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. 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. 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. 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. 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? 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. 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. 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. 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. You do still, still have the same phenomenon that when active MPF gets increased it's going to lead to even more activation of MPF. But, it occurs through this intermediate protein called Cdc25. 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, 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. As a second example, what about the reverse process? What about conversion of active MPF back to the inactive MPF or the pre-MPF form. 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. 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. 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. 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. 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. 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. Let's 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. 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. 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. 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. 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. [BLANK_AUDIO] 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. Of course, what we're discussing with the, the Tyson 1991 model and the Novak Tyson 1993 model, these are things that happened more than 20 years ago at this point. 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. 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. 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. 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. 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. 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. A more mechanistic representation is to say that, well the reason B increases when A increases is because A phosphorylates B. 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. 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. [BLANK_AUDIO]