axis, you see that it jumps up over here on the right, at Ta, what they're calling T activation, but then when it jumps down, it jumps down at a different level. It jumps down to this level on the left called T inhibition. This was a novel model prediction. People didn't know whether it was right at the time they ran these simulations. [BLANK_AUDIO] The simulations that were run by Novak and Tyson and the predictions that were made in these simulations, inspired new experiments to test whether this hysteresis in the [cyclin]-[MPF] relationship actually existed. And in 2003, two, two independent studies, confirmed that these investigators were right about this prediction. I'm not going to go into details of, of these studies. But the idea is that this is checking the the entry into mitosis as a function of cyclin concentration, and then exit from mitosis as a function of cyclin concentration. So the key thing this, that they're looking for here, is that as you add more non-degradable cyclin, when you see these spindles start to form. And when they're adding more and more cyclin they see that it the threshold where you get this entry into mitosis that occurs somewhere between 32 and 40. So this would be the level at which you have an increase in MPF activity when you add cyclin. And then they did some other experiments where they tested to see how you'd get this to exit. And they saw this exiting somewhere between 16 and, and 24. So if you look down here, you can see here you have these spindles, and then here, you know longer have these spindles forming. So, this is how Sha et al concluded that Novak and Tyson were correct about their prediction. That the threshold for an, an increase in MPF activity, or entry into mitosis, is different from the threshold to induce exit from mitosis, and this threshold is, has to be higher than this threshold for this to have hysteresis. There was another similar study performed the same year by Pomerening et al, that they showed the same thing and this was even expressed in the, in the form of a biofication diagram similar to those what we have seen when we talked about bistable system. This is plotting MPF activity versus non-degradable cyclin concentration, and sure enough they saw that the amount, where it jumps up on the way up is different and it's higher than where it jumps down on the way down. So if you look at the solid symbols here this is MPF versus cyclin on the way up and then the open symbols here are MPF versus cyclin on the way down. Confirming this prediction of hysteresis in the relationship. A second prediction that was made by Novak and Tyson that was later confirmed experimentally, was that unreplicated DNA will change the location of the bifurcation. In other words, unreplicated DNA will change where you get a sudden jump in MPF activity, as you add non degradable cyclin. And I'm not going to go into the details of this experiment to explain CH acts, and APH ecetra. The reference is given down here for those of you who might be very interested in this. But the overall message here is that, your looking for where you have this transition between no spindles forming, and spindles forming here. And that gives you some sense of what you're, you're bifurcation is. This is an indicative of a jump in MPF activity as you end at non degradable Cyclin. And then when you add a drug, like, that increases the amount of unreplicated DNA in this replication, you see that this threshold occurs at higher level. So here the threshold is somewhere between 80 and 100. Here the threshold is somewhere between zero and 40, and that's due to the presence of the unreplicated DNA. One other thing I should mention is that we showed some simulations that unreplicated DNA will will change, will slow down the speed of oscillations. Novak and Tyson took that analysis further and showed that unreplicated DNA does do that by changing the location at which you get the sudden increase in MPF activity. That is what they did in 1993, when the published their model and then 10 years later, Sha et al confirmed that, that novel prediction of the dynamical mathematical model. So to summarize, 1993 Novak and Tyson cell cycle model has truly become a classic. It was a very important study, in the field of understanding of mechanism in the cell cycle. But I think it's also a very interesting paper to discuss in this sort of context because this model illustrates several of the steps that are involved in the dynamical modeling study. One is that when you build the model, it very much helps if you can match you're simulations to data that are obtained in a simplified system. And we saw an example of that. How experiments were done using non-degradable cyclin, and those experiments were really critical for them to be able to pick the parameter values in their, in their model by matching the data that they obtained by matching the results they obtained in the simulations to the data that were, that were obtained in the simplified system. It also illustrates a second step that's important in a dynamic modeling studies. You have to validate the model against, norm results. So we saw how, one of the things that give them confidence in their model was the idea that when you add unreplicated DNA it will slow down, the speed of cell cycle, cell cycling and sure enough, their model showed that. That gave them confidence in their, in their model. And then probably the most important step is down here, number three is, you don't want to just use your mathematical model to confirm things that you already know. You want to use it to generate nobel predictions and hopefully, you can present those in a way that these are predictions that can subsequently be tested experimentally, and that was one of the ways in which the Novak and Tyson model was extremely successful. [BLANK_AUDIO]