0:16

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

this fourth lecture on dynamical systems is

how we analyze stability of ODE systems.

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

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

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

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

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

And we're going to to talk about how

nullclines can be used to identify fixed points.

And then, how we can mathematically identify

stable fixed points and unstable fixed points.

0:53

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

And bifurcation means, the place where you're

having an abrupt change in system behavior.

We're

1:14

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

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

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

With phosphofructokinase being the most important of those enzymes.

1:31

Consumption of ATP by ATPases are all lumped together.

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

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

1:55

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

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

These are the default values for these three parameters.

And then last time we varied the fourth parameter.

Which is the Km for the action of the ATPases.

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

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

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

These start oscillating, but the

oscillations get progressively smaller and then

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

So then the question that we, we asked and

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

we understand he qualitatively different behavior in these two cases

with K m equal 13 and K m equal 20.

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

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

Then the direction that our system is travelling

in the phase plane is determined by the derivative

of ATP with the respect of time and

the derivative of glucose with the respect of time.

So, the two derivatives form a vector.

3:16

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

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

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

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

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

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

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

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

So this is to review a couple of

the concepts that we introduced in the previous lecture.

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

3:54

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

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

What do we mean by that?

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

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

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

And these can usually be calculated analytically.

So what you do is you say the, change in

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

We set this equal to zero.

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

want to solve for glucose as a function of ATP.

Or conversely solve for ATP as a function of glucose.

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

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

So solving for ATP as a function of

glucose in this case is not really trivial.

But solving for glucose as a function of ATP is

relatively easy because glucose only appears in one term here.

So we can solve this algebraic equation here.

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

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

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

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

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

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

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

But we want to keep these two consistent with one

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

5:35

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

All right, so the red plot here is

plotting this equation that we've derived over here.

And then the black line is plotting this equation, that

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

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

That means that the derivative of glucose with respected time

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

So glucose is not changing and ATP is not changing.

6:26

Now, when we start plotting direction arrows in the phase

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

Remember, then, the two-dimensional phase plane.

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

That is calculated as the derivative of ATP with respect

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

7:16

This term for in the derivative for ATP is

going to keep getting bigger and bigger and bigger.

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

So this negative term is going to saturate.

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

7:36

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

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

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

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

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

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

8:01

Therefore, we can draw an arrow up in this

region of the phase plane that looks like this.

Right, ATP is pointing to the right.

Because its derivative is positive.

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

8:19

Well, when is this going to switch direction?

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

Because, in order for the derivative to go from

being positive to negative, it has to cross 0.

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

nullclines very, very very useful in this time.

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

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

8:46

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

Well, we crossed the ATP nullcline.

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

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

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

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

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

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

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

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

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

9:30

So when we cross the glucose nullcline we

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

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

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

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

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

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

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

So, we've crossed the glucose nullcline.

Therefore, this arrow, instead of pointing down and to

the left is now pointing up and to the left.

And then, finally, we cross the ATP

nullcline again to go into this region here.

10:11

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

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

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

10:23

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

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

And this is why the nullclines are very useful, is

that nullclines will divide the phase space into discrete regions.

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

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

10:59

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

And we pick a combination of ATP and

glucose that's not exactly at the fixed point.

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

11:13

Well if we run this simulation numerically what we see

is that glucose looks flat and the ATP looks flat.

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

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

And then eventually you have these large stable oscillations.

11:42

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

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

And then it oscillates forever.

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

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

12:11

Now let's consider the non-oscillating system.

When we, we saw before numerically that when we

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

What if we take this non-oscillating system and

we start with the initial condition somewhere over here.

Very far away from the fixed point.

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

The glucose oscillations will, will go away

and the ATP oscillations will go away.

They'll, you'll start with some

small oscillations, but they're damped oscillations.

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

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

And then you spiral along, and you spiral,

and then you eventually converge on the fixed point.

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

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

Therefore, we conclude this is a stable fixed point

13:09

Next, we want to address how we

can understand stable and unstable fixed points mathematically.

And how we can calculate for a given fixed

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

This is a somewhat advance topic for this class.

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

But I do think it is worth going through

it just so just for the sake of completeness.

But mostly, what I want to teach you in this

class is graphical methods for

understanding stability or, or instability.

And these graphical and numerical techniques for assessing this.

14:02

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

It's called Jacobian.

And this matrix consists of the partial derivatives of

these two functions with respect to the two state variables.

So the first element is a partial derivative of the

first function f with respect to the first variable ATP.

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

Partial derivative of little g with respect to ATP.

Partial derivative of little g with respect to glucose.

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

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

Those are the four elements that read

your Jacobian matrix for a two-dimensional system.

And in that case, we get this term here.

And you can verify these partial derivatives just by

going back to the rules that you learned in calculus.

14:55

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

As defined by a particular combination of glucose and

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

And this is where analytical computations can become somewhat difficult.

15:11

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

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

Where both derivatives are equal to 0 ,but in many

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

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

15:33

So you first you you evaluated Jacobian matrix

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

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

And then what we want to do is just we

want to calculate the eigenvalues values of this Jacobian matrix.

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

16:38

As we just said, the eigenvalues of the

Jacobian matrix evaluated at the fix point, determines stability.

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

Sometimes it is impossible to do it analytically.

But it is possible to do it numerically.

16:53

And I wrote a script called bier_stability

and that uses MATLAB's function called eig.

Which computes eigenvalues, as a way to

calculate the Eigenvalues of the fixed point.

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

0.004 plus or minus 0.1132 times i.

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

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

So this is our, mathematical confirmation of

what we saw with the numerical simulations.

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

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

can conclude based on the fact that

these are complex eigenvalues with positive real parts.

17:42

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

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

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

And the negative real part of the complex eigenvalues indicates that

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

So the complex eigenvalues, in either

case, indicate that we have periodic oscillations.

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

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

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

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

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

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

18:28

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

A bifurcation in general, is somewhere

that the system qualitatively changes behavior.

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

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

18:48

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

It the system switched from having steady oscillations to having damped

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

But what about all the different values in

between or slightly higher values or slightly lower values.

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

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

And with each simulation I, I taught I simulated it

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

I calculated the minimum value of glucose and the

maximum value of glucose over those last 500 minutes.

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

between the minimum value of glucose and

the maximum value of glucose over 500 minutes.

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

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

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

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

Because we have an unstable fixed, unstable

fixed point in a stable limit cycle.

20:07

Well here we have the opposite.

We have negative real parts of our

eigenvalues and we have a stable fixed point.

And so at Km is approximately equal to 16, this

is where the fixed point that was unstable becomes stable.

20:22

And this is what we would identify as a bifurcation.

Somewhere around Km equals 16, this is where the system

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

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

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

20:49

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

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

therefore, fixed points are defined by intersections of nullclines.

So in the two-dimensional systems we looked at where

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

21:12

In phase space, each time a nullcline is

crossed, one of the directions of the system changes.

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

And each time we crossed a nullcline it either flipped

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

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

21:46

And finally, bifurcations are locations where

dynamical systems exhibit qualitative changes in behavior.

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

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

22:13

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

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

Where the A nullcline is the red one here.

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

And the B nullcline is plotted in black.

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

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

22:42

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

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

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

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

22:59

And again, this is probably a good idea

for you to pause the recording think about it.

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

[BLANK_AUDIO]

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

23:22

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

We're crossing the, the B nullcline.

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

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

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

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

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

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

So we're still pointing to the left on the

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

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

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

into the left rather than down into the left.

24:10

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

still going to be pointing positive with respect to B.

But now instead of be, being negative in respect to

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

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

24:25

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

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

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

0 and is therefore pointing down instead of up.

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

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

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

[BLANK_AUDIO]