0:17

In the last lecture in part two of the Dynamical Systems Lecture Series, we

Â discussed how you would numerically solve systems

Â of ODE's specifically we discussed using Euler's method.

Â Now in this lecture we're going to talk about not just

Â how you solve them numerically but how you analyze these ODE

Â systems to get, to understand general properties of their system, and

Â we're going to focus on this concept of stability of ODE systems.

Â So first we'll discuss what we mean in general when we talk about stability.

Â 0:45

Then I'm going to show a couple of one-dimensional examples

Â so you can understand how, how stability works in that context.

Â And then we're going to introduce

Â phase-plane techniques for analyzing two-dimensional systems.

Â Although most of what we explained about

Â these phase-plane techniques are going to be

Â covered in more depth in the, in the fourth part of this series of lectures.

Â And then I'm going to briefly introduce an example that we're going to deal

Â with in in part four which is

Â a mathematical model of glycolytic oscillations in yeast.

Â 1:21

So to introduce what we mean when we talk about stability, let's go

Â back once again to this generic three

Â component repressive network that we've seen before.

Â Where you have three protein species, a, b, and c,

Â that could either be dephosphorylated or they could be phosphorylated.

Â And a, b, and c regulate either the

Â dephosphorylation or the phosphorylation reactions of the other species.

Â 1:44

And as we have discussed, this scheme implies a set of differential

Â equations, which we've gone through and looked at as follows over here.

Â What we haven't discussed as much are the model parameters in the system.

Â And when we look at this we can see several constants in here.

Â That are that are model parameters that can be varied.

Â And that can govern the behavior of the system.

Â 2:09

One is we have Kcat and km's for the phosphorylation reactions.

Â And those are the ones that are over here for, labeled as kinases.

Â And Kcats are the little k's in the numerator, and then the

Â km's are the big k's for

Â the phosphorylation reactions in the denominator.

Â Similarly we have Kcats and Kms for the dephosphorylation reactions, those

Â are the analogous parameters over here on these left, these leftmost terms.

Â And then the other things we have, the other parameters we have our total amounts

Â of A B and C that is A total B total and C total here.

Â 2:44

And as we have discussed, these

Â equations are solved using standard numerical techniques.

Â And I'll just give a couple examples of what you

Â can see when you vary the parameters in this case.

Â 2:54

There are two two general categories of solutions to this system.

Â With one set of parameters, we can see that A

Â increases to a steady level, B goes to a different steady

Â level that's much lower than the steady level of A,

Â and then C goes to a level that's very, very low.

Â But if you continue to run this for longer and longer periods

Â of time, it would continue to stay at this very steady level.

Â Once A gets to this level here, right around

Â 0.9, it stays there, and it will stay there forever.

Â 3:24

But with a different parameter set, we see this sort of behavior.

Â We can see the sustained oscillations able to go

Â up and then go down and similarly B and

Â C will also increase and decrease as a function

Â of time and these oscillations will go, go on forever.

Â So what we conclude from this in general is

Â that the parameter values can greatly influence the system behavior.

Â We have qualitatively different behavior over here on the left.

Â Where A, B, and C go to steady levels.

Â Compared to on the right, where A, B, and C continue to oscillate.

Â How do we understand these different behaviors?

Â Well, that's where the tools of dynamical systems come in.

Â 3:59

And that's how we can, we can understand and

Â categorize the different types of behaviors that we observe.

Â [SOUND]

Â To illustrate how the tools of dynamical systems can be

Â used to analyze stability, we will consider a one-dimensional example.

Â This is a standard mathematical model of an

Â isolated cardiac myocyte, where different ionic currents are responsible

Â for changing the, the cells transmembrane potential, the

Â details of these ionic currents are not important here.

Â What is important here is that the

Â differential equation describing voltage across the cell membrane.

Â DV, dt is a negative of the ionic current.

Â That means the sum of, of the current through all of

Â these different channels, and pumps and transporters, divided by the capacitance.

Â 4:53

Then we instantaneously change the voltage.

Â When we instantaneously change the voltage we

Â calculate the instantaneous ionic current, I ion.

Â And if we, once we calculate the instantaneous ionic current, then

Â we know what the instantaneous change is in the in the derivative.

Â So, we can fit, we can vary voltage to

Â whatever we want it to be instantaneously from rest.

Â And then we want to calculate what's the resulting

Â derivative going to be, what's the dV/dt going to be.

Â And in that case we get a curve that looks like this.

Â 5:39

plus 58 millivolts dV/dt is, is negative.

Â And then for voltages above minus 58 millivolts dV/dt is positive again.

Â So what happens when this crosses zero, right.

Â There are two, there are two voltages here

Â for which dV/dt instantaneously will be equal to zero.

Â And if dV/dt is equal to zero, that means that if you change voltage and you

Â put it at exactly that voltage right there

Â where it's not changing, then it's going to stay there.

Â And because these are voltages where the derivative of the voltage is zero.

Â These are, are what are known as fixed points.

Â So the fixed points in this case you can represent in this one dimensional example

Â by plotting with derivative on the y axis, tying the variable on the x axis.

Â And then seeing where the curve crosses zeri, right and so where

Â they, where, where this is zeri is the same where derivative is zero.

Â That means that if the voltage is at this level, and the derivative

Â at that, in that case is is zero, then it's going to stay there.

Â 6:47

But then what we want to do is we want to say, well, what's going to happen

Â to the voltage when we move it away from one of these fixed points.

Â And this is where it gets more complicated and, and, and more interesting.

Â Right, if we start at minus 85 and we go negative.

Â This is a dV/dt is positive, that means

Â that voltage is going to move to the right.

Â Voltage is going to go up because the derivative is positive.

Â So when we change to minus 85 millivolts we have a positive dV/dt.

Â If we were to change something like minus

Â 70, in this case the derivative is negative.

Â And so therefore we can draw the arrow this way.

Â And then if we were to go from minus 85 all the way to like minus

Â 55, we would have a positive dV/dt, and so we would draw the arrow this way.

Â So that's why it's actually useful to, to draw this kind of plot, where you

Â have the derivative on the the y axis, and the variable on the x axis.

Â 7:40

Is because knowing if the derivative is positive or negative in particular

Â ranges will tell you, well which way is the system going to evolve.

Â Which way are things going to change when we when we

Â move the variable, in this case, voltage to that particular location.

Â So here we have an arrow going to the

Â right, indicating that the volt, the derivative is positive.

Â Here we have an arrow moving to

Â the left, indicating that the derivative is negative.

Â And here we have an arrow moving to

Â the right again, indicating that the derivative is positive.

Â 8:13

Why is it useful to plot the the arrows this way to show how the system

Â evolves when we change the when we change the variable, in this case the voltage.

Â Well this is a way that we can tell whether

Â our fix points are stable fix points or unstable fix points.

Â 8:30

What do we mean by that?

Â Well, this fix point where the derivative

Â crosses zero has an arrow pointing towards it.

Â And an arrow pointing on the left, and an arrow pointing towards it on the right.

Â That means that we start at minus 85 and we move away from it, it's

Â going to go back to minus 85, because that's the way that it's pointed, right?

Â Similarly, if we go from minus 85 to minus 75, the arrow is

Â pointing negative, it's pointing to the left, it's pointing back toward minus 85.

Â And so, the voltage as we go back is going to go back to minus 85.

Â So, we can now delineate this fixed point as a stable fixed point because

Â deviations away from that fixed point will always push the system back towards it.

Â 9:13

This fixed point at minus 58, in contrast, is an unstable fixed point.

Â What happens if you move negative to minus 58, well the derivative in that case

Â is going to be negative, and that means

Â that it's voltage is going to continue moving negative.

Â Similarly, if you're at minus 58 and you

Â get nudged a little bit positive, the derivative

Â is going to be positive and that means

Â it's going to continue to move away from that.

Â So, this fixed point has arrows pointing away from it.

Â That's how we identify it as an unstable fixed point.

Â And this fixed point has arrows pointing towards it.

Â That's how we identify it as a stable fixed point.

Â 9:48

And we can perform this experiment numerically by changing voltage and then

Â allowing the system to evolve, then

Â integrating the equations numerically using Euler's Method.

Â And we see exactly what we can see over here on the left in schematic format.

Â 10:05

We're starting at minus 85 millivolts.

Â When we hyper-polarize the cell, when we move the voltage down

Â to minus 95 millivolts, that's the black curve, what do we see?

Â It comes back to minus 85.

Â When we go up to minus 75, which is a Wrenn

Â curve, we see, we instantaneously change it and then it goes back.

Â 10:23

Same thing with minus 65 which is the green curve.

Â it goes up to that instantaneously and it goes back.

Â And we can identify mi, minus 75 and minus 65.

Â On this, on this plot, and see that yes, the system

Â will go back to minus 85, when we're in that regime.

Â But if we start at minus 85, and we move the

Â cell all the way to minus 55, what do we see?

Â The voltage continues to go up.

Â And then it reaches a peak, and then it

Â continues to evolve, and that's because there are multiple

Â 10:51

differential equations in this model.

Â There's somewhere around the order of

Â eight differential equations in this model.

Â But we can see that small deviations from minus

Â 85, the system will come back to minus 85 millivolts.

Â But a large deviation, if you cross this unstable fixed point, then you're going to

Â move away from that unstable fixed point and move away from minus 85 millivolts.

Â 11:12

And so by doing this numerical equation here, we can

Â confirm what we see graphically over here on the left.

Â That between mi, between minus 85 and minus 58, or something that's negative

Â to minus 85 is going to move back to the resting state, to minus 85.

Â So therefore that's a stable fixed point.

Â Small deviations away from minus 58 are going to cause action potentials or

Â return to the states and therefore this one is an unstable fixed point.

Â 11:41

What we're going to do next is we're going to learn to

Â analyze these stable fixed points and unstable fixed points more rigorously.

Â The example I just showed of of a cardiac

Â action potential, we were plotting everything as the derivative

Â of voltage, minus voltage but it is in fact

Â a model that has eight differential equations in that case.

Â And, what's happening with eight differential equations

Â you can't, you cannot visualize everything graphically.

Â So now what we are going to do is we're going to

Â look at a two variable model where you can analyze things graphically.

Â And that's going to help us develop the tools we need

Â of dynamical systems, to be able to analyze these models.

Â And the two dimensional example we're going to use

Â 12:32

And here's an, an example from a, from a pretty famous paper showing what happens.

Â There's an experimental procedure that you have to undertake in order

Â to get to, in order to observe these oscillations, where you condition

Â the cells, you starve them, and then when you add glucose, if

Â you look at oxygen consumption in this case, what do you see?

Â You see it goes up and down, up and down, up and down.

Â And these oscillations will continue for a very long time.

Â And this is a phenomenon that's been, that's

Â been known to occur for quite a while.

Â And in fact, many mathematical models of this process

Â have been developed and have been published over the years.

Â And, we're going to analyze one that was published

Â in, in the year 2000 by Bier et al,

Â and the citation for this for this model

Â that we're going to consider is given right here.

Â 13:43

The Bier et al model has two variables so the two variables that are computed that

Â are evolved with respect to time are glucose

Â inside the cell and ATP inside the cell.

Â Now we'll go through each step in the process.

Â Model simulates transport of glucose from the outside of the cell

Â to the inside of the cell through this rate called, called VN.

Â Then once glucose is inside the cell it can

Â get converted into ATP and this step here represents glycolysis.

Â in, in real life glycolysis repre, is occurs through the action

Â of several different enzymes, there are many different steps in this process.

Â But to keep things simple Bier et al represented lumped all of

Â these steps into, into one, and used the single rate constant k1.

Â 14:31

The most important enzyme in this process, the rate limiting step, is

Â an enzyme that some of you may have heard of called phosphofructokinase.

Â So, this this rate k1, again it represents the

Â action of several steps in the process, several different enzymes.

Â But it can be generally considered primarily

Â the action of phosphofructokinase activity within the cells.

Â 14:53

Once ATP is produced, it can get consumed.

Â And again, this is another place where, where Bier et al made an approximation.

Â There are several different ATPases that act,

Â that act within cells, each one of which

Â has, has different activity to keep things simple

Â Bier et al lumped all the ATPases together.

Â 15:13

And then there's one more aspect of this model that's really

Â important and that's this dashed arrow, right, I drew down here.

Â Glycolysis is an interesting reaction in that it produces ATP

Â from glucose, but it also requires ATP to be initiated.

Â So when ATP goes up there are a couple of actions.

Â One is that ATP goes up, then ATP gets consumed, but the other thing

Â that happens is that this reaction here,

Â this production of ATP occurs more quickly.

Â And so we draw this as a regulatory arrow as ATP goes up.

Â Then you get more conversion of glucose into ATP.

Â 15:54

So, we can look at the equations for the differential

Â equations for ATP and the differential equations for glucose and

Â we can understand the four terms here as the action

Â as either the steps that increase glucose or decrease glucose.

Â The steps that increase ATP or decrease ATP.

Â We'll look at glucose, dG dt first this ODE for glucose, initially.

Â Right, if you have more transport of glucose

Â into the cell, then glucose concentration's going to go up.

Â And then this term here represents conversion

Â of glucose into AT, ATP through glycolysis,

Â through the action of several different enzymes,

Â but phosphofructokinese being the most important one.

Â Now if we look at the things that

Â can either produce or consume ATP, this positive

Â term here represents conversion of glucose into ATP,

Â and again it depends on both glucose and ATP.

Â And then this term here represents consumption

Â of ATP through the action of ATPases.

Â And there are four four primary parameters in the Bier et

Â al model and the default values of three of them VN, K one and KP are given here.

Â And there is a fourth model, a

Â fourth model parameter that we haven't really discussed

Â yet, and that is the the KM for the action of the, of the ATPases.

Â And let's look at what happens in the, in the Bier et al model when we

Â change this fourth parameter, the Km for the

Â ATPases and the kls constants for the ATPases.

Â 17:26

When Km is equal to 13, we see

Â these sustained oscillations of both glucose and ATP.

Â Glucose here is plotted in black.

Â ATP is plotted in red.

Â And, and you see that glucose goes up and goes

Â down, goes up and goes down, and this continues indefinitely.

Â Similarly, ATP exhibits oscillations, and they continue indefinitely.

Â 17:47

What happens if we set Km equal to 20 instead of 13.

Â Well at the very beginning of this simulation, we

Â see oscillations, but what do we notice about these oscillations?

Â As time goes on the oscillations keep getting smaller, and smaller, and smaller.

Â And this is what we would refer to as,

Â as damped oscillations and with Km equal to 20.

Â And then finally what we notice is the

Â oscillations get so small you can't see them anymore.

Â So eventually glucose settles in at a constant

Â value and ATP settles in at a constant value.

Â So just like what we saw with the three component repressive network,

Â we can see qualitatively different behavior

Â as we change one of the parameters.

Â We see sustained oscillations over here, with Km equals 13, and we see

Â damped oscillations, and eventual settling at steady

Â levels of glucose and ATP over here.

Â So what we want to do next is we want to ask, how

Â can we understand the qualitatively different behavior

Â that we see in these two cases.

Â [SOUND] What we want to introduce now are

Â what we call phase plane techniques for analyzing two dimensional systems of ODEs.

Â 18:58

And what we mean by phase plane is that

Â we're not going to plot glucose and ATP versus time.

Â Instead of plotting glucose versus time, and ATP

Â versus time, we want to plot glucose versus ATP.

Â So, glucose will be on the y axis in this case, and ATP will be on the x axis.

Â So let's take the time courses that we simulated on the last slide, for

Â Km equals 13 on the left, and for Km equals 20 on the right.

Â And plot them in the phase plane where ATP is

Â on the x axis and glucose is on the y axis.

Â And what we see with ATP what we see in

Â the phase plane with Km equals 13 is this loop.

Â 19:39

And this loop makes sense intuitively because we saw that glucose

Â would oscillate, and ATP would oscillate and those would continue indefinitely.

Â And when you have two variables that are, that are both oscillating, what you see

Â when you plot one versus the other is

Â something like this that looks like a loop.

Â 19:56

What about for case of Km equals 20

Â where you saw these damped oscillations, where you

Â saw these small fluctuations in the beginning but

Â eventually they settle down to a steady level.

Â Well, what this trajectory looks like in the phase plan is a something that

Â looks like this, it starts here and then it goes like this and it spirals.

Â And it goes down to some level and it eventually, it keeps spiraling and then

Â eventually it gets to some steady level where

Â ATP stays constant and glucose stays constant forever.

Â 20:34

In contrast on the right, we see glucose and ATP that converge to a stable fixed

Â point and in the next lecture we will

Â discuss these, this terminology in, in more detail.

Â 20:59

Well, one of the reasons why the the plotting things in the phase plane

Â is useful is that what you care about is which direction the system is moving.

Â And the direction is determined by the derivative of ATP with

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

Â 21:19

And so at any given location in your phase

Â plane and any combination of ATP and glucose, you

Â can calculate the root of ATP with respect to

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

Â And this defines a vector in the phase plane,

Â and this vector tells you how the system is

Â moving, how ATP is changing with respect to time

Â and how glucose is changing with respect to time.

Â 21:40

So we can understand this directionality of our of

Â our two, of our two dimensional derivative vector, like this.

Â This is our differential equation for glucose,

Â this is our differential equation for ATP.

Â And what we saw was with Km equals 13 we, we saw the stable limit cycle.

Â 22:01

One question becomes well, which way is it moving around this stable limit cycle?

Â Is it going clockwise?

Â Like I've drawn it here with the arrows,

Â or is it, would it be going counterclockwise?

Â Well, we can understand that by looking at these differential equations here.

Â 22:29

And glucose and ATP are multiplied together, in this case,

Â on this term, that has a negative in front of it.

Â So eventually, this term would become greater than this term Vn.

Â And glucose would be, would would be decreasing with respect to time.

Â We'd have a negative derivative.

Â So that's how we know that this arrow for glucose when we are up

Â here at the top right corner of this phase plane, should be pointed down.

Â 22:58

Well, when glucose and ATP both get large this term here, this first

Â term keeps getting bigger and bigger, this positive term gets bigger and bigger.

Â This term here, which has a negative in

Â front of it, doesn't continue to get bigger.

Â As ATP grows this term is eventually going to saturate.

Â And so this term is going to go up for low levels of ATP but

Â then it'll get to a point where it cannot continue to go up any more.

Â So clearly this term is going

Â to eventually overcome this negative term here.

Â So the change in ATP with respect to time for

Â large values of glucose and ATP is going to be positive.

Â So, just by looking at these differential equations qualitatively

Â we can draw this arrow up here for large levels

Â of glucose and ATP, and we know that it's pointing

Â to the right because the derivative of ATP is positive.

Â And we know that it's pointing down because

Â the derivative of glucose in this case is negative.

Â 23:51

And so now that we know that this arrow is pointing this way,

Â we can then deduce how things are going through the rest of the system.

Â And we could have made the same sort of argument if

Â we had picked very low levels of glucose and ATP, for instance.

Â In the next lecture, we'll we'll talk about how we

Â can deduce when these arrows are going to switch directions.

Â Why does it go from pointing down and to the right

Â over here, to pointing down and to the left over here?

Â So, we'll talk about that in the next lecture.

Â But for now, we just want to introduce this concept of plotting,

Â of looking at systems and how they involve in the phase plane.

Â Which means you have one variable on the y

Â axis, and another variable here on the x axis.

Â 24:47

A fixed point can be stable.

Â Which means that small perturbations away from that fixed point will

Â cause the system to evolve, and return back to that fixed point.

Â Conversely, as we saw in the one dimensional

Â example, we can also have unstable fixed points.

Â Which means that if you have a small perturbation away from an

Â unstable fixed point, it will continue to move away from that fixed point.

Â