[BLANK_AUDIO] Hello everyone. Welcome back. [COUGH] In the last set of lectures, we discussed some basic concepts of, of dynamical systems and how we can analyze systems of ordinary differential equations in terms of some of their dynamic properties. In this next set of lectures, we're going to discuss a an, an important concept related to dynamical systems, that's important in a lot of biological processes. And that the process we're going to discuss is called bistability. So what we are going to discuss in this first lecture is some biological background, we're going to talk about the biological importance of bistability. When is bistability needed or when is it a desirable property of a biological system and then we are going to talk in very qualitative terms about the requirements for bistability. And then in subsequent lectures we're going to learn how to analyze systems in a much more quantitative way. What we mean when we use the term bistability? In a formal mathematical sense we mean this is a situation in which two possible steady states are both stable and I'll show examples of what I mean by that a little bit later. And it's important to note in general these correspond to a low activity state or a high activity state. So to illustrate what we mean by this let's consider a classic experiment that was performed about 15 years ago by Ferrel and Machleder. And the complete reference for their classic study is given down here. And what these authors did in this, in this study, is, they added progesterone to Xenopus oocytes. And Xenopus is a particular species of frog. So these were frog oocytes. And they added progesterone. And in response to progesterone, the added activity of a particular interest show that the kinase called map kinase, and it stands for Mitogen Activated Protein Kinase. And it's listed here. And what did they see when they added progesterone to these Xenopus oocytes? Well, they saw that the, that the map Kinase activity increased. And if you applied the population response. If you applied a Progesterone con, concentration in the X-axis. And you plotted MAP kinase activity on the y-axis. You saw this nice smooth curve. This was nothing surprising at all here. As progesterone goes up, MAP kinase activity goes up. So at the population level, they saw a gradual increase of MAP kinase with progesterone. So far, this is typical. But what we're going to see next because, when they looked at this on a cell by cell basis, they saw something very surprising. [BLANK_AUDIO] On a previous slide we, we saw that map kinase activity increased when [INAUDIBLE] con-, concentration increased. Why would we talk about this study as a way to illustrate the concept of bistability? Well, we can see how this helps us understand bistability When we consider that these authors measured map kinase activity, not just on a population level, but at each cell. And this figure shows what Ferrel & Machleder saw when they looked at map kinase activity on a cell by cell basis. So each horizontal graph here is a historgram of MAP kinase, activity for every cell in the population. And they saw something extremely interesting when they, analyzed the data and, and, acquired the data in this way. You can see that if you go from zero micromolar progesterone, from the lowest concentration, to the highest concentration, it, the population switches from almost no MAP kinase activity, all, all your cells are at, at essentially zero. Map kinase activity to all your cells being essentially 100% map kinase activity. But what happens in between? Does your distribution of map kinase activity gradually increase from 0% over here to 100% over here? No, that's not what they saw. [COUGH] What they saw was that you, your, your percentage of cells that were in the low-activity state or the high-activity state changed, so here you have about half and half. Here you have a, a low percentage at, at the low concentration and a greater percentage at higher concentration. Here the percentage at, at the high level, high activity increases further. But they saw almost no cells with any map kinase activity in the middle here. So what happens is that when you increase progesterone activity, oocytes switch from the low state to the high state. And further more, when they looked at intermediate progesterone concentrations, both high activity state and low activity state were present. So if you look at this one, 0.01 progesterone, you saw about half your cells in the low activity state and about half your cells in the high activity state. So this is different might, might have naively expected. What we might have naively expected is that, as your progesterone increases, you move from the top here to the bottom, maybe you saw a gradual shift In the procje, in the map kinase activity from the, from the lowest level through the highest level. But what they saw instead was you were always either in this very low level of map kinase activity, or it is high level of map kinase activity. But what changed when progesterone concentration was altered was the percentage of cells that were either in the low activity state over here or the high activity state over here. Further more as noted, for several of these intermediate concentrations, you could have, you could have either, right? You could have cell being in the low activity state or the high activity state. Either one was possible. This behavior observed when looking at map kinase activities in [INAUDIBLE] sites is different from, from what we're used to thinking about with biology. Most of what we learn in biology is what we could call monostable rather than bistable, and analog rather than digital. This graph here just shows three hypothetical examples of things that we can, that we can measure and what we would, what we would see. For instance if we add epinephrine in the bloodstream, if we increase epinephrine concentration we'll increase cardiac output. Or, if, at the, at the more cellular level we can say if we increase blood glucose you'll have insulin production in, in the beta cells of your pancreas. Or, at the more molecular level, we can just say that when the enzyme concentration increases. We get more conversion of, of substrate to product. And in either one of these cases, we can plot the response on the y axis which would be cardiac output, insulin production, or just conversion of substrate to product. And on the x axis we can plot our, our stimulus. In either one of these cases, we see that the response, the the output goes up smoothly as a function of, of X. And it doesn't really matter what the the curve looks like, if it's nice and linear or if it's hyperbolic, or if it is a sinoid shape. The point in any one of these cases is the response depends directly on the level of stimulus. And further more in each one of these cases, when the stimulus is removed, the response returns to its, prior level. Sometimes, this occurs with the delay. Insulin production doesn't shut off immediately, and cardiac output doesn't shut off immediately, but, eventually, you're going to go back to where you were. So this is the way we're used to thinking of biology. Where things are monostable, you always have one particular output for a particular level of input and they're analog, where you can get a nice, smooth curve, a nice smooth response curve of, of your response in when you change your stimulus. As we just noted, we're used to thinking of biological processes that are monostable and analog. Now we want to ask, when is an analog response not good enough? What I mean by this is, what are the sort of biological processes for which it's not okay for the system to respond by saying, I'm going to have a, I'm going to be 10% on, or I'm going to be 20% on. Or I'm going to be fifty percent on. What are some processes where in terms of the biological function it has to either turn on or be turned off. These might not be obvious to you right away, but when we talk about some of these processes and think about them, hopefully it will become apparent. Why certain biological process don't work in an analog way. Some examples of this are fertilization, cell division, cell differentiation, action potentials in the nervous system. Apoptosis which is a form of program cell death and this last one is arguable but I think many people would agree that learning is something that has some bi-stable elements in it. So what I mean is that when when a cell decides it's going to divide It's not, it doesn't decide that it's going to divide, divide 10% or divide 50% or divide 70%. It makes a decis, a commitment, it makes a decision that it's going to divide, and you can identify, cells in an entire field and say, yes, that cells is dividing, or no, that other cell is not dividing. So cell division is something that it either, either happens or it, it's not happening. You can't really describe it in analog terms. You either descr, you describe it in digital terms. Yes it's dividing, not it's not dividing. Apoptosis, as I noted, is a form of programmed cell death. It works the, the same way. A cell commits to, to undergoing apoptosis, it commits to dying, or, or it doesn't. Action potentials in the nervous system, we're going to talk about this, a little bit later. When we discussed how to mathematically model action potentials in the system. But cells don't undergo a quarter of an action potential or a third of an action potential or half an action potential. It either, a cell either commits to an action potential or it doesn't. So for each of these examples, a graded response is inadequate. These are things that you can describe as they're either happening, either they've turned on or, or they're not happening, or they're off. And furthermore, as, as we're discussing more detail, these are phenomena that, that require persistence. Generally, when a cell commits to undergoing apoptosis, it makes that commitment and then you can take the stimulus for apoptosis away. And most of the time the cell will continue to to undergo apoptosis. Learning is something where you can see that, that persistence would be desirable, right? You want to, it, it would be nice to learn something, and then not forget it immediately. It's nice if you could, if you maintain it. And so each one of these, processes not only is it, not adequate to describe it in analog terms But these are also processes for which persistence is, is either desirable or else it's required. Now we want to ask, is the biochemical level, how does bistability arise? And I'm going to show a figure that come from a nice review article. Written by Ferrell about a decade ago. You can get bistability if you have mutual activation. What we mean by that is that if you have a protein A that activates a protein B as delineated by this arrow, and then protein B also Activates protein A, this is a case where you can get this is called mutual activation and this is a case where you can get bistability. So the way that we're going to imagine this is that trigger one will turn on protein A and trigger two will turn off protein B. So what happens if you, so at, at the beginning, time zero here, at the beginning, both A and B are at essentially zero level. They're, they're both turned off. So what happens if we give a short trigger to turn on protein A, to activate protein A? Well A's going to Then activate B. B's going to activate A, etcetera, etcetera. And so both A and B are going to rise to some level of close to a 100% activity because as soon as, coz, the increase in A's going to make the increase in B greater, and vice-versa. The increase in B is going to increase, make the, activity of A greater. Now what happens if we take away trigger one? Well A is going to continue to activate B and B is going to continue to activate A. So even though we've taken away the trigger, both A and B are going to stay at this high level here. So A and B go up initially in response to the trigger but then even when trigger one gets taken away. A, B stayed at this high level here. Now we have been as if we give a second trigger here, it's going to inhibit B. Well, the second trigger is going to cause B to, to decrease, but what's going to happen to A? Well, the reason A stayed at the, the high level is because it was being activated by protein B. Once B decreases, A is going to decrease as well, so when we apply trigger 2, both A and B are going to decline to this very low level here. And now when we take away trigger two, A and B are going to stay at this low level of essentially 0% activity, because they're no longer activating each other. We can do a similar type of analysis if we consider A and B are inhibiting one another, rather than activating one another. And I'm not going to go through this one in as much depth. The difference here is that, here you can get a state where A's at a high level and B's at a low level, or you can have B at a low level and A's at a high level. So initially if you turn on A with trigger one. A's going to go up, but when A goes up, A's going to turn off B, because remember that A's inhibiting B. And if B shuts off, if B goes down to zero, then A's going to, then B's going to no longer be able to inhibit A, and so therefore, A's going to stay at this high level. Even after you take away trigger one. So you can get bi-stability through this process of mutual activation or mutual inhibition but in the former case, both A and B are going to be either at, together at the same high level or together at the same low level. Over here, you're going to get a case where either A is high and B is low, Or vice versa or B is high and, and A is low. Now we can see how these are, are bistability are, are these are bistable. You can either get this steady state where they're both at the high level, or this steady state where they're both at the low level. And the way that you can determine which steady state you end up in, you end up in this one where they're both high or one where they're both low. Well, it depends on the history, right? In this case, it depends on which trigger has been given first, trigger one or trigger two. Or, which trigger has been given most recently, rather? Trigger one or trigger two? The same way over here. After trigger one, you're going to end up with A high, B low, and then after the second trigger, you're going to end up with B at the high level and A at the low level. It's important to note that these are the types of circuits that can produce bistability, but these circuits do not guarantee bistability. For instance, if this activation of B to A was extremely weak we can see how you might not necessarily get both A and B at, at a very high level. And the fact that these circuits can produce bistability but do not guarantee bistability, is why we need quantitative analyses. And that's what we're going to go into in the subsequent lectures in this section. Now let's think about what we mean by bistability. When we talk about the dynamical behavior of a biological system. Recall what we talked about in the last set of lectures on dynamical systems. In those lectures we encountered stable and unstable fixed points. And we saw stable limit cycles. For instance we had a model of yeast glycolysis. And for one value of the, the Km for the ATPases for the value of 13, we saw a continuous oscillations of glucose and ATP concentration. And what we said in this case was the fixed point was unstable and the oscillation that was observed is what we term the stable limit cycle. And then if you recall from those lectures when we had a different value of the KM of the [UNKNOWN] we saw oscillations at the beginning but then they damped out and then we eventually saw. A return to a steady level and what we said in this case, was that we had a stable fixed point. When we talk about the dynamical behavior of a bistable system it's a little different from what we encountered previously. One of the examples we're going to show is of what's called the Lac Operon And bacteria. And one of our outputs in this case is going to be expression of a gene that we're going to call Lac Y. And what we going to see in this case is whe, when this system is in a monostable regime, what that means is it doesn't matter where you start your initial conditions. LacY could start down here like the red, green and blue curves or it can start right up here with the the black curve. And it's always going to come down to this, to a particular level, at the output. So, this system is monostable because it doesn't matter where you start, you're always going to end up at the same, with the same steady state in the end. But, this system can also bi-bistable. And what that means in terms of dynamical behavior is this, you can start at a high level of LacY and end up at a high level of LacY or you can start at a low level of LacY and end up at a low level of LacY and this is what we mean by Bistability. Where this high steady state is stable and its low steady state is also stable. And then one of the other characteristics that we'll see of, of bistable systems is this. If you look at the difference between the red curve and the green curve in this case, the difference in starting value of LacY is, is really small. These can be really close to one another, but you still see that one of them Brings you down to the, the low steady-state level of LacY, and the other one brings you back to the high steady-state level of LacY. So what this means in terms of dynamical behaviors is that multiple steady-states are possible. Which steady-state are you going to reach in a given case? Well, that depends in part on the initial condition. To summarize this first lecture in our series on bistability biochemical systems, we require the following. One, bistability can be a useful property for biological processes that require persistence. Second, bistability means that a biological response will be essentially, essentially digital. That means all or none rather than graded. So when we talked about the need for bi-stability in the beginning, we talked about processes for which it's not good enough to have a 10% response, 20% response, 50% response. If you want to talk about something like cell division, fertilization or apoptosis then you are talking about something where either you want to do it, or you want to not do it? And then finally, in the language of dynamical systems, bistability means that two fixed points are possible, and initial conditions will determine which fixed point is reached. So what we're going to look at in the next series of lectures is how to analyze a system to determine whether or not it's going to be bistable. And how we can simulate these systems to, in order to determine which initial conditions are going to lead to one steady state and which are going to lead us to another steady state. [BLANK_AUDIO]
