Hello, now we're going to, we're going to move on to a new topic, which is mathematical models that have been developed to describe the cell cycle. This lecture is going to have four parts to it and this is part one. I'm going to start by reviewing some of the concepts we learned in, in the previous lectures. In the lectures on dynamical systems, when we discussed the mathematical model of glycolysis in yeast, we saw these sustained oscillations of glucose in ATP as a function of time. And then we, we plotted these oscillations and, in the phase plane. And what we saw in this yeast glycolysis model was that the system could either oscillate or it could settle to a stable value. So, we had a fixed point here. And this fixed point could be, could be stable, in which case we saw damped oscillations of glucose and ATP. Or this fixed point could be unstable, in which case we saw these stable or sustained oscillations of glucose and ATP. And then in another series of lectures when we discuss bi-stability, we, we discussed this Lac operon model that could either, there be monostable in which case you always ended up at the low steady state or the high steady state. Or this system could be bistable. And if we compare the red and the green ones in this case, we see this, an example of bistability where the system can even move to this high steady state or at low steady state depending on conditions. And we analyze this property of bistability by plotting nullclines in the, in the phase plane. So these are some of the concepts that we saw in previous lectures. Now we want to ask, what is different about the cell cycle compared with what we saw previously? On thing that's different is that now with the cell cycle we are going to consider a system of more than two variables. And what this, what the presence of more than two variables does is it allows the system to exhibit both stable oscillations and bistability. So one of the results that we are going to see is oscillations of these species called cyclin. And NPF is a function of time. These are stable oscillations similar to what we saw with this glycolysis. And then we're also going to generate and, and look at plots that look like this where we have one variable on the y axis, another variable on the x axis. And this is characteristic of, of, of bistability. And these terms that I've just shown are, are going to be defined. Pre-MPF, MPF cyclin, et cetera are all going to be defined a little bit later. But what we look at, what we see here is that, because we have more than two variables we don't just have a choice between either stable oscillations or a stable fixed point. And we don't only have other choice between mono-stability of bi-stability. With multiple variables, we can both see both bi-stability and we can see the stable oscillations. Now, the outline of the, of this first lecture on the cell cycle. In this first lecture, we're not going to talk any, about any particular mathematical models. We're just going to provide the biological background that is needed to understand this mathematicals of the cell cycle. And what we're going to focus on today in terms of biological background is the importance of this species called Maturation-Promoting Factor or MPF. MPF can also be defined as mitosis-promoting factor or M-phase promoting factor. And we are going to stress the importance of MPF in a cell cycle, and we also, then we also have to define the different ways that MPF activation can be regulated in cells that are that are either undergoing the cell cycle or not undergoing the cell cycle. [NOISE]. Let's introduce some of the basics of the cell cycle. One is that, there's a, a transition from a phase called G2 to a phase called M. M in this case is for Mitosis. The G2 to M transition is driven by an increase in something called MPF. As we've defined, MPF is Maturation Promoting Factor. [NOISE]. MPF was originally discovered in experiments in what you might call the pa, the pre-molecular biology age. So that they, they new that there was a, a factor that was important in this G2 to M transition, but they didn't originally know what it was. That's why they called it maturation promoting factor, very sort of generic term describing what it did. Then in later studies they were actually able to discover what MPF was at the molecular level. And what they found was that MPF was a dimer consisting of two proteins that were bound together; one called CycB, cyclin B in this case, and another called Cdk, for cyclin-dependent kinase. Furthermore what was disc, what was discovered was that MPF was active over here on the right, activated MPF. If you look at what's different between the right and the left, it's these two phosphate groups at threonine 14 and tyrosine 15 when there phosphoralated over here on the left. This is the inactive form where, or what we're going to call pre MPF. And when these phosphate groups are gone over here on the right, that's, that's what make MPF move from the inactive form to the active form. So this CDK as we you can see from the definition it depend, it's called cyclin-dependent kinase. So what that means is that it's a, it's a kinase. It will phosphorylate other targets in the cell. But it also means that it needs to be bound to cyclin in order to, to be activated. And when it is bound to cyclin, and when these these two residues here are, are not phosphorylated, that's when it's to the active form. So what we can deduce from this definition of MPF is that there is two obvious ways to activate activity of the cyclin dependent kinase or to activate a activity of MPF. One is by either producing or degrading cyclin. If you don't have cyclin, the cyclin dependent kinase would not be active. And then on the other hand if your cell bloods producing a lot of cyclin, then that would be a way of, of activating the cyclin dependent kinase. And then the second obvious way that you can regulate cyclin-dependent kinase activity is by phosphorylation or, or dephosphorylation of Cdk. When you take this phosphates groups off, you are going to promote the activation of MPF. And when you put these phosphate groups on, you're going to promote the inactive form of MPF. And as we're going to see subsequently, both of these ways of regulating Cdk activity are used extensively in regulation of the cell cycle. Another important concept that's necessary, is necessary to understand the cell cycle in general terms is that this protein cyclin that we, that we just discussed in the context of MPF, cyclin is alternately synthesized and degraded. You can probably already deduce that from the name. The reason they called it cyclin is because it cycles. If it weren't alternately synthesized and degraded, they, they would have called it something else. And this is another diagram from the classic textbook, Molecular Biology of the Cell by Alberts, et al. It's showing that when you have cyclin, which is the green protein here, bound to your Cdk which is the red one, then this is what's going to trigger the mitosis machine, and this is what's going to cause the cell to enter the end phase. Then cyclin gets degraded. You see that the green one gets broken up into pieces here. Then you have Cdk by itself. It binds to a different type of cyclin in order to undergo this transition from the G1 phase to the S phase. This other type of cyclin, this S cyclin gets degraded again, then you have Cdk by itself, you get synthesis of this M-cyclin which is a green one, and entered into mitosis again. Something that's important to note is that in the mathematical model we're going to discuss, we're not going to simulate every single phase here. We're not going to simulate binding of S-cyclin, degradation of S-cyclin, and then synthesis and binding of M-cyclin, also. We're going to only focus on the M-cyclin. And were going to ignore these steps down here, from G1 phase to S phase. So the way to think about the mathematical model we're going to discuss is you can have synthesis of M-cyclin, triggering of mitosis, degradation of M-cyclin. And then we're going to skip over these steps here. Almost a direct transition here to here. And many experiments have shown that when cyclin gets produced and gets degraded and it gets produced and it gets degraded during the cell cycle. And again that's why they call it cyclin rather than some other name. Now let's talk about some of the ways that MPF activity can be regulated during the cell cycle. One important aspect of MPF regulation is that it involves positive feedback. In this diagram, again, from Alberts' textbook shows how this occurs. Remember that MPF is one of the subunits of MPF is cycle independent kinase, so when this is active, it's kinase, it's going to phosphorylate targets. And one of the targets it phosphorylates is this protein called Cdc25. Now, Cdc25 is not a, it's not a kinase, it's a phosphotase. But it's the phosphatase that gets activated when it has a phosphate group on it. So Cdk or MPF phosphoaliate Cdc25 is going to activate Cdc25 and what's one on the targets of Cdc25? One of the targets of Cdc25? One of the targets of Cdc25 is, is preembia, it's Cdk. So when Cdc25 gets activated, it's going to take off this inhibitory phosphate on Cdk subunit. So what this means overall is that greater MPF activity is going to lead to greater Cdc25 activity. And then a greater Cdc25 activity by removing this inhibitory phosphate is going to lead to greater MPF activity. This is an example of, of mutual activation. MPF increases Cdc25 activity, Cdc25 activity increases MPF activity as we have seen in the lectures on bistability, that's one of the consequences of mutual activation. It can lead to bistability. Regulation of MPF by phosphorylation and dephosphorylation has been more complicated because of a protein called wee1. This diagram here illustrates the basics. Pre, this is pre-MPF over here on the left. Cyclin bound to Cdk with this inhibitory phosphate indicated in yellow. And then over here on the right, you have the active MPF, where the inhibitory phosphate has been removed. As we just saw on the last slide, this dephosphorilation reaction that turns pre-MPF into active MPF occurs by a protein called CDC25. The reverse reaction where a phosphate, where the inhibitory phosphate gets put on the CDK sub unit, occurs through a protein called WEE1, and a lot of review articles, such as this one here, just show the basics of this where CDC25 takes the phosphate off wee1, puts the phosphate back on. But it's more complicated than that, because of a couple of regulatory steps, and one of the important ones is that MPF inhibits wee1. So really theum, the, more complete way to draw this would be to draw an arrow here indicating what showed on the last slide. MPF activity will increase CDC25 activity. But then, there's this step here as well. MPF activity will decrease [UNKNOWN] activity if we want. So MPF inhibits wee1. Therefore we can conclude that MPF regulates both its own activation, because it regulates the dephosphorylation reaction which occurs through CDC25, and it regulates its own inactivation, because it regulates the the phosphorylation reaction that occurs through wee1. Now we have to consider what happens with cyclin degradation, and to make the picture even more complicated, cyclin degradation is initiated, again, by MPF. So, we talked about how how MPF is a very important regulatory node, and that's because it's involved in all these different reactions that can regulate the entry into this cell cycle. This is part of a diagram again from the, from the Alberts book, showing that when you have active MPF in this case, we, we discussed how MPF will lead to, the [UNKNOWN] cells to enter the mitosis phase of the cell cycle. And that's why these arrows are shown discussing some of the things that happen during the cell cycle, during cell division. Spindle assembly, chromosome condensation, and breakdown of the nuclear envelope. But then the the one we want to, the step we want to focus on in this particular slide is that when MPF activity increases it activates a cyclin protease and then cyclin will get degraded. So up here you have Cdc 2 which is another another name for Cdk in this case. Cdc 2 bound to cyclin B, and then down here you only have Cdc2 by itself. So what's happened is that cyclin has been degraded. And the degradation of cyclin has, has been triggered through MPF itself. And this occurs through complex called the anaphase promoting complex, APC. And that's how it's going to be represented in the mathematical model we're going to discuss. We can summarize some of these reactions then as false. MPF positively regulates MPF activation, that occurs through cdc25. It negatively regulates MPF de-activation, that occurs through wee1. And it positively regulates MPF destruction, and that occurs through the anaphase promoting complex. So there are several regulatory steps here that involve MPF, one way or the other. And I would argue that it's difficult or impossible to make predictions in this case using only intuition. If you have this positive regulation of MPF activation, but you also have this positive regulation of MPF destruction, you, you don't know just though intuition which of these is going to be more powerful, and which is going to be dominant under, under different conditions. And these are the sorts of systems for which dynamical mathematical models are really required to to get good understanding and to make quantitative predictions about what's going to happen if you were to say you know, change the rate of, of activation or change the rate of destruction. You can't reason through these things just using your intuition, and that's why these types of mathematical models that we're going to discuss are really critical for understanding this process. [BLANK_AUDIO] Before we summarize, I just want to mention one brief and, and possibly confusing aside. Some of you may be familiar with the cell cycle, some of you may have even re, ma, may have even read the relevant chapter in the textbook, such as the Albert's textbook, and you might see a diagram that looks like this. Where you have the inactive CDK here. The inactive CDK binds to cyclin, and it's only partly active here, then a phosphate group gets put on, and here it's fully active. And this seems to go against what we were discussing before, where we were saying it has to be a dephosphorylation with the CDK that makes MPF active. And this diagram seems to contradict that saying that it's putting on the phosphate that makes the complex fully active. And the answer to this is it there are multiple phosphorylation sites on Cdk. This particular phosphorylation site. It's shown in this diagram here, is on threonine 161 or 167, and it depends on whether you're talking about the, the residue in, in yeast cdk or whether you're talking about it in cdk for vertebrates. So, that's why a lot of times we'll have, we'll have two numbers here to reflect the different species. This particular threonine is, is activating, so when this phosphate has to get put on either 161 or 167 in order for the Cdk to be active. And this occurs through a through a kinase called cdk-activating kinase or cak. So this, it's true that this particular phosphate has to get put on for the CDK to, to be active. But, this step is not regulated. So that's why we're going to, most of the mathematical models ignore this stuff. It's the, it's phosphorylation at other sites at threonine14 and tyrosine 15, these are the inhibitory ones. And these are also the ones that are, that are regulated. So when you see a diagram like this, it talks about CAK, the CDK-activating kinase, and it says it putting on this phosphate group makes the protein active. This is referring to phosphate on threonine 161 or 167, and all those is required for function. It's not regulated, in the mathematical models we're going to discuss next we're going to ignore this phosphate. [BLANK_AUDIO] Now we would like to summarize this lecture, we've seen that MPF is the most important regulatory element in the cell cycle. MPF seems to be involved in, in everything. And when MPF activity goes up, that's what causes cells to, to switch from into the mitosis phase of the cell cycle. What we've seen is that MPF activity can be regulated by synthesis of cyclin. Cyclin needs to be present for MPF to be to be active. So you can, so if synthesizing cyclin is a way of regulating MPF activity. Dephosphorylation of cdk by a protein called cdc25 is something that's important for regulating MPF activity. Conversely, phosphorylating cdk is also important for regulating activity. Remember, this dephosphorylation reaction is making an increase in MPF activity. And then this phosphorylation reaction that occurs through wee1 is going to decrease MPF active, act, MPF activity. And then MPF activity can also be regulated by cyclin degradation, and what's, what makes this interesting and complicated is that cyclin degradation is initiated by MPF itself. And because these steps are so complicated, mathematical models can be of great use to help make sense of these, these complex regulatory interactions. And that's what we're going to to discuss in the next couple of lectures. Where we discuss some mathematical models of the cell cycle that were developed by the group of John Heist. [BLANK_AUDIO]
