0:16

So what we're going to discuss in more depth in this fourth part of the,

Â this fourth lecture on dynamical systems is

Â how we analyze stability of ODE systems.

Â We talked about stability a little bit in the third lecture.

Â And we're going to discuss it in more depth this time.

Â The example that we introduced last time that we are going to

Â come back to is a mathematic model of glycolytic oscillations in yeast.

Â We are going to introduce this new concept here called nullcline.

Â And we're going to to talk about how

Â nullclines can be used to identify fixed points.

Â And then, how we can mathematically identify

Â stable fixed points and unstable fixed points.

Â 0:53

And then, finally, we're going to introduce this concept of bifurcations.

Â And bifurcation means, the place where you're

Â having an abrupt change in system behavior.

Â We're

Â 1:14

Remember that the process is simulated by the, the Bier et al model.

Â Our transported glucose from the outside of the cell to the inside of the cell.

Â Production of ATP from glucose through the action of many enzymes.

Â With phosphofructokinase being the most important of those enzymes.

Â 1:31

Consumption of ATP by ATPases are all lumped together.

Â And then remember this feedback, this important feedback, that when ATP

Â goes up, it makes the production of ATP occur more quickly.

Â 1:55

Remember also last time that we said, you can, you have

Â four important parameters in the Bier model, Vin, K1, and kp.

Â These are the default values for these three parameters.

Â And then last time we varied the fourth parameter.

Â Which is the Km for the action of the ATPases.

Â And what we saw with Km equal to 13 we saw sustained oscillations of glucose and ATP.

Â With glucose here plotted in black, ATP plotted in red.

Â But then when we change Km from 13 to 20, we saw what we call damped oscillation.

Â These start oscillating, but the

Â oscillations get progressively smaller and then

Â you eventually settle in, to stable values of glucose and, and ATP.

Â So then the question that we, we asked and

Â that we're going to address in this lecture is how can

Â we understand he qualitatively different behavior in these two cases

Â with K m equal 13 and K m equal 20.

Â Another review of a concept from last time is that when we plot in the, the

Â 2D phase plane where we have glucose on one axis and ATP on the other axis.

Â Then the direction that our system is travelling

Â in the phase plane is determined by the derivative

Â of ATP with the respect of time and

Â the derivative of glucose with the respect of time.

Â So, the two derivatives form a vector.

Â 3:16

So at any given location the derivatives define a vector in the phase plane.

Â And we can plot a trajectory of how glucose evolves with respect

Â I mean, how glucose and ATP evolve with respect to one another.

Â ATP on the x-axis here and glucose on the y-axis.

Â And we can see that this, are system is traveling around

Â in what's known as a stable limits cycle in this case.

Â Where the glucose and, and ATP are oscillating, with a respect of

Â time, so it travels in this loop over and over and over again.

Â So this is to review a couple of

Â the concepts that we introduced in the previous lecture.

Â Now, we're going to discuss some of these concepts in a little more depth.

Â 3:54

When you're goal is to analyze stability of an ODE system,

Â it is useful to plot something that's known as a nullcline.

Â What do we mean by that?

Â The nullcline is a set of points for which one of the derivatives is equal to 0.

Â So, it's either the set of points for which the change in glucose with respect

Â to time is 0 or the change in ATP with respect to time equals 0.

Â And these can usually be calculated analytically.

Â So what you do is you say the, change in

Â glucose with respect to time is equal to this differential equation.

Â We set this equal to zero.

Â And then, what you want to do, is you either

Â want to solve for glucose as a function of ATP.

Â Or conversely solve for ATP as a function of glucose.

Â And if you look at this equation here, ATP appears in both terms.

Â And here, it appears in the numerator and the denominator.

Â So solving for ATP as a function of

Â glucose in this case is not really trivial.

Â But solving for glucose as a function of ATP is

Â relatively easy because glucose only appears in one term here.

Â So we can solve this algebraic equation here.

Â And say you know, glucose equals Kp over 2 times K1 times the

Â sum of ATP in and K m, and that's our, ATP nullcline.

Â These are the set of points for which the derivative of ATP is 0.

Â We can do the same thing over here changing glucose to the respect of time.

Â Is this, differential equation we set this equal to 0.

Â In this case, its equally easy to solve for glucose as a

Â function of ATP or solve for ATP as a function of glucose.

Â But we want to keep these two consistent with one

Â another, so we can plot them on the same axis.

Â 5:35

the glucose nullcline here in black and the ATP nullcline in red.

Â All right, so the red plot here is

Â plotting this equation that we've derived over here.

Â And then the black line is plotting this equation, that

Â we derived here by setting the, glucose ODE equal to 0.

Â Now what happens when you, the two nullclines intersect with one another?

Â That means that the derivative of glucose with respected time

Â is 0, derivative of ATP with respected time is 0.

Â So glucose is not changing and ATP is not changing.

Â 6:26

Now, when we start plotting direction arrows in the phase

Â plane, we can see how plotting nullclines can be very useful.

Â Remember, then, the two-dimensional phase plane.

Â The direction that the system in traveling is defined by the vector.

Â That is calculated as the derivative of ATP with respect

Â to time and the derivative of glucose with respect to time.

Â 7:16

This term for in the derivative for ATP is

Â going to keep getting bigger and bigger and bigger.

Â In the second term is not going to keep getting bigger and magnitude.

Â So this negative term is going to saturate.

Â This positive term is going to keep getting bigger and bigger and bigger.

Â 7:36

This term Vm is a constant so it's not going to change at all.

Â However, this negative term as glucose increases and as ATP increases it's

Â going to, it's going to keep getting bigger and bigger and bigger.

Â So we can conclude that when ATP is very large and glucose is

Â very large, the change in ATP with respect to time is greater than 0.

Â The change in glucose with respect to time is less than 0.

Â 8:01

Therefore, we can draw an arrow up in this

Â region of the phase plane that looks like this.

Â Right, ATP is pointing to the right.

Â Because its derivative is positive.

Â And, glucose is pointing down, it, it's, because its derivative is negative.

Â 8:19

Well, when is this going to switch direction?

Â It's only going to switch direction when you cross one of the nullclines.

Â Because, in order for the derivative to go from

Â being positive to negative, it has to cross 0.

Â So that's real, that's what makes poly

Â nullclines very, very very useful in this time.

Â In this case, is that each time you cross a nullcline, you're going to change

Â either the direction with respect to x or the direction with respect to y.

Â 8:46

So, let's consider what happens when we move from here to here.

Â Well, we crossed the ATP nullcline.

Â So, instead of being pointed to the, to the right

Â where the change in ATP in respect to time is positive.

Â We're going to cross where ATP, the change in ATP in respect to time is 0.

Â So, it's going to to switch from being positive here to negative here.

Â Therefore, we have to flip this arrow with respect to x.

Â But we haven't crossed the glucose nullcline, so we

Â don't flip which way it goes with respect to y.

Â It's still pointing down, but now it's pointing down and

Â to the left, rather than down and to the right.

Â 9:30

So when we cross the glucose nullcline we

Â still have a, we're still pointing to the left.

Â The change in ATP with respect to time is still negative.

Â But now, we've crossed the, the region of points where

Â the there's no change in glucose with respect to time.

Â So, we've gone from a decrease in glu, glucose, with respect to time.

Â The cross where, dg/dt is equal to 0.

Â Now, we have an increase in glucose with respect to time.

Â So, we've crossed the glucose nullcline.

Â Therefore, this arrow, instead of pointing down and to

Â the left is now pointing up and to the left.

Â And then, finally, we cross the ATP

Â nullcline again to go into this region here.

Â 10:11

And now, we flip direction with respect to, with respect to x.

Â So now, instead of pointing up and to the left it's pointing up

Â and to the right and that's because we've crossed the ATP nullcline again.

Â 10:23

So, this tells us that the system is going to proceed in the, clockwise direction.

Â It's going to be going this way and this way, this way, and this way.

Â And this is why the nullclines are very useful, is

Â that nullclines will divide the phase space into discrete regions.

Â What we don't know yet is whether this fixed point

Â here represents a stable fixed point or an unstable fixed point.

Â 10:59

We, what if we have Km equal to, equal to 13?

Â And we pick a combination of ATP and

Â glucose that's not exactly at the fixed point.

Â Where these two nullclines intersect, but close to the fixed point here.

Â 11:13

Well if we run this simulation numerically what we see

Â is that glucose looks flat and the ATP looks flat.

Â But over time, they start to, start to oscillate a

Â little bit and the oscillations get bigger and bigger and bigger.

Â And then eventually you have these large stable oscillations.

Â 11:42

So what happens in this case is the system, even though we started it

Â very close to the fix point over time it moves away from the fixed point.

Â And then it oscillates forever.

Â From this numerical simulation, we conclude that the fixed point is unstable.

Â And the oscillation, in this case, is what we would call a stable-limit cycle.

Â 12:11

Now let's consider the non-oscillating system.

Â When we, we saw before numerically that when we

Â set K m equal to 20, we didn't get oscillations.

Â What if we take this non-oscillating system and

Â we start with the initial condition somewhere over here.

Â Very far away from the fixed point.

Â Well, we saw this previously when we taught it, with respect to time.

Â The glucose oscillations will, will go away

Â and the ATP oscillations will go away.

Â They'll, you'll start with some

Â small oscillations, but they're damped oscillations.

Â And therefore, you end of with a, with a fixed value.

Â What this looks like in the phase space, is, you start here.

Â And then you spiral along, and you spiral,

Â and then you eventually converge on the fixed point.

Â And in this case, no matter what initial conditions you start

Â with this system is always going to move towards the fixed point.

Â Therefore, we conclude this is a stable fixed point

Â 13:09

Next, we want to address how we

Â can understand stable and unstable fixed points mathematically.

Â And how we can calculate for a given fixed

Â point, whether it's going to be stable or unstable.

Â This is a somewhat advance topic for this class.

Â This is not going to be required for you to do well in this class and to pass it.

Â But I do think it is worth going through

Â it just so just for the sake of completeness.

Â But mostly, what I want to teach you in this

Â class is graphical methods for

Â understanding stability or, or instability.

Â And these graphical and numerical techniques for assessing this.

Â 14:02

Next, what we need to do is compute a matrix.

Â It's called Jacobian.

Â And this matrix consists of the partial derivatives of

Â these two functions with respect to the two state variables.

Â So the first element is a partial derivative of the

Â first function f with respect to the first variable ATP.

Â Then you have partial derivative of little f with respect to glucose.

Â Partial derivative of little g with respect to ATP.

Â Partial derivative of little g with respect to glucose.

Â So, first equation, first variable, first equation second variable, second equation

Â with respect to first variable,second equation with respect to second variable.

Â Those are the four elements that read

Â your Jacobian matrix for a two-dimensional system.

Â And in that case, we get this term here.

Â And you can verify these partial derivatives just by

Â going back to the rules that you learned in calculus.

Â 14:55

Next, what we need to do is we need to evaluate this at the fixed point.

Â As defined by a particular combination of glucose and

Â ATP, which we will define as capital G* and ATP*.

Â And this is where analytical computations can become somewhat difficult.

Â 15:11

Because you need to plug-in, G* and ATP*, sometimes you can get,

Â you know, simple equations describing where your null clients cross.

Â Where both derivatives are equal to 0 ,but in many

Â cases you can't get, you can't get those clearer numbers.

Â But if you knew HTG* and ATP*, what would you do?

Â 15:33

So you first you you evaluated Jacobian matrix

Â at the fixed point defined by [G]*, [ATP]*.

Â So we going to, we plug in [G]* and [ATP]* and just Jacobian matrix.

Â And then what we want to do is just we

Â want to calculate the eigenvalues values of this Jacobian matrix.

Â And the eigenvalues of the Jacobian determine the stability of our system.

Â 16:38

As we just said, the eigenvalues of the

Â Jacobian matrix evaluated at the fix point, determines stability.

Â And I said in describing the last slide, this is very, very difficult.

Â Sometimes it is impossible to do it analytically.

Â But it is possible to do it numerically.

Â 16:53

And I wrote a script called bier_stability

Â and that uses MATLAB's function called eig.

Â Which computes eigenvalues, as a way to

Â calculate the Eigenvalues of the fixed point.

Â And what we saw in this case when Km equals 13, we saw eigenvalues of

Â 0.004 plus or minus 0.1132 times i.

Â So these are complex eigenvalues and what you notice here is that

Â the positive, is that the real part of these eigenvalues is positive.

Â So this is our, mathematical confirmation of

Â what we saw with the numerical simulations.

Â But this is going to be a fixed point that it's unstable.

Â And in this case, we're going to have a stable limit cycle which we

Â can conclude based on the fact that

Â these are complex eigenvalues with positive real parts.

Â 17:42

What happens if we have Km equals 20, in this case?

Â We still have complex eigenvalues, but the real part in this

Â case, of these, of these complex eigenvalues is going to be negative.

Â And the negative real part of the complex eigenvalues indicates that

Â this is a stable fixed point rather than an unstable fixed point.

Â So the complex eigenvalues, in either

Â case, indicate that we have periodic oscillations.

Â Remember that we have these stable oscillations for Km equals 13.

Â And then we have the damped oscillations for Km equals 20.

Â But, for Km equals 13, it's an unstable fixed

Â point, illustrated by the positive real part of the eigenvalues.

Â And for Km equals 20, we have a stable fixed point, which we can

Â then, which we can see from the negative real parts of the item values.

Â 18:28

The final topic we want to discuss with this lecture, is that of a bifurcation.

Â A bifurcation in general, is somewhere

Â that the system qualitatively changes behavior.

Â And one way we can illustrate this with the Bier et al

Â model is to it simulate from many different values of of value Km.

Â 18:48

We saw in our two, in our examples that when we switched Km from 13 to 20.

Â It the system switched from having steady oscillations to having damped

Â oscillations and then eventually settling into a, a stable fixed point.

Â But what about all the different values in

Â between or slightly higher values or slightly lower values.

Â And so what I did in the simulation is I simulated values of of Km ranging

Â from 10 to 25 with a relatively a small space in between the different values.

Â And with each simulation I, I taught I simulated it

Â for a long time and then over the last 500 minutes.

Â I calculated the minimum value of glucose and the

Â maximum value of glucose over those last 500 minutes.

Â And for little values of Km we have, we see that there's a large difference

Â between the minimum value of glucose and

Â the maximum value of glucose over 500 minutes.

Â And so this is where it's oscillating between the high

Â value and the low value, high value and the low value.

Â Furthermore, we can compute the we can do this Jacobian analysis and

Â compute the eigenvalues and this is where we have positive real parts.

Â Because we have an unstable fixed, unstable

Â fixed point in a stable limit cycle.

Â 20:07

Well here we have the opposite.

Â We have negative real parts of our

Â eigenvalues and we have a stable fixed point.

Â And so at Km is approximately equal to 16, this

Â is where the fixed point that was unstable becomes stable.

Â 20:22

And this is what we would identify as a bifurcation.

Â Somewhere around Km equals 16, this is where the system

Â switches from having stable oscillations to having a stable fixed point.

Â And you can identify this by, where this curve shifts from having, you

Â know, one branch of the curve to having two different branches of the curve.

Â 20:49

In summary, what we've learned in this lecture is that a nullcline of a dynamical

Â system is a set of points where one of the derivatives is equal to 0.

Â therefore, fixed points are defined by intersections of nullclines.

Â So in the two-dimensional systems we looked at where

Â the two, two nullclines intersected that determined our fixed points.

Â 21:12

In phase space, each time a nullcline is

Â crossed, one of the directions of the system changes.

Â So, we were re, plotting a direction vector in 2-D phase space.

Â And each time we crossed a nullcline it either flipped

Â with respect to x or it flipped with respect to y.

Â That's what we mean when we say one of the directions of the system changes.

Â 21:46

And finally, bifurcations are locations where

Â dynamical systems exhibit qualitative changes in behavior.

Â For instance, a shift from stable oscillations to damped oscillations.

Â In the example that we saw with the Bier et al model.

Â 22:13

And in the phase space, we're plotting A on the x-axis, and B on the y-axis.

Â And furthermore, we're plotting A and B nullclines.

Â Where the A nullcline is the red one here.

Â That's represented as the set of points for which d[A]/dt equals 0.

Â And the B nullcline is plotted in black.

Â This is the set of points for which d[B]/dt equals 0.

Â And we can deduce it in this region of the phase space here [SOUND].

Â 22:42

Where A is [SOUND] decreasing and B is decreasing,

Â [SOUND] so it's pointing down and to the left.

Â Now what we want to do is we, we want to determine which direction

Â is this system travelling in, in this region, and this region, and this region.

Â 22:59

And again, this is probably a good idea

Â for you to pause the recording think about it.

Â Figure out what your answer is and then come back and, and I'll show the answer.

Â [BLANK_AUDIO]

Â Okay, let's move on to the answer of this one.

Â 23:22

What happens when we move from this region here to this region here?

Â We're crossing the, the B nullcline.

Â We're crossing the set of points for which d[B]/dt is equal to 0.

Â So here we have a system that is decreasing with respect to B.

Â And then we cross, when we cross the B nullcline we have to

Â switch from decreasing with respect to B to increasing with respect to B.

Â But we haven't cross the A nullcline, so this

Â is still decreasing, in the, on the x-axis here.

Â So we're still pointing to the left on the

Â x-axis, cause the derivative of A is still negative.

Â But the derivative of B has moved from being negative to being positive.

Â So that's why this region, we're pointing up

Â into the left rather than down into the left.

Â 24:10

So now, we're still going to be pointing up we're

Â still going to be pointing positive with respect to B.

Â But now instead of be, being negative in respect to

Â A we're going to be going positive in respect to A.

Â So now we're going up and to the right in this region.

Â 24:25

And finally as we cross the B nullcline again, we go from

Â pointing to the right and up to pointing to the right and down.

Â because we've crossed the, the region for which d[B]/dt equals

Â 0 and is therefore pointing down instead of up.

Â So these are the four directions that the system will be traveling which

Â we can deduce if we know if we start with one of em.

Â We can deduce what's going to happen in any of the other three regions.

Â [BLANK_AUDIO]

Â