WEBVTT

00:00:00.060 --> 00:00:02.500
The following content is
provided under a Creative

00:00:02.500 --> 00:00:04.019
Commons license.

00:00:04.019 --> 00:00:06.360
Your support will help
MIT OpenCourseWare

00:00:06.360 --> 00:00:10.730
continue to offer high quality
educational resources for free.

00:00:10.730 --> 00:00:13.330
To make a donation or
view additional materials

00:00:13.330 --> 00:00:17.236
from hundreds of MIT courses,
visit MIT OpenCourseWare

00:00:17.236 --> 00:00:17.861
at ocw.mit.edu.

00:00:20.654 --> 00:00:22.070
PROFESSOR: Today,
what we're going

00:00:22.070 --> 00:00:25.130
to do is, first, introduce
this idea of oscillations.

00:00:25.130 --> 00:00:26.740
It might be useful.

00:00:26.740 --> 00:00:29.980
A fair amount of the day will
be spent discussing this paper

00:00:29.980 --> 00:00:32.119
by Michael Elowitz
and Stan Leibler

00:00:32.119 --> 00:00:35.285
that you read over the last few
days, which was the first, kind

00:00:35.285 --> 00:00:37.682
of, experimental
demonstration that you

00:00:37.682 --> 00:00:39.140
could take these
random components,

00:00:39.140 --> 00:00:42.270
put them together, and generate
oscillatory gene networks.

00:00:42.270 --> 00:00:45.320
And finally, it's
likely we're going

00:00:45.320 --> 00:00:46.960
to run out of time around here.

00:00:46.960 --> 00:00:49.780
But if we have time, we'll talk
about other oscillator designs.

00:00:49.780 --> 00:00:51.980
In particular, these
relaxation oscillators

00:00:51.980 --> 00:00:55.930
that are both
robust and tunable.

00:00:55.930 --> 00:00:59.130
It's likely we're going to
discuss this on Tuesday.

00:01:04.360 --> 00:01:06.840
All right, so I want to
start by just thinking

00:01:06.840 --> 00:01:11.026
about other oscillator designs.

00:01:11.026 --> 00:01:12.400
But before we get
into that, it's

00:01:12.400 --> 00:01:14.730
worth just asking a question.

00:01:14.730 --> 00:01:18.250
Why is it that we might want
to design an oscillator?

00:01:18.250 --> 00:01:20.480
What do we like
about oscillations?

00:01:23.229 --> 00:01:24.520
Does anybody like oscillations?

00:01:24.520 --> 00:01:26.820
And if so, why?

00:01:26.820 --> 00:01:27.320
Yes.

00:01:27.320 --> 00:01:28.763
AUDIENCE: You can make clocks.

00:01:28.763 --> 00:01:30.089
And clocks are really--

00:01:30.089 --> 00:01:30.880
PROFESSOR: Perfect.

00:01:30.880 --> 00:01:31.504
Yes, all right.

00:01:31.504 --> 00:01:34.760
So two part answer.

00:01:34.760 --> 00:01:36.060
You can make clocks.

00:01:36.060 --> 00:01:37.111
And clocks are useful.

00:01:37.111 --> 00:01:37.610
All right.

00:01:37.610 --> 00:01:41.310
OK, so this is a fine statement.

00:01:41.310 --> 00:01:43.890
So oscillators are, kind of,
the basis for time keeping.

00:01:43.890 --> 00:01:50.180
And indeed, classic ideas of
clocks, like a pendulum clock.

00:01:50.180 --> 00:01:52.670
The idea is that
you have this thing.

00:01:52.670 --> 00:01:53.850
It's going back and forth.

00:01:53.850 --> 00:01:55.420
And each time that
it goes, it let

00:01:55.420 --> 00:01:57.630
allows some winding
mechanism to move.

00:01:57.630 --> 00:02:00.230
And that's what
the clock is based.

00:02:00.230 --> 00:02:02.782
And even modern clocks
are based on some sort

00:02:02.782 --> 00:02:03.740
of oscillatory dynamic.

00:02:03.740 --> 00:02:05.910
It might be a very
high frequency.

00:02:05.910 --> 00:02:09.699
But in any case, the
basic idea of oscillations

00:02:09.699 --> 00:02:14.110
as a mechanism for time keeping
is why we really care about it.

00:02:14.110 --> 00:02:16.520
Of course, just from a
dynamical systems perspective,

00:02:16.520 --> 00:02:19.930
we also like oscillations
because they're

00:02:19.930 --> 00:02:22.230
interesting from a
dynamical standpoint.

00:02:22.230 --> 00:02:24.230
And therefore, we'd
like to know how

00:02:24.230 --> 00:02:27.180
we might be able to make them.

00:02:27.180 --> 00:02:31.860
Can anybody offer an example
of an oscillator in a G network

00:02:31.860 --> 00:02:34.280
in real life?

00:02:34.280 --> 00:02:34.780
Yes.

00:02:34.780 --> 00:02:35.940
AUDIENCE: Circadian.

00:02:35.940 --> 00:02:37.070
PROFESSOR: The
circadian oscillator.

00:02:37.070 --> 00:02:37.611
That's right.

00:02:37.611 --> 00:02:40.636
So the idea there
is that there's

00:02:40.636 --> 00:02:43.130
a G network within
many organizations

00:02:43.130 --> 00:02:45.480
that actually keeps
track of the daily cycle

00:02:45.480 --> 00:02:47.910
and, indeed, is entrained
by the daily cycle.

00:02:47.910 --> 00:02:50.545
So of course, the
day, night cycle.

00:02:50.545 --> 00:02:51.420
That's an oscillator.

00:02:51.420 --> 00:02:52.086
It's on its own.

00:02:52.086 --> 00:02:53.550
And it goes without us, as well.

00:02:53.550 --> 00:02:55.199
But it's often
useful for organisms

00:02:55.199 --> 00:02:57.490
to be able to keep track of
where in the course the day

00:02:57.490 --> 00:02:58.440
it might be.

00:02:58.440 --> 00:03:01.792
And the amount of light
that the organism is getting

00:03:01.792 --> 00:03:03.250
at this particular
moment might not

00:03:03.250 --> 00:03:05.690
be a faithful indicator
of how much light there

00:03:05.690 --> 00:03:09.290
will be available in an
hour because it could just

00:03:09.290 --> 00:03:11.719
be that there's a cloud
crossing in front of the sun.

00:03:11.719 --> 00:03:13.260
And you don't want--
as an organism--

00:03:13.260 --> 00:03:13.990
to think that it's night.

00:03:13.990 --> 00:03:15.740
And then, you shut
down all that machinery

00:03:15.740 --> 00:03:17.510
because, after
that cloud passes,

00:03:17.510 --> 00:03:19.135
you want to be able
to get going again.

00:03:19.135 --> 00:03:21.150
So it's often useful
for an organism

00:03:21.150 --> 00:03:28.070
to know where in the morning,
night, evening cycle one is.

00:03:28.070 --> 00:03:31.237
And we will not be talking
too much about the circadian

00:03:31.237 --> 00:03:32.320
oscillators in this class.

00:03:32.320 --> 00:03:34.607
Although, I would say to
the degree of your interest

00:03:34.607 --> 00:03:36.660
in oscillations, I
strongly encourage

00:03:36.660 --> 00:03:39.820
you to look up that literature
because it's really beautiful.

00:03:39.820 --> 00:03:42.030
In particular, in some
of these oscillators,

00:03:42.030 --> 00:03:45.390
it's been demonstrating you can
get the oscillations in vitro.

00:03:45.390 --> 00:03:46.920
I.e, outside of the cell.

00:03:46.920 --> 00:03:49.810
Even in the absence of any
gene expression, in some cases,

00:03:49.810 --> 00:03:52.560
you can still get oscillations
of just those protein

00:03:52.560 --> 00:03:53.980
components in a test tube.

00:03:53.980 --> 00:03:55.720
This was quite a
shocking discovery

00:03:55.720 --> 00:03:59.370
when it was first published.

00:03:59.370 --> 00:04:02.910
But we want to start out
with some simpler ones.

00:04:02.910 --> 00:04:04.690
In particular, I want
to start by thinking

00:04:04.690 --> 00:04:08.050
about auto repression.

00:04:08.050 --> 00:04:10.710
So if you have an auto
regulatory loop where

00:04:10.710 --> 00:04:16.240
some gene is repressing
itself, the question

00:04:16.240 --> 00:04:19.380
is does this thing oscillate.

00:04:19.380 --> 00:04:22.300
And indeed, it's
reasonable that it

00:04:22.300 --> 00:04:25.960
might because we can
construct a verbal argument.

00:04:25.960 --> 00:04:28.400
Starts out high.

00:04:28.400 --> 00:04:30.490
Then, it should
repress itself so you

00:04:30.490 --> 00:04:32.740
get less new x being made.

00:04:32.740 --> 00:04:35.440
So the concentration falls.

00:04:35.440 --> 00:04:39.830
So maybe I'll give you
a plot to add to it.

00:04:39.830 --> 00:04:42.000
Concentration of x is
a function of time.

00:04:42.000 --> 00:04:44.370
You can imagine just
starting somewhere high.

00:04:44.370 --> 00:04:46.590
That means it's a
repressing expression.

00:04:46.590 --> 00:04:48.210
So it's going to fall.

00:04:48.210 --> 00:04:50.560
But then, once it falls too
much, then all of a sudden,

00:04:50.560 --> 00:04:53.350
OK, well we're not
repressing ourselves anymore.

00:04:53.350 --> 00:04:56.600
So maybe then we
get more expression.

00:04:56.600 --> 00:04:58.370
More of this x is being made.

00:04:58.370 --> 00:05:00.260
So it should come back up.

00:05:00.260 --> 00:05:02.500
And then, now we're
back where we started.

00:05:02.500 --> 00:05:06.561
So this is a totally
reasonable statement.

00:05:06.561 --> 00:05:07.060
Yes?

00:05:07.060 --> 00:05:10.357
AUDIENCE: [INAUDIBLE]?

00:05:10.357 --> 00:05:11.810
PROFESSOR: Well I don't know.

00:05:11.810 --> 00:05:14.660
I mean, I didn't introduce
any damping in here.

00:05:14.660 --> 00:05:16.437
The amplitude is
the same everywhere.

00:05:16.437 --> 00:05:18.520
AUDIENCE: So you're saying
that you could actually

00:05:18.520 --> 00:05:19.444
have something--

00:05:19.444 --> 00:05:24.467
PROFESSOR: Well I guess what
I'm really trying to say

00:05:24.467 --> 00:05:26.800
is that just because you can
construct a verbal argument

00:05:26.800 --> 00:05:29.870
that something happens does not
mean that a particular equation

00:05:29.870 --> 00:05:31.420
is going to do that.

00:05:31.420 --> 00:05:32.870
Part of the value
of equations is

00:05:32.870 --> 00:05:35.580
that they force you to
be explicit about all

00:05:35.580 --> 00:05:37.092
the assumptions
that you're making.

00:05:37.092 --> 00:05:38.550
And then what you're
going to do is

00:05:38.550 --> 00:05:40.850
you're going to ask,
well, a given equation is

00:05:40.850 --> 00:05:43.717
a mathematical manifestation of
the assumptions you're making.

00:05:43.717 --> 00:05:45.800
And then, you're going to
ask does that oscillate.

00:05:45.800 --> 00:05:46.570
Yes/no?

00:05:46.570 --> 00:05:47.200
And then you're going
to say, OK, well

00:05:47.200 --> 00:05:48.741
what would we need
to change in order

00:05:48.741 --> 00:05:50.380
to introduce oscillations?

00:05:50.380 --> 00:05:53.660
And I'll just-- OK.

00:05:53.660 --> 00:05:56.230
So this is definitely
an oscillation.

00:05:56.230 --> 00:05:58.630
The question is, should
you find this argument

00:05:58.630 --> 00:06:00.030
I just gave you convincing?

00:06:00.030 --> 00:06:04.910
And what I'm, I guess, about
to say is that you shouldn't.

00:06:04.910 --> 00:06:08.900
But then, we need to be clear
about what's going on and why.

00:06:08.900 --> 00:06:11.640
And just because you can make
a verbal argument for something

00:06:11.640 --> 00:06:13.430
doesn't mean that
it actually exist.

00:06:13.430 --> 00:06:16.190
I mean, that's a guide
to how you might want

00:06:16.190 --> 00:06:19.450
to formalize your thinking.

00:06:19.450 --> 00:06:21.200
And in particular,
the simplest way

00:06:21.200 --> 00:06:24.540
to think about oscillations
that might be induced

00:06:24.540 --> 00:06:27.150
in this situation would
be to just say, all right,

00:06:27.150 --> 00:06:33.080
well the simplest model
we have for an auto

00:06:33.080 --> 00:06:35.510
regulatory loop that's
negative is we say,

00:06:35.510 --> 00:06:46.670
OK, well there's some alpha
1 plus protein and minus p.

00:06:46.670 --> 00:06:48.570
So this is, kind of,
the simplest equation

00:06:48.570 --> 00:06:51.460
you can write that captures
this idea that this protein

00:06:51.460 --> 00:06:56.000
p is negatively regulating
itself in a cooperative fashion

00:06:56.000 --> 00:06:57.930
maybe.

00:06:57.930 --> 00:07:02.321
Now it's already in a
non-dimensionalize version.

00:07:02.321 --> 00:07:02.820
Right?

00:07:02.820 --> 00:07:06.400
And what you can see is
that, within this realm,

00:07:06.400 --> 00:07:09.150
there are only two things
that can possibly be changing.

00:07:09.150 --> 00:07:11.930
There's how cooperative
that repression

00:07:11.930 --> 00:07:16.530
is-- n-- and then, the
strength of the expression

00:07:16.530 --> 00:07:19.220
in the absence of repression.

00:07:19.220 --> 00:07:21.995
And as we discussed
on Tuesday, alpha

00:07:21.995 --> 00:07:26.680
is capturing all these
dynamics of the actual strength

00:07:26.680 --> 00:07:31.530
of expression together with
the lifetime of the protein

00:07:31.530 --> 00:07:33.425
together with the binding.

00:07:33.425 --> 00:07:35.500
You know, the
binding affinity k.

00:07:35.500 --> 00:07:40.230
So all those things get wrapped
up in this a or alpha rather.

00:07:40.230 --> 00:07:44.310
All right, so this is,
indeed, the simplest model

00:07:44.310 --> 00:07:47.500
you can write down to
describe such a negative auto

00:07:47.500 --> 00:07:49.360
regulatory loop.

00:07:49.360 --> 00:07:52.910
Now the question is now
that we've done this,

00:07:52.910 --> 00:07:58.020
we want to know does
this thing oscillate.

00:07:58.020 --> 00:08:02.590
And even without
analyzing this equation,

00:08:02.590 --> 00:08:04.940
there's something that's very
strong, which you can say.

00:08:08.140 --> 00:08:10.710
So in theory we're going to ask
is it possible for this thing

00:08:10.710 --> 00:08:13.320
to oscillate.

00:08:13.320 --> 00:08:14.190
All right.

00:08:14.190 --> 00:08:15.210
Possible.

00:08:15.210 --> 00:08:18.690
Your oscillations, we'll
say oscillations possible.

00:08:18.690 --> 00:08:22.712
And this time, referring
to mathematically possible.

00:08:22.712 --> 00:08:24.170
So maybe this thing
does oscillate.

00:08:24.170 --> 00:08:24.690
Maybe it doesn't.

00:08:24.690 --> 00:08:26.440
But in particular,
without analyzing it,

00:08:26.440 --> 00:08:29.815
is there anything that you
can say without analyzing it?

00:08:29.815 --> 00:08:31.440
We're just going to
say is it possible.

00:08:31.440 --> 00:08:33.940
Yes or no?

00:08:33.940 --> 00:08:35.679
If you say no, you
have to be prepared

00:08:35.679 --> 00:08:37.924
to give an argument for
why this thing is not

00:08:37.924 --> 00:08:38.799
allowed to oscillate.

00:08:38.799 --> 00:08:40.200
I'm talking about this equation.

00:08:43.320 --> 00:08:46.360
Do you don't you
understand the question

00:08:46.360 --> 00:08:48.785
that I'm trying to ask?

00:08:48.785 --> 00:08:50.410
And we haven't analyzed
this thing yet.

00:08:50.410 --> 00:08:53.210
But the question is,
even before analyzing it,

00:08:53.210 --> 00:08:56.320
can we say anything about
whether it's mathematically

00:08:56.320 --> 00:08:57.843
allowed to oscillate?

00:09:02.190 --> 00:09:04.300
I'll give you 10 seconds
to think about it.

00:09:12.690 --> 00:09:14.600
And if you say no, you
get to tell me why.

00:09:14.600 --> 00:09:15.660
All right, ready?

00:09:15.660 --> 00:09:17.530
Three, two, one.

00:09:20.400 --> 00:09:23.100
All right, so we got a
smattering of things.

00:09:23.100 --> 00:09:27.164
So I think this is not,
obviously, a priori.

00:09:27.164 --> 00:09:28.830
But it turns out that
it's not actually.

00:09:28.830 --> 00:09:30.610
It's just mathematically
impossible for this hing

00:09:30.610 --> 00:09:31.300
to oscillate.

00:09:31.300 --> 00:09:33.266
And can somebody say
why that might be?

00:09:33.266 --> 00:09:35.099
AUDIENCE: Because it
might be you could only

00:09:35.099 --> 00:09:36.918
have one value of p dot?

00:09:36.918 --> 00:09:38.120
PROFESSOR: Perfect OK.

00:09:38.120 --> 00:09:40.210
So for a given
value of p, there's

00:09:40.210 --> 00:09:43.410
only some value of p
dot that you can have.

00:09:43.410 --> 00:09:47.480
And in a particular-- so p here
is like a concentration of x.

00:09:47.480 --> 00:09:52.420
So I'm going to pick some
value, randomly, here of p.

00:09:52.420 --> 00:09:54.130
And what you're
pointing out is this

00:09:54.130 --> 00:09:57.300
is a differential
equation in which

00:09:57.300 --> 00:10:01.860
if you give me or I give you
the p, you can give me p dot.

00:10:01.860 --> 00:10:04.030
And there's a single
value p dot for each p.

00:10:06.780 --> 00:10:12.140
And in this oscillatory
scheme, is that statement true?

00:10:12.140 --> 00:10:12.730
No.

00:10:12.730 --> 00:10:14.890
What you can see is
that, over here, this is

00:10:14.890 --> 00:10:20.922
x slash p concentration of x.

00:10:20.922 --> 00:10:22.880
We're using p here because
we're about to start

00:10:22.880 --> 00:10:26.400
talking about mRNA So I want to
keep the notation consistent.

00:10:26.400 --> 00:10:29.297
What you see is that the
derivative here is negative.

00:10:29.297 --> 00:10:30.630
The derivative here is positive.

00:10:30.630 --> 00:10:32.410
Negative, positive.

00:10:32.410 --> 00:10:36.720
So any oscillation that you're
going to be able to imagine

00:10:36.720 --> 00:10:41.940
is going to have multiple
values for the derivative

00:10:41.940 --> 00:10:44.160
as a function of that
value just because you

00:10:44.160 --> 00:10:45.470
have to come back and forth.

00:10:45.470 --> 00:10:48.700
You have to cross that
point multiple times.

00:10:48.700 --> 00:10:52.277
So what this is saying is that
since this is a differential

00:10:52.277 --> 00:10:53.860
equation-- and it's
actually important

00:10:53.860 --> 00:10:55.150
that it's a differential
equation rather

00:10:55.150 --> 00:10:57.030
than a difference equation
where you have discrete values.

00:10:57.030 --> 00:10:59.290
But given that this is a
differential equation where

00:10:59.290 --> 00:11:02.410
time is taking little,
little, little steps

00:11:02.410 --> 00:11:05.180
and you have a single variable,
it just can't oscillate.

00:11:10.380 --> 00:11:13.120
So for example, if you're
talking about the oscillations

00:11:13.120 --> 00:11:15.420
the harmonic oscillator
the important thing there

00:11:15.420 --> 00:11:18.260
is a you have both the
position in the velocity

00:11:18.260 --> 00:11:20.660
see these two dynamical
variables that are interacting

00:11:20.660 --> 00:11:25.020
in some way because you have
momentum, in that case, that

00:11:25.020 --> 00:11:29.521
allows for the oscillations in
the case of a mass on a spring,

00:11:29.521 --> 00:11:30.020
for example.

00:11:32.740 --> 00:11:33.240
Question.

00:11:33.240 --> 00:11:34.734
AUDIENCE: I'm still
not understanding.

00:11:34.734 --> 00:11:36.234
So the value of p
can not oscillate?

00:11:38.718 --> 00:11:39.500
PROFESSOR: Right.

00:11:39.500 --> 00:11:43.610
So we're saying is that, right,
p simply cannot oscillate

00:11:43.610 --> 00:11:46.910
in this situation where we
have a differential equation

00:11:46.910 --> 00:11:52.790
describing p with-- if we just
have p dot as a function of p

00:11:52.790 --> 00:11:54.210
and we don't have
a second order.

00:11:54.210 --> 00:11:56.090
A p double dot, for example.

00:11:56.090 --> 00:12:02.640
So if we just have a single
derivative with respect to time

00:12:02.640 --> 00:12:04.850
and some function of p
over here, what that means

00:12:04.850 --> 00:12:10.264
is that, if p is specified,
then p dot is specified.

00:12:10.264 --> 00:12:12.430
And that's inconsistent
with any sort of oscillation

00:12:12.430 --> 00:12:15.340
because any oscillation's going
to require that, at this is

00:12:15.340 --> 00:12:17.580
given value of p-- this
concentration of p--

00:12:17.580 --> 00:12:20.020
in this case, the
concentration's going down.

00:12:20.020 --> 00:12:21.890
Here, it's going up.

00:12:21.890 --> 00:12:24.290
So here, this is--
from that standpoint--

00:12:24.290 --> 00:12:26.775
a multi valued function.

00:12:26.775 --> 00:12:27.274
OK?

00:12:30.417 --> 00:12:32.125
And other questions
about this statement?

00:12:35.840 --> 00:12:38.810
Even if I just written down some
other function of p over here,

00:12:38.810 --> 00:12:40.268
this statement
would still be true.

00:12:43.050 --> 00:12:47.860
And it's valuable to be able to
have some intuition about what

00:12:47.860 --> 00:12:51.180
are the essential ingredients
to get this sort of oscillation.

00:12:51.180 --> 00:12:53.710
And for simple harmonic
motion, right there we

00:12:53.710 --> 00:12:56.110
have the second derivative,
first derivative,

00:12:56.110 --> 00:13:00.190
and that's what allows
oscillations there.

00:13:00.190 --> 00:13:03.240
OK, so we can, maybe, write
down a more complicated model

00:13:03.240 --> 00:13:04.490
of a negative auto regulation.

00:13:04.490 --> 00:13:06.560
And then, try to
ask the same thing.

00:13:06.560 --> 00:13:08.095
Might this new model oscillate?

00:13:15.220 --> 00:13:17.230
And this looks a little
bit more complicated.

00:13:17.230 --> 00:13:19.063
But we just have to be
a little bit careful.

00:13:19.063 --> 00:13:22.589
All right, so this is, again,
negative auto-regulation.

00:13:22.589 --> 00:13:24.130
What we're going to
do is we're going

00:13:24.130 --> 00:13:31.034
to explicitly think about the
concentration of the mRNA.

00:13:31.034 --> 00:13:32.440
OK.

00:13:32.440 --> 00:13:37.050
And that's just because when a
gene is initially transcribed,

00:13:37.050 --> 00:13:38.080
it first makes mRNA.

00:13:38.080 --> 00:13:40.820
And then, the mRNA is
translated into protein.

00:13:40.820 --> 00:13:41.870
Right?

00:13:41.870 --> 00:13:43.670
So what we can do
is we can write down

00:13:43.670 --> 00:13:44.961
something that looks like this.

00:13:44.961 --> 00:13:48.660
M dot derivative of m
with respect to time.

00:13:48.660 --> 00:14:07.060
It's going to be--
all right, so this

00:14:07.060 --> 00:14:09.740
is the concentration of mRNA.

00:14:09.740 --> 00:14:12.266
And p is the
concentration of protein.

00:14:12.266 --> 00:14:12.766
OK?

00:14:27.644 --> 00:14:29.060
All right, and
what you can see is

00:14:29.060 --> 00:14:32.810
that the protein
is now repressing

00:14:32.810 --> 00:14:35.910
expression of the mRNA.

00:14:35.910 --> 00:14:38.161
mRNA is being degraded.

00:14:38.161 --> 00:14:40.160
But then, down here, this
is a little bit funny.

00:14:40.160 --> 00:14:43.750
But what you can see is
that, if you have more mRNA,

00:14:43.750 --> 00:14:46.380
then that's going to lead to
the production of protein.

00:14:46.380 --> 00:14:49.450
Yet, we also have a degradation
term for the protein.

00:14:54.557 --> 00:14:55.539
Yes?

00:14:55.539 --> 00:15:00.449
AUDIENCE: Why are we
multiplying the degradation

00:15:00.449 --> 00:15:04.377
rate of the protein
times some beta, as well?

00:15:04.377 --> 00:15:07.190
PROFESSOR: That's
a good question.

00:15:07.190 --> 00:15:10.670
OK, you're wondering why
we've pulled out this beta.

00:15:10.670 --> 00:15:12.980
In particular-- right.

00:15:12.980 --> 00:15:13.530
OK, perfect.

00:15:13.530 --> 00:15:13.810
OK, yeah.

00:15:13.810 --> 00:15:14.930
This is very important.

00:15:14.930 --> 00:15:16.670
And actually, this
gets in-- once again--

00:15:16.670 --> 00:15:19.940
to this question of these
non-dimensional versions

00:15:19.940 --> 00:15:21.580
of equations.

00:15:21.580 --> 00:15:22.550
Mathematically, simple.

00:15:22.550 --> 00:15:24.140
Biologically, very complicated.

00:15:24.140 --> 00:15:29.070
Well, first of all, what is that
we've used as our unit of time

00:15:29.070 --> 00:15:30.230
in these equations?

00:15:32.991 --> 00:15:34.116
AUDIENCE: The life of mRNA.

00:15:34.116 --> 00:15:34.850
PROFESSOR: Right.

00:15:34.850 --> 00:15:38.055
So it's based on the
lifetime of the mRNA

00:15:38.055 --> 00:15:39.680
because we can see
that there's nothing

00:15:39.680 --> 00:15:41.060
sitting in front of this m.

00:15:41.060 --> 00:15:43.100
And if we want to, then,
allow for a difference

00:15:43.100 --> 00:15:45.150
in the lifetime
mRNA and protein,

00:15:45.150 --> 00:15:48.030
then we have to introduce
some other thing, which

00:15:48.030 --> 00:15:50.340
we're calling beta.

00:15:50.340 --> 00:15:54.890
So beta is the ratio of--
well which one's more stable?

00:15:54.890 --> 00:15:56.307
mRNA or protein,
often, typically?

00:15:56.307 --> 00:15:57.056
AUDIENCE: Protein.

00:15:57.056 --> 00:15:59.080
PROFESSOR: Proteins are,
typically, more stable.

00:15:59.080 --> 00:16:00.580
So does that mean
that beta should

00:16:00.580 --> 00:16:03.346
be larger or smaller than 1?

00:16:03.346 --> 00:16:05.470
OK, I'm going to let you
guys think about this just

00:16:05.470 --> 00:16:08.900
make sure we're all--
OK, so the question

00:16:08.900 --> 00:16:12.380
is beta, A, greater than 1?

00:16:12.380 --> 00:16:13.930
Typically, much greater.

00:16:13.930 --> 00:16:20.805
Or is it, B, much less than
1, given what we just said?

00:16:20.805 --> 00:16:22.680
All right, you think
about it for 10 seconds.

00:16:28.572 --> 00:16:31.030
All right.

00:16:31.030 --> 00:16:33.520
Are you ready?

00:16:33.520 --> 00:16:36.670
Three, two, one.

00:16:36.670 --> 00:16:39.040
All right, so most
people are saying B.

00:16:39.040 --> 00:16:42.390
So indeed, beta should
be much less than 1.

00:16:42.390 --> 00:16:48.700
And that's because beta is
the ratio of the lifetime.

00:16:48.700 --> 00:16:52.150
So you can see, if
beta gets larger,

00:16:52.150 --> 00:16:56.020
that increases the degradation
rate of the protein.

00:17:00.180 --> 00:17:02.370
What do I want to say?

00:17:02.370 --> 00:17:05.400
So beta is the ratio
of the lifetime

00:17:05.400 --> 00:17:08.149
in the mRNA through the
lifetime of the protein.

00:17:08.149 --> 00:17:08.649
Yes?

00:17:08.649 --> 00:17:09.802
AUDIENCE: So I get
why we have to--

00:17:09.802 --> 00:17:10.619
PROFESSOR: Yeah.

00:17:10.619 --> 00:17:11.069
No, I understand.

00:17:11.069 --> 00:17:11.760
No, I understand you.

00:17:11.760 --> 00:17:12.968
I'm getting to your question.

00:17:12.968 --> 00:17:14.930
First, we have to
make since of this

00:17:14.930 --> 00:17:18.149
because the next thing
is actually even weirder.

00:17:18.149 --> 00:17:19.690
But I just want to
be clear that beta

00:17:19.690 --> 00:17:28.170
is defined as the
lifetime of mRNA

00:17:28.170 --> 00:17:30.800
over the lifetime
of the protein.

00:17:30.800 --> 00:17:35.030
What's interesting
is, actually, there's

00:17:35.030 --> 00:17:38.190
a typo or mistake in the
elements paper, actually.

00:17:38.190 --> 00:17:47.032
So if you look at
figure 1B or so-- yeah,

00:17:47.032 --> 00:17:47.990
so figure 1B, actually.

00:17:47.990 --> 00:17:50.890
It says that beta is the protein
lifetime divided by the mRNA

00:17:50.890 --> 00:17:51.570
lifetime.

00:17:51.570 --> 00:17:55.339
So you can correct
that, if you like.

00:17:55.339 --> 00:17:57.880
So beta's is the mRNA divided
by the lifetime of the protein.

00:18:04.940 --> 00:18:06.500
OK, so I think
that we understand

00:18:06.500 --> 00:18:07.458
why that term is there.

00:18:07.458 --> 00:18:09.910
But the weird
thing is that we're

00:18:09.910 --> 00:18:12.300
doing p minus m over here.

00:18:12.300 --> 00:18:13.830
Right?

00:18:13.830 --> 00:18:18.410
And it feels, somehow, that
that can't be possible.

00:18:18.410 --> 00:18:22.350
You know, that it shouldn't
be beta times m over here

00:18:22.350 --> 00:18:25.590
because it feels like
it's under determined.

00:18:25.590 --> 00:18:27.370
Right?

00:18:27.370 --> 00:18:28.920
OK.

00:18:28.920 --> 00:18:32.630
So it's possible
I just screwed up.

00:18:32.630 --> 00:18:38.520
But does anybody want to
defend my equation here?

00:18:38.520 --> 00:18:41.282
How might it be possible that
this makes any sense that you

00:18:41.282 --> 00:18:43.720
can just have the one beta
here that you pull out,

00:18:43.720 --> 00:18:45.530
and it's just p
minus m over here?

00:18:45.530 --> 00:18:48.076
AUDIENCE: I think it's an
assumption of the model where

00:18:48.076 --> 00:18:51.426
they choose the lifetime
of the protein and the mRNA

00:18:51.426 --> 00:18:52.470
to be similar.

00:18:52.470 --> 00:18:54.400
PROFESSOR: Well no
because, actually, we

00:18:54.400 --> 00:18:56.740
have this term beta, which
is the lifetime of mRNA

00:18:56.740 --> 00:18:57.596
divided by lifetime of protein.

00:18:57.596 --> 00:18:59.689
So we haven't assumed
anything about this beta.

00:18:59.689 --> 00:19:01.480
It could be, in principle,
larger than one.

00:19:01.480 --> 00:19:02.229
Smaller, actually.

00:19:02.229 --> 00:19:07.445
So it's true that given typical
facts about life in the cell,

00:19:07.445 --> 00:19:09.570
it's true that you expect
beta be much less than 1.

00:19:09.570 --> 00:19:11.010
But we haven't made
any assumption.

00:19:11.010 --> 00:19:11.801
Beta is just there.

00:19:11.801 --> 00:19:13.230
It could be anything.

00:19:13.230 --> 00:19:14.100
Right?

00:19:14.100 --> 00:19:17.630
So yeah, it's possible we've
made some other assumption.

00:19:17.630 --> 00:19:18.660
But what is going on.

00:19:18.660 --> 00:19:19.160
Yes?

00:19:19.160 --> 00:19:22.184
AUDIENCE: Is it
the concentration

00:19:22.184 --> 00:19:24.960
is scaled by the
amount of necessary--

00:19:24.960 --> 00:19:28.150
PROFESSOR: Yes, that's
right because, remember,

00:19:28.150 --> 00:19:31.530
you can only choose
one unit for time.

00:19:31.530 --> 00:19:34.360
And we've already chosen that
to get this to be just minus m

00:19:34.360 --> 00:19:34.990
here.

00:19:34.990 --> 00:19:37.240
But you get to choose what's
the unit of concentration

00:19:37.240 --> 00:19:40.870
for, both, mRNA and for protein.

00:19:40.870 --> 00:19:43.630
Can somebody remind us what
the unit of concentration

00:19:43.630 --> 00:19:46.070
is for protein?

00:19:46.070 --> 00:19:49.214
AUDIENCE: The dissociation
constant of the protein

00:19:49.214 --> 00:19:51.006
to the--

00:19:51.006 --> 00:19:52.030
PROFESSOR: That's right.

00:19:52.030 --> 00:19:53.890
So it's this
dissociation constant.

00:19:53.890 --> 00:19:56.630
And more generally, it's
the protein concentration,

00:19:56.630 --> 00:20:00.750
which you get half
maximal repression.

00:20:00.750 --> 00:20:02.522
And depending on
the detailed models,

00:20:02.522 --> 00:20:03.730
it could be more complicated.

00:20:03.730 --> 00:20:07.480
But in this phenomenological
realm, if p is equal to 1,

00:20:07.480 --> 00:20:09.310
you get half repression.

00:20:09.310 --> 00:20:12.270
And that's our definition
for what p equal to 1 means.

00:20:12.270 --> 00:20:15.290
So we've rescaled out that k.

00:20:15.290 --> 00:20:17.530
So what we've really
done is that there's

00:20:17.530 --> 00:20:20.350
some unit for the
concentration of mRNA

00:20:20.350 --> 00:20:22.190
that we were free to choose.

00:20:22.190 --> 00:20:26.020
And it was chosen so that
you could just say p minus m.

00:20:26.020 --> 00:20:32.330
But what that means is that it
requires a genius to figure out

00:20:32.330 --> 00:20:35.064
what m equal to 1 means, right?

00:20:35.064 --> 00:20:36.480
It doesn't quite
require a genius.

00:20:36.480 --> 00:20:38.605
But what do you guys think
it's going to depend on?

00:20:49.601 --> 00:20:50.100
Yes?

00:20:50.100 --> 00:20:52.092
AUDIENCE: It's going to depend
on this ratio of lifetimes,

00:20:52.092 --> 00:20:52.590
as well.

00:20:52.590 --> 00:20:53.506
PROFESSOR: Yes, right.

00:20:53.506 --> 00:20:55.510
So beta is going
to appear in there.

00:20:55.510 --> 00:20:56.926
So I'll give you
a hint, there are

00:20:56.926 --> 00:21:00.119
three things that determine it.

00:21:00.119 --> 00:21:02.410
AUDIENCE: Transcription, or
the speed of transcription.

00:21:02.410 --> 00:21:03.650
PROFESSOR: Translation, yes.

00:21:03.650 --> 00:21:04.960
So the translation efficiency.

00:21:04.960 --> 00:21:08.920
So each mRNA, it's going to
lead to some rate of protein

00:21:08.920 --> 00:21:09.420
synthesis.

00:21:09.420 --> 00:21:12.450
So yeah, the translation rate
or efficiency is going to enter.

00:21:19.281 --> 00:21:21.280
There aren't that many
other things it could be.

00:21:21.280 --> 00:21:22.870
But yeah, I mean,
this is tricky.

00:21:26.230 --> 00:21:28.230
And it's OK if you can't
just figure it out here

00:21:28.230 --> 00:21:32.580
because this, I think,
is pretty subtle.

00:21:32.580 --> 00:21:35.710
It turns out it also
depends on that k parameter

00:21:35.710 --> 00:21:39.710
because there's some sense
that-- m equal to 1-- what it's

00:21:39.710 --> 00:21:43.300
saying is that that's
the amount of mRNA

00:21:43.300 --> 00:21:49.220
that you need so that, if the
protein concentration where 1,

00:21:49.220 --> 00:21:52.630
you would not get any change
in the protein concentration.

00:21:52.630 --> 00:21:58.200
And given that now I
had to invoke p in there

00:21:58.200 --> 00:22:00.820
and p is scaled by k,
so then k also ends up

00:22:00.820 --> 00:22:02.675
being relevant for this mRNA.

00:22:02.675 --> 00:22:04.730
So you can, if
you'd like, go ahead

00:22:04.730 --> 00:22:07.720
and start with a original,
reasonable set of equations.

00:22:07.720 --> 00:22:09.274
And then, get back to this.

00:22:09.274 --> 00:22:10.690
But I think, once
again, this just

00:22:10.690 --> 00:22:13.480
highlights that these
non-dimensional versions

00:22:13.480 --> 00:22:14.760
of the equations are great.

00:22:14.760 --> 00:22:17.760
But you have to be careful.

00:22:17.760 --> 00:22:19.780
You don't know what means what.

00:22:19.780 --> 00:22:22.162
All right?

00:22:22.162 --> 00:22:24.370
Are there any questions
about what we've said so far?

00:22:31.110 --> 00:22:32.250
OK.

00:22:32.250 --> 00:22:35.530
Now what we've done is we have
now a protein concentration.

00:22:35.530 --> 00:22:36.820
We have mRNA concentration.

00:22:36.820 --> 00:22:40.695
And what I'm going to ask
for now is, for these sets

00:22:40.695 --> 00:22:42.560
of equations, is it
mathematically possible

00:22:42.560 --> 00:22:46.930
that they could,
maybe, oscillate?

00:22:46.930 --> 00:22:47.450
Yes.

00:22:47.450 --> 00:22:50.550
I mean, we're going to find
that the answer is that these

00:22:50.550 --> 00:22:51.870
actually don't oscillate.

00:22:51.870 --> 00:22:54.600
But have to actually
do the calculation

00:22:54.600 --> 00:22:55.850
if you want to determine that.

00:22:55.850 --> 00:22:59.110
You can't just say that
it's impossible based

00:22:59.110 --> 00:23:00.230
on the same argument here.

00:23:00.230 --> 00:23:02.730
And that's because, if
you think about this

00:23:02.730 --> 00:23:06.620
in the case of there's
some mRNA concentration.

00:23:06.620 --> 00:23:08.860
Some protein concentration.

00:23:08.860 --> 00:23:12.560
What we want to know is do
things oscillate in this space.

00:23:12.560 --> 00:23:15.170
And they could.

00:23:15.170 --> 00:23:18.535
I mean, I could
certainly draw a curve.

00:23:18.535 --> 00:23:20.660
It ends up not being true
for these particular sets

00:23:20.660 --> 00:23:21.201
of equations.

00:23:21.201 --> 00:23:25.690
But you can't a priori, kind
of, dismiss the possibility.

00:23:25.690 --> 00:23:26.458
Yes?

00:23:26.458 --> 00:23:28.450
AUDIENCE: That's like a
differential equation.

00:23:28.450 --> 00:23:30.442
But if you write down the
stochastic model of that,

00:23:30.442 --> 00:23:30.942
would that--

00:23:30.942 --> 00:23:33.280
PROFESSOR: OK, this is
a very good question.

00:23:33.280 --> 00:23:37.350
So this is the differential
equation format of this

00:23:37.350 --> 00:23:40.120
and that we're
assuming that there

00:23:40.120 --> 00:23:41.550
are no stochastic fluctuations.

00:23:41.550 --> 00:23:45.600
And indeed, there is a large
area of excitement, recently,

00:23:45.600 --> 00:23:49.460
that is trying to understand
cases in which you can have,

00:23:49.460 --> 00:23:51.530
so-called, noise
induced oscillations.

00:23:51.530 --> 00:23:54.970
So you can have cases that the
deterministic equations do not

00:23:54.970 --> 00:23:55.770
oscillate.

00:23:55.770 --> 00:24:00.560
But if you do the full
stochastic treatment,

00:24:00.560 --> 00:24:01.860
then that could oscillate.

00:24:01.860 --> 00:24:05.507
In particular, if you do a
master equation type formalism.

00:24:05.507 --> 00:24:07.923
And actually, I don't know,
for this particular equations.

00:24:11.094 --> 00:24:12.830
Yeah, I don't know for this one.

00:24:12.830 --> 00:24:14.740
But towards the end
of the semester,

00:24:14.740 --> 00:24:17.630
we will be talking about
explicit models in which,

00:24:17.630 --> 00:24:21.069
predator prey systems, in
which the differential equation

00:24:21.069 --> 00:24:22.110
format doesn't oscillate.

00:24:22.110 --> 00:24:24.651
But then, if you do the master
equation stochastic treatment,

00:24:24.651 --> 00:24:26.046
then it does oscillate.

00:24:26.046 --> 00:24:28.629
Yeah, so we will be talking
about this in other contexts.

00:24:28.629 --> 00:24:30.420
But I don't know the
answer for this model.

00:24:37.140 --> 00:24:40.420
All right, so let's
go and, maybe, try

00:24:40.420 --> 00:24:42.320
to analyze this a little bit.

00:24:42.320 --> 00:24:46.457
And this is useful to
do, partly because some

00:24:46.457 --> 00:24:49.040
of the calculations are going
to be very similar to what we're

00:24:49.040 --> 00:24:52.200
about to do next, which
is look at stability

00:24:52.200 --> 00:24:56.321
analysis of a repressilator
kind of system.

00:24:56.321 --> 00:24:56.820
All right.

00:25:03.350 --> 00:25:10.800
So this thing here is some
function f of m and p.

00:25:10.800 --> 00:25:13.260
And this guy here
is indeed, again,

00:25:13.260 --> 00:25:15.960
some other function
g of m and p.

00:25:15.960 --> 00:25:18.590
And we're going to be taking
derivatives of these functions

00:25:18.590 --> 00:25:20.420
around the fixed point.

00:25:24.260 --> 00:25:25.960
And maybe I will
also say there's

00:25:25.960 --> 00:25:29.431
going to be some stable point.

00:25:29.431 --> 00:25:30.930
We should just
calculate what it is.

00:25:30.930 --> 00:25:33.230
I'm sorry I'm making
this go up and down.

00:25:33.230 --> 00:25:33.970
Don't get dizzy.

00:25:37.440 --> 00:25:40.640
So first of all, it's always
good to know whether there

00:25:40.640 --> 00:25:43.430
are fixed points in
any sort of equations

00:25:43.430 --> 00:25:46.280
that you ever look at.

00:25:46.280 --> 00:25:48.190
So let's go ahead and see that.

00:25:48.190 --> 00:25:54.520
First of all, is m equal
to 0, p equal to 0?

00:25:54.520 --> 00:25:58.030
Is that a fixed
point in the system?

00:25:58.030 --> 00:25:58.550
No.

00:25:58.550 --> 00:25:59.049
Right?

00:25:59.049 --> 00:26:02.310
So if m and p are 0, then
this is a fixed point.

00:26:02.310 --> 00:26:05.000
But that one's
not because we get

00:26:05.000 --> 00:26:07.470
expression of the mRNA in
the absence of the protein.

00:26:07.470 --> 00:26:10.492
So the origin is
not a fixed point.

00:26:10.492 --> 00:26:11.950
Now to figure out
the fixed points,

00:26:11.950 --> 00:26:13.900
we just set these
things equal to 0.

00:26:13.900 --> 00:26:17.740
So if m dot is equal
to 0, we have 0.

00:26:17.740 --> 00:26:25.260
That's alpha 1 plus
p to the n minus m.

00:26:25.260 --> 00:26:27.474
Again, 0 is this.

00:26:30.530 --> 00:26:34.160
So what you can see is
that, at equilibrium, we

00:26:34.160 --> 00:26:42.130
have a condition here
where m is equal to p.

00:26:42.130 --> 00:26:45.772
So from this, we
get m equilibrium

00:26:45.772 --> 00:26:46.855
is equal to p equilibrium.

00:26:51.920 --> 00:26:55.930
So m equilibrium over here has
to be equal to p equilibrium,

00:26:55.930 --> 00:26:56.914
we just said.

00:26:56.914 --> 00:26:58.330
And that's equal
to this guy here.

00:26:58.330 --> 00:27:04.204
It's alpha 1 plus p
equilibrium to the n.

00:27:04.204 --> 00:27:05.070
All right.

00:27:05.070 --> 00:27:07.625
And the condition
for this equilibrium

00:27:07.625 --> 00:27:09.250
is then something
that looks like this.

00:27:16.170 --> 00:27:21.570
Now this is maybe
not so intuitive.

00:27:21.570 --> 00:27:24.560
But alpha is this non
dimensional version

00:27:24.560 --> 00:27:27.980
of the strength of expression.

00:27:27.980 --> 00:27:33.742
And what this is saying is
that, broadly, it's not obvious

00:27:33.742 --> 00:27:34.950
how to solve this explicitly.

00:27:34.950 --> 00:27:37.950
But as the strength
of expression goes up,

00:27:37.950 --> 00:27:42.000
the equilibrium here-- and
I'm saying equilibrium.

00:27:42.000 --> 00:27:43.910
And that's, maybe, a
little bit dangerous.

00:27:43.910 --> 00:27:46.885
We might even want
to just call it--

00:27:46.885 --> 00:27:48.600
it's a fixed point
in concentration,

00:27:48.600 --> 00:27:50.550
so it doesn't have to be stable.

00:27:50.550 --> 00:27:53.414
So if we don't want
to bias our thinking,

00:27:53.414 --> 00:27:55.830
different people argue about
whether equilibrium should be

00:27:55.830 --> 00:27:57.430
a stable or require a stable.

00:27:57.430 --> 00:27:59.920
We could just call
it some p 0 if that

00:27:59.920 --> 00:28:02.940
makes you less likely to
bias our thinking in terms

00:28:02.940 --> 00:28:05.831
of whether this concentration
should be a stable or unstable

00:28:05.831 --> 00:28:06.330
fixed point.

00:28:09.480 --> 00:28:13.320
But for example, if we
have that, in these units,

00:28:13.320 --> 00:28:18.080
if alpha is around
10, n might 2.

00:28:18.080 --> 00:28:19.590
Then, this thing
gives us something.

00:28:19.590 --> 00:28:25.520
It's in the range
of a couple or 2, 3.

00:28:25.520 --> 00:28:28.780
I mean, you can calculate
what it should be.

00:28:28.780 --> 00:28:32.780
2, 4, maybe even exactly 2.

00:28:32.780 --> 00:28:34.640
Did that-- yeah.

00:28:34.640 --> 00:28:36.015
All right, so yes.

00:28:36.015 --> 00:28:37.140
I'm just giving an example.

00:28:37.140 --> 00:28:39.720
If alpha were 10, then this
equilibrium concentration

00:28:39.720 --> 00:28:41.320
or this fixed
point concentration

00:28:41.320 --> 00:28:43.910
would be 2 if n were equal
to 2 to give you, kind

00:28:43.910 --> 00:28:46.320
of, some sense of the numbers.

00:28:46.320 --> 00:28:52.192
And this is 2 in units of that
binding affinity k, right.

00:28:52.192 --> 00:28:54.150
Now the question is,
well, what does this mean?

00:28:54.150 --> 00:28:55.020
Why did we do this?

00:28:55.020 --> 00:28:59.180
Why do we care at all about the
properties of that fix point?

00:28:59.180 --> 00:29:02.760
OK, so this might be some p 0.

00:29:02.760 --> 00:29:07.380
And this is, again, m 0 is
equal to p 0 in these units.

00:29:07.380 --> 00:29:11.190
So there's some fixed point
somewhere in the middle there.

00:29:11.190 --> 00:29:15.170
Now it turns out that the
stability of that fixed point

00:29:15.170 --> 00:29:17.980
is very important in determining
whether there are oscillations

00:29:17.980 --> 00:29:20.390
or not.

00:29:20.390 --> 00:29:23.500
Now the question
of the generality

00:29:23.500 --> 00:29:25.250
or what can you say
that's universally

00:29:25.250 --> 00:29:26.870
true about when you
get oscillations

00:29:26.870 --> 00:29:29.370
and when you don't, this
is, in general, a very hard

00:29:29.370 --> 00:29:31.130
mathematical
problem, particularly

00:29:31.130 --> 00:29:32.990
in higher numbers of dimensions.

00:29:32.990 --> 00:29:35.910
But for two dimensions,
there's a very nice statement

00:29:35.910 --> 00:29:40.800
that you can make based on the
Poincare-Bendixson criterion.

00:29:40.800 --> 00:29:43.690
I cannot remember
how to spell that.

00:29:43.690 --> 00:29:47.105
I'm probably
mispronouncing it, as well.

00:29:47.105 --> 00:29:48.730
So Poincare-Bendixson,
what they showed

00:29:48.730 --> 00:29:52.140
is that if, in two
dimensions, you

00:29:52.140 --> 00:29:57.150
can draw some box here such
that all of the trajectories

00:29:57.150 --> 00:29:58.150
are, kind of, coming in.

00:30:00.710 --> 00:30:03.260
And indeed, in this
case, they do come in

00:30:03.260 --> 00:30:06.600
because the trajectories
aren't going to cross 0.

00:30:06.600 --> 00:30:09.160
If you have some
mRNA, then you're

00:30:09.160 --> 00:30:11.920
going to start making protein.

00:30:11.920 --> 00:30:13.880
If you have just
protein, no mRNA,

00:30:13.880 --> 00:30:15.530
you're going to start
making some mRNA.

00:30:15.530 --> 00:30:17.988
And we know that trajectories
have to come in from out here

00:30:17.988 --> 00:30:20.190
because if the
concentration of mRNA

00:30:20.190 --> 00:30:22.610
and the concentration of
protein are very large then,

00:30:22.610 --> 00:30:24.110
eventually, the
degradation is going

00:30:24.110 --> 00:30:25.260
to start pulling things in.

00:30:25.260 --> 00:30:28.170
So if you come out far
enough, eventually, you're

00:30:28.170 --> 00:30:29.670
going to get
trajectories coming in.

00:30:29.670 --> 00:30:31.230
So now we have
there is some domain

00:30:31.230 --> 00:30:32.890
where all the trajectories
are going to come in.

00:30:32.890 --> 00:30:34.306
Now you can imagine
that, somehow,

00:30:34.306 --> 00:30:37.510
the stability of this thing is
very important because in two

00:30:37.510 --> 00:30:41.080
dimensions here when you
have a differential equation,

00:30:41.080 --> 00:30:45.120
trajectories cannot
cross each other.

00:30:45.120 --> 00:30:47.270
So I'm not allowed
in any sort of space

00:30:47.270 --> 00:30:51.490
like this to do something that
looks like this because this

00:30:51.490 --> 00:30:55.010
would require that, at some
concentration of m and p,

00:30:55.010 --> 00:30:57.482
I have different values
for m dot and p dot.

00:30:57.482 --> 00:30:59.940
So it's similar to this argument
we made for one dimension.

00:30:59.940 --> 00:31:02.360
But it's just generalized
to two dimensions.

00:31:02.360 --> 00:31:06.250
So we're not allowed
to cross trajectories.

00:31:06.250 --> 00:31:09.250
Well if you have a differential
equation in any dimensions,

00:31:09.250 --> 00:31:09.750
that's true.

00:31:09.750 --> 00:31:11.340
But the thing is
that this constraint

00:31:11.340 --> 00:31:13.445
is a very strong constraint
in two dimensions.

00:31:13.445 --> 00:31:15.820
Whereas, in three dimensions,
everything kind of goes out

00:31:15.820 --> 00:31:17.936
the window because in
the three dimensions,

00:31:17.936 --> 00:31:19.060
you have another axis here.

00:31:19.060 --> 00:31:21.480
And then, these lines can do
all sorts of crazy things.

00:31:21.480 --> 00:31:24.080
And that's actually, basically,
why you need three dimensions

00:31:24.080 --> 00:31:27.400
in order to get chaos in
differential equations

00:31:27.400 --> 00:31:30.540
because this thing about
the absence of crossing

00:31:30.540 --> 00:31:33.150
is just such a strong
constraint in two dimensions.

00:31:35.081 --> 00:31:37.080
Other questions about
what I'm saying right now?

00:31:37.080 --> 00:31:38.538
I'm a little bit
worried that I'm--

00:31:42.380 --> 00:31:45.364
All right, so the trajectories
are not allowed to cross.

00:31:45.364 --> 00:31:47.280
And that's really saying
something very strong

00:31:47.280 --> 00:31:49.210
because we know that,
here, trajectories are

00:31:49.210 --> 00:31:53.930
going to come out of the axis.

00:31:53.930 --> 00:31:56.824
And mRNA, we don't
know which direction

00:31:56.824 --> 00:31:57.740
they're going to come.

00:31:57.740 --> 00:32:00.557
But let's figure out,
if it were to oscillate,

00:32:00.557 --> 00:32:03.140
would the trajectories be going
clockwise or counterclockwise?

00:32:05.115 --> 00:32:06.490
And actually,
there's going to be

00:32:06.490 --> 00:32:08.779
some sense of the
trajectories even

00:32:08.779 --> 00:32:10.070
in the absence of oscillations.

00:32:10.070 --> 00:32:14.400
But broadly, is there kind of
a counterclockwise or clockwise

00:32:14.400 --> 00:32:18.690
kind of motion to
the trajectories?

00:32:18.690 --> 00:32:19.810
Counterclockwise, right?

00:32:19.810 --> 00:32:24.100
And that's because
mRNA leads to protein.

00:32:24.100 --> 00:32:26.320
So things are going
to go like this.

00:32:26.320 --> 00:32:29.020
And the question is is
it going to oscillate.

00:32:29.020 --> 00:32:31.940
And in two dimensions,
actually-- Poincare-Bendixson--

00:32:31.940 --> 00:32:36.330
what they say is
that, if there's just

00:32:36.330 --> 00:32:38.504
one fixed point here,
then the question

00:32:38.504 --> 00:32:40.670
of whether it oscillates
is the same as the question

00:32:40.670 --> 00:32:43.130
of whether this is stable.

00:32:43.130 --> 00:32:50.026
So if it's stable, then
there's no oscillations.

00:32:50.026 --> 00:32:51.400
If it's unstable,
than there are.

00:32:59.990 --> 00:33:02.600
We'll just say no
oscillations and oscillations.

00:33:02.600 --> 00:33:04.850
And that's because if it's
a stable point and all

00:33:04.850 --> 00:33:08.440
the trajectories are coming in,
then it just looks like this.

00:33:08.440 --> 00:33:12.520
So it spirals, maybe, into
a state of coexistence.

00:33:12.520 --> 00:33:15.667
Well it spirals to
this point of m and p.

00:33:15.667 --> 00:33:17.750
Whereas, if it's unstable,
then those trajectories

00:33:17.750 --> 00:33:19.235
are, somehow, being pushed out.

00:33:19.235 --> 00:33:20.860
If it's unstable,
then the trajectories

00:33:20.860 --> 00:33:23.870
are coming out of
that fixed point.

00:33:23.870 --> 00:33:25.460
In which case, then
that's actually

00:33:25.460 --> 00:33:27.710
precisely the situation in
which you get a limit cycle

00:33:27.710 --> 00:33:28.900
oscillations.

00:33:28.900 --> 00:33:32.710
So if the fixed
point were unstable,

00:33:32.710 --> 00:33:36.930
it looks like this
because we have some box.

00:33:36.930 --> 00:33:40.760
The trajectories are all
coming in, somehow, in here.

00:33:40.760 --> 00:33:43.260
But if we have one
fixed point here

00:33:43.260 --> 00:33:44.867
and the trajectories
are coming out,

00:33:44.867 --> 00:33:46.950
that means we have something
that looks like this.

00:33:46.950 --> 00:33:47.825
It kind of comes out.

00:33:47.825 --> 00:33:49.741
And given that these
trajectories can't cross,

00:33:49.741 --> 00:33:51.830
the question is, well,
what can happen in between?

00:33:51.830 --> 00:33:54.470
And the answer
is, basically, you

00:33:54.470 --> 00:33:57.110
have to get a limit
cycle oscillation.

00:33:57.110 --> 00:33:58.830
There are these
strange situations

00:33:58.830 --> 00:34:02.044
where you can get a path that
is an oscillation that's,

00:34:02.044 --> 00:34:03.460
kind of, stable
from one direction

00:34:03.460 --> 00:34:05.250
and unstable from another.

00:34:05.250 --> 00:34:08.030
We're not going to
worry about that here.

00:34:08.030 --> 00:34:11.050
But broadly, if this
thing is coming out,

00:34:11.050 --> 00:34:13.040
then you end up,
in both directions,

00:34:13.040 --> 00:34:15.343
converging to a stable
limit cycle oscillation.

00:34:17.949 --> 00:34:22.020
So it's a unstable
fixed point, then

00:34:22.020 --> 00:34:25.010
this is the exact situation,
which you get a limit cycle

00:34:25.010 --> 00:34:27.600
oscillation.

00:34:27.600 --> 00:34:29.154
OK.

00:34:29.154 --> 00:34:30.570
So that means that,
what we really

00:34:30.570 --> 00:34:34.690
want to do if we want to ask--
let's try to back up again.

00:34:34.690 --> 00:34:37.630
We have this pair of
differential equations.

00:34:37.630 --> 00:34:42.070
We want to know will this
negative auto regulatory loop

00:34:42.070 --> 00:34:43.659
oscillate.

00:34:43.659 --> 00:34:46.350
Now what I'm telling you
is that that question

00:34:46.350 --> 00:34:49.100
for two dimensions is analogous
to the question of figuring out

00:34:49.100 --> 00:34:52.732
whether this fixed
point is stable or not.

00:34:52.732 --> 00:34:54.690
If it's stable, then we
don't get oscillations.

00:34:54.690 --> 00:34:57.506
If it's unstable, then we do.

00:35:00.790 --> 00:35:02.190
Any questions about this?

00:35:05.280 --> 00:35:07.660
So let's see what is is.

00:35:07.660 --> 00:35:10.300
On Tuesday, what we do is we
talked about stability analysis

00:35:10.300 --> 00:35:13.470
for linear systems.

00:35:13.470 --> 00:35:16.020
We got what I hope is
some intuition about that.

00:35:16.020 --> 00:35:18.237
And of course, what
we need to do here

00:35:18.237 --> 00:35:20.320
is try to understand how
to apply linear stability

00:35:20.320 --> 00:35:25.060
analysis to this non-linear
pair of differential equations.

00:35:25.060 --> 00:35:28.320
And to do that,
what we need to do

00:35:28.320 --> 00:35:33.370
is we need to linearize
around that fixed point.

00:35:36.830 --> 00:35:42.200
So what we have is we have
these two functions, f and g.

00:35:42.200 --> 00:35:44.650
And what we want to know is
around that fixed point--

00:35:44.650 --> 00:35:52.810
so we can define some m tilde,
which is m minus this m 0.

00:35:52.810 --> 00:35:57.620
And some p tilde,
which is p minus p 0.

00:35:57.620 --> 00:36:01.070
So when m tilde and
p tilde are around 0,

00:36:01.070 --> 00:36:06.490
that's telling us that we're
close to that fixed point.

00:36:06.490 --> 00:36:09.550
And we want to know, if we just
go a little away from the fixed

00:36:09.550 --> 00:36:11.670
point, do we get pushed
away or do we come back

00:36:11.670 --> 00:36:12.503
to where we started?

00:36:15.520 --> 00:36:19.963
Well we know that m tilde dot,
which is actually equal to m

00:36:19.963 --> 00:36:23.890
dot, as well because m
0 and p 0 are the same.

00:36:23.890 --> 00:36:26.710
p tilde dot.

00:36:26.710 --> 00:36:32.460
We can linearize by taking
derivatives around the fixed

00:36:32.460 --> 00:36:32.960
points.

00:36:36.559 --> 00:36:38.100
And in particular,
what we want to do

00:36:38.100 --> 00:36:44.980
is we want to take the
derivative of f with respect

00:36:44.980 --> 00:36:46.400
to m.

00:36:46.400 --> 00:36:49.340
Evaluate at the fixed point.

00:36:49.340 --> 00:36:53.590
That derivative is,
indeed, just minus 1.

00:36:53.590 --> 00:36:57.750
So in general, in these
situations, what we have is

00:36:57.750 --> 00:37:01.460
we have derivatives
m, m dot p dot,

00:37:01.460 --> 00:37:04.655
and we have partial of this
first function f with respect

00:37:04.655 --> 00:37:06.060
to m.

00:37:06.060 --> 00:37:07.340
Partial of g.

00:37:07.340 --> 00:37:07.840
Oh, no.

00:37:07.840 --> 00:37:10.160
So this is still f.

00:37:10.160 --> 00:37:13.300
Respect to p.

00:37:13.300 --> 00:37:16.470
Down here is derivative
g with respect to m.

00:37:16.470 --> 00:37:19.670
Derivative g with respect to p.

00:37:19.670 --> 00:37:22.910
And this is all evaluated
around the fixed point m 0 p 0.

00:37:27.900 --> 00:37:29.790
So we want to take
these derivatives

00:37:29.790 --> 00:37:34.270
and evaluate at the fixed point.

00:37:34.270 --> 00:37:36.530
And if we do that,
we get minus 1 here,

00:37:36.530 --> 00:37:41.070
derivative m with respect
to m times m tilde.

00:37:41.070 --> 00:37:46.020
This other guy, when
you take the derivative,

00:37:46.020 --> 00:37:50.870
you get a minus sign
with respect to p.

00:37:50.870 --> 00:37:53.800
So we get a minus sign because
this is in the denominator.

00:37:53.800 --> 00:37:56.110
And then, we have to
take derivative inside.

00:37:56.110 --> 00:38:02.650
So we get n alpha p
0 to the n minus 1.

00:38:02.650 --> 00:38:08.530
And down, we get a
1 plus p 0 squared.

00:38:08.530 --> 00:38:12.190
So we took the derivative of
this term with respect to p.

00:38:12.190 --> 00:38:16.214
And we evaluated at
the fixed point p 0.

00:38:16.214 --> 00:38:18.160
Did I do that right?

00:38:18.160 --> 00:38:20.820
But we still have
to add a p tilde

00:38:20.820 --> 00:38:23.100
because this is
saying how sensitive

00:38:23.100 --> 00:38:24.590
is the function to
changes in where

00:38:24.590 --> 00:38:28.270
you are times how far you've
gone away from the fixed point.

00:38:28.270 --> 00:38:30.990
And then, again, over here,
we take the derivatives

00:38:30.990 --> 00:38:31.510
down below.

00:38:31.510 --> 00:38:34.970
So derivative g
with respect to m.

00:38:34.970 --> 00:38:40.400
That gives us a beta m tilde.

00:38:40.400 --> 00:38:45.420
And then, we have a
minus beta p tilde.

00:38:48.535 --> 00:38:51.380
All right, so this
is just an example

00:38:51.380 --> 00:38:55.646
of linearizing those equations
around that fixed point.

00:38:59.360 --> 00:39:02.050
So ultimately, what we
care about is really

00:39:02.050 --> 00:39:07.172
this matrix that's
specifying deviations

00:39:07.172 --> 00:39:08.130
around the equilibrium.

00:39:08.130 --> 00:39:08.800
Right?

00:39:08.800 --> 00:39:12.984
So it's useful to just write it
in matrix format because we get

00:39:12.984 --> 00:39:14.275
rid of some of the M's and P's.

00:39:20.980 --> 00:39:24.580
Indeed, so this matrix that we
either call A or the Jacobean

00:39:24.580 --> 00:39:31.155
depending on-- so what
we have is a minus 1.

00:39:35.950 --> 00:39:39.970
And we're going
to call this thing

00:39:39.970 --> 00:39:47.474
x because it's going to pop up
a lot is this minus n alpha p 0.

00:39:54.120 --> 00:40:02.134
So it's an x beta
and minus beta.

00:40:02.134 --> 00:40:03.550
And then, we have
our simple rules

00:40:03.550 --> 00:40:06.330
for determining whether this
thing is going to be stable

00:40:06.330 --> 00:40:07.210
or not.

00:40:07.210 --> 00:40:08.240
It depends on the trace.

00:40:08.240 --> 00:40:10.360
And it depends on
the determinant.

00:40:10.360 --> 00:40:12.866
So the trace should be negative.

00:40:12.866 --> 00:40:13.990
And is this trace negative?

00:40:17.180 --> 00:40:17.980
Yes.

00:40:17.980 --> 00:40:22.827
Yes because beta-- does anybody
remember what beta was again.

00:40:22.827 --> 00:40:24.035
AUDIENCE: Ratio of lifetimes.

00:40:24.035 --> 00:40:25.150
PROFESSOR: Ratio of lifetimes.

00:40:25.150 --> 00:40:26.108
Lifetimes are positive.

00:40:26.108 --> 00:40:27.100
So beta is positive.

00:40:27.100 --> 00:40:35.210
All right, so the trace is
equal to minus 1 minus beta.

00:40:35.210 --> 00:40:37.080
This is, indeed, less than 0.

00:40:37.080 --> 00:40:39.410
So this is consistent
for stability.

00:40:39.410 --> 00:40:41.760
Does prove that it's stable?

00:40:41.760 --> 00:40:43.980
No.

00:40:43.980 --> 00:40:49.070
But we also need to know
about the determinant of a,

00:40:49.070 --> 00:40:53.960
which is going to be
beta, this times this,

00:40:53.960 --> 00:40:55.710
minus this times this.

00:40:55.710 --> 00:40:59.910
So that's minus.

00:40:59.910 --> 00:41:02.530
And this is a beta
times what x was.

00:41:02.530 --> 00:41:04.160
So this gives us--
we can write this

00:41:04.160 --> 00:41:08.015
all down just so that it's clear
that it has to be positive.

00:41:12.900 --> 00:41:14.480
So beta is positive.

00:41:14.480 --> 00:41:16.830
Positive, positive,
positive, positive, positive.

00:41:16.830 --> 00:41:19.240
Everything's positive.

00:41:19.240 --> 00:41:23.730
So this thing has to
be greater than 0.

00:41:23.730 --> 00:41:27.630
So what does this mean about
the stability of Ethics Point?

00:41:27.630 --> 00:41:28.130
Stable.

00:41:30.960 --> 00:41:31.970
Fixed point stable.

00:41:31.970 --> 00:41:33.856
And what does that mean
about oscillations?

00:41:33.856 --> 00:41:35.864
It means there are
no oscillations.

00:41:35.864 --> 00:41:36.655
Fixed point stable.

00:41:39.580 --> 00:41:40.910
Therefore, no oscillations.

00:41:47.860 --> 00:41:51.950
So what this is saying is that
the original, kind of simple,

00:41:51.950 --> 00:41:54.260
equation we wrote down for
negative auto regulation,

00:41:54.260 --> 00:41:58.650
that thing was not allowed
to oscillate mathematically.

00:41:58.650 --> 00:42:01.612
But that doesn't mean that, if
you explicitly model the mRNA,

00:42:01.612 --> 00:42:02.570
it could go either way.

00:42:02.570 --> 00:42:07.040
But still, that's insufficient
to generate oscillations.

00:42:07.040 --> 00:42:10.989
However, maybe if you
included more steps,

00:42:10.989 --> 00:42:12.030
maybe it would oscillate.

00:42:12.030 --> 00:42:12.620
Question?

00:42:12.620 --> 00:42:14.762
AUDIENCE: So just to double
check-- when you said,

00:42:14.762 --> 00:42:17.210
no oscillations, you
mean stable oscillations?

00:42:17.210 --> 00:42:18.700
PROFESSOR: That's right, sorry.

00:42:18.700 --> 00:42:21.070
When I mean no oscillations,
what I mean are indeed,

00:42:21.070 --> 00:42:22.760
no limit cycle oscillations.

00:42:22.760 --> 00:42:25.250
AUDIENCE: This is
like a dampened--

00:42:25.250 --> 00:42:26.840
PROFESSOR: Yeah.

00:42:26.840 --> 00:42:30.870
Yeah, so we, actually,
have not solved

00:42:30.870 --> 00:42:32.280
exactly what it looks like.

00:42:32.280 --> 00:42:34.729
And I've drawn this is a
pretty oscillatory thing.

00:42:34.729 --> 00:42:36.520
But it might just look
like this, depending

00:42:36.520 --> 00:42:38.220
on the parameters and so forth.

00:42:38.220 --> 00:42:41.070
And indeed, we haven't
even proven that this thing

00:42:41.070 --> 00:42:43.770
has complex eigenvalues.

00:42:43.770 --> 00:42:46.190
But certainly, there are no
limit cycle oscillations.

00:42:46.190 --> 00:42:49.510
And I'd say it's really limit
cycle oscillations that people

00:42:49.510 --> 00:42:58.130
find most exciting as because
limit cycle oscillations have

00:42:58.130 --> 00:43:00.259
a characteristic amplitude.

00:43:00.259 --> 00:43:01.800
So it doesn't matter
where you start.

00:43:01.800 --> 00:43:04.316
The oscillations go
to some amplitude.

00:43:04.316 --> 00:43:06.190
And they have a
characteristic period, again,

00:43:06.190 --> 00:43:08.250
independent of your
starting condition.

00:43:10.940 --> 00:43:13.748
So a limit cycle
oscillation has a feeling

00:43:13.748 --> 00:43:15.581
similar to a stable
fixed point in the since

00:43:15.581 --> 00:43:16.460
that it doesn't matter
where you start.

00:43:16.460 --> 00:43:17.460
You always end up there.

00:43:17.460 --> 00:43:18.980
So they're the ones
that are really

00:43:18.980 --> 00:43:21.411
what you would
call mathematically

00:43:21.411 --> 00:43:22.160
nice oscillations.

00:43:26.530 --> 00:43:28.950
And when I say this,
I'm, in particular,

00:43:28.950 --> 00:43:32.660
comparing them to
neutrally stable orbits.

00:43:32.660 --> 00:43:39.060
So there are cases in
which, in two variables,

00:43:39.060 --> 00:43:40.640
you have a fixed point here.

00:43:40.640 --> 00:43:45.990
And at least in the case
of linear stability,

00:43:45.990 --> 00:43:49.220
if you have purely
imaginary eigenvalues,

00:43:49.220 --> 00:43:53.450
what that means is that
you have orbits that

00:43:53.450 --> 00:43:54.890
go around your fixed point.

00:43:57.990 --> 00:44:01.570
And we'll see some cases
that look like this later on.

00:44:01.570 --> 00:44:04.190
And this is, indeed, the
nature of the oscillations

00:44:04.190 --> 00:44:07.424
in the Lotka-Volterra model
for predator prey oscillations.

00:44:07.424 --> 00:44:09.340
They're not actually
limit cycle oscillations.

00:44:09.340 --> 00:44:13.190
They're of this kind that are
considered less interesting

00:44:13.190 --> 00:44:16.050
because they're less robust.

00:44:16.050 --> 00:44:18.900
Small changes in the model
can cause these things

00:44:18.900 --> 00:44:25.870
to either go away, to turn into
this kind of stable spiral,

00:44:25.870 --> 00:44:28.530
or to turn into limit
cycle oscillations.

00:44:28.530 --> 00:44:32.950
So we'll talk about this
more in a couple months.

00:44:32.950 --> 00:44:34.460
These are neutrally
stable orbits.

00:44:48.810 --> 00:44:51.710
OK, but what I
wanted to highlight,

00:44:51.710 --> 00:44:56.880
though, is that just because
the original, simple, protein

00:44:56.880 --> 00:44:59.832
only model didn't oscillate
and this protein mRNA together

00:44:59.832 --> 00:45:01.540
doesn't oscillate does
not mean that it's

00:45:01.540 --> 00:45:04.530
impossible to get oscillations
using negative auto

00:45:04.530 --> 00:45:07.060
regulation, either
experimentally

00:45:07.060 --> 00:45:10.080
or computationally.

00:45:10.080 --> 00:45:12.050
And the question
is, what might you

00:45:12.050 --> 00:45:13.440
need to do to get oscillations?

00:45:22.120 --> 00:45:24.570
AUDIENCE: So in the paper
they talk about leakage

00:45:24.570 --> 00:45:25.550
in the negative--

00:45:30.446 --> 00:45:31.590
PROFESSOR: OK, right.

00:45:31.590 --> 00:45:33.590
So in the paper, they
talk about various things,

00:45:33.590 --> 00:45:34.964
including things
such as leakage.

00:45:34.964 --> 00:45:37.390
It terms out that
leakage in an expression

00:45:37.390 --> 00:45:39.800
only inhibits
oscillations though.

00:45:39.800 --> 00:45:43.360
So in some sense, if you're
trying to get oscillations,

00:45:43.360 --> 00:45:45.880
leakage is a problem, actually.

00:45:45.880 --> 00:45:48.420
And that's why they use
this especially tight-- well

00:45:48.420 --> 00:45:50.030
we're going to talk about
that in a few minutes.

00:45:50.030 --> 00:45:52.321
They use an especially tight
version of these promoters

00:45:52.321 --> 00:45:55.920
to have low rates of
leakage in a synthesis.

00:45:55.920 --> 00:45:58.410
But what might you need in
order to get oscillations

00:45:58.410 --> 00:45:59.740
in negative autoregulation?

00:45:59.740 --> 00:46:02.000
Did you have-- have delay.

00:46:02.000 --> 00:46:02.740
Yes indeed.

00:46:02.740 --> 00:46:05.480
And that's something that they
mentioned in the Elowitz paper

00:46:05.480 --> 00:46:09.150
is if you add explicit delay.

00:46:09.150 --> 00:46:14.761
So for example, if instead of
having the repression depend

00:46:14.761 --> 00:46:16.260
on--OK, I already
erased everything.

00:46:16.260 --> 00:46:22.164
But instead of having
the protein, for example,

00:46:22.164 --> 00:46:24.330
being a function of the
mRNA now, maybe if you said,

00:46:24.330 --> 00:46:27.190
oh, it's a function of
the mRNA five minutes ago.

00:46:27.190 --> 00:46:29.860
And that's just because maybe it
takes time to make the protein.

00:46:29.860 --> 00:46:31.276
Or it takes time
for this or that.

00:46:31.276 --> 00:46:34.560
You could introduce an
explicit delay like that.

00:46:34.560 --> 00:46:37.470
Or you could even,
instead, have a model

00:46:37.470 --> 00:46:40.610
where you just have more steps.

00:46:40.610 --> 00:46:42.730
So what you do is you
say, oh, well yeah, sure.

00:46:42.730 --> 00:46:45.165
What happens is that,
first, the mRNA is made.

00:46:45.165 --> 00:46:46.540
But then, after
the mRNA is made,

00:46:46.540 --> 00:46:49.960
then you have to make
the peptide chain.

00:46:49.960 --> 00:46:53.010
Then, the that peptide
chain has to fold.

00:46:53.010 --> 00:46:56.050
And then, maybe, those
proteins have to multimerize.

00:46:56.050 --> 00:46:58.740
Indeed, if you right
down such a model

00:46:58.740 --> 00:47:00.310
then, for some
reasonable parameters,

00:47:00.310 --> 00:47:03.630
you can get oscillations just
with negative auto regulation.

00:47:03.630 --> 00:47:06.250
And indeed, I would say
that over the last 10 years,

00:47:06.250 --> 00:47:09.890
probably, the reigning
king of oscillations

00:47:09.890 --> 00:47:12.260
in the field of system
synthetic biology

00:47:12.260 --> 00:47:14.230
is Jeff Hasty at San Diego.

00:47:14.230 --> 00:47:17.880
And he's written a whole train
of beautiful papers exploring

00:47:17.880 --> 00:47:21.991
how you can make these
oscillators in simple G

00:47:21.991 --> 00:47:22.490
network.

00:47:22.490 --> 00:47:24.420
So he's been
focusing in E. coli.

00:47:24.420 --> 00:47:26.420
There's also been great
work in higher organisms

00:47:26.420 --> 00:47:27.320
in this regard.

00:47:27.320 --> 00:47:30.450
But let's say, Hasty's
work stands out

00:47:30.450 --> 00:47:32.770
in terms of really being
able to take these models

00:47:32.770 --> 00:47:35.015
and then implement them
in cells and, kind of,

00:47:35.015 --> 00:47:35.890
going back and forth.

00:47:35.890 --> 00:47:39.180
And he's shown that you can
generate oscillations just

00:47:39.180 --> 00:47:40.730
using negative auto
regulation if you

00:47:40.730 --> 00:47:45.120
have enough delays in that
negative feedback loop.

00:47:48.510 --> 00:47:51.654
Are there any questions
about where we are right now?

00:47:51.654 --> 00:47:53.320
I know that we're
supposed to be talking

00:47:53.320 --> 00:47:54.319
about the repressilator.

00:47:54.319 --> 00:47:57.283
But we first have to make sure
we understand the negative auto

00:47:57.283 --> 00:47:57.782
regulation.

00:48:00.580 --> 00:48:05.670
So everything that we've said,
so far, in terms of the models

00:48:05.670 --> 00:48:07.231
was all known.

00:48:07.231 --> 00:48:09.730
But what Michael wanted to do
is ask whether he could really

00:48:09.730 --> 00:48:12.710
construct an oscillator.

00:48:12.710 --> 00:48:17.570
And he did this using these
three mutual reppressors.

00:48:17.570 --> 00:48:21.100
We'll say x, y,
and z just for now.

00:48:21.100 --> 00:48:27.740
x represses y, represses
z, represses x.

00:48:27.740 --> 00:48:31.500
And has a nice
model of this system

00:48:31.500 --> 00:48:34.380
that helped him guide the
design of his circuits.

00:48:34.380 --> 00:48:36.920
So experiments--
as most of us who

00:48:36.920 --> 00:48:39.570
have done them know--
experiments are hard.

00:48:39.570 --> 00:48:43.370
So if you can do
a week of thinking

00:48:43.370 --> 00:48:46.890
before you do a year of
experimental biology,

00:48:46.890 --> 00:48:49.180
then you should do that.

00:48:49.180 --> 00:48:52.680
And what were the
lessons that he

00:48:52.680 --> 00:48:56.854
learned from the modeling
that guided his construction

00:48:56.854 --> 00:48:57.520
of this circuit?

00:49:01.484 --> 00:49:01.984
Yeah?

00:49:01.984 --> 00:49:03.412
AUDIENCE: Lifetime of mRNA.

00:49:03.412 --> 00:49:04.120
PROFESSOR: Right.

00:49:04.120 --> 00:49:05.619
So you want to have
similar lifetime

00:49:05.619 --> 00:49:08.300
of the mRNA and the protein.

00:49:08.300 --> 00:49:10.740
And this is, somehow,
similar to this idea

00:49:10.740 --> 00:49:13.875
that you need more delay
elements because if you

00:49:13.875 --> 00:49:17.030
have very different lifetimes,
then the more rapid process,

00:49:17.030 --> 00:49:19.920
somehow, doesn't count.

00:49:19.920 --> 00:49:23.020
It's very hard to increase the
lifetime of the mRNA that much

00:49:23.020 --> 00:49:24.220
in bacteria.

00:49:24.220 --> 00:49:26.500
So instead, what he
did is he decreased

00:49:26.500 --> 00:49:29.960
the lifetime of the proteins
of the transcription factors.

00:49:29.960 --> 00:49:31.130
In this case, x, y, and z.

00:49:33.914 --> 00:49:36.080
And you mentioned the other
thing that he maybe did.

00:49:36.080 --> 00:49:37.580
AUDIENCE: He
introduced the leakage,

00:49:37.580 --> 00:49:39.510
but he didn't
mention that that was

00:49:39.510 --> 00:49:40.510
PROFESSOR: That's right.

00:49:40.510 --> 00:49:44.820
So I guess, he knew that leakage
was going to be a problem.

00:49:44.820 --> 00:49:47.910
I.e, that you want
tight repression.

00:49:47.910 --> 00:49:52.610
So he used these
synthetic promoters

00:49:52.610 --> 00:49:55.910
that both had high
level of expression when

00:49:55.910 --> 00:49:57.875
on but then very low
level of expression

00:49:57.875 --> 00:49:58.750
when being repressed.

00:50:07.800 --> 00:50:10.580
He made this thing.

00:50:10.580 --> 00:50:15.190
And in particular, he
looked at it in a test tube.

00:50:15.190 --> 00:50:18.720
He was able to use, in this
case, IPDG to synchronize them.

00:50:18.720 --> 00:50:22.860
And he looked at the
fluorescence in the test tube.

00:50:22.860 --> 00:50:27.020
So the fluorescence is reporting
on one of the proteins.

00:50:27.020 --> 00:50:29.890
We can call it x if we'd like.

00:50:29.890 --> 00:50:32.850
But fluorescence is kind
of telling about the state.

00:50:32.850 --> 00:50:37.390
And if it starts out, say,
here, he saw a single cycle.

00:50:37.390 --> 00:50:40.550
Damped oscillations, maybe.

00:50:40.550 --> 00:50:44.870
So the question is,
why did this happen?

00:50:54.920 --> 00:50:56.560
So why is it that,
in the test tube,

00:50:56.560 --> 00:51:00.030
he didn't see something
that looked very nice?

00:51:00.030 --> 00:51:02.660
Oscillations.

00:51:02.660 --> 00:51:03.250
Noise.

00:51:03.250 --> 00:51:05.210
And in particular,
what kind of noise?

00:51:05.210 --> 00:51:06.040
Or what's going on?

00:51:11.510 --> 00:51:13.351
Desynchronization, exactly.

00:51:13.351 --> 00:51:15.100
So the idea is that,
even if you start out

00:51:15.100 --> 00:51:17.980
with them all synchronized--
you give it IPDG pulls,

00:51:17.980 --> 00:51:20.105
and they're synchronized
in some way--

00:51:20.105 --> 00:51:21.980
it may be that, at the
beginning, all of them

00:51:21.980 --> 00:51:23.980
are oscillating in
phase with each other.

00:51:23.980 --> 00:51:27.420
But over time, random
noise, phase drift,

00:51:27.420 --> 00:51:31.320
and the different oscillators
leads to some of them

00:51:31.320 --> 00:51:33.320
come down and come back up.

00:51:33.320 --> 00:51:34.644
And then, others are slower.

00:51:34.644 --> 00:51:36.560
You start averaging all
these things together.

00:51:36.560 --> 00:51:42.230
And it leads to damped
oscillations at the test tube

00:51:42.230 --> 00:51:44.530
level within the bulk.

00:51:44.530 --> 00:51:45.100
Yes?

00:51:45.100 --> 00:51:46.555
AUDIENCE: So what do you
mean in the test tube?

00:51:46.555 --> 00:51:48.500
Like, you just take all
these components and put it--

00:51:48.500 --> 00:51:50.624
PROFESSOR: Sorry, when I
say test tube, what I mean

00:51:50.624 --> 00:51:52.100
is that you have all the cells.

00:51:52.100 --> 00:51:54.690
So they still are intact cells.

00:51:54.690 --> 00:51:56.069
But it's just many cells.

00:51:56.069 --> 00:51:58.110
So then, the signal that
you get the fluorescence

00:51:58.110 --> 00:52:00.484
is some average over
all or sum overall.

00:52:00.484 --> 00:52:02.400
The fluorescence you get
from all those cells.

00:52:06.200 --> 00:52:07.850
So there's a sense
that this is really

00:52:07.850 --> 00:52:11.172
what you expect given
the fact that they're

00:52:11.172 --> 00:52:12.130
going to desynchronize.

00:52:12.130 --> 00:52:14.171
Of course, the better the
oscillator in the sense

00:52:14.171 --> 00:52:16.730
that the lower the
phase drift, then maybe

00:52:16.730 --> 00:52:21.110
you can see a slower rate of
this kind of desynchronization.

00:52:21.110 --> 00:52:23.730
But this is really what
you, kind of, expect.

00:52:23.730 --> 00:52:24.230
All right.

00:52:24.230 --> 00:52:26.760
So that's what, maybe,
led him to go and look

00:52:26.760 --> 00:52:28.870
at the single cell
level where he put down

00:52:28.870 --> 00:52:31.240
single cells on this
agar pad and just

00:52:31.240 --> 00:52:34.090
imaged as the cells
oscillated and divided.

00:52:37.170 --> 00:52:40.330
Now there are a
few features that

00:52:40.330 --> 00:52:43.340
are important to
note from the data.

00:52:43.340 --> 00:52:45.480
The first is that
they do oscillate.

00:52:48.590 --> 00:52:51.030
That's a big deal
because this was, indeed,

00:52:51.030 --> 00:52:53.320
the first demonstration
of being able to put

00:52:53.320 --> 00:52:55.070
these random components
together like that

00:52:55.070 --> 00:52:56.520
and generate oscillation.

00:52:56.520 --> 00:52:59.200
But they didn't
oscillate very well.

00:52:59.200 --> 00:53:01.850
So they said, oh, maybe 40%
of the cells oscillated.

00:53:01.850 --> 00:53:07.470
And I have no idea what the
rest of the cells were doing.

00:53:07.470 --> 00:53:09.720
But also, even the cells
that were oscillating, there

00:53:09.720 --> 00:53:12.640
was a fair amount of
noise to the oscillation.

00:53:12.640 --> 00:53:15.110
And the latter
half of this paper

00:53:15.110 --> 00:53:17.620
has a fair amount of discussion
of why that might be.

00:53:17.620 --> 00:53:22.410
And they allude to the ideas
that had been bouncing around

00:53:22.410 --> 00:53:25.150
and from the theoretical
computational side

00:53:25.150 --> 00:53:28.380
demonstrating that it may
be that the low numbers

00:53:28.380 --> 00:53:31.140
of proteins, genes involved
here could introduce

00:53:31.140 --> 00:53:34.900
stochastic noise into the
system and, thus, lead

00:53:34.900 --> 00:53:38.060
to this kind of phase drift that
was observed experimentally.

00:53:38.060 --> 00:53:39.970
I think that this
basic observation

00:53:39.970 --> 00:53:42.900
that Michael had that
he got oscillations,

00:53:42.900 --> 00:53:43.780
but they were noisy.

00:53:43.780 --> 00:53:45.363
That is probably
what led him to start

00:53:45.363 --> 00:53:48.570
thinking more and more about
the role of noise in G networks

00:53:48.570 --> 00:53:51.410
and so forth and led,
later, to another hugely

00:53:51.410 --> 00:53:54.790
influential paper that is
not going to be a required

00:53:54.790 --> 00:53:56.420
reading in this
class but is listed

00:53:56.420 --> 00:53:59.780
under the optional reading,
if you're interested.

00:53:59.780 --> 00:54:03.360
But we'll really get into this
question of noise more a couple

00:54:03.360 --> 00:54:04.752
weeks from now.

00:54:08.520 --> 00:54:12.890
Were there any other questions
about the experimental side

00:54:12.890 --> 00:54:15.560
of this paper?

00:54:15.560 --> 00:54:19.666
I wanted to analyze maybe a
little bit of simple model

00:54:19.666 --> 00:54:20.540
of the repressilator.

00:54:26.782 --> 00:54:28.240
So the model that
they used to help

00:54:28.240 --> 00:54:31.870
them design this experiment
involved all three proteins,

00:54:31.870 --> 00:54:32.720
all three mRNAs.

00:54:32.720 --> 00:54:35.220
And what that means is that,
when you go and you do a model,

00:54:35.220 --> 00:54:37.830
you're going to end up
with a six by six matrix.

00:54:37.830 --> 00:54:40.164
And I don't have boards
that are big enough.

00:54:40.164 --> 00:54:41.580
So what I'm going
to do instead is

00:54:41.580 --> 00:54:45.860
I'm going to analyze just
the protein only version

00:54:45.860 --> 00:54:47.000
model of the repressilator.

00:54:52.300 --> 00:54:52.800
All right.

00:55:11.630 --> 00:55:15.080
So what we have here
is three proteins.

00:55:15.080 --> 00:55:18.030
p1 2 3 p1 dot.

00:55:18.030 --> 00:55:20.885
And we have degradation
of this protein.

00:55:20.885 --> 00:55:23.540
And we're going to analyze
the symmetric version, just

00:55:23.540 --> 00:55:25.430
like what Michael did.

00:55:25.430 --> 00:55:28.070
So that means we're assuming
that all the proteins are

00:55:28.070 --> 00:55:28.570
equivalent.

00:55:28.570 --> 00:55:31.919
I'm sure that's not true because
these are different promoters

00:55:31.919 --> 00:55:32.960
and different everything.

00:55:32.960 --> 00:55:35.210
But this gives us the intuition.

00:55:35.210 --> 00:55:35.920
So it's minus p1.

00:55:35.920 --> 00:55:40.870
And this is protein 1 is
repressed by trajectory protein

00:55:40.870 --> 00:55:41.370
3.

00:55:47.050 --> 00:55:50.400
Protein 2 is going to be
repressed by protein 1.

00:56:00.130 --> 00:56:05.360
And then protein 3 is going
to be repressed by protein 2.

00:56:15.790 --> 00:56:18.105
So this is what you would
call the protein only model

00:56:18.105 --> 00:56:18.980
of the repressilator.

00:56:21.610 --> 00:56:23.630
Now just as before,
the fixed points

00:56:23.630 --> 00:56:28.480
are when the pi
dots are equal to 0.

00:56:28.480 --> 00:56:33.370
And we get the same
equation that we, basically,

00:56:33.370 --> 00:56:37.160
had before where the
equilibrium or the fixed point,

00:56:37.160 --> 00:56:43.030
again, is going to be given by
something that looks like this.

00:56:43.030 --> 00:56:47.020
So it's the same requirement
that we had before.

00:56:51.470 --> 00:56:55.500
Now the question is, how
can we get the stability

00:56:55.500 --> 00:56:56.840
of that internal fixed point?

00:56:56.840 --> 00:57:00.280
It's worth mentioning here that
now we have three proteins.

00:57:00.280 --> 00:57:04.360
So the trajectories are in
this three dimensional space.

00:57:04.360 --> 00:57:08.260
So from a mathematical
standpoint,

00:57:08.260 --> 00:57:11.010
determining the stability of
that internal fixed point is

00:57:11.010 --> 00:57:13.410
actually not sufficient to
tell you that there has to be

00:57:13.410 --> 00:57:15.830
oscillations or there cannot
be oscillations because these

00:57:15.830 --> 00:57:18.121
trajectories are, in principal,
allowed to do all sorts

00:57:18.121 --> 00:57:20.170
of crazy things in
three dimensions.

00:57:20.170 --> 00:57:23.210
But it turns out
that it still ends up

00:57:23.210 --> 00:57:28.465
being true here that when this
internal fixed point is stable,

00:57:28.465 --> 00:57:29.590
you don't get oscillations.

00:57:29.590 --> 00:57:32.200
And when it's unstable, you do.

00:57:32.200 --> 00:57:33.740
But that, sort of,
didn't have to be

00:57:33.740 --> 00:57:38.720
true from a
mathematical standpoint.

00:57:41.430 --> 00:57:41.930
All right.

00:57:41.930 --> 00:57:46.740
Now since this is now going
to be a 3 by 3 matrix,

00:57:46.740 --> 00:57:52.630
we're going to have to
calculate those eigenvalues.

00:57:52.630 --> 00:57:56.690
Now how many eigenvalues
are there going to be?

00:57:56.690 --> 00:57:57.930
Three?

00:57:57.930 --> 00:57:58.430
OK.

00:58:01.140 --> 00:58:05.840
So this thing I've written
in the form of a matrix

00:58:05.840 --> 00:58:07.630
to help us out a little bit.

00:58:07.630 --> 00:58:10.610
But in particular, we're
going to get the same thing

00:58:10.610 --> 00:58:14.410
that we had before,
which is the p1 tilde.

00:58:14.410 --> 00:58:20.421
So these are deviations,
again, from the fixed point.

00:58:20.421 --> 00:58:22.670
And we got this matrix that's
going to look like this.

00:58:22.670 --> 00:58:24.510
Minus 1, again, 0.

00:58:24.510 --> 00:58:29.234
It's the same x that we had
before conveniently still

00:58:29.234 --> 00:58:29.775
on the board.

00:58:39.610 --> 00:58:43.820
So this is just after we
take these derivatives.

00:58:43.820 --> 00:58:52.710
And then, we have p1 tilde,
p2 tilde, and p3 tilde.

00:58:52.710 --> 00:58:55.440
Now what we need to know
is, for this Jacobian,

00:58:55.440 --> 00:58:58.920
what are going to
be the eigenvalues?

00:58:58.920 --> 00:59:01.318
For this thing to be
stable, it requires what?

00:59:08.300 --> 00:59:12.090
What's the requirement for
stability of that fixed point?

00:59:12.090 --> 00:59:14.704
That p0?

00:59:14.704 --> 00:59:17.010
AUDIENCE: [INAUDIBLE].

00:59:17.010 --> 00:59:17.920
PROFESSOR: OK.

00:59:17.920 --> 00:59:18.500
Right.

00:59:18.500 --> 00:59:22.030
For two dimensions, this trace
and determinant condition

00:59:22.030 --> 00:59:22.920
works.

00:59:22.920 --> 00:59:25.680
It's important to say that that
only works for two dimensions,

00:59:25.680 --> 00:59:28.035
actually, the rule about
traces and determinants.

00:59:31.980 --> 00:59:33.889
So be careful.

00:59:33.889 --> 00:59:35.430
So what's the more
general statement?

00:59:35.430 --> 00:59:35.930
Yeah.

00:59:35.930 --> 00:59:38.009
AUDIENCE: Negative eigenvalues.

00:59:38.009 --> 00:59:38.800
PROFESSOR: Exactly.

00:59:38.800 --> 00:59:41.695
So in order for that
fixed point to be stable,

00:59:41.695 --> 00:59:44.360
it requires that
all the eigenvalues

00:59:44.360 --> 00:59:48.094
have real parts less than 0.

00:59:48.094 --> 00:59:50.510
So in order to determine the
stability of the fixed point,

00:59:50.510 --> 00:59:55.100
we need to ask what are the
eigenvalues of this matrix.

00:59:55.100 --> 00:59:58.710
And to get the
eigenvalues, what we do

00:59:58.710 --> 01:00:01.480
is we calculate this
characteristic equation,

01:00:01.480 --> 01:00:05.570
this thing that we learned about
in linear algebra and so forth.

01:00:05.570 --> 01:00:08.670
What we do is we take-- all
right, this is the matrix A,

01:00:08.670 --> 01:00:09.550
we'll say.

01:00:09.550 --> 01:00:12.410
This is matrix A. And
what we want to do

01:00:12.410 --> 01:00:22.100
is we want to ask
whether the determinant

01:00:22.100 --> 01:00:30.540
of the matrix A minus some
eigenvalue times the identity

01:00:30.540 --> 01:00:31.230
matrix.

01:00:31.230 --> 01:00:33.600
We want this thing
to be equal to 0.

01:00:33.600 --> 01:00:40.120
So this is how we determine
what the eigenvalues are.

01:00:40.120 --> 01:00:43.510
And this is not as bad as it
could be for general three

01:00:43.510 --> 01:00:47.180
by three matrices because a
lot of these things are 0.

01:00:47.180 --> 01:00:50.090
So this thing is just
this is the determinant

01:00:50.090 --> 01:00:51.430
of the following matrix.

01:00:54.270 --> 01:00:57.970
So we have minus
1 minus lambda 0.

01:00:57.970 --> 01:01:01.550
This thing x that's,
in principle, bad.

01:01:01.550 --> 01:01:04.870
Minus 1 minus lambda 0.

01:01:04.870 --> 01:01:09.070
Getting 0 x minus
1 minus lambda.

01:01:09.070 --> 01:01:10.860
Now to take the
determinant three by three

01:01:10.860 --> 01:01:13.250
matrix, remember, you can
say, well, this determinant

01:01:13.250 --> 01:01:16.700
is going to be equal
to-- we have this term.

01:01:16.700 --> 01:01:23.040
So this is a minus 1 plus a
lambda times the determinant

01:01:23.040 --> 01:01:24.950
of this matrix.

01:01:24.950 --> 01:01:27.030
And then, we just
have that's this.

01:01:27.030 --> 01:01:29.030
The product of these minus
the product of these.

01:01:29.030 --> 01:01:31.080
So this just gives
us this thing again.

01:01:31.080 --> 01:01:36.710
So this is actually just
minus 1 plus lambda cubed.

01:01:36.710 --> 01:01:37.729
Next term, this is 0.

01:01:37.729 --> 01:01:38.270
That's great.

01:01:38.270 --> 01:01:39.686
We don't need to
worry about that.

01:01:39.686 --> 01:01:41.390
The next one, we get plus.

01:01:41.390 --> 01:01:43.470
We have an x.

01:01:43.470 --> 01:01:45.640
Determining here, we
get, again, x squared.

01:01:45.640 --> 01:01:48.814
So this is just an x cubed.

01:01:48.814 --> 01:01:49.980
We want the same equal to 0.

01:01:49.980 --> 01:01:52.780
So we actually get a
very simple requirement

01:01:52.780 --> 01:01:56.280
for the eigenvalues,
which is that 1

01:01:56.280 --> 01:02:02.335
plus the eigenvalues cubed is
equal to this thing x cubed.

01:02:02.335 --> 01:02:03.710
Now be careful
because, remember,

01:02:03.710 --> 01:02:08.310
x is actually a negative number.

01:02:08.310 --> 01:02:11.230
So watch out.

01:02:11.230 --> 01:02:13.840
So I think that the best
way to get a sense of what

01:02:13.840 --> 01:02:15.670
this thing is is to plot it.

01:02:28.380 --> 01:02:29.940
Of course, it's a
little bit tempting

01:02:29.940 --> 01:02:31.740
here to just say,
all right, well,

01:02:31.740 --> 01:02:34.360
can we just say that 1
plus lambda is equal to x?

01:02:40.210 --> 01:02:41.139
No.

01:02:41.139 --> 01:02:42.430
So what's the matter with that?

01:02:46.960 --> 01:02:51.790
I mean, it's, sort of,
true, maybe, possibly.

01:02:51.790 --> 01:02:52.290
Right.

01:02:52.290 --> 01:02:55.030
So the problem
here is that we're

01:02:55.030 --> 01:02:58.740
supposed to be getting
three different eigenvalues.

01:02:58.740 --> 01:03:00.220
Or at lease, it's
possible to get

01:03:00.220 --> 01:03:02.520
three different eigenvalues.

01:03:02.520 --> 01:03:05.040
So this is really specifying
the solution for 1

01:03:05.040 --> 01:03:07.400
plus lambda on
the complex plane.

01:03:07.400 --> 01:03:12.010
So the solution
for 1 plus lambda

01:03:12.010 --> 01:03:14.350
we can get by
thinking about this

01:03:14.350 --> 01:03:17.920
is the real part
of 1 plus lambda.

01:03:17.920 --> 01:03:22.720
And this is the imaginary
part of 1 plus lambda.

01:03:22.720 --> 01:03:34.015
And we know that one solution
is going to be out here at x.

01:03:37.830 --> 01:03:40.090
This distance here is
the magnitude of x.

01:03:44.870 --> 01:03:46.880
Now the others,
however, are going

01:03:46.880 --> 01:03:52.349
to be around the complex plane
similar distances where we get

01:03:52.349 --> 01:03:53.640
something that looks like this.

01:03:57.620 --> 01:04:01.270
So these are, like,
30, 60, 90 triangle.

01:04:01.270 --> 01:04:03.440
So this is 30 degrees
here because what

01:04:03.440 --> 01:04:06.370
you see is that, for each of
these three solutions for 1

01:04:06.370 --> 01:04:14.320
plus lambda, if you cube
them, you end up with x cubed.

01:04:14.320 --> 01:04:18.060
So this guy, you square it.

01:04:18.060 --> 01:04:18.560
Cube.

01:04:18.560 --> 01:04:20.640
You end up back here.

01:04:20.640 --> 01:04:24.050
This one, if you cube
it, you start out here,

01:04:24.050 --> 01:04:26.530
squared, and then cubed
comes back out here.

01:04:26.530 --> 01:04:29.896
Same thing and this
goes around somehow.

01:04:29.896 --> 01:04:31.570
All right, so there
are three solutions

01:04:31.570 --> 01:04:33.280
to this 1 plus lambda.

01:04:33.280 --> 01:04:37.127
And there are these points here.

01:04:37.127 --> 01:04:38.710
Now, of course, it's
not 1 plus lambda

01:04:38.710 --> 01:04:40.293
that we actually
wanted to know about.

01:04:40.293 --> 01:04:42.490
It was lambda.

01:04:42.490 --> 01:04:46.730
But if we know what
1 plus lambda is,

01:04:46.730 --> 01:04:48.850
then we can get what lambda is.

01:04:48.850 --> 01:04:50.800
What do we have to do?

01:04:50.800 --> 01:04:52.560
Right, we have to
slide it to the left.

01:04:52.560 --> 01:04:54.910
So this is the real axis.

01:04:54.910 --> 01:04:56.310
This is the imaginary axis.

01:04:56.310 --> 01:04:58.830
1.

01:04:58.830 --> 01:05:02.600
So we have to move
everything over 1.

01:05:02.600 --> 01:05:04.530
Now remember, the
requirement for stability

01:05:04.530 --> 01:05:07.540
was that all of the
eigenvalues have

01:05:07.540 --> 01:05:09.071
real parts that were negative.

01:05:09.071 --> 01:05:11.570
That means the requirement for
stability of that fixed point

01:05:11.570 --> 01:05:13.350
is that all three of
these fixed points

01:05:13.350 --> 01:05:16.470
are in the left
half of the plane.

01:05:16.470 --> 01:05:19.420
So what you can see is
that, in this problem,

01:05:19.420 --> 01:05:21.695
the whole question of
stability and whether we

01:05:21.695 --> 01:05:25.220
get oscillations boils down
to how big this thing is.

01:05:25.220 --> 01:05:26.310
What's this distance?

01:05:26.310 --> 01:05:29.950
If this distance is more
than 1, then we subtract 1,

01:05:29.950 --> 01:05:35.931
we don't get it into the
left part of the plane.

01:05:35.931 --> 01:05:38.370
OK, I can't remember
which case I just gave.

01:05:38.370 --> 01:05:42.119
But yeah, we need to know
whether this thing is

01:05:42.119 --> 01:05:43.160
larger or smaller than 1.

01:05:45.820 --> 01:05:48.795
And that has to do with
the magnitude of x.

01:05:51.600 --> 01:05:56.535
So if the magnitude
of x-- do you

01:05:56.535 --> 01:05:59.872
guys remember your geometry
for a 30, 60, 90 triangle?

01:06:03.099 --> 01:06:06.550
All right, so if the
magnitude of x-- and this

01:06:06.550 --> 01:06:08.760
is indeed the magnitude of x.

01:06:08.760 --> 01:06:16.320
This short edge on a 30, 60, 90
is half the long edge, right?

01:06:16.320 --> 01:06:21.220
So what we can say is that
this fixed point stable, state

01:06:21.220 --> 01:06:30.610
fixed point, is if and only
if the magnitude of x is what?

01:06:35.630 --> 01:06:37.840
Lesson two.

01:06:37.840 --> 01:06:38.930
OK.

01:06:38.930 --> 01:06:41.140
That's nice.

01:06:41.140 --> 01:06:44.100
And if we want, we could plug
in-- just to ride this out.

01:06:44.100 --> 01:06:48.320
This is n alpha p 0 n minus 1.

01:07:05.100 --> 01:07:08.000
So it's useful, once you
get to something like this,

01:07:08.000 --> 01:07:12.332
to try to just ask, for
various kind of values,

01:07:12.332 --> 01:07:13.290
how does this play out?

01:07:13.290 --> 01:07:18.062
What does the
requirement end up being?

01:07:18.062 --> 01:07:19.770
And a useful limit is
to think about what

01:07:19.770 --> 01:07:22.070
happens in the limit of
very strong expression?

01:07:22.070 --> 01:07:24.110
So strong expression
corresponds to what?

01:07:32.696 --> 01:07:34.127
AUDIENCE: Big alpha?

01:07:34.127 --> 01:07:35.080
PROFESSOR: Big alpha.

01:07:35.080 --> 01:07:35.780
Yes, perfect.

01:07:38.750 --> 01:07:44.730
And it turns out, big
alpha is a little bit-- OK,

01:07:44.730 --> 01:07:47.340
and remember we have to
remember what p0 was.

01:07:47.340 --> 01:07:50.766
p0 was this p0 times 1 plus p0.

01:07:54.877 --> 01:07:56.460
All right, so this
is the requirement.

01:08:00.860 --> 01:08:04.840
And actually, if you
play with these equations

01:08:04.840 --> 01:08:06.300
just a little bit,
what you'll find

01:08:06.300 --> 01:08:09.920
is that, if alpha is
much larger than 1,

01:08:09.920 --> 01:08:18.340
then this requirement
is that n is less than 2

01:08:18.340 --> 01:08:19.359
or less than around 2.

01:08:24.080 --> 01:08:26.479
This is saying,
on the flip side,

01:08:26.479 --> 01:08:32.500
the fixed point is stable if
you don't have very strong

01:08:32.500 --> 01:08:33.920
cooperativity and repression.

01:08:33.920 --> 01:08:36.740
And the flip side is, if you
have strong cooperativity

01:08:36.740 --> 01:08:39.830
of repression, then you
can get oscillations

01:08:39.830 --> 01:08:42.029
because this interior fixed
point becomes unstable.

01:08:44.910 --> 01:08:49.754
So this is also saying that
n greater than around 2

01:08:49.754 --> 01:08:50.670
leads to oscillations.

01:09:14.370 --> 01:09:16.590
And this maybe
makes sense because,

01:09:16.590 --> 01:09:19.140
when you have strong
productivity in the repression

01:09:19.140 --> 01:09:21.390
there, what that's
telling you is that it's

01:09:21.390 --> 01:09:22.979
a switch like response.

01:09:22.979 --> 01:09:28.039
And in that regime,
it maybe becomes more

01:09:28.039 --> 01:09:29.580
like a simple Boolean
kind of network

01:09:29.580 --> 01:09:31.663
where, if you just write
down the ones and zeroes,

01:09:31.663 --> 01:09:34.494
you can convince yourself that
this thing maybe, in principle,

01:09:34.494 --> 01:09:35.160
could oscillate.

01:09:38.010 --> 01:09:41.540
Now if you look at the
Elowitz repressilator paper,

01:09:41.540 --> 01:09:46.270
you'll see that he gives
some expression for what

01:09:46.270 --> 01:09:48.516
this thing should be like.

01:09:48.516 --> 01:09:51.710
And it looks vaguely similar.

01:09:51.710 --> 01:09:56.540
Of course, there he's
including the mRNAs, as well.

01:09:56.540 --> 01:09:59.420
But if you think that this
was painful to do in class,

01:09:59.420 --> 01:10:01.460
then including the
mRNAs is more painful.

01:10:07.680 --> 01:10:14.764
Are there any questions
about this idea?

01:10:14.764 --> 01:10:15.264
Yeah.

01:10:15.264 --> 01:10:17.847
AUDIENCE: So in the paper, did
they also only do the stability

01:10:17.847 --> 01:10:20.104
analysis to determine the--

01:10:20.104 --> 01:10:23.390
PROFESSOR: I think they
did simulations, as well.

01:10:23.390 --> 01:10:27.330
So the nature of simulations is
that you can convince yourself

01:10:27.330 --> 01:10:31.790
that their exist places that do
oscillate or don't oscillate.

01:10:31.790 --> 01:10:33.800
Although, you'll notice
that they have a very,

01:10:33.800 --> 01:10:36.850
kind of, enigmatic
sentence in here,

01:10:36.850 --> 01:10:39.160
which is that it is
possible that, in addition

01:10:39.160 --> 01:10:41.770
to simple oscillations, this
and more realistic models

01:10:41.770 --> 01:10:45.220
may exhibit other complex
types of dynamic behavior.

01:10:45.220 --> 01:10:47.640
And this is just
a way of saying,

01:10:47.640 --> 01:10:49.810
well, you know, I don't know.

01:10:49.810 --> 01:10:52.990
Maybe someday
because once you talk

01:10:52.990 --> 01:10:55.870
about six dimensional
system, you never

01:10:55.870 --> 01:10:58.750
know if you've explored
all of the parameter space.

01:10:58.750 --> 01:11:00.170
I mean, even for
fixed parameters,

01:11:00.170 --> 01:11:02.720
you don't know if you started
at all the right locations.

01:11:02.720 --> 01:11:06.047
You can kind develop
some sense that, oh,

01:11:06.047 --> 01:11:08.380
this thing seems to oscillate
or seems to not oscillate.

01:11:08.380 --> 01:11:10.171
And it does correspond
to these conditions.

01:11:10.171 --> 01:11:13.080
But you don't know.

01:11:13.080 --> 01:11:15.270
I mean, it could be
that, in some regions,

01:11:15.270 --> 01:11:16.590
you get chaos or other things.

01:11:16.590 --> 01:11:17.340
Right?

01:11:17.340 --> 01:11:20.650
So it's funny because I've
read this paper many times.

01:11:20.650 --> 01:11:22.905
But it was only last night
when I was re-reading it

01:11:22.905 --> 01:11:24.529
that I kind of thought
about that sense

01:11:24.529 --> 01:11:28.600
like, yeah, I'm not sure either
what this model could possibly

01:11:28.600 --> 01:11:29.361
do.

01:11:29.361 --> 01:11:29.860
Yes?

01:11:29.860 --> 01:11:31.348
AUDIENCE: In this
linear analysis

01:11:31.348 --> 01:11:33.780
the three x's are the same.

01:11:33.780 --> 01:11:34.780
PROFESSOR: That's right.

01:11:34.780 --> 01:11:36.529
AUDIENCE: Because
they're non-dimensional?

01:11:39.375 --> 01:11:40.250
PROFESSOR: All right.

01:11:40.250 --> 01:11:42.370
So the reason that the
three X's are the same

01:11:42.370 --> 01:11:46.250
is because we've
assumed that this really

01:11:46.250 --> 01:11:52.067
is the symmetric version
of the repressilator

01:11:52.067 --> 01:11:54.150
because we're assuming
that all of the alphas, all

01:11:54.150 --> 01:11:55.608
the ends, all the
K's, everything's

01:11:55.608 --> 01:11:57.090
the same across
all three of them.

01:11:57.090 --> 01:11:58.650
So given that
symmetry, then you're

01:11:58.650 --> 01:12:02.190
always going to end up with
a symmetric version of this.

01:12:02.190 --> 01:12:05.690
So I think if it were
asymmetric and then you

01:12:05.690 --> 01:12:07.880
made the non-dimensional
versions of things,

01:12:07.880 --> 01:12:10.555
I think you still won't end
up getting the same X's just

01:12:10.555 --> 01:12:12.680
because, if it's asymmetric,
then and something has

01:12:12.680 --> 01:12:13.388
to be asymmetric.

01:12:16.382 --> 01:12:16.882
Yes?

01:12:16.882 --> 01:12:21.489
AUDIENCE: [INAUDIBLE] so
large alpha leads to--

01:12:21.489 --> 01:12:22.280
PROFESSOR: Yes, OK.

01:12:22.280 --> 01:12:24.430
We can go ahead and do this.

01:12:37.020 --> 01:12:41.350
So for large alpha,
this fixed point

01:12:41.350 --> 01:12:48.360
is going to be-- p0 is going
to be much larger than 1.

01:12:48.360 --> 01:12:51.190
So this is about
p0 to the n plus 1.

01:12:51.190 --> 01:12:53.615
We can neglect the
1 for large alpha.

01:12:53.615 --> 01:12:54.990
And then and then
what we can say

01:12:54.990 --> 01:13:00.580
is that, over here, for
example, if we multiply

01:13:00.580 --> 01:13:07.520
both sides by-- p0 squared, p0
squared, so multiply it by one.

01:13:07.520 --> 01:13:11.650
Then this down here is
definitely alpha squared.

01:13:11.650 --> 01:13:13.350
And then, up here,
what we have is

01:13:13.350 --> 01:13:17.190
p0 to the n plus 1, which
we decided was around

01:13:17.190 --> 01:13:20.980
alpha for a strong alpha.

01:13:20.980 --> 01:13:26.770
So that gives us alpha times
alpha divided by alpha squared.

01:13:26.770 --> 01:13:30.410
So this actually all goes
away for large alpha.

01:13:30.410 --> 01:13:33.630
So then, you're just
left with n less than 2.

01:13:33.630 --> 01:13:34.130
Did that--

01:13:34.130 --> 01:13:35.030
AUDIENCE: Sorry.

01:13:35.030 --> 01:13:36.380
Where did that top right
equation come from?

01:13:36.380 --> 01:13:38.280
PROFESSOR: OK, so this
equation here is this

01:13:38.280 --> 01:13:42.140
is the solution for where
that fixed point is.

01:13:42.140 --> 01:13:47.780
So in this space of the p0's,
if you set the equations

01:13:47.780 --> 01:13:49.770
for p1, p2, p3, if you
set that equal to 0,

01:13:49.770 --> 01:13:54.880
this is the expression always
for a large alpha, small.

01:13:54.880 --> 01:13:59.180
So this is a need be
location of that fixed point.

01:13:59.180 --> 01:14:01.020
And it's just, as
alpha is large,

01:14:01.020 --> 01:14:04.240
then we get that
p0 to the n plus 1

01:14:04.240 --> 01:14:05.920
is approximately equal to alpha.

01:14:05.920 --> 01:14:08.326
And this is for alpha
much greater than 1.

01:14:08.326 --> 01:14:11.400
And in that case, all of
these things just go away.

01:14:11.400 --> 01:14:14.890
And you're just left
with n less than 2.

01:14:14.890 --> 01:14:20.370
So for example, as alpha
goes down in magnitude,

01:14:20.370 --> 01:14:23.090
then you end up getting a
requirement that oscillations

01:14:23.090 --> 01:14:24.158
require a larger n.

01:14:32.110 --> 01:14:33.640
We'll give you practice on this.

01:14:33.640 --> 01:14:46.680
All right, so I think I wrote
another-- if I can find my--

01:14:46.680 --> 01:14:48.430
you can ask for
alpha equal to 2.

01:14:51.180 --> 01:14:54.090
What n required
for oscillations.

01:15:05.589 --> 01:15:07.130
I'll let you start
playing with that.

01:15:07.130 --> 01:15:09.912
And I will make sure
that I've given you

01:15:09.912 --> 01:15:10.870
the right alpha to use.

01:15:36.110 --> 01:15:37.910
So in this case,
what we're asking is,

01:15:37.910 --> 01:15:41.740
instead of having really strong
maximal expression, if instead

01:15:41.740 --> 01:15:44.465
expression is just not quite
as strong, then what we'll find

01:15:44.465 --> 01:15:45.840
is that you actually
need to have

01:15:45.840 --> 01:15:49.170
a more cooperative repression
in order to get oscillations.

01:15:49.170 --> 01:15:51.500
And that's just because,
if alpha is equal to 2,

01:15:51.500 --> 01:15:53.922
then we can, kind of, figure
out what p0 is equal to.

01:15:57.370 --> 01:15:57.950
1.

01:15:57.950 --> 01:15:58.860
Right.

01:15:58.860 --> 01:16:00.190
Great.

01:16:00.190 --> 01:16:03.570
So the fixed point is at one.

01:16:03.570 --> 01:16:05.660
That's great because this
we can then figure out.

01:16:05.660 --> 01:16:06.560
Right?

01:16:06.560 --> 01:16:08.230
So this is 1 plus 1 square.

01:16:08.230 --> 01:16:09.880
That's a four.

01:16:09.880 --> 01:16:10.760
1.

01:16:10.760 --> 01:16:11.260
2.

01:16:13.800 --> 01:16:16.380
So this tells us
that, in this case,

01:16:16.380 --> 01:16:19.710
we need to have very
cooperative repression.

01:16:19.710 --> 01:16:22.660
We have to have an n
greater than around 4

01:16:22.660 --> 01:16:26.980
in order to get oscillations
in this protein only model.

01:16:31.330 --> 01:16:31.830
Yes?

01:16:31.830 --> 01:16:33.538
AUDIENCE: It is kind
of strange that even

01:16:33.538 --> 01:16:37.650
for a really big alpha you
still need n greater than sum.

01:16:37.650 --> 01:16:39.064
PROFESSOR: Yeah, right, right.

01:16:39.064 --> 01:16:40.480
So this is an
interesting question

01:16:40.480 --> 01:16:45.130
that you might think that
for a very large expression

01:16:45.130 --> 01:16:49.020
that you wouldn't need to have
cooperative repression at all.

01:16:49.020 --> 01:16:49.750
Right?

01:16:49.750 --> 01:16:54.880
And I can't say that I have any
wonderful intuition about this

01:16:54.880 --> 01:16:58.440
because it, somehow,
has to do with just

01:16:58.440 --> 01:17:01.507
the slopes of those curves
around that fixed point.

01:17:01.507 --> 01:17:02.715
And it's in three dimensions.

01:17:06.050 --> 01:17:08.900
But I think that this
highlights that it's

01:17:08.900 --> 01:17:11.160
a priori if you
go and say, oh, I

01:17:11.160 --> 01:17:14.905
want to construct
this repressilator,

01:17:14.905 --> 01:17:17.640
it's maybe not even obvious that
you want it to be more or less.

01:17:17.640 --> 01:17:19.640
I mean, you might not
even think about this idea

01:17:19.640 --> 01:17:20.870
of cooperative repression.

01:17:20.870 --> 01:17:25.240
You might be tempted to
think that any chain of three

01:17:25.240 --> 01:17:27.330
proteins repressing each
other just, kind of,

01:17:27.330 --> 01:17:28.314
has to oscillate.

01:17:28.314 --> 01:17:29.980
I mean, there's a
little bit of a sense.

01:17:29.980 --> 01:17:34.180
And that's the logic that
you get at if you just do

01:17:34.180 --> 01:17:34.980
0's and 1's.

01:17:34.980 --> 01:17:36.030
If you say, oh, here's x.

01:17:36.030 --> 01:17:37.000
Here's y.

01:17:37.000 --> 01:17:37.880
Here's z.

01:17:37.880 --> 01:17:39.310
And they're
repressing each other.

01:17:39.310 --> 01:17:40.062
Right?

01:17:40.062 --> 01:17:45.470
And you say, oh, OK, well if
I start out at, say, 0 1 0

01:17:45.470 --> 01:17:47.950
and you say, OK,
that's all fine.

01:17:47.950 --> 01:17:49.400
But OK, so this is repressing.

01:17:49.400 --> 01:17:51.608
And it's OK, but this guy
wasn't repressing this one.

01:17:51.608 --> 01:17:54.616
So now we get a 1, 1, 0, maybe.

01:17:54.616 --> 01:17:55.490
Then you say, oh, OK.

01:17:55.490 --> 01:17:57.870
Well now this guy starts
repressing this one.

01:17:57.870 --> 01:18:00.170
So now it gives us a 1 0 0.

01:18:00.170 --> 01:18:05.255
And what you see is that, over
these two steps, the on protein

01:18:05.255 --> 01:18:05.755
has shifted.

01:18:05.755 --> 01:18:07.120
And indeed, that's
going to continue

01:18:07.120 --> 01:18:08.250
going all the way around.

01:18:08.250 --> 01:18:12.720
So from this Boolean
logic kind of perspective,

01:18:12.720 --> 01:18:15.930
you might think that any
three proteins mutually

01:18:15.930 --> 01:18:18.332
repressing each other
just has to oscillate.

01:18:18.332 --> 01:18:20.290
And it's only by looking
at things a little bit

01:18:20.290 --> 01:18:21.998
more carefully that
you say, oh, well, we

01:18:21.998 --> 01:18:25.520
have to actually worry
about this that you really

01:18:25.520 --> 01:18:30.110
have to think about you want
to choose some transcription

01:18:30.110 --> 01:18:33.176
factors that are multimerizing
and cooperatively repressing

01:18:33.176 --> 01:18:36.930
the next protein just to have
some reasonable shot at having

01:18:36.930 --> 01:18:40.925
this thing actually oscillate.

01:18:40.925 --> 01:18:44.010
AUDIENCE: So in this, we
might still be able-- I mean,

01:18:44.010 --> 01:18:47.700
oscillations like this
might still [INAUDIBLE]

01:18:47.700 --> 01:18:50.188
but just not like,
maybe, oscillations

01:18:50.188 --> 01:18:51.938
around some stable fix
point or something.

01:18:51.938 --> 01:18:56.577
Like, they're just not
limit cycle oscillations.

01:18:56.577 --> 01:18:57.994
Do you think that
in a [INAUDIBLE]

01:18:57.994 --> 01:19:00.285
there would probably still
be some kind of oscillations

01:19:00.285 --> 01:19:00.960
somewhere.

01:19:00.960 --> 01:19:04.346
Just not this beautiful
limit cycle kind.

01:19:04.346 --> 01:19:05.720
PROFESSOR: Yeah,
my understanding

01:19:05.720 --> 01:19:09.820
is that in, for example,
this protein only model

01:19:09.820 --> 01:19:13.610
of the repressilator
that if you do not

01:19:13.610 --> 01:19:16.280
have cooperative repression,
then it really just

01:19:16.280 --> 01:19:17.904
goes to that stable fixed point.

01:19:17.904 --> 01:19:19.570
Of course, you have
to worry about maybe

01:19:19.570 --> 01:19:22.200
these noise-induced
oscillation ideas.

01:19:22.200 --> 01:19:24.690
But at least within the
realm of the deterministic,

01:19:24.690 --> 01:19:28.130
differential equations,
then the system

01:19:28.130 --> 01:19:30.900
just goes to that internal fixed
point that's specified by this.

01:19:34.181 --> 01:19:34.680
Question?

01:19:34.680 --> 01:19:41.750
AUDIENCE: Can we think like
that the cooperation, sort of,

01:19:41.750 --> 01:19:44.570
introduced delay?

01:19:44.570 --> 01:19:47.130
PROFESSOR: That's an
interesting question.

01:19:47.130 --> 01:19:50.290
Whether cooperativity, maybe,
is introducing a delay.

01:19:50.290 --> 01:19:52.400
And that's because, after
the proteins are made,

01:19:52.400 --> 01:19:55.290
maybe it takes some extra
time to dimer and so forth.

01:19:55.290 --> 01:19:58.000
So that statement may be true.

01:19:58.000 --> 01:19:59.950
But it's not relevant.

01:19:59.950 --> 01:20:00.530
OK?

01:20:00.530 --> 01:20:02.220
And I think this
is very important.

01:20:02.220 --> 01:20:06.170
This model has certainly
not taken that into account.

01:20:06.170 --> 01:20:09.200
So the mechanism that's here
is not what you're saying.

01:20:09.200 --> 01:20:11.440
But it may be true that,
for any experimental system,

01:20:11.440 --> 01:20:14.935
such delay from dimerization
is relevant and helps

01:20:14.935 --> 01:20:15.810
you get oscillations.

01:20:15.810 --> 01:20:16.310
Right?

01:20:16.310 --> 01:20:19.180
But at least within the
realm of this model,

01:20:19.180 --> 01:20:21.910
we have very much not included
any sort of delay associated

01:20:21.910 --> 01:20:23.400
with dimerization or anything.

01:20:23.400 --> 01:20:25.340
So that is very much
not the explanation

01:20:25.340 --> 01:20:28.300
for why dimerization leads
to oscillations here.

01:20:28.300 --> 01:20:30.700
And I think this
is a wider point

01:20:30.700 --> 01:20:33.460
that it's very important
always to keep track

01:20:33.460 --> 01:20:35.815
of which effects you've
included in any given analysis

01:20:35.815 --> 01:20:37.900
and which ones are not.

01:20:37.900 --> 01:20:39.970
And it's very, very common.

01:20:39.970 --> 01:20:41.470
There are many
things that are true.

01:20:41.470 --> 01:20:44.390
But they may not actually be
relevant for the discussion

01:20:44.390 --> 01:20:44.985
at hand.

01:20:44.985 --> 01:20:46.360
And I think, in
those situations,

01:20:46.360 --> 01:20:51.810
it's easy to get mixed up
because it still is true,

01:20:51.810 --> 01:20:54.220
even if it's not what's
driving the effect that is

01:20:54.220 --> 01:20:57.530
being, in this case, analyzed.

01:20:57.530 --> 01:20:58.280
We're out of time.

01:20:58.280 --> 01:20:59.190
So we should quit.

01:20:59.190 --> 01:21:02.750
On Tuesday, we'll start by
wrapping up the oscillation

01:21:02.750 --> 01:21:06.610
discussion by talking about
other oscillator designs that

01:21:06.610 --> 01:21:09.460
allow for robustness
and tunability.

01:21:09.460 --> 01:21:11.010
OK?