Hello. Welcome back. We are now in part five of our series of lectures on bistability in biochemical signaling models. In the last lecture, we talked about bistability in two variable systems. And we discussed a couple of methods for predicting where bistability may or may not be present in these two variable systems. Now we're going to follow that up and define where bistability may or may not be present in more rigorous terms. We ended the last lecture by talking about how you could plot nullclines in the phase plane in a few variable system. That's the first step in determining whether bistability is present. Now, we're going to discuss the mathematically vigorous way of determining whether or not bistability is there, which is computing a Jacobian matrix, and computing the eigenvalues of that. And then we're also going to discuss a more qualitative and, and graphical method for determining whether bistability is present, and that involves plotting directions arrows in the phase plane. [NOISE]. As a reminder, the model we were working with the last lecture, this is generic example of mutual repression that was described in this very nice review article by Tyson, published in 2003. Again, as a reminder, when R the response goes up, that moves your enzyme E from the unphosphorylated form to the phosphorylated form. So when R goes up you are going to decrease the level of unphosphorylated E. Conversely when E goes up, E catalyzes degradation of R here. So when E goes up you are going to have a lower, lower value of R because it's going to enhance this process of degradation of R. So, an increase in R is going to lead to a decrease in E, whereas an increase in E is going to lead to a decrease in R, that's why we call this mutual, mutual repression. And the ordinary differential equations that describe this system are listed here. One of the things we saw last time is that if you apply a steady state R versus stimulus strength here, you could either end up with always at a low value of R for low values of stimulus, always at a high value of R for high values of stimulus, but in the middle range, you could either end up at a high value of R or a low value of R, depending on whether you started with low initial conditions or with high initial conditions. This is just to remind you of the model that we were working with before and some of the behavior that it shows. Second thing we saw in the previous lecture is that for some values of this stimulus S nullclines can intersect 3 times. Last time we calculated the equation for the R nullcline, this is R is a function of, of these other points for which d R dt equals 0. And these are the points for which d E dt equal 0. In either case, we solved for R as a function of E after setting the ordinary differential equation equal to 0. And what we saw was that for S equals 6, the E nullcline looks like this, it's this red one that, that dips and stays flat and then dips again. And then the R nullcline looks like this, this is the one that declines more gradually. And because of the peculiar shape of this E nullcline, you're going to have an intersection up here, an intersection here, and an intersection here. What we want to address now is how can we tell if these fixed points are stable and unstable. So each time the three, the two nullclines intersect, this is a a fixed point. Remember the definition of nullcline is when one of the variables is equal to 0. So when, when the two nullclines intersect, that means that both of the derivatives are, are equal to 0. But in the one variable system we saw that you could have a stable fixed point, or an unstable fixed point. We also saw this in the lectures on Dynamical Systems with [UNKNOWN] model. So, that's what we want to address now. We have three fixed points here, but are all three of them stable? Are all three of them unstable? For something more complicated than that. If you wish to determine whether or not a particular fixed point is stable, a sort of crude, brute-force way to do it is to start your start your ODE system with initial conditions that are close to the fixed point and run a simulation. Integrate the differential equations and see what happens. And that's what we've done here. We're focusing on this middle fixed point here. And I started the system of ODEs with this red point here with R equal to 5.4 and E equal to 0.3. This 0 subscript here means we're talking about the initial conditions. So, the red point is for R is 5.4, and E is 0.3, and then the black point is for R is 5.2 and E is 0.5. And both of these points in the phased plane are relatively close to this middle fixed point here. If this were a stable fixed point, you would expect that the red one and the black one would both evolve the same steady state, meaning that this, this middle fixed point. What we can see actually happens when we integrate these differential equations with respect to time is the opposite. The red one goes up to a high value of R, the black one declines to a low value of R. So just by writing these simulations, this suggests but does not conclusively prove that the middle fixed point is, is unstable. Again if, if this were a stable fixed point, we would expect that for initial conditions close to the fixed point, the system would evolve back to that fixed point, but instead the, the system evolves in opposite directions, even though we started with initial conditions that are relatively close to the fixed points. [BLANK_AUDIO] Now we want to ask, how can we understand stable and unstable fixed points in more mathematical terms? We're going to review a calculation that we talked about in lectures on dynamical systems to show you how it can be applied to this particular example of mutual repression. We can take our differential equation for E with respect to time, d E dt, and define this as some term f, and then our second differential equation for R with respect to time, and define this as some, some variable g. And we can compute the Jacobian matrix. The Jacobian matrix, remember, consists of the all of your, your functions defining your differential equations that partial derivatives with those functions, with respect to the the different variables. So this first row, first column is our first differential equation f, partial with respect to the first variable, E. First equation with respect to the second variable. Second equation with respect to the first variable. Second equation with respect to second variable. And we're not going to go through why each of these, these terms makes sense, but rest assured that they do. This is what you get with the Jacobian matrix. Partial derivative of f with respect to E, partial derivative of f with respect to R, partial derivative of g with respect to E and then partial of g with respect to R, down here on the bottom right. And if we want to determine whether a given fixed point, is stable or unstable, we have to evaluate this Jacobian, Jacobian at the fixed points, that are defined by E star, and R star. And this is where analytical computations become really difficult, but numerical computations are, are still possible. [SOUND]. Once you've computed the Jacobian, you need, now you need to evaluate the Jacobian matrix, at the fixed points that are defined by E star, and R star. So wherever we have E and R in our Jacobian matrix, we plug in E star and R star. And we can recall from our lectures on dynamical systems that the eigenvalues of the Jacobian at the fixed points determine the stability of a given fixed point. If the real part of either fixed point is positive, that means we have an unstable fixed point. And if the real parts of both are negative, then the fixed point is unstable. [SOUND]. Remember this is our, these are nullclines for R nullcline and our E nullclines showing that the two nullclines intersect at three points for a value of, a middle value or stimulus of six. And I wrote a script called repression_stability. It computes it computes the eigenvalues of the Jacobian at these three fixed points. And what we see is that at this upper fixed point here, for a low value of R, and a high value of E, the eigenvalues are minus 59, and minus 0.0983. Therefore, this is a stable fixed point, because both eigenvalues are negative. [SOUND]. Similarly, up here, we also have two negative eigenvalues minus 21, and minus 0.0915. So this fixed point here, with a high value of R, and a low value of E is also stable. What about the middle one? Given that dynamic simulations that we showed a couple of slides ago you can probably already guess what's going to happen. This has, also has two eigenvalues, one of which is negative, but the other one is positive. Because, one of our eigenvalues is positive that tells us that this middle fixed point is unstable, therefore we delineate this with an open circle rather than a closed circle. So the last couple of slides summarized the mathematically vigorous way determining whether or not the given fixed point is stable or unstable. [SOUND]. Next we want to consider a more qualitative, graphical way of determining whether given particular fixed points are stable or unstable. And this again shows the benefit of acquiring things in the phases plane where you have one variable on one axis and another variable on the other axis. Remember that in a two dimensional phase plane the direction that you are travelling is determined by your two derivatives. D E dt tells you which direction you are travelling with respect to the, to x and d R dt tells you which direction you are travelling with respect to y. And we can calculate the d E dt and d R dt because we have the differential equations that describe them. So let's first consider what happens when we have a very, very big value of E and a very big value of R. If we look at this first differential equation here, d E dt, this first term which is negative is going to be saturating this, this going to be the saturating value and as R continues to go up this one's going to keep getting bigger and bigger. So this term is going to be big. What about the second term here as E gets big, then E TOTAL minus E gets small, right? So the second term here, the positive term is going to become essentially 0, as E gets bigger and bigger and approaches E TOTAL. So this term is going to be, this positive term is going to be 0, this negative term here is going to keep getting bigger and bigger, therefore E is going to be decreasing when you have a high value of E. So d E dt is going to be less than 0. What about this second term here? Well, this term, this first term here, which is positive, is going to be a constant. The second term here again it depends on S, but for a given value of S this is also going to be a constant. So these two terms are going to be, are going to be limited. They're going to not continue to increase. But as E goes up and as R goes up this product here is going to get bigger and bigger and bigger. This product here has a negative in front of it. So as this term gets bigger and bigger and bigger, eventually this, this negative term is going to overcome these two positive terms which are both constant. So that's how we can conclude that d R dt is also going to be less than 9 for large values of E and large values of R. So in this region of the phase plane here, above the E nullcline and above the R nullcline, we know that both we know that we're pointed down and to the left because the d E dt is less than 0, and d R dt is also less than 0. And remember something that we learned in our lecture on dynamical systems, which is that you only flip an arrow each time you cross a nullcline. Anywhere in this, in this region of the phase space here, anywhere up here, the actual angle that your arrow is pointing is going to be a little bit different, but in general it's going to be plotted, pointing down because d R dt is less than 0, and it's going to be pointed in general to the left rather than to the right, because the d E dt is less than 0. In order for it to start pointing to the right or to start pointing up or both; we're going to have to cross nullclines. So we can say as we cross the E nullclines, we cross the, the red line here we're going to flip it horizontally. Instead of pointing to the left, it's going to point to the right, but it's still pointing down, because we haven't crossed the R nullcline yet. Conversely, if we go from this region here to this region here, if we only cross the R nullcline and don't cross the E nullcline, then we're pointed up rather than down, but we're pointing to the left rather than to the right. So if we go from here to here, we only cross the R nullcline, we haven't crossed the E nullcline. Now we're pointing up and to the left rather than down and to the left. And if we cross both, we move from this region here to this region down here, we've crossed both the E nullcline and the R nullcline. Now we're pointed up and to the right, rather than down and to the left. And with these simple rules we can often determine stability. For instance, let's look at this point here, our our leftmost fixed point. Well, we've got an arrow pointing towards it in this region, and arrow pointing towards it in this region, this arrow's pointing toward it, and this arrow is pointing toward it. So because all of our arrows are pointing towards this fixed point, that gives us a strong indication that this fixed point is probably stable. Conversely what about this middle fixed point? Well if we move from this middle fixed point into this region here, suddenly we're pointing away from it. Similarly if we go from here to this region here, again, we're pointing away from the fixed point. That gives us a strong indication that this middle fixed point is unstable and we can use similar logic to tell us that this right most fixed point is probably going to be stable rather than unstable. [BLANK_AUDIO] There is a variation on this technique that we can use for our qualitative graphical analysis of bistability, stability, which is that flat arrows on the nullclines rather than plotting arrows in the rest of the region. For instance, we know that if we're anywhere on the E nullcline then d E dt is going to be equal to 0. So we're not going to be moving to the left or to the right if we're on this blue curve here, because by definition it's a nullcline, therefore there's no change in enzyme with respect of time. So we're not moving left or right, we're only moving up or down. On the other hand, when we're on the blue curve, when we're on the R nullcline, we're not moving up or down, we're only moving left or right. So when you're on the red curve, you're only moving up or down, you're not moving left or right and when you're on the R nullcline, when you're on the blue curve, you're only moving it to either go left or to the right. Not moving up or down. And then furthermore like we saw before the direction changes any time you cross the nullcline. And with these simple rules, we can usually determine stability. What we want to look at here is for instance on the E nullcline, we're only going up or down. But are we going down over on this side, and up over on this side of the R nullcline or is it the other way around, where we're pointed up over here, and we're pointing down over here? And by analyzing the system this way, we can usually determine whether our given, our particular phase points are stable or unstable. [BLANK_AUDIO] Now we want to see how we can apply this analysis, applying direction arrows on the nullclines. These are our differential equations again, d E dt and d R dt. And on the E nullcline, well, we know that d E dt is equal to 0, we're only going to plot d R dt. We're going to plot whether it's, the system is moving up or down. And then conversely, when we're on the R nullcline, we want to plot d E dt, we want to see if the system is moving left or right. Well one thing we can conclude is that when we're on the E nullcline and we're above the R nullcline, then d R dt is less than 0. Let's consider some points here. Let's consider this point here on the E nullcline. We know that d E dt is equal to 0, so we only need to determine is this system moving up or down. What's our value of d R dt. Well if we were down here, d R dt would be equal to 0. Alright? So, what's, what's different between this point here, and this point here. R has gotten bigger. What happens to d R dt when R gets bigger? When R gets bigger this term here which is negative gets bigger. And since this negative term has gotten bigger, we move from a, a point where d R dt is equal to 0 for this point here. And this point d R dt must be less than 0, because this negative term has gotten bigger. So we can conclude that these arrows here are pointed down. We don't have, luckily, we don't have to do this analysis for every point in the space, because now, what, now as we move from here to here, we're crossing the R nullcline, because we're crossing the R nullcline, our direction arrow must flip. So now they have to be pointed up sit it down. Over here, they're pointed down. Over here, the arrow is pointed up again. [BLANK_AUDIO] We can do a similar analysis for the d E dt. What happens when we're on the R nullcline but we're above the E nullcline? What if we're at a point, we're at a point over here, where we move from this this point here, where you're on the E nullcline, the d E dt is equal to 0, and then you move up here, where the only thing that's different is that R has gotten bigger. And, and now you're at a point that's on the R nullcline, but above the E nullcline, so you move from this location here to this location here. Well luckily this isn't so hard to evaluate because R only appears at one, at one place, our differential equation for E. And when R gets bigger this term gets bigger, and again this term is negative just like it is in the, differential equation for R. So because this negative term has gotten bigger, therefore d E dt must be less than 0. So in these locations here, we're pointing to the left, and now we crossed the E nullcline, so here we must be pointing to the right. We cross the E nullcline again, here we're pointed to the left. And we cross the E nullcline moving from this location to this location, so here we're pointed to the right. And these considerations suggest that our middle steady-state is unstable, and our left and right steady-states are stable. We can see here that the arrows are pointed towards the steady-state, towards it, towards it, towards it, towards it, and here, especially, this is especially apparent on the, on the blue curve on the R nullcline, our arrows are pointed away from this middle steady-state and away from this middle steady-state. So this is another way that we can point errors in the phased plane. And get a good example, get a good understanding of whether particular fixed points are stable or unstable. To summarize this lecture, part five on bistability and biochemical signaling models. In two-variable systems, we can get bistability through either mutual activation, or mutual repression. We've also seen that when nullclines intersect 3 times, bistability may be present. Now what we've addressed in this lecture is, how can we determine whether given fixed points are stable or unstable. What we've seen is that stability of fixed points can be determined graphically by plotting direction arrows for extreme values of the two variables. And then you flip arrows in one direction each time a nullcline is crossed, and by doing this you can fill in the arrows in all the different regions of your phased plane, and by observing which way the arrows are pointing in different regions you can conclude whether particular fixed points are stable or unstable. [BLANK_AUDIO]