Synthetic Biology and Stability Analysis in the Toggle Switch

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visit MIT OpenCourseWare at PROFESSOR: So today
the basic idea is to try to understand
the toggle switch. How such a thing can be made,
why it represents a memory module.

But then we'll
relatively quickly get into two themes
that are going to be useful throughout
the rest of the semester. First is these
dimensionless equations that often pop up in the
analysis of these gene circuits. And it's absolutely
essential that you understand how to get to the
dimensionless equations, and also what these
parameters end up meaning. And then we'll, at the end,
talk about stability analysis. How is it that you can determine
whether a particular set of interacting pieces in
a cell or in an ecosystem whatnot– once you get it in
the form of an equation, how's is it you can determine whether
a particular fixed point or a particular
location is going to be stable to perturbations? This is going to
be useful for us to determine where
the gene network is going to kind of move towards. Also it'll be useful for us
to determine whether a gene network is going to oscillate.

And later, it'll be relevant
in the context of predator prey oscillations, and
a bunch of things. So can somebody
maybe explain briefly the idea of the toggle switch? Yes, please. AUDIENCE: You have these two
genes that repress expression to each other. So when one of them is
large, the other one is not expressed. [INAUDIBLE]. PROFESSOR: Perfect. So you'll have two genes that
are going to mutually repress each other. So in this case, each of them
will be a transcription factor of some sort that will
bind to the other promoter and repress it.

So there are various
levels of abstraction that we might use to
describe such things. So we might, for example,
just say A repressing B, B repressing A. Of course
when you write it that way, it doesn't have to be in the
context of a gene network. This A's and B's could
just be chemicals, they could be species
eating each other. It could be almost anything. Now in this framework,
to get a basic sense of why this thing might have
two alternative stable states, is that often we like to take
the Boolean approximation. This thing will not always
work, but it's a useful thing to do to just kind of first get
a sense of what might possibly be happening.

So we might say, 0 corresponds
to some sort of low. 1 might correspond to high. And of course, these
things have to be put it in quotes, because we haven't
specified what we mean by this. But it's useful
to just make sure that we're all thinking
about the same things. And then in the
context of A and B, we can just say, well, there's
a number of different states it could possibly be in. And you can ask whether this
assignment of logic values keeps everybody happy. And so you might ask, well, is
0, 0 a mutually happy state? And of course then
you have to say well, if you're in
the 0 state, you're not repressing the
other guy, but maybe 0 is sort of your
equilibrium anyway.

So what we have to do is we
have to make the assumption that when you're not being
repressed, in that case the promoter will be
actively making that protein. So then you'll go to some
sort of high or 1 state. So in that case, you say, if you
start out with both repressed, maybe both of them
should start trying to increase their levels. So this, in some ways,
is not a stable state. Similarly here, this is
also not a stable state. Because in this
case, they're both going to be trying to
repress one another. So then they'll both
start coming down and then the situation may
resolve into one of these two. So this is just where
either A or B is on, and repressing the other one. And so for example, in
context of the repressilator, on Thursday, this
is just a useful way to start imagining how
this A repressing B, repressing C repressing
A– how such a loop can lead to oscillations. This kind of analysis
does not at all prove that there are
oscillations in any given manifestation of this thing.

But it's useful
to just make sure that you're roughly getting the
idea of what the system might be doing. Now in many cases, we'll want to
be a little bit more explicit, and draw the gene
network in more detail. And there are multiple
manifestations of the Thomas Switch. There are many of them
that have been made. So the important thing
is not too necessarily keep track of exactly
what the components are, but in one case for example,
we might have A corresponding to something here. It's coming back and
repressing again. Expression of this
B. And this might all be on one piece of DNA. Whereas this B here will
come back and repress A. Now one thing that is– and
I just want to mention here, they also have a GFP. And this actually
is a case where those two can be expressed
off of a single promoter. So this is often
done in bacteria where there's a
single promoter– so RNA polymerase actually will
transcribe both of these genes. This repressor B, as well
as this fluorescent protein.

So there are going to be
alternative loading sites for the ribosome in that case. And eukaryotes typically
do not do this. Yeah? AUDIENCE: [INAUDIBLE]. PROFESSOR: That's right. So– AUDIENCE: And it will
transcribe everything until– PROFESSOR: Exactly. So here comes– so an
RNA polymerase down here made this whole thing. And now you might have two
separate locations where the ribosome loads and
makes this protein B. And then a different
ribosome would make this GFP. AUDIENCE: And when does it stop? It just keeps going down? PROFESSOR: Yes. So there's a
termination sequence. AUDIENCE: No, no, but
it would B and then GFP, and then it would
just keep going? PROFESSOR: No. The ribosome is told to
basically start here and end here. So then it just
makes the B protein. And then another
ribosome binds here. AUDIENCE: If you had more
proteins on the same strand after GFP– PROFESSOR: And when
you say protein, you're referring
to the ribosome.

[INTERPOSING VOICES] AUDIENCE: I mean G's right? PROFESSOR: Oh! OK, you're saying if
there were another gene? AUDIENCE: No, if you
have more genes coded, if you're coding for
more proteins after GFP– if you have more genes on a [INTERPOSING VOICES] –it will just keep going
for an arbitrarily long– [INAUDIBLE] PROFESSOR: Arbitrary is
always a dangerous word. But they can be more than two. And actually at the biophysics
retreat that some of you guys were at just last two days,
there was a great talk by Gene-Wei Li, who's going
to be a new incoming biology faculty member. And he was talking about
the FO F1 ATP synthase. So it's the thing
responsible for making ATP. He analyzes process work. And there are many sub units. So there are half a dozen or so. And so it's a very
long transcript. And then what he showed
is that actually you have different
rates of synthesis of the different genes
on this one transcript. And in some cases you want
actually more copies of one of the subunits
than another one. And so then actually
if the final protein, if it needs 12 of these,
only one of these, then actually you can make 12
times as much of this, because you just
have more translation here than you did here.

And then it's great,
because then you have all the right
ratios, all the components to make the protein. So you can actually have
additional regulation even at that stage. It's possible not everybody
followed that discussion, and my apologies. But feel free to just
erase it from your brain if you're too confused. But what you need to keep track
of here is the level of GFP is going to be perhaps
proportional to the level of B. Because they're being
expressed at the same time. Now in order for this thing
to be a memory module, you also want to be
able to reset the state. So if you were in
this state, you'd like to be able to somehow
get it to move to this state instead. Does anybody remember
what the inputs were in the context of
the sample toggle switch, that it was in that review? AUDIENCE: [INAUDIBLE].

PROFESSOR: Right. OK, so there are multiple
versions of toggle switch. And indeed in one case this was
just a small molecule, IPTG, and that's because this was
in that case, the lac I. And this then represses
the repression. So in many contexts
this class, you'll have to remember that a minus,
minus is equal to a plus. And in different toggle
switches you indeed have different ways of
inhibiting this repression. So this could be another
small molecule, ATCN. In the example that
they had in this review, it was actually
heat that did this.

But that's just because this
transcription factor was a temperature sensitive mutant. So above some temperature,
it could no longer repress. So for example, if
you start out in high GFP– so in this case,
where they– all right. If the cells are in
high GFP, and you want to switch it into
the alternative state, what stimulus do you
want to apply here? So I'll give you a guess. It's going to be
either heat or IPTG. I just want to make sure that
we can all read these diagrams.

This is to switch
from the high GFP, and we want to go
to the low GFP. Give you 15 seconds to just
try to read off this diagram. Do you need more time? No. Ready. Three, two, one. So we have a majority are
B. So a majority of people are saying, well, in this
case, you have a lot of GFP. Means you have a
lot of this protein B. In this case, the lac I. And we don't have very
much of this other protein.

So that means that if we
want to switch the state, and get a lot of A, we have
to stop this repression. So we have to add IPTG. Any questions about what we
mean by the various symbols up on this board? So after we had IPTG, then
indeed the GFP should go down. How long is it going to
take for it to go down? Does this switch go
down immediately? After you add the
IPTG, how long do you think it's going to take
for this repressor, lac I to fall off of that promoter? Do you think it's going
to be seconds or hours? It's actually
seconds, and that's because this IPTG rapidly
can go across the membrane, it'll rapidly bind to
the inhibitor lac I, and then lac I will fall off.

So in this case, lac I is
actually still present, so it's going to take hours,
actually for lac I to go away. But it takes seconds for the
lac I to become inactive, ineffective. So this is the separation
of time scales idea. But how long is it going to
take for the concentration of– well, how long is it going
to take for GFP to go away maybe? Is that going to be
seconds or hours? AUDIENCE: [INAUDIBLE]. PROFESSOR: Yeah, In this
environment the bacterium– they might be dividing
every half hour. That's the characteristic
time– if these are stable, that would be the characteristic
time scale for example, for GFP to go down. And of course, that's
even after it's inhibited. After you stop making GFP. In this case, once you
had the IPTG, the lac I is going to fall
off of this promoter, and then we have
to first make A.

And only after we make A
will we start repressing expression of the B and GFP. That's the process that
you expect to take, hours, based on the generation times. But this is a memory
module because after we've added the IPTG, now we can, in
principle, take the IPTG away, and it'll stay low. So in principle,
this IPTG signal can be a transient signal.

And the cells will remember
that they encountered IPTG. That's what makes this a memory. This input can be transient. Transient signal is remembered. Now a big part of why this
why this paper was important is because this toggle
switch was constructed out of components that previously
in principle, had not even ever seen each other before. So maybe that lac I and this
promoter had been put together, but this whole system,
this whole gene network was composed of
individual components that they were synthetic. And were put together
because they thought, oh this should kind of work. And it's led to, I think
a real flowering of, this intersection between
modeling and experiment. This thing was built based
on a model that told them that maybe if we do
this, these things they multiplarize in order
to repress and so forth.

So there was a real sense in
which the modeling– and then we're going to talk more
about the modeling– on how it was essential to get an
idea of how we should construct this thing, and
to guide our work. Because if any of you
do work in the lab, you'll know that
things are hard. And often components
don't behave the way you think they
should and so forth. Now modeling can't save
you from all that pain, but at least it can guide
you in the right direction so that it limits the number
of things you have to try. Are there any questions about
this element, before we switch over to some of
the dimensionless equations that we're
going to be using? So the reading for
today was composed of a review that
these two pieces, and one of which I think
might be hard for some people, and the other one might
be hard for other people.

For those of you with maybe more
limited experimental experience in biology, maybe the
review was a challenge to try to understand
all of the nomenclature and the words, whereas
those of you that have not played with differential
equations as much recently, may have found the reading
on modeling the toggle switch and stability analysis
to be more challenging. We will, in some
cases, do this where we have two different hopefully
shorter kinds readings. And hopefully they're
not both hard for you, because then you'll spend
a lot of time reading. But then it will
be good for you. So don't run away quite yet. But from my standpoint,
it's essential that you develop intuition
behind these ideas. So once you have this equation
that somebody gives you, you have to be
able to figure out what assumptions have they made,
and what should the behavior be roughly, before you go
off and you do simulations and full mathematical analysis.

So a lot of what we're
going to do in this class, and in particular
during the lectures, is to try to work
on that intuition. And the first thing
you need to do is make sure that you know if
you don't understand something. Sometimes things
look really simple, especially these
dimensionless equations. Part of what's attractive about
them is that they are simpler. But the connection
to experiments can be quite challenging. And we'll kind of
see some of this. So these dimensionless
equations– dimensionless
equations– I think that they're both good and bad. The good is that
you can figure out what are the essential
features of the model. So you can focus your attention
on the essential mathematical features. The disadvantage is
that you might not even know which of the parameters
change when you do something experimentally.

So the problem is
that the connection to the biology or experiments
is obscured in some cases. Connection to experiments. And as an example of
this, what we want to do is look at these equations
for the toggle switch. I know that you guys
just did reading on how to get to these
final pair of equations. It's important to remember that
in the context of the paper, they had a model. Here are the
dimensionless equations that describe our system. And we use them to
design the toggle switch. But just from that,
you don't necessarily realize what's been done. So we want to make
sure we understand it. So the equations that we
want to be comfortable with are the following. So it's u dot du dt. Now here they use alpha
as the rate of expression. So beware, in the
past, we sometimes used alpha as a
rate of degradation. So fair warning. Alpha 1 1 plus that's v beta.
v to the beta minus u v dot is equal to– here is an
alpha 2 divided by 1 plus u to the gamma minus v.
Now in many, many cases, we're going to
get– equations that look like this are going to
pop up time and time again over the rest of this
semester, and you just have to be intimately
familiar with them.

So first of all,
can somebody say why this might be capturing the
dynamics of a toggle switch? I can say something. Yes, please. AUDIENCE: There's some more
of each u or [INAUDIBLE]. PROFESSOR: That's right. So the more v you have,
the less production you're going to have of
u, and vice versa. Now in many cases,
beta and gamma are going to be something
that's larger than 1. This is capturing some
element of cooperativity in the repression on each side. So this thing is indeed just a u
repressing a v, and vice versa. Now these things are
wonderfully simple equations. See that there are four
parameters that are completely specifying the dynamics.

Now you'll notice that
the world is always to be more complicated
than the four parameters. Now there are two ways in which
these complications went away. One is that we are
modeling a simple– it's a simple model
of a complex system. But the other is that
even the simple model has been simplified by going
to this dimensionless version. So I want to make sure that
you understood the reading to the point where at
least you understand what units of concentrations,
times, and so forth are in this model. So first I want to ask about the
effective lifetimes of proteins u and v. Of u verses v. So
we'll maybe call this tau u and tau v. And I want
to know which one– and how these things are
related to each other. Tau u greater than [INAUDIBLE]. DK is again, don't know. Now of course in principle,
the lifetimes of u and v can be anything. The question is, once we've
written down that equation, have we actually said
anything about that? Have we already made
in assumption or not? Do you need more time? I see a fair number of
quizzical faces, which may mean that even with extra
time, the quizzical faces would not go away.

Question. Yes? AUDIENCE: Are you asking
about the lifetime of individual [INAUDIBLE]? PROFESSOR: OK. All right. Yeah. So when I say
effective lifetime, that's because if
it's a stable protein, then the lifetime of
that individual protein is maybe infinite, but you get
an effective lifetime because of this dilution effect. So when I say effective
lifetime in general, in this class, what
I'm referring to is the sum of two effects of
dilution to the cell growth, as well as actual degradation. AUDIENCE: And you're
saying [INAUDIBLE]. PROFESSOR: No. No, I'm saying given that the
Collins Lab– in their paper, wrote down those
set of equations. I'm asking have they
already specified anything about the effective lifetimes
of these two proteins? AUDIENCE: So in those
equations, [INAUDIBLE]. Your question asks in units
of that dimensionless time? PROFESSOR: Yeah, I
mean we can compare two times in some
dimensionless– yeah, we're going to also
talk about that. Maybe I should have
done that first. But this is a nice
question because it's highlighting that you
get a pair of equations, they look obvious, but then
some of those basic things about the system are
some how not quite clear.

So I'm going talk about in
this units of whatever time is being– however
time– we're going to discuss how time is being
measured in a moment as well. But however time
is being measured, is there some
relationship here or not? Let's go ahead and vote. And it's fine, if
you really do not know what I'm talking about,
you can say E and that's fine. Let's see where we are. Ready, three, two, one. So I would say that it's
split between B's and D's. And that's great,
because that means we have something to talk about. There's broad agreement
that it's one of these two. So turn your neighbor. You should be able
to find somebody that disagrees with you. If you can't find anybody
that disagrees with you, you could think
about how parameters are going to change as you
vary other experimental things.

go ahead and reconvene? I just want to see where we are. So let's get our cards ready. And we're still working on this
one, so you can ignore this. Is it B or D? Ready? Three, two, one. So I'd say there's some
migration towards B, but not 100% still. So I'm going to
side with this here. Can somebody volunteer
why they're saying that? AUDIENCE: Well, if I remember
correctly then they [INAUDIBLE] time by multiplying time
by degradation rate.

And they do that with
the same degradation rate for both [INAUDIBLE]. PROFESSOR: Right. So the answer here is yeah– so
the way that we got to this non dimensional time, that we're
going to discuss in a moment, is multiplying by– or divided
by some degradation rate. And you remember the derivation. You remember that
it was [INAUDIBLE]. But in many cases you don't
get to read the derivation before you have to answer it. In many cases you just
get these equations, and you have to figure out
what the authors have assumed. So I think your answer
is very much correct, but it might be–
I'm glad that you're using the pre-class reading,
but at the same time we have to be able to
answer this question just from these equations.

Because it is contained there. Yeah? AUDIENCE: So if you remove
production [INAUDIBLE]. PROFESSOR: Perfect. AUDIENCE: If you
solve that equation, it's exponential to E,
and the [INAUDIBLE]. PROFESSOR: Yes. That's right. So the statement
here is that it's nice to just imagine that
we shut down production. So these first terms go away. All right now we just have
u dot is equal to minus u. So the concentration of u and
B will both fall exponentially. And they're going to fall the
same rate in whatever units of time– and we'll
discuss this in a moment– but however time
is being measured, they're going to
fall at the same rate because it's whatever
appears in front of these two that determines that rate.

So we've already assumed
that the effective lifetime of u and v are the
same once we've written down these equations. Now that doesn't
have to be true. That means that if
experimentally, you want to make a toggle
switch with two proteins, with different st
abilities, then you can't use these equations. You have to add a delta on
one of the two equations to capture that dynamic that
they have different lifetimes.

So these equations are
simple because we've already combine things, but they've
also already assumed some things to make it look more
simple and more symmetric. So for example,
they're allowing for different effective
cooperativities, beta gamma, for the two repressors. They didn't have to do that
if they didn't want to. If they wanted to,
they could have just had beta in both equations. And that case they
would be assuming that the effective
cooperativity of the repression is the same for the two. So depending on
what you write down, you're making different
assumptions about the system. And so you have to be able
to look at these equations and figure out what
assumptions have been made. Yes, question. AUDIENCE: I'm having
trouble seeing what the first term is doing. So the second one
is [INAUDIBLE]. PROFESSOR: So broadly, the
first term in these equations is the production
rate of the protein. And the second term is some
sort of effective degradation, but it could be due to dilution.

The nice thing about just
ignoring the production term is that it's a way of focusing
only on the effective lifetime portion. This effective lifetime
is it's captured here, irrespective of whether
there's production or not. Of course the
production is going to affect what the equilibrium
is that we go to and so forth, but remember for example, that
this effective lifetime that's set for time scale both to
come up to some equilibrium, as well as come down
to some equilibrium. That's telling us about
this effective lifetime is essential to tell
us about the rate that concentration is going
to change within the cell, whether you're going up or down.

Did that answer your
question a little bit? Yes? AUDIENCE: That
last thing you said is not quite true because– PROFESSOR: Because
of the toggle. I agree. What I was really referring to
there was if we get rid of one, and we're just talking
about where we just manually set the production
rate to one thing or another. The actual dynamics of this are
more complicated, certainly. So is everybody
happy with this idea that just by writing down
those equations, we've made some assumptions constraining
relationships between u and v in some ways, but
not in other ways. Yeah? So we've kind of already danced
around this other question, but I want to make sure
that we address it head on. We want to know what
is this unit of time. If I say oh, at time T
[INAUDIBLE] 1 versus time [INAUDIBLE] to 2– so if delta
t– what is 1 in this case? Are we referring to one second? CGS. Or is it one hour
corresponding to another unit? Cell generation time, effective
lifetime, or don't know? Ten seconds, since we've
already kind of said this, but it's important
enough to make sure.

Ready? No? Yes? I'll give you another ten
seconds, just to make sure. Do you need more time? Let's vote. Ready? Three, two, one. OK. A majority of the group
is agreeing in this case it's the effective lifetime. And it's not the cell
generation time necessarily because if we actually
have a degradation tag on this protein, if it's
not a stable protein then the effective lifetime
is going to be shorter than the cell generation time.

So we've normalized
time by dividing out that– whatever that delta thing
would have been on the right. Now this is great because it
makes the equations simple. But remember what that
means is that if we change the degradation rate,
then all of a sudden it's not obvious
what's going to happen. Experimentally
we're always allowed to affect the degradation rate. But the question is, which
parameter or parameters change if you add
a degradation tag? So if you increase
the degradation rate. That's what I want to do. We'll do that in
just moment there. I have another question
that I wanted to do. Do you guys remember
the equations up there? Maybe I'll come over
to the other side just so that I– it's useful
to be able to stare at those equations as we discuss. So the question is,
what is the unit of concentration of protein u? And of course, these
are dimensionless. What I really mean is
what does u equal to 1 mean in real units? I'll just write that. What does u equal to 1 mean? What does u equal
to 1 correspond to? In something that
is recognizable to an experimentalist in the lab? You guys understand what I mean? So if in this model I
say, u is equal to 1, or u is equal to 10.

What do those numbers–
what do they mean? So are there any questions
about my question first of all? Yes? AUDIENCE: Are we [INAUDIBLE]
solely on seeing those equations? Or based on the
reading of [INAUDIBLE]? PROFESSOR: I would say
that the reading actually has a somewhat more
complicated model, and involves multiple steps. So this is really just
looking at those equations, given that we can
see that there is some statement about how u and
v are repressing each other.

So we're saying that there is
some function that describes– it's a phenomenological
function– describes the input output relationship in terms of
some concentration of u leads to some repression
of v and vice versa. From that actually, we should
be able to say something about what we've already
assumed in these equations. Do you need more time? Yeah, question. AUDIENCE: So the k of u promoter
is– that's defined as the– PROFESSOR: That's the binding of
v to the u promoter– the piece of DNA in front of the u gene. Do we need more time, or
shall we give it a go? Ready? Three, two, one.

All right. So we got some A, B, C, D's. No E's. At least it means that your
neighbor has an opinion. He or she cannot say that– Turn
to your neighbor and discuss. [CLASSROOM CHATTER] Did you guys all decide
you're comfortable? Why don't we go
ahead and reconvene. It seems like it may be we're
coming to consensus here. Ready? Three, two, one. So now we got a clear majority
agreeing that it should be D. So what happened was that over
here in the original equations we had a real concentration
of u divided by some k. And that was the k for
repressing v. Because v is this guy that is over here. And what we did is we then
just turned u divided by this k for binding this v
promoter into just u.

So in particular, when
that concentration of u is equal to its
associated k, that corresponds to half repression. Same thing here. One way to think about this is
just that when u is equal to 1, we're getting half of
maximum possible repression. Similarly, that's what
v equal to 1 means. U and v, do they have to
have the same– I mean, does u equal to 1
and v equal to 1 mean the same thing
in terms of the number of the proteins in the cell? No. Not necessarily. So we've allowed
for the possibility that those things are measured
in different real units. But in both cases,
it's telling us about how the strength
of repression. And u and v equal to 1 tells
us about that crossing point where the other promoter
is half repressed.

Are there any questions
about what happened there? This is especially confusing
because it doesn't enter into the equations at all. Things just went away. But you know that this is
the dimensionless versions of the equations
because we have u here, and we're adding 1 to it. Are we allowed to add
things with different units? No. Never. What that means is that–
since they're being added, that means that we
already know that we made– this is the dimensionless
version of the equation. And it actually then
immediately tells us what u equal to 1 means. So now what we want
to do is we want to make sure that our intuition
on this is tip top shape. In particular, we
want to know if I want to change the
dynamics of the system– so let's say that we go
and we spend lots of time calculating the fixed
point stability, and we do everything
on these equations. And then we know
what we need to do is we need to change
some parameter in order to get say, a toggle switch.

We need to know how we do that. We need to know how the
parameters in real life, or even in the context
of the model, how is it that the parameters you can
actually change experimentally, how is it that the affect or not
the parameters in this model? So the question is, let's say
we increase the degradation rate of these two
transcription factors u and v, which of the parameters
are going to change? So I'll give you 30
seconds to think about what should be happening here. Do you need more time? Let's see where we are. Ready? Three, two, one. And this is the
possibility where you can put up two things. Remember our
fabulous card system? So I think most people are
saying it's going to be A and B are going to change.

So beta and gamma are capturing
how cooperative that transition is. And the cooperativity
is not affected by the questions of the exact
concentration, or time scale and so forth. Because that has to do
with the molecular nature of the interactions
with the promoter. So in some ways beta gamma
are the simplest things in this system. Now the question is, do
alpha 1– do they go up or do they go down? Now that I've told
you that they change. If degradation rate goes
up, alpha 1 and alpha 2, do they go up or
do they go down? I'll give you 10 seconds
to think about it. Do you need more time? Let's see it. Ready? Three, two, one. It's a majority are
saying it's going go down. Can somebody offer up
an intuitive explanation for why this might be? [INAUDIBLE] AUDIENCE: The
degradation rate goes up, it means the times
scale goes down. As you produce a fixed
number of per unit time, the unit time gets
smaller and produces less. PROFESSOR: Right, so if the
degradation rate goes up, that's kind of reducing
this unit of time. So you just are not going
to make as much protein in that unit of time.

The way that I like
to think about this maybe is that if the
degradation rate goes up, then the real concentration
of the protein should go down. And that means
that the repression should be less effective. And then wait, is
this helping me? No, now that I'm saying
this– I don't like to think about it that way. [INTERPOSING VOICES] That's right. Yeah, so– that's right. You should decrease
the sort of– So if the degradation rate
goes up, you decrease the real concentration,
which it means indeed that you are at steady state. Say a less effective repressor. That means the
concentration goes down in these units of how
repressive are you. I think the way of thinking
about it was correct, just the words were not. You can also then go– and it's
useful in the context of when you actually go and you do
the math of removing all these parameters, just to
make sure that– because it's easy to do this thing where
you just divide everything out, and you're happy, and
whistling and so forth.

But at the end of the
day, you really just have no sense of what happened. Of how the parameters
that come out of the model are affected by the real
things that you can change. And in some ways what's
funny about these equations is that essentially
everything is in the alphas. Because beta and gamma, that's
this cooperativity parameter. That's what it is. That's all it is. So that means that everything
else ends up in the alphas. So the strength of the
promoter, the concentration you need to repress,
the lifetime, everything rolls up in these alphas. And this is the beauty of
the dimensionless equations. Is it's telling you that you
don't have all these different, separate knobs. You can't change one
and change another, and go into some funny regime. Because it's all rolled into
one fundamental parameter, these alphas. That's telling you that
once you understand how these equations behave,
then you in principle understand everything that could possibly
happen in that simple model.

But that's what's
wonderful about the dimensionless equations. But the problem is
that you sometimes lose track of how it's related
to the real experimental things. So I would say that I very
much like these dimensionless equations. They clarify things for you. But you have to spend
the time to make sure that you understand where
all of reality went.

Because it all ends up in
this mathematical equation, and that simplifies things. You're not floating in a
sea of symbols anymore, but it's really easy to
lose track of the connection to real measurements. And that's why we
want to do modeling so we can make that connection. So do the dimensionless
equations, but make sure that you play with them
a bit so you know what changes when you change what. And we'll actually see a
bit later, some other cases where it's actually quite tricky
to figure out what's happening. On exams I always want to
just ask a equation like this, if this parameter goes off–
and then the TAs always say, it's too hard of an question. I feel like it's like the most
basic thing you would want from an equation like this. Is that if you increase
the strength of this– But it actually is– it's
surprisingly difficult. So maybe this year
I'll convince the TAs that it's an OK question. Are there any questions
about where we are right now? So what I want to do
for the last half hour is talk about
stability analysis.

The first context in
which stability comes up in this class is indeed
in this toggle switch. But it's not– I would say–
the most satisfying application of it in some ways. Just because you end up with
equations that all you can do is plot them. And it's useful to be
able to recapitulate, to understand how the figures
from that paper come about. But at the same time
it's not always– after you find the solution,
you don't feel so happy about it either. Because in terms of complexity
as a function of time, things always start out simple. And then in the course
of the calculations things get complicated.

And the problems
that are fun to solve are the cases where
it comes simple again. This one– it never
quite converges. I do want to talk about
the stability analysis so that you can understand
the calculation that was in the notes. But also so that you can
just get some more intuition about some of the other
problems that we're going to be solving
in the next few weeks. Just to make sure that we're all
talking about the same thing, it's useful to start
by just making sure that in a one
dimensional problem, we understand what
we mean by stability. We're going to be fast. X equals 0 is stable
if and only if what? I'll give you 10 seconds. Ready? Three, two, one. All right.

So this is always–
So there's actually a fair number of answers here. And this is tricky because
the temptation is always to jump into the two
dimensional stability analysis, or the n dimensional
stability analysis. And the thing is that
we have to make sure that we are completely
comfortable with a one– talked about one dimension before you
talk about multiple dimensions, because then everything is lost. I'm not going to have
you guys discuss, the but it's going to
be C. Many people are saying A or other things. There is a context in
which this guy comes in. But let's just make sure. So x equal to 0. Now first of all, is this
thing– is x equals 0, is it always a fixed
point of the system? Yes. So fixed point means that
if you go right there, then in principle you
don't move off it.

So it doesn't say anything
about whether it's stable or unstable. But if x equals zero, then
indeed x dot is equal to 0. So that's a fixed point. But if you have positive x–
in order for that to be stable, you have to have a
negative change in x. So if you talk about the
behavior as a function of time, this function of x here. If you start out above 0,
the definition of stable is if you go a little bit
away, you should come back.

And that means that x
dot has to be negative. So if positive x– and
similarly if x is negative, you want it to be
stable, then you need the x dot to be positive. So this is– A less than 0. Now there is a context in
which a stability condition looks something a little
bit more like this. And can somebody say when
it is that you get something to look– condition around 1? Yes? AUDIENCE: Discreet. PROFESSOR: Yeah, in
discrete maps then indeed the condition for stability
looks something like this.

If you have something that
looks the x of t plus 1, is equal to axt, then
if the condition for x equal to 0 being stable is for
indeed a to be less than 1. Or it's in this case it's really
the magnitude of a being less than 1. Because in that case
you might get hopping. But then the condition
is around the 1 thing, whereas here 0
[INAUDIBLE] indeed. Solution is different– it's
going to go exponentially to– so x is a function
of time is just going to be some x naught
e to the– and is it at, or a is less than 0. It's just at, right? And in this case, we
just have one dimension.

So the eigenvector
[INAUDIBLE] is really just x. There's just one eigenvalue,
and a is indeed the eigenvalue. So the stability for x
equals 0 to be stable is that all– and in
general, for n dimensions, is that you need all the
eigenvalues to be less than 0. And in this case, a is
indeed the eigenvalue. And you need that
to be less than 0.

So if you're not
comfortable with this, or you got something
other than C, then I think it's essential
that you take the time now to go over all of these ideas. First in one dimension. Make sure you're all
comfortable with that. But then to look again
at these two dimensional, high dimensional
stability analyses. Because if these things
you're finding tricky just because it's been
a while since you looked at this stuff, that's fine. But if you don't spend
the time to iron out now, then everything gets
much more painful later. I highly recommend Strogatz's
book on dynamical systems. It's a beautiful
book with just as clear as a textbook could be. So that book is available at
various libraries and reading rooms and so forth. So it's a great reference. Now we want to do is
generalize this idea to think about in n dimensions. So now what we have a vector x.

And there's going to be some
matrix A that– oh, and x dot. So the change in
this vector x is going to be described by
a linear set of equations, so that they can be
specified by some matrix. To determine the
stability, we're going to then look at the
eigenvalues of this matrix. Now in many cases in this
class, what we're going to do is we're going to find that
some set of non-linear equations has a fixed point
somewhere, and then we're going to linearize
around that fixed point to convert into a linear
problem that looks like this. But for now what
we're going to do– the general statement
is for n n dimensions.

You want all the eigenvalues–
lambda I in particular– to be the real part
to be less than 0. And that has to be true
for all the eigenvalues. We're going to be talking
about the conditions in a two dimensional system
where there are in principles and shortcuts,
but it's all the same thing. It's all that you need. The real part of all the
eigenvalues be less than 0 for the fixed point be stable. So for now what we're
going to do is we're just going to imagine
that we have– assume we have two dimensional problem.

This vector x, we're
going to write as x and y. So we can write the
dynamics like this. This is x. This is x dot, y dot. And this is really
the same thing as saying that x dot can be
described by some ax plus by, and y dot is some cx plus dy. And so we want to be as
comfortable as we can with all the
possible things that could happen in these two linear
and differential equations. And particularly we want to
know what the condition for 0, 0 being stable? 0, 0 is stable if and
only if– now there's a rule that you
found in your reading having to do with the
trace and the determinant. So trace is the
sum of these two. Determinant is product
minus the product here.

Now this is not
the kind of thing that you have to memorize. But it's useful to know
that there is a simple rule, because it allows you to quickly
determine whether something could possibly be stable. And you don't have
to memorize it, but you should be able to
figure it back out later. So what we have is the trace of
this thing being less than 0– so I'm going to give
you some options. Now of course, you can
always look at your notes and that is not
going to help you. You're not being graded
on your answers right now. I'm going to encourage you
to try and think about it and see if you can recapitulate
what this thing should be. So it's less than,
greater than– So for example you
can start to think about– you should be able
to write down some equation that you know is stable. And figure out, OK, what
conditions would it satisfy. That's a common, useful trick
to be able to do in life. To dredge these
things out of memory. Do you understand the question? I'm going to give you
30 seconds because it's well worth trying to figure
out what this rule should be.

So from the reading
you know there's some rule about the trace
and the determinant. And indeed they
both have to be true in order– this is an and sign. May be an and sign. Do you need more time? It's OK if you haven't actually
been able to recapitulate this. But it's useful
to think about it. Should we go ahead
and vote just to see? I'm just curious where we are. Ready? Three, two, one. So we have a lots of B's. And can somebody give me
an example of a matrix A that really ought to be stable? Yeah? AUDIENCE: Negative identity. PROFESSOR: Negative
identity, OK. So if the matrix A is
something that's minus 1 here. 0, 0 minus 1. Then x and y are uncoupled.

X decays exponentially,
y decays exponentially. Now we can just directly
say, well, this, the trace is
definitely negative, and the determinant's positive. It's 1 minus 0. So that gets us here. Because I think there are
many situations in life that are like this, where you
know that there's some rule and you can't
remember what it is. You don't need to
actually remember, once you know that
there's the rule, then you can figure out
what it had to have been. This is not a proof. The proof is only a few lines. You can do it, but the point
is that this kind of situation comes up a lot.

It's useful to just
have simple things that you know what
the answer has to be, and then that allows you to
figure out where things were. And indeed what you'll
find is that for a two dimensional system,
this condition that the trace of the
matrix is less than 0 and the determinant
is larger, that is equivalent to the
statement that the real part of both eigenvalues
is less than 0. And there's the derivation is
simple, and it's in your notes. Are there any questions
about where we are right now? So what I want to just
say a few more things about the trajectories here. Can somebody explain
the notion of what's an intuitive statement
about the eigenvectors that you get out here? Why do we like eigenvectors? What are they useful for? AUDIENCE: Decoupling the
differential equations.

PROFESSOR: Right, so
they're decoupling, and there's a sense
that– I guess the way I like to think
about this is that you have a fixed point say, like this. Now if these things
are stable, that means the arrows are
all say coming in. Are these eigenvalues–
are they purely real, or are they complex? Reminder, somebody? Real and negative. There's no imaginary component. This is a stable fixed point
with real negative eigenvalues. Now the idea of
these eigenvectors is that these are
the two directions in which if you start out on one
of them, you'll stay on them.

And in general then you
can– any other trajectory you can decompose as a
combination of the pads on the two. The position as function
of time can always described as the sum
of the eigenvectors where you grow or
you shrink along each eigenvector exponentially. Now for this thing to be
stable, all these lambda I's are going to be negative. We have this property
that– that the dynamics of this matrix kind of keep
you along the direction of the eigenvector. What this is saying is that
if you start somewhere random, that is off one
the eigenvectors, then you can decompose
the trajectory along each. But I just want to
highlight that it's often useful to draw what these
things end up looking like. Now I want to make sure
I get this one correct. I'm a little bit worried I'm
going to do something funny. So here– we're going to say
this is v1 and this is v2. So this direction
is one eigenvector, this direction is the other. So let's imagine that the
trajectories look like this.

The question is which
eigenvalue is closer to 0? Is it A eigenvector v1, or B
eigenvector– So this is really lambda 1 verses lambda 2. 1 is closer to 0 than is 2. Which of the eigenvalues
is closer to 0? I'll give you 20 seconds to
think about this because it's useful to be able to
extract these things from the trajectories. Do you need more time? And it's fine if
you don't really understand how you can get to
this question from this figure, then go ahead and flash C, D, or
E, just so I know where we are. Let's vote. Ready? Three, two, one. All right, so there's
at least a majority are saying that it's
going to be lambda 2. Can somebody offer why that is? Yes? AUDIENCE: I sort of
see it as whichever one is bigger is
going to be more effective at squeezing
things toward the origin along that axis. PROFESSOR: OK and bigger– and
you're saying more negative, in this case, you're saying? Yes, right. So yeah that's right. So there's some notion
that lambda 1 has to be more negative
than lambda 2 because you first collapse along
the direction of eigenvector 1, and then you slowly come
in along the direction of eigenvector 2.

That's saying that you have
these two exponential decays, and the directional
on the eigenvector 1 is more rapid than the
directional on eigenvector 2. I think in many of these
cases in dynamical systems, differential equations,
it's hugely valuable to be able to draw
the trajectories. So in many cases,
what we're going to do over the course
of the semester is we're going to have a simple
pair of differential equations that are going to
be two proteins, or they're going to be
rabbits and foxes or whatever. So we're going to locate
where the fixed points are. And then we're
going to figure out the stabilities and
the eigenvectors. And then we can really
understand the entire dynamics of the system without
solving the full thing, without using a computer.

But just by figuring out
where the fixed points are, and then the dynamics
around there. Then you can
basically understand all the dynamics of the system. But you have to
make sure that you develop intuition of how systems
behave near their fixed points. Now this is a case where both
of the eigenvalues were real. But of course, if you
have complex eigenvalues, if you do the
calculations, you'll see that the
eigenvalues are going to be a complex
conjugates of each other. And there you get spirals. So trajectories
might look like so. So there are two qualitatively
different ways that the fixed point can be stable.

You can have spiraling through
a state of the fixed point, or you can come in via
these straight lines. Of course, the specific
trajectories in some cases, can be curved. And so if you look at a
particular trajectory, sometimes even these can
look a little bit spirally, so be careful. Those are the two
basic ways it works. There's a simple way
of getting a sense of the dynamics of a system,
as a function of the trace and the determinant. So what we have–
something here. If you go ahead and you do
the calculation, what you find is that the two eigenvalues are
going to be described by this. And then the basic dynamics
are going to be the following. So down here, you have a case
where the eigenvalues are real, and they're of opposite sign.

And in this case, is
that stable or unstable? Unstable. So in order for it to be
stable, all the eigenvalues have to have negative
real components. So if they have opposite
sign, one of them is going to be positive. So this is all
unstable down here. Over here you have a case
where the eigenvalues are real, greater than 0. So in this case,
the trajectories are also coming out somehow. All right, here, this is the
case where they are real, but now both less than 0. So this is again, this is
like what we drew here, where everything is stable coming in. And up here is where
we get the spirals. But over here, it's the
real part is less than 0. So this is where we have
the spirals coming in, whereas over here, the
real part is greater than 0 and we have spirals coming out. Oh, I'm sorry. Uh, that would have been useful. So this is in the trace of A
and this is in the determinant.

So these are the variety
of possible outcomes when you have a two
dimensional system that's already a linear two
dimensional system. And this is indeed
consistent with it, to have a stable
fixed point, you need to have the
trace less than 0, and the determinant
greater than 0. So it's in this quadrant. Now it's– finally it's useful
to– so let's just for now, stick with the linear system. And just imagine a case
where we have in this A– so remember it's A,
B, C, and D. So we saw one way in which this
thing could be stable. Was that if b and
c were both equal to 0, so there's no
cross interaction, but a and d were both
less than 0, then it's all trivially stable. Now the question
is, if something that looks like this, a
is equal to minus 2 and d is equal to plus 1. Questions is, could
such a system be stable? So I'm seeing some nods. And on the face of this,
you say, oh, that's a little bit surprising.

Because d being plus
1, what that's saying is that y on its
own is unstable. So if you start out
with no x and no y, and you add a little bit of y,
y starts growing exponentially. So y on its own is unstable,
but x is stable its own. And the trace
being less than 0– that's saying that there's
some sense in which if y is unstable on its own,
then x has to somehow be more stable than y is unstable. Because the sum of those things
still has to be negative. Now that was necessary,
but not sufficient, in order to have stability. Because we also need to have a
condition on the determinant. Now so the trace of a
here– that's minus 1. That's less than 0. OK, that's great. Now the determinant– now
it's going to be a times d. That's minus 2. But then we also have to
say minus b times c, right? And this thing has
to be greater than 0.

So what you see here
is that b and c– they have to have opposite signs
order for this thing to work, in order for the
origin to be stable. And the product has
to somehow be strong. Now this makes sense, because
of course, if b and c, or even for matter,
just one of them were 0, then it would be impossible
to get the stability. But for example, if we
have some situation where we have some x that's inhibiting
itself– that's a here– but then y is activating itself. So y here is somehow
on its own, unstable. Then what you need is you
need maybe something that looks like this. Some cross activation
and or repression. And what's interesting is
that it doesn't matter which of the two, b or c is negative.

You can get actually, the origin
to be stable in either way. But in this situation, you need
the direction of the regulation to be in opposite directions. And if you'd like,
you could then play with– you could
think for example about the directions of these
trajectories around the origin, and so forth. But it's useful, I think, to
play with the simplest toy systems that you can imagine,
just so that you can get a sense of what are the
basic ingredients that you need in order to get stability
in something like this.

Because once you start doing
the whole linearization around the fixed point,
and then calculating traces and determinants,
you're not going to have it. You're going to lose all
your intuition about that about things at that stage. So it's useful to make
sure that you nail things down in this context. We are out of time,
so I'll let you go..

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