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JEFF GORE: Today, what we want to do is focus more explicitly
on bacterial chemotaxis.
Of course, a discussion that we had
on Thursday about "Life at Low Reynolds Numbers"
is certainly very relevant to this question
of how bacteria are maybe able to find food
or what constraints they're faced in trying
to solve this problem.
Today, we're going to discuss in more depth
this idea of the biased random walk that runs and tumbles
that we talked about on Thursday, that allow bacteria
to swim towards attractants and away from repellents.
But in particular, there's something
that is rather subtle about the particular biased random walk
that bacteria implement.
Now, the way that you might kind of naively
imagine that this thing would work
is that you would just have this tumbling frequency be
a function of the concentration of the attractant, for example.
And that indeed would allow you to swim or to execute
a biased random walk towards, say, food sources,
towards attractants, but it would not
be effective over a very wide range of concentrations.
However, if you experimentally ask, how well can bacteria
swim towards attractants, it turns out
they can respond over five orders of magnitude
of concentration of these attractants, which is really
quite incredible, if you think about the engineering challenge
that these little one-micron cells are able to overcome.
And the basic way that they do this is they
implement what's essentially equivalent to integral feedback
in the context of engineering, where it turns out
that the steady-state tumbling frequency, so the frequency
of which they'd execute one of these
tumbles that randomizes their motion, that displays what's
known as perfect adaptation.
It's not a function of the concentration of-- if you have
a constant concentration of an attractant,
then it's not a function of that.
Of course, changes in concentrations it responds to,
but somehow, E. coli and many other microorganisms
are able to implement this very clever thing where
the steady-state frequency of tumbling, of changing
direction, is somehow not a function
of the overall attractant concentration.
Now, that's already an interesting, I think,
phenomenon.
And in some ways, you can think of it
as some kind of robustness, because there
is some way in which this tumbling frequency is
robust against constant changes in the level of attractants.
And it's that aspect, I think, that
makes this example both rather subtle but also
quite confusing, because it's the phenomenon
of perfect adaptation that already
has some aspects of robustness, but it's
this phenomenon of perfect adaptation
that is robust against changes in concentrations of proteins,
for example, the concentration of the protein [? key ?]
R that we're going to talk about.
So I think that this is in some ways maybe
the prettiest example that we have
of this principle of robustness, but I
think it's also in some ways the most
tricky to wrap your head around, because it's sort of robustness
of a robustness in some ways.
All right.
So our goal for today is going to be to make sure
that we understand the challenge that E. coli are facing
and then to try to understand the genetic circuit
that they use in order to overcome this challenge.
So what I'm going to do for the next hour and 15 minutes
is we're going to leave this network up on the board
so that any time that you're confused about what
is R, B, Z, Y, so forth, you can kind of
look up here and remind yourself what this thing is.
But hopefully, the reading from last night
will help you in following what's going on.
There are a fair number of letters, I will admit it.
So first, I just want to make sure
that we're all kind of remembering
the basic phenomenon that we're trying
to study, which is this idea of consecutive runs and tumbles.
So this random walk is really composed of what you might call
or what we do call runs, where the bacteria goes sort
of straight and then tumbles, which
they randomize their motion.
So this is runs and tumbles, where bacteria, they
go semi-straight for of order a second-- order one second--
and this is the run.
Then, they have this tumbling that
might last about one second, so a 0.1 second tumble,
over which the motion is sort of random in this axis
of the orientation.
And then, they kind of go in a new direction
and then tumble again and then new direction and so forth.
So it's runs followed by tumbles.
Now, the thing that changes depending
on whether the cells are sensing that they're
moving up or down, say, an attractant gradient
is the frequency of those tumbles, i.e.
how long are the runs?
I said that they're around one second,
but this will vary depending upon whether bacteria
sense that things are getting better
or things are getting worse.
Can somebody remind us how the bacteria actually execute
one of these tumble motions?
Yes?
AUDIENCE: They have this other motors-- they have mini-motors
in their flagella and they usually
all spin one way, which I think is counterclockwise.
JEFF GORE: That's right.
I always have trouble-- so the runs are indeed
when these things are rotating counterclockwise.
AUDIENCE: Yeah.
And one of them can decide to switch
and start turning clockwise.
And what it does, it just sort of-- I don't know.
I just imagine it throws the whole motor off.
JEFF GORE: Exactly.
And the flagella in the context of-- so we have our E. coli.
The velocity is of order, say, 30 microns per second.
Now, they have these flagella that are actually
distributed-- the motors are actually distributed
across the entire cell.
So it's not just on the back.
But then, these individual filaments kind of
come together towards the back and then
they have this corkscrew shape.
When these things are all rotating
in the counterclockwise direction,
that corresponds to a run, when there's directed motion.
But then if one of them goes clockwise,
the bundle falls apart.
And indeed, there are some very nice movies online
where you can see that when they're going clockwise,
you see that the flagella are kind of doing crazy things
and that causes this thing to kind of tumble
in a random orientation.
So it randomizes its direction.
Now, one thing that we did not talk
about in the context of this low Reynolds number motion
was the question of how hard you would
have to pull in order to get a cell
to go 30 microns per second.
Now, remember, we're in the low Reynolds number regime,
where the force you have to apply
is in general proportional to this velocity.
And this thing is very much not a sphere,
but if it were a sphere, for a sphere, the proportionality
constant is this 6 pi eta av, where a here is again
the radius.
Now, regardless of the precise shape,
you'll always get for the low Reynolds regime something that
looks vaguely like this, where this is going to the longest
linear dimension and this here depends on the precise geometry
of the object.
And so just-- it's useful to try to get
a sense of order of magnitude.
How large are these forces that we're talking about?
In particular, how hard would you
have to pull a cell in order to get
it to go 30 microns per second?
Now, of course then, there's a question.
What's roughly the scale that we should be thinking about?
Right now, a newton is the scale that's for macroscopic objects.
So you'd say, OK, probably not going to be up that large.
Then, on the other scale, we can think
about the forces that can be applied or exerted
by individual molecular motors.
Has anybody studied this at all?
AUDIENCE: [INAUDIBLE]
JEFF GORE: Right, order of picanewtons.
So force for a molecular motor is going
to be of order picanewtons.
Now, there's a simple way to kind of
get at why this might actually be, because what is it
that powers many of these molecular motors?
AUDIENCE: [INAUDIBLE]
JEFF GORE: Yeah, so it's often-- you're
thinking of maybe the motors that operate in a membrane.
And then, there might be some difference
in a gradient, so a difference in concentration
across that membrane and that can actually
power, for example, rotation.
We're going to talk a little bit more
about this in the context of the flagella motor,
because this is one of only a few known rotary motors.
I'm just going to write this down so that I don't forget
to say something about it.
But in this case--
AUDIENCE: ATP?
JEFF GORE: ATP, right?
So in many cases, what happens is that you have,
for example, kinesin or myosin.
These are motors that walk along some track and in many cases,
each step corresponds to a single ATP being hydrolyzed.
And so given that, there's some maximal force
that you can imagine such a motor applying.
So for example-- and what might it depend upon?
AUDIENCE: The energy of ATP hydrolysis [INAUDIBLE]
JEFF GORE: Ah, very good.
So we have delta G for say, ATP.
And there's going to be a length scale, right?
And what's going to be the relevant length
scale in the case of, for example, a motor like kinesin?
AUDIENCE: The flagella, I guess?
JEFF GORE: Well, kinesin is not walking along the flagella,
but in this case, kinesin is walking along microtubules,
for example.
So you're right that we need a length scale,
because this is going to have some units of say, picanewton
nanometers-- oops, that's an Nm--
and we want to get a picanewton so we're going to need a length
scale to divide by, right?
AUDIENCE: So how long you move each time?
JEFF GORE: That's right.
It's kind of the distance that this thing's moving, right?
So indeed, what happens in the case of kinesin
is that they take steps with what are called "heads"
but we can think of them as "feet" if we want.
So each step is in this case eight nanometers.
For example, delta L for kinesin is equal to eight nanometers.
You don't need to know this, but just it's
useful to have some sense of scale.
Now, of course, delta G of ATP, this
depends upon the concentrations of the reactants,
of the products, and so forth.
But this thing might be of order 100 picanewton nanometers.
Now, depending on conditions, maybe it's
70, but of order here.
And indeed, what this tells us is
that just based on what we've said right now, even
without building any fancy microscopes to measure
how much force these things can apply,
you can see that the maximum force it could possibly apply
would be something on the order of delta G divided
by the length that it's pulling.
Otherwise, you could make a perpetual motion machine.
So this-- well, maybe what we'll do
if we want to make our math simpler,
we could say this is around 80.
It's of order there.
But the point is that this is of order 10 picanewtons,
so it's kind of picanewton scale.
All right.
So given this, it's interesting to ask, how hard would you
have to pull?
So how big is a kinesin model going to be, this protein?
AUDIENCE: About eight nanometers.
JEFF GORE: Yeah, right, about eight nanometers.
Of course, it's kind of long, spindly things
like my skinny legs, right, but let's
indeed going to be around that scale,
whereas the cell is one micron wide, multiple microns long, so
much, much larger.
We'll draw a little-- here's a kinesin in there.
It's even smaller than that.
Kinesin's small.
Now, the question is, how much force would you
have to apply in order to pull an E. coli at 30 microns
a second?
Force to pull E. coli at the speed
that it's actually observed to go-- now,
you're unlikely to be able to do the calculation right here.
It's actually a simple calculation.
We could do it in a moment, but it's
useful to just kind of imagine what scale might it be.
All right, this is a way of-- we can go up as high as we want.
You can continue it on and this is all in units of picanewtons.
Once again, it's useful to make guesses
about your intuition on these things
just so you have some notion of where we might be.
And of course, in this case, nobody wants to guess anything,
because they feel-- all right, I'll
give you 10 seconds just to make your best guess.
Somebody's forcing you to do it.
In this case, it's me.
No reason that you actually should necessarily get this.
Pulling through water.
AUDIENCE: At 30 microns.
JEFF GORE: At 30 microns per second.
AUDIENCE: Could you give us the viscosity
of water in picanewtons?
JEFF GORE: Yeah, it's 10 to the minus nine
in the units of picanewton nanometer second something.
AUDIENCE: Great.
JEFF GORE: Right.
Now, I've given you enough time.
You should have just been able-- now,
I'm going to be disappointed if you don't get it right.
No.
Let's just see where we are.
Ready?
Three, two, one.
OK, so I'd say that we have a bunch of-- I'd say
the mean is kind of a C-D-ish.
We got some A-B's.
All right, so I'd say it's somewhere in the sense of,
yeah, maybe it's 100,000.
So maybe if you had 100 of these kinesins pulling you along,
then you'd be able to go 30 microns a second, although you
have to be careful because kinesin can't actually
go that fast.
But these are details, right?
No, there's a reasonable question.
How hard would you have to pull.
Of course, if we want, we could actually just
do the equations here, right?
The force, it's around 20.
In units of picanewton nanometer seconds,
I told you that the eta is around 10 to the minus 9.
All right.
And the radius, what radius do you want to use?
AUDIENCE: A micron.
JEFF GORE: A micron.
So do we write a 1 here?
Do we write what?
10 to the 3 because I already told you
that this was in units where everything is picanewtons
nanometers seconds.
And velocity, this is again per second.
So this is three times 10 to the 4, because that's how many
nanometers per second.
And this gets us to around-- that's 10 to the seven.
We're going to have to divide by 100,
so this is 60 divided by 100-- so less than a picanewton.
So this is really pretty surprising.
And again, this is a reflection of the wonders of low Reynolds
numbers kind of behavior.
So it's somewhere here.
Yes?
AUDIENCE: So since there are many flagella,
is it telling us-- [INAUDIBLE] is it telling us
that they're very inefficient compared to the [INAUDIBLE]
JEFF GORE: Yeah, it's a good question, right?
So I think there are several ways.
And this is one aspect maybe of what
we read about in the "Life at Low Reynolds Numbers."
The comparison was to a Datsun in Saudi Arabia.
Of course, I think that this was written in the '70s where
that meant something.
But I think that a Datsun, I guess, is a fuel efficient car?
Well, that was my inference from that.
And Saudi Arabia has a lot of oil, still true.
And the saying was that it might only be 1% efficient or so,
right?
But if you don't have to apply very much force there,
then maybe that's not a disaster.
The swimming speed is not actually
a super strong function of the number of flagella
that are there, as far as my understanding.
So I'm actually a little bit-- there are cases
where you do get somewhat higher speed with more flagella,
although I have to confess, just from the geometry
of the multiple flagella I find totally mystifying, because I
would've thought they would get tangled up,
because each of them goes in and they form a corkscrew.
I don't know.
It just doesn't seem that it should work.
But it does, so I'm not going to argue that it doesn't.
AUDIENCE: Probably it has to do with the low Reynolds number.
So you imagine swimming, sort of flying around.
JEFF GORE: Yeah, no, my concern is not even
a matter of the low Reynolds number or not.
It's just a matter of they're each spinning
and then-- this is something really I've never understood,
so I typically avoid talking about it.
I don't know.
I just feel like there's something wrong.
But in any case, you don't need to pull very hard in order
to get even a micron-size object to go rather fast.
Yes?
AUDIENCE: [INAUDIBLE]
JEFF GORE: Oh, I'm not aware of it.
This thing can actually go-- it's 10 microns long,
so this thing is huge, actually.
But I--
AUDIENCE: It's probably just a repeat.
Or even if it is a repeat, you can
[INAUDIBLE] I'd just be surprised
if it wasn't, since it seems like that.
JEFF GORE: Yeah, so I don't know,
but I'm not even sure if that would necessarily
address my concern, which is it seems like they
have to go through each other.
But it's not true, so I don't want to make the argument too
strongly.
Did you have a question?
AUDIENCE: So I was just thinking about this [INAUDIBLE]
Might it be possible that they, if they are all sync up
their rotations, then it doesn't matter how tangled they are?
JEFF GORE: Yeah, no, even-- even if they're all
spinning at the exact same rate, I still
feel that they should get tangled,
but it's a feeling that is apparently not true.
So I don't want to--
AUDIENCE: But if they're always tumbling, maybe the time to--
JEFF GORE: No, no, no, wait-- so it's not-- no,
the tumble is not due to my supposed mechanism.
They're spinning at something like 100 hertz.
The motors are spinning at something like 100 hertz,
right?
So in principle, if they were going to get tangled,
they would've gotten tangled.
AUDIENCE: Seems [INAUDIBLE]
JEFF GORE: That's right, that's right.
I do want to just come back and say
one thing about this question of a rotary motor,
because it's a fascinating question-- oh yes, go ahead.
AUDIENCE: Sorry, I just had a quick question.
What was the units of eta again?
JEFF GORE: Oh, this is the thing.
The units of eta, I always just go back here
and I say, OK, well, it's the units
of a force divided by unit of an area-- or sorry, of a radius,
unit of a velocity, and then I have to plug everything in.
And then, I have to find an equation with a force other
than this equation, because if you put this equation back in,
then you don't get anywhere.
These are useful.
So I always have to actually figure it out fresh each time.
So it turns out that these are indeed rotary motors.
And I think that this was actually the first-- as far
as I'm aware, it's the first example of a rotary motor
in biology that had been demonstrated,
which is a fascinating idea because in human engineering,
rotary motors are everywhere.
Yet somehow, if you look at living things,
they don't seem to have rotary motors.
Now, anybody want to suggest why rotary motors might be rare
in biology, in life?
AUDIENCE: Rotary is just that it rotates, right,
not in terms of a mechanism [INAUDIBLE] because there's
also [INAUDIBLE]
JEFF GORE: Well, I guess I'm thinking of something that
can continuously rotate, right?
It's true that if you look at-- we have ball socket joints
and I can rotate them a little bit, but certainly,
I can't go-- there are limits, right?
And this is a pretty striking difference
between human engineering and biological engineering.
Yeah?
AUDIENCE: If you tried to connect anything across this,
it would be more [INAUDIBLE]
JEFF GORE: That's right.
AUDIENCE: [INAUDIBLE]
JEFF GORE: That's right.
The problem is you just can't have
anything connected across it.
Otherwise, you really do get tangled, right?
Now, the question then is, well, why
is it that it's possible to do that here then?
AUDIENCE: Because we're at such a small scale
that we're not thinking of connecting anything else.
We're individual molecules, anyway.
There's nothing that can go through that.
JEFF GORE: Yeah, it's interesting.
And this is how I think about it that well,
at the molecular scale, you just have some rotary something
and it doesn't have to be attached
via, say, covalent bonds or whatnot.
And you can then force something to rotate.
Of course, it's a little bit funny because this issue
about connecting across this rotation, that
should be a problem for human engineering, as well,
but somehow, we do get around it.
I guess what I would say is I don't have
any comments, except for this was then quite a surprise when
it was discovered that this was a rotor motor, just because we
had not seen any in other macroscopic living things.
So then, it was quite exciting to see it
in the case of the flagellar motor.
And in your book, you'll see that there's
a very nice kind of EM reconstruction of this motor
and you can see how it's hinged and it's
powered by proton gradients that kind of cause this thing
to rotate.
It's really a beautiful, amazing thing.
Does anybody know any other examples of rotary motors?
That's right.
The other really famous one is the F0F1 ATP synthase,
which is, again, something that's across a membrane
and has, once again, a really beautiful structure where
it has this circular thing in the membrane that rotates,
responds to, again, a proton gradient,
and then that drives rotation of this other part
of the protein, F1.
And then, that makes ATP.
And that motor is amazing also because it's reversible,
in that the cell can also burn ATP and drive a proton
gradient.
And indeed, in some single-molecule experiments,
they've even done something where
they attached a little magnetic particle onto this F1
and then rotated themselves.
And they showed that they could actually make ATP.
The efficiency was maybe not great, something like that,
because they're rotating a macroscopic magnet
and then they're making ATPs there.
But it's a pretty remarkable aspect
of these molecular motors.
Now in this class, we're not going
to say anything more about molecular motors,
but I just wanted to mention a few things about them
because they're really fun, beautiful things
and there are other classes that are
at MIT that may give you an opportunity to think about them
some more.
Any question about where we are now?
AUDIENCE: [INAUDIBLE]
JEFF GORE: The common motor?
AUDIENCE: Yeah, [INAUDIBLE]
JEFF GORE: Oh.
Well, I'd say that the other class of motors
that are seen a lot in the context
of these molecular motors are motors that
travel along linear tracks.
So there's kinesin that walks along microtubules.
There are various myosins that walk on actin.
And then, of course, DNA and RNA polymerase,
we don't normally think of them as motors
but indeed, they take an energy fuel
and then they have to walk along the template
as they make either the DNA or the RNA.
So I'd say there are many, many examples of molecular motors
that convert chemical energy into mechanical force
and motion, particularly along one-dimensional tracks.
So those, I think, are the most well-studied examples.
Now, one way to study this run and tumble motion
is, of course, to actually apply a gradient
and then watch the cells as they swim it.
And that has been done.
A lot's been learned from that kind of assay,
but it turns out that there are two other assays that maybe
allowed for more controlled analysis
of this chemotaxis kind of response.
So let's just say-- so studying chemotaxis.
The most obvious thing is to apply a gradient
and then watch.
And indeed, the classic assays where
you add a little pipette with an attractant or a repellent
and watch the bacteria swim toward or away,
that demonstrates there is indeed chemotaxis.
But it's a little bit difficult to quantify
that process in many cases.
What are the assays that maybe you've read about recently?
In [? Yuri's ?] book, how is it that they actually analyzed
this perfect adaptation?
Yes?
AUDIENCE: So you can bind the flagella on the slide and then
[INAUDIBLE]
JEFF GORE: So one thing that you can do is you can--
I may put it here.
All right.
So you can kind of attach the cell to a slide.
And typically attach it to the slide by what?
This isn't it.
And does it matter where you attach the cell to the slide?
AUDIENCE: By the flagella.
JEFF GORE: Yeah, so you typically
have attach it via this hook that's at the end,
so by some part of the flagella-- to the slide
by the flagella, we'll say.
Flagellum?
Flagella-- whatever.
And the nice thing there is that as the cell is doing its thing
and it's spinning either clockwise or counterclockwise,
you can directly visualize that because the whole cell is
moving.
And the cell is both the thing that is doing the work
and processing the signals and everything,
but it's also your marker for what the state
of this little hook is, right?
This is a wonderfully quantitative assay where you
can get high time resolution.
It's easy to do the image analysis.
And then, how do you typically get this cell to change its,
for example, tumbling frequency?
AUDIENCE: Put an attractant.
JEFF GORE: Right, so you can add an attractant.
And indeed, I just wanted to separate this a little bit,
because you can-- even without doing this--
this is kind of the next order step-- you can just
add an attractant and mix, because you don't
want the spatial patterns.
But a nice thing here, this is a gradient in space
and then you can watch the bacteria swim.
But you can also have the gradient in time.
And the cells can't tell the difference.
The nice thing here is that you can then just
add the attractant, mix, and then
you just watch all the cells as they're going.
You don't need to try to follow them or whatnot,
but you can just look to see how the tumbling frequency changes
over time.
And of course, you would typically use this trick
together with this in order to study perfect adaptation and so
forth.
If you collect this sort of data and you plot the tumbling
frequency as a function of time, what you might see
is that it starts out at one per second.
Now, if at this time, I add an attractant,
does the tumbling frequency go up or down?
And we're going to do a verbal answer.
Ready?
Three, two, one.
AUDIENCE: Down.
JEFF GORE: Down.
And that makes sense because the cells
think that they're moving up an attractant gradient.
So over a very short time scale, tumbling frequency goes down.
But then, over a time scale of minutes--
it might be 5, 10 minutes-- that tumbling frequency
goes back to where it started.
I just want to-- this varies, but this
could be order of 10 minutes.
AUDIENCE: Why is it so long?
JEFF GORE: Yeah.
AUDIENCE: 10 minutes is a very long time.
JEFF GORE: Yeah, it's a long time scale.
And here's a question.
Is it because this is the time that it
takes to make new protein?
New protein synthesis, we'll say.
AUDIENCE: You're asking if that's the time
or if that's the reason why is it long?
JEFF GORE: I'm saying is this the explanation for why this
is 10 minutes, because the cell has
to go make new protein in order to do this?
So the cell is presumably going to be making some new proteins,
but is this what you would really
describe as being the causative agent of this thing taking 10,
20 minutes?
Ready?
Three, two, one.
All right.
So I'd say most people are agreeing that actually,
yes, it is.
The answer is no.
This is not.
It may be the case that the cell is making new protein,
but this is not what's setting the time scale there.
Is--
AUDIENCE: More [INAUDIBLE]
JEFF GORE: Right.
You agreed that that was the answer, but you didn't-- well,
so I would say there are several ways you can think about this,
but we're going to go through the model that is supported
by, I think, a fair amount of experimental evidence.
But the key feature there is that in this model,
it works even if all the protein concentrations are
constant over time.
So everything that's happening in this network
is happening as a result of changes
of the states of the protein.
So proteins are either getting methylated or phosphorylated
and these-- right.
But then, of course, it's the question of,
why is it 10 minutes instead of 10 seconds or a minute?
AUDIENCE: You need a [INAUDIBLE] in there
that's on the order of minutes, like 100 minutes, which
is very slow.
Is there--
JEFF GORE: Yeah, well, their one question is, this
is what is called the adaptation time.
Now, is this a robust feature in this model
or in the cells, for that matter?
We'll just add verbal, yes or no.
Ready?
Three, two, one,
AUDIENCE: No.
JEFF GORE: No.
Now, and what that means is that indeed,
different kind of versions of this network
will have different times.
And there is data looking at variation in this
between different cells.
And I guess I don't have a clear feeling for what
would be optimal, in the sense of allowing optimal climbing up
of an attractant gradient.
This thing has to be much longer than the typical times
for a tumble.
Otherwise, it's not even-- well, we
wouldn't have been able to measure it, I guess.
But I agree that it could have been one minute
and I wouldn't have batted an eye,
in the sense that I don't have any feeling for why
it had to have been this or something else.
But somebody who actually studies this
might be able to give a better answer.
All right.
AUDIENCE: But I guess-- sorry, just one last thing.
My question is not why it's 10 minutes.
It's not an evolutionary question,
like why is it useful, right?
It's just somehow, [INAUDIBLE]
JEFF GORE: Yeah I think that there
are different ways of looking at this
and then depending on how you look at it,
you either feel surprised or not.
So it's a little-- now on the other hand,
if you had added a repellent, then
the tumbling frequency would actually go up
and then come back down.
But the key, key thing in this system
that we want to focus our attention on
is the fact that it comes back to where it started.
So it's the fact that I can draw this dashed line that
is this perfect adaptation.
And so what we want to do is understand
where this phenomenon of perfect adaptation
comes from and maybe why it is robust to changes
in, for example, the concentrations of some
of these proteins.
Now, let's just make sure that we're
all on the same page in terms of what was known
about this chemotaxis network.
And I think it's worth mentioning, perhaps, here
that the whole series of studies of bacterial chemotaxis
going back to Howard Berg and company
and then later, the studies that in robustness
that Uri Alon and Stan Leibler and Naama Barkai did.
I think they represent really just a wonderfully beautiful
exploration at the interface between physics and biology.
I think you could teach an entire course just
on bacterial chemotaxis and you could hit pretty much
all the major themes in biophysics over the last 40
years.
It's really amazing to me.
I myself have not done any work in the field,
but from afar, I've really just admired the beauty
of all these studies.
And because you go back to Howard Berg and Purcell
and they're thinking about how simple physics can inform
the challenges that bacteria are facing
and how cells are actually able to do a biased random walk
and get anywhere, limits on sensing both concentrations
and gradients-- and then later, the studies
that they're this topic of robustness,
it's really a wonderful example where
Naama Barkai, when she was a postdoc at Stan Leibler,
they had published a Nature paper in maybe '97 basically
saying, this idea of robustness is really important in biology
and in order to have a robust response of perfect adaptation,
a model has to have these features.
So there's no experiments there, but their model
was guided by previous observations
that people had made.
And then, two years later, Uri, when
he was a postdoc in Stan's lab again,
did kind of the experimental confirmation
of the model, where he went in and he
controlled the concentration of chi R
and showed that thee key predictions of the model, i.e.
the perfect adaptation, would be robust to the concentration
but that the tumbling frequency and the adaptation time, they
would not be robust to chi R concentrations.
They would move in a way predicted
by the model and all that.
It's really amazing that it all kind of holds together,
because in many cases, we do the modeling kind of post facto,
right?
And then, it's kind of explaining our results.
But the case where a model is really useful
is when it makes new predictions that get you
to go make new measurements.
And this is a situation where there
are an infinite number of experiments that you could do,
but only some of them will actually provide you
a deep insight into the mechanisms that
are going on in the system.
And I think this was a real case where the models made
some really clear predictions and that allowed, in this case,
Uri to go and make the strains that allowed him to test
the predictions of the model.
And I think it's really amazing that it all kind of works.
All right.
Now, all of the letters that you see up there,
they're not actually-- well, first of all, the letters
are somehow real.
This is the real names of the protein components
in the chemotaxis network in E. coli
and largely in other organisms.
But in each case, there's a chi that comes in front, right?
So R corresponds to chi R, for example.
And then, there's chi B, chi W, chi A.
And these were all identified by genetics, so then
by researchers looking for mutants that
were defective in chemotaxis.
I don't know what happened to C, D, E, F, G,
because it does seem like we got the first part of the alphabet
and the last part of the alphabet.
I don't know.
All right.
Now, the basic idea in the system
is that you have chi W/A that we often here will refer to as X
just for simplicity.
Now, these are proteins in the membrane,
so they have a binding kind of pocket outside
that will allow binding of attractants or repellents.
There might be say, five different kinds
of these receptor complexes that can sense,
that bind at different rates, different kinds of attractants,
repellents, and then the signal is somehow integrated.
Now, this could be either an attractant or a repellant.
The distinction between these two
is that we get different levels of methylation
onto chi W. Now in this lecture, we're
only going to be talking about the methylated state
versus the unmethylated.
But in reality, depending on the receptor complex they have,
they might have four or five different methylation sites.
And this ability to switch between different methylation
states is really at the heart of the phenomenon
of robust perfect adaptation.
The base feature in it, though, is that chi A can phosphorylate
chi Y and chi Y will then go and yield the output,
which is in this case, increased tumbling frequency.
But there are lots of other bells and whistles
that you can see on here, right?
So of course, it's not just that this phosphorylated
chi Y just kind of comes off on its own,
but rather it's actually done by chi Z.
So there's a constant cycling here
and again, there's a constant cycling here
where the methyl groups are taken off and then put back on.
So if you look at this, you really
do feel that it's rather wasteful,
because there's a huge number of these futile cycles going on.
Chi Y is always kind of being phosphorylated and then
dephosphorylated and this is all going to be costly to the cell.
So you can imagine then the only reason
it's there is because it's doing something useful.
Now, there was a comment in the chapter
about what is the rate limiting step in all this.
And can somebody remember what it was?
Yeah?
AUDIENCE: Is it methylation?
JEFF GORE: So methylation, there is a sense
that that is actually the longest time scale,
but I guess because the methylation
is what results in this thing coming
in the perfect adaptation over this 10 minutes.
I guess what I meant-- so yeah, that is the longest time scale.
But I guess what I was thinking about in rate limiting
is the sense of when the cell finds itself
in a new environment, it changes its state
over a much shorter time scale.
So this thing I drew is almost vertical, right?
So this question is, if the cell finds itself
in a new environment suddenly and then it just really
wants to tumble-- so you find yourself in a crappy bar,
how long does it take for you to get out?
Now, what's going to be rate limiting there?
Yeah?
AUDIENCE: Phospho-- phosphorylation.
JEFF GORE: Phosphorylation, yeah,
although it turns out that's not the rate limiting step.
And this actually comes back a little bit to something
that we talked about in the first part of the class
that in these transcription networks,
the characteristic timescale is what?
AUDIENCE: G.
JEFF GORE: Right.
So the characteristic timescale in the case of transcription
networks is the time it takes for you
to change concentrations of proteins, which is kind of cell
generation time or if you have active degradation,
you might be able to make it faster, whereas all of this
is happening rather quickly, say,
maybe 1/10 tenth of a second.
And a lot of these kinds of processes, binding, unbinding,
and actually even the interactions
between the proteins can take place
even maybe faster than that.
So the actual rate limiting step for when the cell finds itself
in a bad environment for it to start tumbling
is actually due to diffusion.
And that's diffusion of what?
AUDIENCE: Y protein.
JEFF GORE: Yeah, diffusion of the phosphorylated Y, right?
And that's because we have the cells here.
They find themselves in the bad environment.
They rapidly bind the repellent or they quickly
phosphorylate chi Y. But then, Y is
going to be formed at one of the poles,
because actually, there's actually clustering
of these receptors at the poles of a cell
and incidentally, we're not going to talk about that here,
but I think there's strong experimental and theoretical
evidence that this actually increases the sensitivity.
And indeed, people have used simple Ising type models
to try to understand how the coupling between the binding
of repellents on what's essentially almost
like a crystalline array of receptors
can allow the array to better than you
build it to as an individual.
We're not going to get into that here, but in any case,
there's receptors at the poles-- I
don't know if it's both poles or one pole,
but one of the poles of the other--
and that's where chi Y is phosphorylated.
But then, you can see that these flagella are distributed
all around the cell.
So you have to diffuse from say, the pole
to the site of the flagella motor
in order to cause it to go clockwise and then cause
a tumble.
Yeah?
AUDIENCE: So do you need only one motor?
JEFF GORE: Yeah, so you actually only
need one motor to get the tumbling.
And so then, you may not have to diffuse all the way
to the other end, but-- and of course, diffusion is random.
And we all know that 0.1 seconds is around the time
that it takes for a protein-sized object
to diffuse across the volume of a bacterial cell.
We did that calculation a couple weeks ago.
So 0.1 seconds is indeed-- so the rate limiting step
is indeed this step right here, which
is diffusion of chi Y, the phosphorylated version of it.
All right.
Now, we may not get too much into the models
here, because you did read about them.
But I will just kind of sketch out
sort of what you might call the fine-tuned model and then
the key assumption that goes into this robust model.
Now, what both models assume and indeed
what was known from previous work
was that chi R is present at small number.
So chi R, there might be around 100 proteins in the cell.
And what does this mean about the activity of chi R?
What is the other word for it?
It doesn't actually have to quite mean it,
but what is it that-- how--
AUDIENCE: [INAUDIBLE]
JEFF GORE: What's that?
AUDIENCE: [? High ?] saturation.
JEFF GORE: So yeah, something is high.
And what is typically assumed is that chi R acts at saturation,
chi R. But what do we mean by "saturation" in these models?
AUDIENCE: Maximum.
JEFF GORE: Right.
Does it mean that if we add more chi
R than the rate of methylation, it doesn't increase?
And I should have put a little methyl group here.
There are multiple things you might
mean by "acts at saturation."
And in the models that you read about last night,
is what it means that if we increase the number of chi R
that it doesn't change the rate of methylation?
Yes or no?
Ready?
Three, two, one.
AUDIENCE: No.
JEFF GORE: No.
And what they mean is something rather different, which
is that if we plot or if we calculate the change
in the concentration of the methylated X--
now, we're calling this whole thing Xm, methylated X,
and this is just X. The assumption is
that we have X is being methylated
at a rate that is-- there's no Michaelis-Menten term.
If we were to write this as it not being saturated,
what is it acting on?
AUDIENCE: [INAUDIBLE]
JEFF GORE: Yeah, it's not the methylated x.
So indeed, if we were to write this,
we're assuming that this is at saturation, as they say.
But any time that you see something like this,
you have to ask, well, what would
be the alternative, right?
And the alternative would be to include
a term that looks kind of like just X over some K plus X,
because R is acting on the unmethylated X.
We're not including that.
What that means is that we're assuming that this thing is
saturated, that the concentration of X to be acted
on is significantly larger than the Michaelis constant there.
And of course, this is related to the amount of R,
because as we get more and more R,
then eventually, we'll remove some of this X
and then we'll get into the non-saturated regime.
So these two statements are related, but not
the same thing.
Now, there's also some rate that the phosphorylated version of B
removes the methyl groups and that's indeed just going
to be this Michaelis constant.
And this is going to be for the fine-tuned model.
The robust model looks very similar,
but this is the simplest kind of manifestation of this model.
Now, once we're writing this down,
it's useful to make sure that we can
keep track of what's actually happening
over the course of perfect adaptation.
So now, let's imagine first that an attractant arrives.
That's going to change the activity of X.
And when we say "activity," what we mean
is the rate that's it's going to phosphorylate both B and Y.
So let's just make sure that we know this direction.
So we add an attractant.
We'll say "add."
What does this do?
Does it make activity go up or down?
I'll give you 15 seconds to make sure
that you kind of understand the workings of this network.
This is activity of X, this complex X. All right.
Do you need more time?
All right.
Let's see where we are.
Ready?
Three, two, one.
All right.
So we got a majority of the group
is saying that it should go down.
Well, let's just follow the logic.
So we imagine an attractant binding.
If the activity goes down, that means
that we get less of the phosphorylated chi
Y. That means we get less propensity to tumbling,
which means that we keep on going further.
All right, that sounds reasonable.
Any questions about that logic?
Yes?
AUDIENCE: [INAUDIBLE]
JEFF GORE: Right.
So we haven't said anything yet about BM.
That's what going to happen next.
And that's actually the slow time scale.
So what we had in here is that we
were kind of at this steady-state tumbling
frequency of one per second.
We add an attractant.
The tumbling frequency goes down because of what we just said.
But now, there's going to be this longer time scale
process whereby we get recovery, where
we come back to this steady-state tumbling
frequency.
And that's going to involve action on chi B.
All right.
So what happens is that the attractant causes less
of the phosphorylated chi Y. But it's also
going to cause less of the phosphorylated chi B. Now, that
means that we're going to-- and remember,
the phosphorylated chi B is what's
removing the methyl groups.
So if we have less flux going to the left,
but we have the same-- at that moment,
we don't have any change in the flux going to the right.
So chi R is still acting on the same unmethylated X's that it
was operating on before.
So it's the same flux to the right, less flux to the left.
So there's a net accumulation of the methylated receptor, which
we are calling X. So the key thing here is that--
and it's this methylated receptor that
has more activity.
In this model, this unmethylated version actually
doesn't have any activity.
So then, if we get more of the methylated X, then over time,
we get a buildup of the methylated X
and that causes the activity to come back up.
Now, of course, there's a question
of-- I just said that it comes back up,
but I didn't say that it comes back up
exactly to its original tumbling frequency.
I didn't say that it necessarily displays perfect adaptation.
And that's because in this model,
the perfect adaptation arises as a result of what
we call fine-tuning, because it only
happens if all of the parameters are just so.
In Uri's book, he describes a typical condition
where that would be the case.
And the problem is that you can always
fine-tune for some concentrations of everything,
chi R or chi this, chi that.
But then if the concentrations change,
then you're no longer fine-tuned correctly.
You were fine-tuned for a different world
and now you're not-- and that's the definition of being
fine-tuned is that if things change,
you're no longer fine-tuned.
You're just finely off-tune, right?
So this is the problem with the fine-tuned model
is that you can get it right for a given--
for given concentrations of everything,
you can always find what numbers.
This is VB and then there's going
to be-- you can talk about the activities of XM given
the previous attractant concentration and so on and so
forth, but it's not going to be fine-tuned
if you change anything else, like
if you change concentration of R, for example.
And I'm happy to go through the math
and so forth maybe after class if anybody's curious,
but it's really precisely what Uri did,
so maybe I won't get into it now.
But the question then is, well, how
is it that you might change this model in order
to make it robust?
And the change is somehow surprisingly simple,
which is that what you want is chi B, instead of just
acting on any old methylated X, you
want it to act only on the methylated X that
is in sort of in what we would call this active state,
where it's actually able to catalyze
either of those reactions.
So the notion is that if you have this methylated X, then
it's kind of over a very fast time scale.
It's switching between what we call an "active" state
and some inactive one.
And it's really only in the active versions
that chi B is able to act on and remove the methyl group.
And this is on the one hand a clever thing
that allows you to implement this integral feedback.
On the other hand, it's a little bit of like pulling a bunny out
of a hat, because you feel like, well, these things may
be happening over microsecond time scales.
It's hard to know exactly-- how would you actually
experimentally confirm this is precisely what's going on?
And I think that here, it's a little bit subtle because you
could maybe show that indeed the rate of this demethylation
is proportional to the activity here,
but you don't necessarily have access
to all of the molecular dynamics that are taking place
over microsecond time scales.
So I think that you can do measurements that give you
confidence that this is maybe what's going on,
but you can't quite 100% nail it because
of the nature of these molecular fluctuations.
So the idea there is that if we say
that there's this rapid shuttling
between the so-called "active" and "inactive"
methylated guys-- so this is indicating that it's
what we call "active," able to catalyze this and this--
then this ends up being equivalent
to integral feedback, where you'll always
get perfect adaptation.
Yeah?
AUDIENCE: When you put [INAUDIBLE]
JEFF GORE: So the idea is that we imagine that we're this
receptor X. It's chi W, chi A. Now whether it's-- let's say
it's bound to something.
It still bound to an attractant, right?
The way that the attractant influences its activity--
and it has to influence its activity if it's
going to do anything-- what we assume
is the way that it's doing it is that it's changing
the sort of fraction of time that I'm
in some active conformation where I can actually do work,
versus the inactive conformation where I'm taking a break.
So when you get the attractant and you
spend more of your time in this active conformation
where you're, in this case, phosphorylating proteins,
and that's sort of the mechanism through which
an attractant or a repellent or whatnot
actually transmits its signal.
And indeed, it has to do something.
Nothing that we're discussing would work at all
if we don't allow the signal to be transmitted somehow
through this receptor.
So there is this sense that this activity
has to be a function of the things out there.
And the assumption that goes into the perfect adaptation
is really that the rate of demethylation
is proportional to that kind of active fraction.
And then, you can argue about how discrete these entities
have to be in order for the mechanism to work and so forth,
but certainly, there has to be some way
that binding to an attractant leads to what we decided
was less activity.
Yeah?
AUDIENCE: Just a quick question.
So what's the relationship between activity
and methylation?
Was it [INAUDIBLE]
JEFF GORE: So the idea is that we're typically maybe assuming
that the unmethylated guy has no activity,
so it doesn't do any of this phosphorylation,
whereas the methylated guy has some activity.
And you can characterize it by some rate of activity
or some fraction of the time that it
is in this active state that is doing something.
Now, the question-- and indeed this
ends up-- well, you can see here that in this model,
because you're directly acting on the active XM, then
the steady-state activity you can
get from just setting this equal to zero and it's some number.
But then, the question is, how long
does it take to come back to that steady state?
And that's where we get differences
as a function of concentration of chi R,
because what's happening always is
that we have some kind of cycle here where
chi B is removing the methyl groups,
groups chi R is adding them back.
So you can imagine that if you have more chi
R than a steady state, you get more-- when you're moving right
by a steady state, you have to have
the same moving to the left, because at steady state,
it's equal.
So the more chi R you have, the faster this thing
is going around.
And that means that the more chi R that you have,
the more rapidly that you'll get this perfect adaptation.
So the experiment that Uri did that I think is very nice
is he directly modulated the amount of chi R
and he looked at this adaptation time.
And he found this kind of came down,
whereas if you look at the steady-state tumbling
frequency, this came up, whereas the degree
of perfect adaptation, say the ratio or the error
in this thing, perfect adaptation
was always kind of correct, in the sense it always came back
to its original value.
AUDIENCE: Sorry.
You say the [INAUDIBLE] it only phosphorylates
chi B when an attractant [INAUDIBLE]
Y [INAUDIBLE] attractant.
JEFF GORE: No.
So the attractant or repellent can
be combined to either the methylated
or the non-methylated, right?
AUDIENCE: But [INAUDIBLE] only [INAUDIBLE]
phosphorylation of chi B when [INAUDIBLE]
JEFF GORE: No.
So you're talking about-- oh, OK.
So it's really that the methylated state can
phosphorylate either chi B or chi Y,
but this is regardless whether an attractant is bound or not.
The attractant will influence the rate or the activity
that this happens.
And given this model, you can see that you have more chi R,
then at steady state, you're going to have more activity.
More activity corresponds to more phosphorylated chi
y and more tumbling, so an increase
in the tumbling frequency.
Yes?
AUDIENCE: I guess it's not clear to me
where the ligand concentration actually come in.
JEFF GORE: Where which concentration?
AUDIENCE: The ligand concentration.
JEFF GORE: Oh, OK, yeah.
AUDIENCE: Because that's what [INAUDIBLE]
JEFF GORE: Yeah, right.
Yeah, so the idea here is that if we start out
without any attractant-- so first of all,
so let's imagine we're at steady state.
There's no attractant or little attractant
now, where of course, the fluxes to the left and the right
are the same.
So there's some methylated, some not.
AUDIENCE: No, but I agree the mechanics of just in the actual
to write an equation for the ligands
to come in like [INAUDIBLE].
JEFF GORE: Right.
So the idea is that when you bind an attractant,
that's going to change the activity of the methylated.
So it's going to change, for example,
the fraction that are active.
AUDIENCE: And decreases the activity.
JEFF GORE: And it decreases the activity, right.
AUDIENCE: And some sort of signal--
JEFF GORE: Yeah, there's some-- right.
And of course, we haven't specified what that function,
but the idea is that it leads to a rapid decrease in activity,
which corresponds to a rapid decrease in this fraction that
are active XM star.
Does that make sense?
So the last thing I wanted to do is say something
about what this means for individuality.
In particular, let's imagine that we
have a clonal population of bacteria.
And the question is, in what ways
will they be similar or different?
So now, we can just imagine an experiment
where I take a population of cells with exactly
the same genetic code and I go and I measure, for example,
the tumbling frequency across this population.
So we can talk about-- we measure f1, f2, f3, fn--
so these are the tumbling frequencies across n cells.
This is n we'll say genetically identical cells.
The question is, will we get the same tumbling frequency
or should these things be the same?
And if not, why not?
So let's just do our little votes, all right?
We have should they be the same or should they be different.
All right.
Do you understand the question?
Let's vote.
Ready?
Three, two, one.
All right.
Well, OK, so at least the majority of people
are saying they should be different.
And why might that be, somebody?
AUDIENCE: We might have [INAUDIBLE]
JEFF GORE: Right.
For example, we might have variation
in the concentration of chi R. Now,
this variation is sort of a natural result
of just fluctuating this or that, right?
But it's a little bit similar to the experiment Uri did,
where he actually in this case has actually
put chi R under the control of an inducible promoter, where
he could just add IPDG and drive expression
and then just measure the mean of these things
across the population.
But even if you try to have everything be the same,
you'll have some variation, which
we found for small numbers of proteins could be large.
So that variation will transmit itself
into variations in both the adaptation time
and the tumbling frequency.
And this is indeed was observed as early as 1976.
So there's a paper in Nature in 1976
by Jim Spudich and Dan Koshland.
So Spudich went on to study a number
of these molecular motors.
In particular, he studied many of these myosins.
But he wrote this classic paper in '76 called "Non-genetic
individuality, chance in the single cell."
What he did is they looked at many different individual cells
using a few of those techniques that I told you about.
And what they found is that some of the cells
seemed to be what they called kind of "twitchy"
and some of them seemed to be more relaxed.
So the twitchy cells are the guys that had a larger tumbling
frequency.
So they wouldn't swim as far as the others.
So they'd swim just a little bit and then
they'd change their mind, swim a little bit, change their mind,
whereas other cells had much longer kind of runs,
even though they're all nominally identical.
Now, can somebody say the time scale
over which we would expect that personality to persist?
AUDIENCE: Cell generation time.
JEFF GORE: Yeah, cell generation time.
And why is that?
AUDIENCE: [INAUDIBLE]
JEFF GORE: Yeah, right.
So if this is chi R, over time, it's going to do something.
And so the typical autocorrelation time
should be kind of the cell generation time.
So what we call the typical time should be rather generational.
So indeed, this is the same as humans.
We have a well-defined personality
and then we pass on some of our personality to our kids.
The typical time scale is a generation.
So I'm teaching you guys super useful things in this class.
I don't want anybody saying anything else.
But I think that this is a neat example of how
different cells can have what you might describe
as being somehow different personalities,
but it arises for a very particular reason
because this information is being transmitted
through this network.
Now, despite the fact that they all
have different chi R concentrations--
so they might have different tumbling frequencies
and so forth, but they should all
be able to display this phenomenon
of perfect adaptation.
So what we then see is that from the experiments
and from some simple models, you can
get insight into how it is that this phenomenon
of perfect adaptation might be robust to changes
in the concentrations of different things, particularly
chi R in this case.
And chi R is the dominant source of error
because it's present in such small numbers.
Now, in the last two minutes or so,
I just want to remind everybody about the other context
in which we studied robustness, because it's
a much simpler example and it helps
to clarify what we mean by it.
Does anybody remember the other context
that we have talked about robustness?
AUDIENCE: [INAUDIBLE]
JEFF GORE: Negative autoregulation someone
said, maybe?
AUDIENCE: Yes.
JEFF GORE: OK, good, all right.
So if we have some protein that is negatively
autoregulating itself, then this adds robustness.
And this is because we can say, all right,
this is the degradation term, alpha x, the production term.
And the limit of being perfect, sharp, negative autoregulation
looks like this.
So this is the production/degradation.
And here, this might be beta.
This is a k.
Then, what we can talk about is how
the steady-state concentration of x here is going to be k.
And this thing is robust to variations in some things,
but not other things.
For example, it's robust to changes in alpha.
That changes the slope.
It's robust to changes in beta, because that just
brings it up and down, but it's not robust to changes in k.
So I think that any time that if you are confused
about robustness, in particular thinking about robustness
in the context of chemotaxis gets
you confused because of perfect adaptation being confusing,
I think it's always good to come back and think
about this because this is the most clear case where you can
talk about-- this is the robustness
of the steady-state concentration
of some protein against variations in alpha and beta,
but not against k.
So it reminds you that it's not that this level of x
is robust against everything, but it's
robust against some things.
And you can make sense of which things it
should be robust against, which things not.
So with that, I think we're going to quit.
I just want to remind everybody that none
of the work that we're talking about here
in the context of chemotaxis and the genetic network
will appear on the exam, but we may
have a problem on low Reynolds number flow,
maybe something Stokes drag.
Diffusion might make an appearance.
Good luck on the exam.
I'll see you guys on Tuesday.