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PROFESSOR: Today,
what we're going

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to do is, first, introduce
this idea of oscillations.

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00:00:25,130 --> 00:00:26,740
It might be useful.

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00:00:26,740 --> 00:00:29,980
A fair amount of the day will
be spent discussing this paper

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00:00:29,980 --> 00:00:32,119
by Michael Elowitz
and Stan Leibler

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00:00:32,119 --> 00:00:35,285
that you read over the last few
days, which was the first, kind

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of, experimental
demonstration that you

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could take these
random components,

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00:00:39,140 --> 00:00:42,270
put them together, and generate
oscillatory gene networks.

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00:00:42,270 --> 00:00:45,320
And finally, it's
likely we're going

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00:00:45,320 --> 00:00:46,960
to run out of time around here.

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00:00:46,960 --> 00:00:49,780
But if we have time, we'll talk
about other oscillator designs.

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00:00:49,780 --> 00:00:51,980
In particular, these
relaxation oscillators

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00:00:51,980 --> 00:00:55,930
that are both
robust and tunable.

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00:00:55,930 --> 00:00:59,130
It's likely we're going to
discuss this on Tuesday.

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00:01:04,360 --> 00:01:06,840
All right, so I want to
start by just thinking

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00:01:06,840 --> 00:01:11,026
about other oscillator designs.

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00:01:11,026 --> 00:01:12,400
But before we get
into that, it's

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00:01:12,400 --> 00:01:14,730
worth just asking a question.

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00:01:14,730 --> 00:01:18,250
Why is it that we might want
to design an oscillator?

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00:01:18,250 --> 00:01:20,480
What do we like
about oscillations?

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00:01:23,229 --> 00:01:24,520
Does anybody like oscillations?

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00:01:24,520 --> 00:01:26,820
And if so, why?

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00:01:26,820 --> 00:01:27,320
Yes.

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00:01:27,320 --> 00:01:28,763
AUDIENCE: You can make clocks.

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00:01:28,763 --> 00:01:30,089
And clocks are really--

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00:01:30,089 --> 00:01:30,880
PROFESSOR: Perfect.

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00:01:30,880 --> 00:01:31,504
Yes, all right.

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00:01:31,504 --> 00:01:34,760
So two part answer.

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00:01:34,760 --> 00:01:36,060
You can make clocks.

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00:01:36,060 --> 00:01:37,111
And clocks are useful.

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00:01:37,111 --> 00:01:37,610
All right.

40
00:01:37,610 --> 00:01:41,310
OK, so this is a fine statement.

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00:01:41,310 --> 00:01:43,890
So oscillators are, kind of,
the basis for time keeping.

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00:01:43,890 --> 00:01:50,180
And indeed, classic ideas of
clocks, like a pendulum clock.

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00:01:50,180 --> 00:01:52,670
The idea is that
you have this thing.

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00:01:52,670 --> 00:01:53,850
It's going back and forth.

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00:01:53,850 --> 00:01:55,420
And each time that
it goes, it let

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00:01:55,420 --> 00:01:57,630
allows some winding
mechanism to move.

47
00:01:57,630 --> 00:02:00,230
And that's what
the clock is based.

48
00:02:00,230 --> 00:02:02,782
And even modern clocks
are based on some sort

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00:02:02,782 --> 00:02:03,740
of oscillatory dynamic.

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00:02:03,740 --> 00:02:05,910
It might be a very
high frequency.

51
00:02:05,910 --> 00:02:09,699
But in any case, the
basic idea of oscillations

52
00:02:09,699 --> 00:02:14,110
as a mechanism for time keeping
is why we really care about it.

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00:02:14,110 --> 00:02:16,520
Of course, just from a
dynamical systems perspective,

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00:02:16,520 --> 00:02:19,930
we also like oscillations
because they're

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00:02:19,930 --> 00:02:22,230
interesting from a
dynamical standpoint.

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00:02:22,230 --> 00:02:24,230
And therefore, we'd
like to know how

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00:02:24,230 --> 00:02:27,180
we might be able to make them.

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00:02:27,180 --> 00:02:31,860
Can anybody offer an example
of an oscillator in a G network

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00:02:31,860 --> 00:02:34,280
in real life?

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00:02:34,280 --> 00:02:34,780
Yes.

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00:02:34,780 --> 00:02:35,940
AUDIENCE: Circadian.

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PROFESSOR: The
circadian oscillator.

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00:02:37,070 --> 00:02:37,611
That's right.

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00:02:37,611 --> 00:02:40,636
So the idea there
is that there's

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a G network within
many organizations

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00:02:43,130 --> 00:02:45,480
that actually keeps
track of the daily cycle

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00:02:45,480 --> 00:02:47,910
and, indeed, is entrained
by the daily cycle.

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00:02:47,910 --> 00:02:50,545
So of course, the
day, night cycle.

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That's an oscillator.

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It's on its own.

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00:02:52,086 --> 00:02:53,550
And it goes without us, as well.

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00:02:53,550 --> 00:02:55,199
But it's often
useful for organisms

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00:02:55,199 --> 00:02:57,490
to be able to keep track of
where in the course the day

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00:02:57,490 --> 00:02:58,440
it might be.

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00:02:58,440 --> 00:03:01,792
And the amount of light
that the organism is getting

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00:03:01,792 --> 00:03:03,250
at this particular
moment might not

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00:03:03,250 --> 00:03:05,690
be a faithful indicator
of how much light there

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will be available in an
hour because it could just

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00:03:09,290 --> 00:03:11,719
be that there's a cloud
crossing in front of the sun.

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00:03:11,719 --> 00:03:13,260
And you don't want--
as an organism--

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to think that it's night.

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00:03:13,990 --> 00:03:15,740
And then, you shut
down all that machinery

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00:03:15,740 --> 00:03:17,510
because, after
that cloud passes,

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00:03:17,510 --> 00:03:19,135
you want to be able
to get going again.

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00:03:19,135 --> 00:03:21,150
So it's often useful
for an organism

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00:03:21,150 --> 00:03:28,070
to know where in the morning,
night, evening cycle one is.

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00:03:28,070 --> 00:03:31,237
And we will not be talking
too much about the circadian

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00:03:31,237 --> 00:03:32,320
oscillators in this class.

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00:03:32,320 --> 00:03:34,607
Although, I would say to
the degree of your interest

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00:03:34,607 --> 00:03:36,660
in oscillations, I
strongly encourage

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00:03:36,660 --> 00:03:39,820
you to look up that literature
because it's really beautiful.

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00:03:39,820 --> 00:03:42,030
In particular, in some
of these oscillators,

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00:03:42,030 --> 00:03:45,390
it's been demonstrating you can
get the oscillations in vitro.

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00:03:45,390 --> 00:03:46,920
I.e, outside of the cell.

95
00:03:46,920 --> 00:03:49,810
Even in the absence of any
gene expression, in some cases,

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00:03:49,810 --> 00:03:52,560
you can still get oscillations
of just those protein

97
00:03:52,560 --> 00:03:53,980
components in a test tube.

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00:03:53,980 --> 00:03:55,720
This was quite a
shocking discovery

99
00:03:55,720 --> 00:03:59,370
when it was first published.

100
00:03:59,370 --> 00:04:02,910
But we want to start out
with some simpler ones.

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00:04:02,910 --> 00:04:04,690
In particular, I want
to start by thinking

102
00:04:04,690 --> 00:04:08,050
about auto repression.

103
00:04:08,050 --> 00:04:10,710
So if you have an auto
regulatory loop where

104
00:04:10,710 --> 00:04:16,240
some gene is repressing
itself, the question

105
00:04:16,240 --> 00:04:19,380
is does this thing oscillate.

106
00:04:19,380 --> 00:04:22,300
And indeed, it's
reasonable that it

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00:04:22,300 --> 00:04:25,960
might because we can
construct a verbal argument.

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00:04:25,960 --> 00:04:28,400
Starts out high.

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00:04:28,400 --> 00:04:30,490
Then, it should
repress itself so you

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00:04:30,490 --> 00:04:32,740
get less new x being made.

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00:04:32,740 --> 00:04:35,440
So the concentration falls.

112
00:04:35,440 --> 00:04:39,830
So maybe I'll give you
a plot to add to it.

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00:04:39,830 --> 00:04:42,000
Concentration of x is
a function of time.

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00:04:42,000 --> 00:04:44,370
You can imagine just
starting somewhere high.

115
00:04:44,370 --> 00:04:46,590
That means it's a
repressing expression.

116
00:04:46,590 --> 00:04:48,210
So it's going to fall.

117
00:04:48,210 --> 00:04:50,560
But then, once it falls too
much, then all of a sudden,

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00:04:50,560 --> 00:04:53,350
OK, well we're not
repressing ourselves anymore.

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00:04:53,350 --> 00:04:56,600
So maybe then we
get more expression.

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00:04:56,600 --> 00:04:58,370
More of this x is being made.

121
00:04:58,370 --> 00:05:00,260
So it should come back up.

122
00:05:00,260 --> 00:05:02,500
And then, now we're
back where we started.

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00:05:02,500 --> 00:05:06,561
So this is a totally
reasonable statement.

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00:05:06,561 --> 00:05:07,060
Yes?

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00:05:07,060 --> 00:05:10,357
AUDIENCE: [INAUDIBLE]?

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00:05:10,357 --> 00:05:11,810
PROFESSOR: Well I don't know.

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00:05:11,810 --> 00:05:14,660
I mean, I didn't introduce
any damping in here.

128
00:05:14,660 --> 00:05:16,437
The amplitude is
the same everywhere.

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00:05:16,437 --> 00:05:18,520
AUDIENCE: So you're saying
that you could actually

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00:05:18,520 --> 00:05:19,444
have something--

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00:05:19,444 --> 00:05:24,467
PROFESSOR: Well I guess what
I'm really trying to say

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00:05:24,467 --> 00:05:26,800
is that just because you can
construct a verbal argument

133
00:05:26,800 --> 00:05:29,870
that something happens does not
mean that a particular equation

134
00:05:29,870 --> 00:05:31,420
is going to do that.

135
00:05:31,420 --> 00:05:32,870
Part of the value
of equations is

136
00:05:32,870 --> 00:05:35,580
that they force you to
be explicit about all

137
00:05:35,580 --> 00:05:37,092
the assumptions
that you're making.

138
00:05:37,092 --> 00:05:38,550
And then what you're
going to do is

139
00:05:38,550 --> 00:05:40,850
you're going to ask,
well, a given equation is

140
00:05:40,850 --> 00:05:43,717
a mathematical manifestation of
the assumptions you're making.

141
00:05:43,717 --> 00:05:45,800
And then, you're going to
ask does that oscillate.

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00:05:45,800 --> 00:05:46,570
Yes/no?

143
00:05:46,570 --> 00:05:47,200
And then you're going
to say, OK, well

144
00:05:47,200 --> 00:05:48,741
what would we need
to change in order

145
00:05:48,741 --> 00:05:50,380
to introduce oscillations?

146
00:05:50,380 --> 00:05:53,660
And I'll just-- OK.

147
00:05:53,660 --> 00:05:56,230
So this is definitely
an oscillation.

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00:05:56,230 --> 00:05:58,630
The question is, should
you find this argument

149
00:05:58,630 --> 00:06:00,030
I just gave you convincing?

150
00:06:00,030 --> 00:06:04,910
And what I'm, I guess, about
to say is that you shouldn't.

151
00:06:04,910 --> 00:06:08,900
But then, we need to be clear
about what's going on and why.

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00:06:08,900 --> 00:06:11,640
And just because you can make
a verbal argument for something

153
00:06:11,640 --> 00:06:13,430
doesn't mean that
it actually exist.

154
00:06:13,430 --> 00:06:16,190
I mean, that's a guide
to how you might want

155
00:06:16,190 --> 00:06:19,450
to formalize your thinking.

156
00:06:19,450 --> 00:06:21,200
And in particular,
the simplest way

157
00:06:21,200 --> 00:06:24,540
to think about oscillations
that might be induced

158
00:06:24,540 --> 00:06:27,150
in this situation would
be to just say, all right,

159
00:06:27,150 --> 00:06:33,080
well the simplest model
we have for an auto

160
00:06:33,080 --> 00:06:35,510
regulatory loop that's
negative is we say,

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00:06:35,510 --> 00:06:46,670
OK, well there's some alpha
1 plus protein and minus p.

162
00:06:46,670 --> 00:06:48,570
So this is, kind of,
the simplest equation

163
00:06:48,570 --> 00:06:51,460
you can write that captures
this idea that this protein

164
00:06:51,460 --> 00:06:56,000
p is negatively regulating
itself in a cooperative fashion

165
00:06:56,000 --> 00:06:57,930
maybe.

166
00:06:57,930 --> 00:07:02,321
Now it's already in a
non-dimensionalize version.

167
00:07:02,321 --> 00:07:02,820
Right?

168
00:07:02,820 --> 00:07:06,400
And what you can see is
that, within this realm,

169
00:07:06,400 --> 00:07:09,150
there are only two things
that can possibly be changing.

170
00:07:09,150 --> 00:07:11,930
There's how cooperative
that repression

171
00:07:11,930 --> 00:07:16,530
is-- n-- and then, the
strength of the expression

172
00:07:16,530 --> 00:07:19,220
in the absence of repression.

173
00:07:19,220 --> 00:07:21,995
And as we discussed
on Tuesday, alpha

174
00:07:21,995 --> 00:07:26,680
is capturing all these
dynamics of the actual strength

175
00:07:26,680 --> 00:07:31,530
of expression together with
the lifetime of the protein

176
00:07:31,530 --> 00:07:33,425
together with the binding.

177
00:07:33,425 --> 00:07:35,500
You know, the
binding affinity k.

178
00:07:35,500 --> 00:07:40,230
So all those things get wrapped
up in this a or alpha rather.

179
00:07:40,230 --> 00:07:44,310
All right, so this is,
indeed, the simplest model

180
00:07:44,310 --> 00:07:47,500
you can write down to
describe such a negative auto

181
00:07:47,500 --> 00:07:49,360
regulatory loop.

182
00:07:49,360 --> 00:07:52,910
Now the question is now
that we've done this,

183
00:07:52,910 --> 00:07:58,020
we want to know does
this thing oscillate.

184
00:07:58,020 --> 00:08:02,590
And even without
analyzing this equation,

185
00:08:02,590 --> 00:08:04,940
there's something that's very
strong, which you can say.

186
00:08:08,140 --> 00:08:10,710
So in theory we're going to ask
is it possible for this thing

187
00:08:10,710 --> 00:08:13,320
to oscillate.

188
00:08:13,320 --> 00:08:14,190
All right.

189
00:08:14,190 --> 00:08:15,210
Possible.

190
00:08:15,210 --> 00:08:18,690
Your oscillations, we'll
say oscillations possible.

191
00:08:18,690 --> 00:08:22,712
And this time, referring
to mathematically possible.

192
00:08:22,712 --> 00:08:24,170
So maybe this thing
does oscillate.

193
00:08:24,170 --> 00:08:24,690
Maybe it doesn't.

194
00:08:24,690 --> 00:08:26,440
But in particular,
without analyzing it,

195
00:08:26,440 --> 00:08:29,815
is there anything that you
can say without analyzing it?

196
00:08:29,815 --> 00:08:31,440
We're just going to
say is it possible.

197
00:08:31,440 --> 00:08:33,940
Yes or no?

198
00:08:33,940 --> 00:08:35,679
If you say no, you
have to be prepared

199
00:08:35,679 --> 00:08:37,924
to give an argument for
why this thing is not

200
00:08:37,924 --> 00:08:38,799
allowed to oscillate.

201
00:08:38,799 --> 00:08:40,200
I'm talking about this equation.

202
00:08:43,320 --> 00:08:46,360
Do you don't you
understand the question

203
00:08:46,360 --> 00:08:48,785
that I'm trying to ask?

204
00:08:48,785 --> 00:08:50,410
And we haven't analyzed
this thing yet.

205
00:08:50,410 --> 00:08:53,210
But the question is,
even before analyzing it,

206
00:08:53,210 --> 00:08:56,320
can we say anything about
whether it's mathematically

207
00:08:56,320 --> 00:08:57,843
allowed to oscillate?

208
00:09:02,190 --> 00:09:04,300
I'll give you 10 seconds
to think about it.

209
00:09:12,690 --> 00:09:14,600
And if you say no, you
get to tell me why.

210
00:09:14,600 --> 00:09:15,660
All right, ready?

211
00:09:15,660 --> 00:09:17,530
Three, two, one.

212
00:09:20,400 --> 00:09:23,100
All right, so we got a
smattering of things.

213
00:09:23,100 --> 00:09:27,164
So I think this is not,
obviously, a priori.

214
00:09:27,164 --> 00:09:28,830
But it turns out that
it's not actually.

215
00:09:28,830 --> 00:09:30,610
It's just mathematically
impossible for this hing

216
00:09:30,610 --> 00:09:31,300
to oscillate.

217
00:09:31,300 --> 00:09:33,266
And can somebody say
why that might be?

218
00:09:33,266 --> 00:09:35,099
AUDIENCE: Because it
might be you could only

219
00:09:35,099 --> 00:09:36,918
have one value of p dot?

220
00:09:36,918 --> 00:09:38,120
PROFESSOR: Perfect OK.

221
00:09:38,120 --> 00:09:40,210
So for a given
value of p, there's

222
00:09:40,210 --> 00:09:43,410
only some value of p
dot that you can have.

223
00:09:43,410 --> 00:09:47,480
And in a particular-- so p here
is like a concentration of x.

224
00:09:47,480 --> 00:09:52,420
So I'm going to pick some
value, randomly, here of p.

225
00:09:52,420 --> 00:09:54,130
And what you're
pointing out is this

226
00:09:54,130 --> 00:09:57,300
is a differential
equation in which

227
00:09:57,300 --> 00:10:01,860
if you give me or I give you
the p, you can give me p dot.

228
00:10:01,860 --> 00:10:04,030
And there's a single
value p dot for each p.

229
00:10:06,780 --> 00:10:12,140
And in this oscillatory
scheme, is that statement true?

230
00:10:12,140 --> 00:10:12,730
No.

231
00:10:12,730 --> 00:10:14,890
What you can see is
that, over here, this is

232
00:10:14,890 --> 00:10:20,922
x slash p concentration of x.

233
00:10:20,922 --> 00:10:22,880
We're using p here because
we're about to start

234
00:10:22,880 --> 00:10:26,400
talking about mRNA So I want to
keep the notation consistent.

235
00:10:26,400 --> 00:10:29,297
What you see is that the
derivative here is negative.

236
00:10:29,297 --> 00:10:30,630
The derivative here is positive.

237
00:10:30,630 --> 00:10:32,410
Negative, positive.

238
00:10:32,410 --> 00:10:36,720
So any oscillation that you're
going to be able to imagine

239
00:10:36,720 --> 00:10:41,940
is going to have multiple
values for the derivative

240
00:10:41,940 --> 00:10:44,160
as a function of that
value just because you

241
00:10:44,160 --> 00:10:45,470
have to come back and forth.

242
00:10:45,470 --> 00:10:48,700
You have to cross that
point multiple times.

243
00:10:48,700 --> 00:10:52,277
So what this is saying is that
since this is a differential

244
00:10:52,277 --> 00:10:53,860
equation-- and it's
actually important

245
00:10:53,860 --> 00:10:55,150
that it's a differential
equation rather

246
00:10:55,150 --> 00:10:57,030
than a difference equation
where you have discrete values.

247
00:10:57,030 --> 00:10:59,290
But given that this is a
differential equation where

248
00:10:59,290 --> 00:11:02,410
time is taking little,
little, little steps

249
00:11:02,410 --> 00:11:05,180
and you have a single variable,
it just can't oscillate.

250
00:11:10,380 --> 00:11:13,120
So for example, if you're
talking about the oscillations

251
00:11:13,120 --> 00:11:15,420
the harmonic oscillator
the important thing there

252
00:11:15,420 --> 00:11:18,260
is a you have both the
position in the velocity

253
00:11:18,260 --> 00:11:20,660
see these two dynamical
variables that are interacting

254
00:11:20,660 --> 00:11:25,020
in some way because you have
momentum, in that case, that

255
00:11:25,020 --> 00:11:29,521
allows for the oscillations in
the case of a mass on a spring,

256
00:11:29,521 --> 00:11:30,020
for example.

257
00:11:32,740 --> 00:11:33,240
Question.

258
00:11:33,240 --> 00:11:34,734
AUDIENCE: I'm still
not understanding.

259
00:11:34,734 --> 00:11:36,234
So the value of p
can not oscillate?

260
00:11:38,718 --> 00:11:39,500
PROFESSOR: Right.

261
00:11:39,500 --> 00:11:43,610
So we're saying is that, right,
p simply cannot oscillate

262
00:11:43,610 --> 00:11:46,910
in this situation where we
have a differential equation

263
00:11:46,910 --> 00:11:52,790
describing p with-- if we just
have p dot as a function of p

264
00:11:52,790 --> 00:11:54,210
and we don't have
a second order.

265
00:11:54,210 --> 00:11:56,090
A p double dot, for example.

266
00:11:56,090 --> 00:12:02,640
So if we just have a single
derivative with respect to time

267
00:12:02,640 --> 00:12:04,850
and some function of p
over here, what that means

268
00:12:04,850 --> 00:12:10,264
is that, if p is specified,
then p dot is specified.

269
00:12:10,264 --> 00:12:12,430
And that's inconsistent
with any sort of oscillation

270
00:12:12,430 --> 00:12:15,340
because any oscillation's going
to require that, at this is

271
00:12:15,340 --> 00:12:17,580
given value of p-- this
concentration of p--

272
00:12:17,580 --> 00:12:20,020
in this case, the
concentration's going down.

273
00:12:20,020 --> 00:12:21,890
Here, it's going up.

274
00:12:21,890 --> 00:12:24,290
So here, this is--
from that standpoint--

275
00:12:24,290 --> 00:12:26,775
a multi valued function.

276
00:12:26,775 --> 00:12:27,274
OK?

277
00:12:30,417 --> 00:12:32,125
And other questions
about this statement?

278
00:12:35,840 --> 00:12:38,810
Even if I just written down some
other function of p over here,

279
00:12:38,810 --> 00:12:40,268
this statement
would still be true.

280
00:12:43,050 --> 00:12:47,860
And it's valuable to be able to
have some intuition about what

281
00:12:47,860 --> 00:12:51,180
are the essential ingredients
to get this sort of oscillation.

282
00:12:51,180 --> 00:12:53,710
And for simple harmonic
motion, right there we

283
00:12:53,710 --> 00:12:56,110
have the second derivative,
first derivative,

284
00:12:56,110 --> 00:13:00,190
and that's what allows
oscillations there.

285
00:13:00,190 --> 00:13:03,240
OK, so we can, maybe, write
down a more complicated model

286
00:13:03,240 --> 00:13:04,490
of a negative auto regulation.

287
00:13:04,490 --> 00:13:06,560
And then, try to
ask the same thing.

288
00:13:06,560 --> 00:13:08,095
Might this new model oscillate?

289
00:13:15,220 --> 00:13:17,230
And this looks a little
bit more complicated.

290
00:13:17,230 --> 00:13:19,063
But we just have to be
a little bit careful.

291
00:13:19,063 --> 00:13:22,589
All right, so this is, again,
negative auto-regulation.

292
00:13:22,589 --> 00:13:24,130
What we're going to
do is we're going

293
00:13:24,130 --> 00:13:31,034
to explicitly think about the
concentration of the mRNA.

294
00:13:31,034 --> 00:13:32,440
OK.

295
00:13:32,440 --> 00:13:37,050
And that's just because when a
gene is initially transcribed,

296
00:13:37,050 --> 00:13:38,080
it first makes mRNA.

297
00:13:38,080 --> 00:13:40,820
And then, the mRNA is
translated into protein.

298
00:13:40,820 --> 00:13:41,870
Right?

299
00:13:41,870 --> 00:13:43,670
So what we can do
is we can write down

300
00:13:43,670 --> 00:13:44,961
something that looks like this.

301
00:13:44,961 --> 00:13:48,660
M dot derivative of m
with respect to time.

302
00:13:48,660 --> 00:14:07,060
It's going to be--
all right, so this

303
00:14:07,060 --> 00:14:09,740
is the concentration of mRNA.

304
00:14:09,740 --> 00:14:12,266
And p is the
concentration of protein.

305
00:14:12,266 --> 00:14:12,766
OK?

306
00:14:27,644 --> 00:14:29,060
All right, and
what you can see is

307
00:14:29,060 --> 00:14:32,810
that the protein
is now repressing

308
00:14:32,810 --> 00:14:35,910
expression of the mRNA.

309
00:14:35,910 --> 00:14:38,161
mRNA is being degraded.

310
00:14:38,161 --> 00:14:40,160
But then, down here, this
is a little bit funny.

311
00:14:40,160 --> 00:14:43,750
But what you can see is
that, if you have more mRNA,

312
00:14:43,750 --> 00:14:46,380
then that's going to lead to
the production of protein.

313
00:14:46,380 --> 00:14:49,450
Yet, we also have a degradation
term for the protein.

314
00:14:54,557 --> 00:14:55,539
Yes?

315
00:14:55,539 --> 00:15:00,449
AUDIENCE: Why are we
multiplying the degradation

316
00:15:00,449 --> 00:15:04,377
rate of the protein
times some beta, as well?

317
00:15:04,377 --> 00:15:07,190
PROFESSOR: That's
a good question.

318
00:15:07,190 --> 00:15:10,670
OK, you're wondering why
we've pulled out this beta.

319
00:15:10,670 --> 00:15:12,980
In particular-- right.

320
00:15:12,980 --> 00:15:13,530
OK, perfect.

321
00:15:13,530 --> 00:15:13,810
OK, yeah.

322
00:15:13,810 --> 00:15:14,930
This is very important.

323
00:15:14,930 --> 00:15:16,670
And actually, this
gets in-- once again--

324
00:15:16,670 --> 00:15:19,940
to this question of these
non-dimensional versions

325
00:15:19,940 --> 00:15:21,580
of equations.

326
00:15:21,580 --> 00:15:22,550
Mathematically, simple.

327
00:15:22,550 --> 00:15:24,140
Biologically, very complicated.

328
00:15:24,140 --> 00:15:29,070
Well, first of all, what is that
we've used as our unit of time

329
00:15:29,070 --> 00:15:30,230
in these equations?

330
00:15:32,991 --> 00:15:34,116
AUDIENCE: The life of mRNA.

331
00:15:34,116 --> 00:15:34,850
PROFESSOR: Right.

332
00:15:34,850 --> 00:15:38,055
So it's based on the
lifetime of the mRNA

333
00:15:38,055 --> 00:15:39,680
because we can see
that there's nothing

334
00:15:39,680 --> 00:15:41,060
sitting in front of this m.

335
00:15:41,060 --> 00:15:43,100
And if we want to, then,
allow for a difference

336
00:15:43,100 --> 00:15:45,150
in the lifetime
mRNA and protein,

337
00:15:45,150 --> 00:15:48,030
then we have to introduce
some other thing, which

338
00:15:48,030 --> 00:15:50,340
we're calling beta.

339
00:15:50,340 --> 00:15:54,890
So beta is the ratio of--
well which one's more stable?

340
00:15:54,890 --> 00:15:56,307
mRNA or protein,
often, typically?

341
00:15:56,307 --> 00:15:57,056
AUDIENCE: Protein.

342
00:15:57,056 --> 00:15:59,080
PROFESSOR: Proteins are,
typically, more stable.

343
00:15:59,080 --> 00:16:00,580
So does that mean
that beta should

344
00:16:00,580 --> 00:16:03,346
be larger or smaller than 1?

345
00:16:03,346 --> 00:16:05,470
OK, I'm going to let you
guys think about this just

346
00:16:05,470 --> 00:16:08,900
make sure we're all--
OK, so the question

347
00:16:08,900 --> 00:16:12,380
is beta, A, greater than 1?

348
00:16:12,380 --> 00:16:13,930
Typically, much greater.

349
00:16:13,930 --> 00:16:20,805
Or is it, B, much less than
1, given what we just said?

350
00:16:20,805 --> 00:16:22,680
All right, you think
about it for 10 seconds.

351
00:16:28,572 --> 00:16:31,030
All right.

352
00:16:31,030 --> 00:16:33,520
Are you ready?

353
00:16:33,520 --> 00:16:36,670
Three, two, one.

354
00:16:36,670 --> 00:16:39,040
All right, so most
people are saying B.

355
00:16:39,040 --> 00:16:42,390
So indeed, beta should
be much less than 1.

356
00:16:42,390 --> 00:16:48,700
And that's because beta is
the ratio of the lifetime.

357
00:16:48,700 --> 00:16:52,150
So you can see, if
beta gets larger,

358
00:16:52,150 --> 00:16:56,020
that increases the degradation
rate of the protein.

359
00:17:00,180 --> 00:17:02,370
What do I want to say?

360
00:17:02,370 --> 00:17:05,400
So beta is the ratio
of the lifetime

361
00:17:05,400 --> 00:17:08,149
in the mRNA through the
lifetime of the protein.

362
00:17:08,149 --> 00:17:08,649
Yes?

363
00:17:08,649 --> 00:17:09,802
AUDIENCE: So I get
why we have to--

364
00:17:09,802 --> 00:17:10,619
PROFESSOR: Yeah.

365
00:17:10,619 --> 00:17:11,069
No, I understand.

366
00:17:11,069 --> 00:17:11,760
No, I understand you.

367
00:17:11,760 --> 00:17:12,968
I'm getting to your question.

368
00:17:12,968 --> 00:17:14,930
First, we have to
make since of this

369
00:17:14,930 --> 00:17:18,149
because the next thing
is actually even weirder.

370
00:17:18,149 --> 00:17:19,690
But I just want to
be clear that beta

371
00:17:19,690 --> 00:17:28,170
is defined as the
lifetime of mRNA

372
00:17:28,170 --> 00:17:30,800
over the lifetime
of the protein.

373
00:17:30,800 --> 00:17:35,030
What's interesting
is, actually, there's

374
00:17:35,030 --> 00:17:38,190
a typo or mistake in the
elements paper, actually.

375
00:17:38,190 --> 00:17:47,032
So if you look at
figure 1B or so-- yeah,

376
00:17:47,032 --> 00:17:47,990
so figure 1B, actually.

377
00:17:47,990 --> 00:17:50,890
It says that beta is the protein
lifetime divided by the mRNA

378
00:17:50,890 --> 00:17:51,570
lifetime.

379
00:17:51,570 --> 00:17:55,339
So you can correct
that, if you like.

380
00:17:55,339 --> 00:17:57,880
So beta's is the mRNA divided
by the lifetime of the protein.

381
00:18:04,940 --> 00:18:06,500
OK, so I think
that we understand

382
00:18:06,500 --> 00:18:07,458
why that term is there.

383
00:18:07,458 --> 00:18:09,910
But the weird
thing is that we're

384
00:18:09,910 --> 00:18:12,300
doing p minus m over here.

385
00:18:12,300 --> 00:18:13,830
Right?

386
00:18:13,830 --> 00:18:18,410
And it feels, somehow, that
that can't be possible.

387
00:18:18,410 --> 00:18:22,350
You know, that it shouldn't
be beta times m over here

388
00:18:22,350 --> 00:18:25,590
because it feels like
it's under determined.

389
00:18:25,590 --> 00:18:27,370
Right?

390
00:18:27,370 --> 00:18:28,920
OK.

391
00:18:28,920 --> 00:18:32,630
So it's possible
I just screwed up.

392
00:18:32,630 --> 00:18:38,520
But does anybody want to
defend my equation here?

393
00:18:38,520 --> 00:18:41,282
How might it be possible that
this makes any sense that you

394
00:18:41,282 --> 00:18:43,720
can just have the one beta
here that you pull out,

395
00:18:43,720 --> 00:18:45,530
and it's just p
minus m over here?

396
00:18:45,530 --> 00:18:48,076
AUDIENCE: I think it's an
assumption of the model where

397
00:18:48,076 --> 00:18:51,426
they choose the lifetime
of the protein and the mRNA

398
00:18:51,426 --> 00:18:52,470
to be similar.

399
00:18:52,470 --> 00:18:54,400
PROFESSOR: Well no
because, actually, we

400
00:18:54,400 --> 00:18:56,740
have this term beta, which
is the lifetime of mRNA

401
00:18:56,740 --> 00:18:57,596
divided by lifetime of protein.

402
00:18:57,596 --> 00:18:59,689
So we haven't assumed
anything about this beta.

403
00:18:59,689 --> 00:19:01,480
It could be, in principle,
larger than one.

404
00:19:01,480 --> 00:19:02,229
Smaller, actually.

405
00:19:02,229 --> 00:19:07,445
So it's true that given typical
facts about life in the cell,

406
00:19:07,445 --> 00:19:09,570
it's true that you expect
beta be much less than 1.

407
00:19:09,570 --> 00:19:11,010
But we haven't made
any assumption.

408
00:19:11,010 --> 00:19:11,801
Beta is just there.

409
00:19:11,801 --> 00:19:13,230
It could be anything.

410
00:19:13,230 --> 00:19:14,100
Right?

411
00:19:14,100 --> 00:19:17,630
So yeah, it's possible we've
made some other assumption.

412
00:19:17,630 --> 00:19:18,660
But what is going on.

413
00:19:18,660 --> 00:19:19,160
Yes?

414
00:19:19,160 --> 00:19:22,184
AUDIENCE: Is it
the concentration

415
00:19:22,184 --> 00:19:24,960
is scaled by the
amount of necessary--

416
00:19:24,960 --> 00:19:28,150
PROFESSOR: Yes, that's
right because, remember,

417
00:19:28,150 --> 00:19:31,530
you can only choose
one unit for time.

418
00:19:31,530 --> 00:19:34,360
And we've already chosen that
to get this to be just minus m

419
00:19:34,360 --> 00:19:34,990
here.

420
00:19:34,990 --> 00:19:37,240
But you get to choose what's
the unit of concentration

421
00:19:37,240 --> 00:19:40,870
for, both, mRNA and for protein.

422
00:19:40,870 --> 00:19:43,630
Can somebody remind us what
the unit of concentration

423
00:19:43,630 --> 00:19:46,070
is for protein?

424
00:19:46,070 --> 00:19:49,214
AUDIENCE: The dissociation
constant of the protein

425
00:19:49,214 --> 00:19:51,006
to the--

426
00:19:51,006 --> 00:19:52,030
PROFESSOR: That's right.

427
00:19:52,030 --> 00:19:53,890
So it's this
dissociation constant.

428
00:19:53,890 --> 00:19:56,630
And more generally, it's
the protein concentration,

429
00:19:56,630 --> 00:20:00,750
which you get half
maximal repression.

430
00:20:00,750 --> 00:20:02,522
And depending on
the detailed models,

431
00:20:02,522 --> 00:20:03,730
it could be more complicated.

432
00:20:03,730 --> 00:20:07,480
But in this phenomenological
realm, if p is equal to 1,

433
00:20:07,480 --> 00:20:09,310
you get half repression.

434
00:20:09,310 --> 00:20:12,270
And that's our definition
for what p equal to 1 means.

435
00:20:12,270 --> 00:20:15,290
So we've rescaled out that k.

436
00:20:15,290 --> 00:20:17,530
So what we've really
done is that there's

437
00:20:17,530 --> 00:20:20,350
some unit for the
concentration of mRNA

438
00:20:20,350 --> 00:20:22,190
that we were free to choose.

439
00:20:22,190 --> 00:20:26,020
And it was chosen so that
you could just say p minus m.

440
00:20:26,020 --> 00:20:32,330
But what that means is that it
requires a genius to figure out

441
00:20:32,330 --> 00:20:35,064
what m equal to 1 means, right?

442
00:20:35,064 --> 00:20:36,480
It doesn't quite
require a genius.

443
00:20:36,480 --> 00:20:38,605
But what do you guys think
it's going to depend on?

444
00:20:49,601 --> 00:20:50,100
Yes?

445
00:20:50,100 --> 00:20:52,092
AUDIENCE: It's going to depend
on this ratio of lifetimes,

446
00:20:52,092 --> 00:20:52,590
as well.

447
00:20:52,590 --> 00:20:53,506
PROFESSOR: Yes, right.

448
00:20:53,506 --> 00:20:55,510
So beta is going
to appear in there.

449
00:20:55,510 --> 00:20:56,926
So I'll give you
a hint, there are

450
00:20:56,926 --> 00:21:00,119
three things that determine it.

451
00:21:00,119 --> 00:21:02,410
AUDIENCE: Transcription, or
the speed of transcription.

452
00:21:02,410 --> 00:21:03,650
PROFESSOR: Translation, yes.

453
00:21:03,650 --> 00:21:04,960
So the translation efficiency.

454
00:21:04,960 --> 00:21:08,920
So each mRNA, it's going to
lead to some rate of protein

455
00:21:08,920 --> 00:21:09,420
synthesis.

456
00:21:09,420 --> 00:21:12,450
So yeah, the translation rate
or efficiency is going to enter.

457
00:21:19,281 --> 00:21:21,280
There aren't that many
other things it could be.

458
00:21:21,280 --> 00:21:22,870
But yeah, I mean,
this is tricky.

459
00:21:26,230 --> 00:21:28,230
And it's OK if you can't
just figure it out here

460
00:21:28,230 --> 00:21:32,580
because this, I think,
is pretty subtle.

461
00:21:32,580 --> 00:21:35,710
It turns out it also
depends on that k parameter

462
00:21:35,710 --> 00:21:39,710
because there's some sense
that-- m equal to 1-- what it's

463
00:21:39,710 --> 00:21:43,300
saying is that that's
the amount of mRNA

464
00:21:43,300 --> 00:21:49,220
that you need so that, if the
protein concentration where 1,

465
00:21:49,220 --> 00:21:52,630
you would not get any change
in the protein concentration.

466
00:21:52,630 --> 00:21:58,200
And given that now I
had to invoke p in there

467
00:21:58,200 --> 00:22:00,820
and p is scaled by k,
so then k also ends up

468
00:22:00,820 --> 00:22:02,675
being relevant for this mRNA.

469
00:22:02,675 --> 00:22:04,730
So you can, if
you'd like, go ahead

470
00:22:04,730 --> 00:22:07,720
and start with a original,
reasonable set of equations.

471
00:22:07,720 --> 00:22:09,274
And then, get back to this.

472
00:22:09,274 --> 00:22:10,690
But I think, once
again, this just

473
00:22:10,690 --> 00:22:13,480
highlights that these
non-dimensional versions

474
00:22:13,480 --> 00:22:14,760
of the equations are great.

475
00:22:14,760 --> 00:22:17,760
But you have to be careful.

476
00:22:17,760 --> 00:22:19,780
You don't know what means what.

477
00:22:19,780 --> 00:22:22,162
All right?

478
00:22:22,162 --> 00:22:24,370
Are there any questions
about what we've said so far?

479
00:22:31,110 --> 00:22:32,250
OK.

480
00:22:32,250 --> 00:22:35,530
Now what we've done is we have
now a protein concentration.

481
00:22:35,530 --> 00:22:36,820
We have mRNA concentration.

482
00:22:36,820 --> 00:22:40,695
And what I'm going to ask
for now is, for these sets

483
00:22:40,695 --> 00:22:42,560
of equations, is it
mathematically possible

484
00:22:42,560 --> 00:22:46,930
that they could,
maybe, oscillate?

485
00:22:46,930 --> 00:22:47,450
Yes.

486
00:22:47,450 --> 00:22:50,550
I mean, we're going to find
that the answer is that these

487
00:22:50,550 --> 00:22:51,870
actually don't oscillate.

488
00:22:51,870 --> 00:22:54,600
But have to actually
do the calculation

489
00:22:54,600 --> 00:22:55,850
if you want to determine that.

490
00:22:55,850 --> 00:22:59,110
You can't just say that
it's impossible based

491
00:22:59,110 --> 00:23:00,230
on the same argument here.

492
00:23:00,230 --> 00:23:02,730
And that's because, if
you think about this

493
00:23:02,730 --> 00:23:06,620
in the case of there's
some mRNA concentration.

494
00:23:06,620 --> 00:23:08,860
Some protein concentration.

495
00:23:08,860 --> 00:23:12,560
What we want to know is do
things oscillate in this space.

496
00:23:12,560 --> 00:23:15,170
And they could.

497
00:23:15,170 --> 00:23:18,535
I mean, I could
certainly draw a curve.

498
00:23:18,535 --> 00:23:20,660
It ends up not being true
for these particular sets

499
00:23:20,660 --> 00:23:21,201
of equations.

500
00:23:21,201 --> 00:23:25,690
But you can't a priori, kind
of, dismiss the possibility.

501
00:23:25,690 --> 00:23:26,458
Yes?

502
00:23:26,458 --> 00:23:28,450
AUDIENCE: That's like a
differential equation.

503
00:23:28,450 --> 00:23:30,442
But if you write down the
stochastic model of that,

504
00:23:30,442 --> 00:23:30,942
would that--

505
00:23:30,942 --> 00:23:33,280
PROFESSOR: OK, this is
a very good question.

506
00:23:33,280 --> 00:23:37,350
So this is the differential
equation format of this

507
00:23:37,350 --> 00:23:40,120
and that we're
assuming that there

508
00:23:40,120 --> 00:23:41,550
are no stochastic fluctuations.

509
00:23:41,550 --> 00:23:45,600
And indeed, there is a large
area of excitement, recently,

510
00:23:45,600 --> 00:23:49,460
that is trying to understand
cases in which you can have,

511
00:23:49,460 --> 00:23:51,530
so-called, noise
induced oscillations.

512
00:23:51,530 --> 00:23:54,970
So you can have cases that the
deterministic equations do not

513
00:23:54,970 --> 00:23:55,770
oscillate.

514
00:23:55,770 --> 00:24:00,560
But if you do the full
stochastic treatment,

515
00:24:00,560 --> 00:24:01,860
then that could oscillate.

516
00:24:01,860 --> 00:24:05,507
In particular, if you do a
master equation type formalism.

517
00:24:05,507 --> 00:24:07,923
And actually, I don't know,
for this particular equations.

518
00:24:11,094 --> 00:24:12,830
Yeah, I don't know for this one.

519
00:24:12,830 --> 00:24:14,740
But towards the end
of the semester,

520
00:24:14,740 --> 00:24:17,630
we will be talking about
explicit models in which,

521
00:24:17,630 --> 00:24:21,069
predator prey systems, in
which the differential equation

522
00:24:21,069 --> 00:24:22,110
format doesn't oscillate.

523
00:24:22,110 --> 00:24:24,651
But then, if you do the master
equation stochastic treatment,

524
00:24:24,651 --> 00:24:26,046
then it does oscillate.

525
00:24:26,046 --> 00:24:28,629
Yeah, so we will be talking
about this in other contexts.

526
00:24:28,629 --> 00:24:30,420
But I don't know the
answer for this model.

527
00:24:37,140 --> 00:24:40,420
All right, so let's
go and, maybe, try

528
00:24:40,420 --> 00:24:42,320
to analyze this a little bit.

529
00:24:42,320 --> 00:24:46,457
And this is useful to
do, partly because some

530
00:24:46,457 --> 00:24:49,040
of the calculations are going
to be very similar to what we're

531
00:24:49,040 --> 00:24:52,200
about to do next, which
is look at stability

532
00:24:52,200 --> 00:24:56,321
analysis of a repressilator
kind of system.

533
00:24:56,321 --> 00:24:56,820
All right.

534
00:25:03,350 --> 00:25:10,800
So this thing here is some
function f of m and p.

535
00:25:10,800 --> 00:25:13,260
And this guy here
is indeed, again,

536
00:25:13,260 --> 00:25:15,960
some other function
g of m and p.

537
00:25:15,960 --> 00:25:18,590
And we're going to be taking
derivatives of these functions

538
00:25:18,590 --> 00:25:20,420
around the fixed point.

539
00:25:24,260 --> 00:25:25,960
And maybe I will
also say there's

540
00:25:25,960 --> 00:25:29,431
going to be some stable point.

541
00:25:29,431 --> 00:25:30,930
We should just
calculate what it is.

542
00:25:30,930 --> 00:25:33,230
I'm sorry I'm making
this go up and down.

543
00:25:33,230 --> 00:25:33,970
Don't get dizzy.

544
00:25:37,440 --> 00:25:40,640
So first of all, it's always
good to know whether there

545
00:25:40,640 --> 00:25:43,430
are fixed points in
any sort of equations

546
00:25:43,430 --> 00:25:46,280
that you ever look at.

547
00:25:46,280 --> 00:25:48,190
So let's go ahead and see that.

548
00:25:48,190 --> 00:25:54,520
First of all, is m equal
to 0, p equal to 0?

549
00:25:54,520 --> 00:25:58,030
Is that a fixed
point in the system?

550
00:25:58,030 --> 00:25:58,550
No.

551
00:25:58,550 --> 00:25:59,049
Right?

552
00:25:59,049 --> 00:26:02,310
So if m and p are 0, then
this is a fixed point.

553
00:26:02,310 --> 00:26:05,000
But that one's
not because we get

554
00:26:05,000 --> 00:26:07,470
expression of the mRNA in
the absence of the protein.

555
00:26:07,470 --> 00:26:10,492
So the origin is
not a fixed point.

556
00:26:10,492 --> 00:26:11,950
Now to figure out
the fixed points,

557
00:26:11,950 --> 00:26:13,900
we just set these
things equal to 0.

558
00:26:13,900 --> 00:26:17,740
So if m dot is equal
to 0, we have 0.

559
00:26:17,740 --> 00:26:25,260
That's alpha 1 plus
p to the n minus m.

560
00:26:25,260 --> 00:26:27,474
Again, 0 is this.

561
00:26:30,530 --> 00:26:34,160
So what you can see is
that, at equilibrium, we

562
00:26:34,160 --> 00:26:42,130
have a condition here
where m is equal to p.

563
00:26:42,130 --> 00:26:45,772
So from this, we
get m equilibrium

564
00:26:45,772 --> 00:26:46,855
is equal to p equilibrium.

565
00:26:51,920 --> 00:26:55,930
So m equilibrium over here has
to be equal to p equilibrium,

566
00:26:55,930 --> 00:26:56,914
we just said.

567
00:26:56,914 --> 00:26:58,330
And that's equal
to this guy here.

568
00:26:58,330 --> 00:27:04,204
It's alpha 1 plus p
equilibrium to the n.

569
00:27:04,204 --> 00:27:05,070
All right.

570
00:27:05,070 --> 00:27:07,625
And the condition
for this equilibrium

571
00:27:07,625 --> 00:27:09,250
is then something
that looks like this.

572
00:27:16,170 --> 00:27:21,570
Now this is maybe
not so intuitive.

573
00:27:21,570 --> 00:27:24,560
But alpha is this non
dimensional version

574
00:27:24,560 --> 00:27:27,980
of the strength of expression.

575
00:27:27,980 --> 00:27:33,742
And what this is saying is
that, broadly, it's not obvious

576
00:27:33,742 --> 00:27:34,950
how to solve this explicitly.

577
00:27:34,950 --> 00:27:37,950
But as the strength
of expression goes up,

578
00:27:37,950 --> 00:27:42,000
the equilibrium here-- and
I'm saying equilibrium.

579
00:27:42,000 --> 00:27:43,910
And that's, maybe, a
little bit dangerous.

580
00:27:43,910 --> 00:27:46,885
We might even want
to just call it--

581
00:27:46,885 --> 00:27:48,600
it's a fixed point
in concentration,

582
00:27:48,600 --> 00:27:50,550
so it doesn't have to be stable.

583
00:27:50,550 --> 00:27:53,414
So if we don't want
to bias our thinking,

584
00:27:53,414 --> 00:27:55,830
different people argue about
whether equilibrium should be

585
00:27:55,830 --> 00:27:57,430
a stable or require a stable.

586
00:27:57,430 --> 00:27:59,920
We could just call
it some p 0 if that

587
00:27:59,920 --> 00:28:02,940
makes you less likely to
bias our thinking in terms

588
00:28:02,940 --> 00:28:05,831
of whether this concentration
should be a stable or unstable

589
00:28:05,831 --> 00:28:06,330
fixed point.

590
00:28:09,480 --> 00:28:13,320
But for example, if we
have that, in these units,

591
00:28:13,320 --> 00:28:18,080
if alpha is around
10, n might 2.

592
00:28:18,080 --> 00:28:19,590
Then, this thing
gives us something.

593
00:28:19,590 --> 00:28:25,520
It's in the range
of a couple or 2, 3.

594
00:28:25,520 --> 00:28:28,780
I mean, you can calculate
what it should be.

595
00:28:28,780 --> 00:28:32,780
2, 4, maybe even exactly 2.

596
00:28:32,780 --> 00:28:34,640
Did that-- yeah.

597
00:28:34,640 --> 00:28:36,015
All right, so yes.

598
00:28:36,015 --> 00:28:37,140
I'm just giving an example.

599
00:28:37,140 --> 00:28:39,720
If alpha were 10, then this
equilibrium concentration

600
00:28:39,720 --> 00:28:41,320
or this fixed
point concentration

601
00:28:41,320 --> 00:28:43,910
would be 2 if n were equal
to 2 to give you, kind

602
00:28:43,910 --> 00:28:46,320
of, some sense of the numbers.

603
00:28:46,320 --> 00:28:52,192
And this is 2 in units of that
binding affinity k, right.

604
00:28:52,192 --> 00:28:54,150
Now the question is,
well, what does this mean?

605
00:28:54,150 --> 00:28:55,020
Why did we do this?

606
00:28:55,020 --> 00:28:59,180
Why do we care at all about the
properties of that fix point?

607
00:28:59,180 --> 00:29:02,760
OK, so this might be some p 0.

608
00:29:02,760 --> 00:29:07,380
And this is, again, m 0 is
equal to p 0 in these units.

609
00:29:07,380 --> 00:29:11,190
So there's some fixed point
somewhere in the middle there.

610
00:29:11,190 --> 00:29:15,170
Now it turns out that the
stability of that fixed point

611
00:29:15,170 --> 00:29:17,980
is very important in determining
whether there are oscillations

612
00:29:17,980 --> 00:29:20,390
or not.

613
00:29:20,390 --> 00:29:23,500
Now the question
of the generality

614
00:29:23,500 --> 00:29:25,250
or what can you say
that's universally

615
00:29:25,250 --> 00:29:26,870
true about when you
get oscillations

616
00:29:26,870 --> 00:29:29,370
and when you don't, this
is, in general, a very hard

617
00:29:29,370 --> 00:29:31,130
mathematical
problem, particularly

618
00:29:31,130 --> 00:29:32,990
in higher numbers of dimensions.

619
00:29:32,990 --> 00:29:35,910
But for two dimensions,
there's a very nice statement

620
00:29:35,910 --> 00:29:40,800
that you can make based on the
Poincare-Bendixson criterion.

621
00:29:40,800 --> 00:29:43,690
I cannot remember
how to spell that.

622
00:29:43,690 --> 00:29:47,105
I'm probably
mispronouncing it, as well.

623
00:29:47,105 --> 00:29:48,730
So Poincare-Bendixson,
what they showed

624
00:29:48,730 --> 00:29:52,140
is that if, in two
dimensions, you

625
00:29:52,140 --> 00:29:57,150
can draw some box here such
that all of the trajectories

626
00:29:57,150 --> 00:29:58,150
are, kind of, coming in.

627
00:30:00,710 --> 00:30:03,260
And indeed, in this
case, they do come in

628
00:30:03,260 --> 00:30:06,600
because the trajectories
aren't going to cross 0.

629
00:30:06,600 --> 00:30:09,160
If you have some
mRNA, then you're

630
00:30:09,160 --> 00:30:11,920
going to start making protein.

631
00:30:11,920 --> 00:30:13,880
If you have just
protein, no mRNA,

632
00:30:13,880 --> 00:30:15,530
you're going to start
making some mRNA.

633
00:30:15,530 --> 00:30:17,988
And we know that trajectories
have to come in from out here

634
00:30:17,988 --> 00:30:20,190
because if the
concentration of mRNA

635
00:30:20,190 --> 00:30:22,610
and the concentration of
protein are very large then,

636
00:30:22,610 --> 00:30:24,110
eventually, the
degradation is going

637
00:30:24,110 --> 00:30:25,260
to start pulling things in.

638
00:30:25,260 --> 00:30:28,170
So if you come out far
enough, eventually, you're

639
00:30:28,170 --> 00:30:29,670
going to get
trajectories coming in.

640
00:30:29,670 --> 00:30:31,230
So now we have
there is some domain

641
00:30:31,230 --> 00:30:32,890
where all the trajectories
are going to come in.

642
00:30:32,890 --> 00:30:34,306
Now you can imagine
that, somehow,

643
00:30:34,306 --> 00:30:37,510
the stability of this thing is
very important because in two

644
00:30:37,510 --> 00:30:41,080
dimensions here when you
have a differential equation,

645
00:30:41,080 --> 00:30:45,120
trajectories cannot
cross each other.

646
00:30:45,120 --> 00:30:47,270
So I'm not allowed
in any sort of space

647
00:30:47,270 --> 00:30:51,490
like this to do something that
looks like this because this

648
00:30:51,490 --> 00:30:55,010
would require that, at some
concentration of m and p,

649
00:30:55,010 --> 00:30:57,482
I have different values
for m dot and p dot.

650
00:30:57,482 --> 00:30:59,940
So it's similar to this argument
we made for one dimension.

651
00:30:59,940 --> 00:31:02,360
But it's just generalized
to two dimensions.

652
00:31:02,360 --> 00:31:06,250
So we're not allowed
to cross trajectories.

653
00:31:06,250 --> 00:31:09,250
Well if you have a differential
equation in any dimensions,

654
00:31:09,250 --> 00:31:09,750
that's true.

655
00:31:09,750 --> 00:31:11,340
But the thing is
that this constraint

656
00:31:11,340 --> 00:31:13,445
is a very strong constraint
in two dimensions.

657
00:31:13,445 --> 00:31:15,820
Whereas, in three dimensions,
everything kind of goes out

658
00:31:15,820 --> 00:31:17,936
the window because in
the three dimensions,

659
00:31:17,936 --> 00:31:19,060
you have another axis here.

660
00:31:19,060 --> 00:31:21,480
And then, these lines can do
all sorts of crazy things.

661
00:31:21,480 --> 00:31:24,080
And that's actually, basically,
why you need three dimensions

662
00:31:24,080 --> 00:31:27,400
in order to get chaos in
differential equations

663
00:31:27,400 --> 00:31:30,540
because this thing about
the absence of crossing

664
00:31:30,540 --> 00:31:33,150
is just such a strong
constraint in two dimensions.

665
00:31:35,081 --> 00:31:37,080
Other questions about
what I'm saying right now?

666
00:31:37,080 --> 00:31:38,538
I'm a little bit
worried that I'm--

667
00:31:42,380 --> 00:31:45,364
All right, so the trajectories
are not allowed to cross.

668
00:31:45,364 --> 00:31:47,280
And that's really saying
something very strong

669
00:31:47,280 --> 00:31:49,210
because we know that,
here, trajectories are

670
00:31:49,210 --> 00:31:53,930
going to come out of the axis.

671
00:31:53,930 --> 00:31:56,824
And mRNA, we don't
know which direction

672
00:31:56,824 --> 00:31:57,740
they're going to come.

673
00:31:57,740 --> 00:32:00,557
But let's figure out,
if it were to oscillate,

674
00:32:00,557 --> 00:32:03,140
would the trajectories be going
clockwise or counterclockwise?

675
00:32:05,115 --> 00:32:06,490
And actually,
there's going to be

676
00:32:06,490 --> 00:32:08,779
some sense of the
trajectories even

677
00:32:08,779 --> 00:32:10,070
in the absence of oscillations.

678
00:32:10,070 --> 00:32:14,400
But broadly, is there kind of
a counterclockwise or clockwise

679
00:32:14,400 --> 00:32:18,690
kind of motion to
the trajectories?

680
00:32:18,690 --> 00:32:19,810
Counterclockwise, right?

681
00:32:19,810 --> 00:32:24,100
And that's because
mRNA leads to protein.

682
00:32:24,100 --> 00:32:26,320
So things are going
to go like this.

683
00:32:26,320 --> 00:32:29,020
And the question is is
it going to oscillate.

684
00:32:29,020 --> 00:32:31,940
And in two dimensions,
actually-- Poincare-Bendixson--

685
00:32:31,940 --> 00:32:36,330
what they say is
that, if there's just

686
00:32:36,330 --> 00:32:38,504
one fixed point here,
then the question

687
00:32:38,504 --> 00:32:40,670
of whether it oscillates
is the same as the question

688
00:32:40,670 --> 00:32:43,130
of whether this is stable.

689
00:32:43,130 --> 00:32:50,026
So if it's stable, then
there's no oscillations.

690
00:32:50,026 --> 00:32:51,400
If it's unstable,
than there are.

691
00:32:59,990 --> 00:33:02,600
We'll just say no
oscillations and oscillations.

692
00:33:02,600 --> 00:33:04,850
And that's because if it's
a stable point and all

693
00:33:04,850 --> 00:33:08,440
the trajectories are coming in,
then it just looks like this.

694
00:33:08,440 --> 00:33:12,520
So it spirals, maybe, into
a state of coexistence.

695
00:33:12,520 --> 00:33:15,667
Well it spirals to
this point of m and p.

696
00:33:15,667 --> 00:33:17,750
Whereas, if it's unstable,
then those trajectories

697
00:33:17,750 --> 00:33:19,235
are, somehow, being pushed out.

698
00:33:19,235 --> 00:33:20,860
If it's unstable,
then the trajectories

699
00:33:20,860 --> 00:33:23,870
are coming out of
that fixed point.

700
00:33:23,870 --> 00:33:25,460
In which case, then
that's actually

701
00:33:25,460 --> 00:33:27,710
precisely the situation in
which you get a limit cycle

702
00:33:27,710 --> 00:33:28,900
oscillations.

703
00:33:28,900 --> 00:33:32,710
So if the fixed
point were unstable,

704
00:33:32,710 --> 00:33:36,930
it looks like this
because we have some box.

705
00:33:36,930 --> 00:33:40,760
The trajectories are all
coming in, somehow, in here.

706
00:33:40,760 --> 00:33:43,260
But if we have one
fixed point here

707
00:33:43,260 --> 00:33:44,867
and the trajectories
are coming out,

708
00:33:44,867 --> 00:33:46,950
that means we have something
that looks like this.

709
00:33:46,950 --> 00:33:47,825
It kind of comes out.

710
00:33:47,825 --> 00:33:49,741
And given that these
trajectories can't cross,

711
00:33:49,741 --> 00:33:51,830
the question is, well,
what can happen in between?

712
00:33:51,830 --> 00:33:54,470
And the answer
is, basically, you

713
00:33:54,470 --> 00:33:57,110
have to get a limit
cycle oscillation.

714
00:33:57,110 --> 00:33:58,830
There are these
strange situations

715
00:33:58,830 --> 00:34:02,044
where you can get a path that
is an oscillation that's,

716
00:34:02,044 --> 00:34:03,460
kind of, stable
from one direction

717
00:34:03,460 --> 00:34:05,250
and unstable from another.

718
00:34:05,250 --> 00:34:08,030
We're not going to
worry about that here.

719
00:34:08,030 --> 00:34:11,050
But broadly, if this
thing is coming out,

720
00:34:11,050 --> 00:34:13,040
then you end up,
in both directions,

721
00:34:13,040 --> 00:34:15,343
converging to a stable
limit cycle oscillation.

722
00:34:17,949 --> 00:34:22,020
So it's a unstable
fixed point, then

723
00:34:22,020 --> 00:34:25,010
this is the exact situation,
which you get a limit cycle

724
00:34:25,010 --> 00:34:27,600
oscillation.

725
00:34:27,600 --> 00:34:29,154
OK.

726
00:34:29,154 --> 00:34:30,570
So that means that,
what we really

727
00:34:30,570 --> 00:34:34,690
want to do if we want to ask--
let's try to back up again.

728
00:34:34,690 --> 00:34:37,630
We have this pair of
differential equations.

729
00:34:37,630 --> 00:34:42,070
We want to know will this
negative auto regulatory loop

730
00:34:42,070 --> 00:34:43,659
oscillate.

731
00:34:43,659 --> 00:34:46,350
Now what I'm telling you
is that that question

732
00:34:46,350 --> 00:34:49,100
for two dimensions is analogous
to the question of figuring out

733
00:34:49,100 --> 00:34:52,732
whether this fixed
point is stable or not.

734
00:34:52,732 --> 00:34:54,690
If it's stable, then we
don't get oscillations.

735
00:34:54,690 --> 00:34:57,506
If it's unstable, then we do.

736
00:35:00,790 --> 00:35:02,190
Any questions about this?

737
00:35:05,280 --> 00:35:07,660
So let's see what is is.

738
00:35:07,660 --> 00:35:10,300
On Tuesday, what we do is we
talked about stability analysis

739
00:35:10,300 --> 00:35:13,470
for linear systems.

740
00:35:13,470 --> 00:35:16,020
We got what I hope is
some intuition about that.

741
00:35:16,020 --> 00:35:18,237
And of course, what
we need to do here

742
00:35:18,237 --> 00:35:20,320
is try to understand how
to apply linear stability

743
00:35:20,320 --> 00:35:25,060
analysis to this non-linear
pair of differential equations.

744
00:35:25,060 --> 00:35:28,320
And to do that,
what we need to do

745
00:35:28,320 --> 00:35:33,370
is we need to linearize
around that fixed point.

746
00:35:36,830 --> 00:35:42,200
So what we have is we have
these two functions, f and g.

747
00:35:42,200 --> 00:35:44,650
And what we want to know is
around that fixed point--

748
00:35:44,650 --> 00:35:52,810
so we can define some m tilde,
which is m minus this m 0.

749
00:35:52,810 --> 00:35:57,620
And some p tilde,
which is p minus p 0.

750
00:35:57,620 --> 00:36:01,070
So when m tilde and
p tilde are around 0,

751
00:36:01,070 --> 00:36:06,490
that's telling us that we're
close to that fixed point.

752
00:36:06,490 --> 00:36:09,550
And we want to know, if we just
go a little away from the fixed

753
00:36:09,550 --> 00:36:11,670
point, do we get pushed
away or do we come back

754
00:36:11,670 --> 00:36:12,503
to where we started?

755
00:36:15,520 --> 00:36:19,963
Well we know that m tilde dot,
which is actually equal to m

756
00:36:19,963 --> 00:36:23,890
dot, as well because m
0 and p 0 are the same.

757
00:36:23,890 --> 00:36:26,710
p tilde dot.

758
00:36:26,710 --> 00:36:32,460
We can linearize by taking
derivatives around the fixed

759
00:36:32,460 --> 00:36:32,960
points.

760
00:36:36,559 --> 00:36:38,100
And in particular,
what we want to do

761
00:36:38,100 --> 00:36:44,980
is we want to take the
derivative of f with respect

762
00:36:44,980 --> 00:36:46,400
to m.

763
00:36:46,400 --> 00:36:49,340
Evaluate at the fixed point.

764
00:36:49,340 --> 00:36:53,590
That derivative is,
indeed, just minus 1.

765
00:36:53,590 --> 00:36:57,750
So in general, in these
situations, what we have is

766
00:36:57,750 --> 00:37:01,460
we have derivatives
m, m dot p dot,

767
00:37:01,460 --> 00:37:04,655
and we have partial of this
first function f with respect

768
00:37:04,655 --> 00:37:06,060
to m.

769
00:37:06,060 --> 00:37:07,340
Partial of g.

770
00:37:07,340 --> 00:37:07,840
Oh, no.

771
00:37:07,840 --> 00:37:10,160
So this is still f.

772
00:37:10,160 --> 00:37:13,300
Respect to p.

773
00:37:13,300 --> 00:37:16,470
Down here is derivative
g with respect to m.

774
00:37:16,470 --> 00:37:19,670
Derivative g with respect to p.

775
00:37:19,670 --> 00:37:22,910
And this is all evaluated
around the fixed point m 0 p 0.

776
00:37:27,900 --> 00:37:29,790
So we want to take
these derivatives

777
00:37:29,790 --> 00:37:34,270
and evaluate at the fixed point.

778
00:37:34,270 --> 00:37:36,530
And if we do that,
we get minus 1 here,

779
00:37:36,530 --> 00:37:41,070
derivative m with respect
to m times m tilde.

780
00:37:41,070 --> 00:37:46,020
This other guy, when
you take the derivative,

781
00:37:46,020 --> 00:37:50,870
you get a minus sign
with respect to p.

782
00:37:50,870 --> 00:37:53,800
So we get a minus sign because
this is in the denominator.

783
00:37:53,800 --> 00:37:56,110
And then, we have to
take derivative inside.

784
00:37:56,110 --> 00:38:02,650
So we get n alpha p
0 to the n minus 1.

785
00:38:02,650 --> 00:38:08,530
And down, we get a
1 plus p 0 squared.

786
00:38:08,530 --> 00:38:12,190
So we took the derivative of
this term with respect to p.

787
00:38:12,190 --> 00:38:16,214
And we evaluated at
the fixed point p 0.

788
00:38:16,214 --> 00:38:18,160
Did I do that right?

789
00:38:18,160 --> 00:38:20,820
But we still have
to add a p tilde

790
00:38:20,820 --> 00:38:23,100
because this is
saying how sensitive

791
00:38:23,100 --> 00:38:24,590
is the function to
changes in where

792
00:38:24,590 --> 00:38:28,270
you are times how far you've
gone away from the fixed point.

793
00:38:28,270 --> 00:38:30,990
And then, again, over here,
we take the derivatives

794
00:38:30,990 --> 00:38:31,510
down below.

795
00:38:31,510 --> 00:38:34,970
So derivative g
with respect to m.

796
00:38:34,970 --> 00:38:40,400
That gives us a beta m tilde.

797
00:38:40,400 --> 00:38:45,420
And then, we have a
minus beta p tilde.

798
00:38:48,535 --> 00:38:51,380
All right, so this
is just an example

799
00:38:51,380 --> 00:38:55,646
of linearizing those equations
around that fixed point.

800
00:38:59,360 --> 00:39:02,050
So ultimately, what we
care about is really

801
00:39:02,050 --> 00:39:07,172
this matrix that's
specifying deviations

802
00:39:07,172 --> 00:39:08,130
around the equilibrium.

803
00:39:08,130 --> 00:39:08,800
Right?

804
00:39:08,800 --> 00:39:12,984
So it's useful to just write it
in matrix format because we get

805
00:39:12,984 --> 00:39:14,275
rid of some of the M's and P's.

806
00:39:20,980 --> 00:39:24,580
Indeed, so this matrix that we
either call A or the Jacobean

807
00:39:24,580 --> 00:39:31,155
depending on-- so what
we have is a minus 1.

808
00:39:35,950 --> 00:39:39,970
And we're going
to call this thing

809
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.

810
00:39:54,120 --> 00:40:02,134
So it's an x beta
and minus beta.

811
00:40:02,134 --> 00:40:03,550
And then, we have
our simple rules

812
00:40:03,550 --> 00:40:06,330
for determining whether this
thing is going to be stable

813
00:40:06,330 --> 00:40:07,210
or not.

814
00:40:07,210 --> 00:40:08,240
It depends on the trace.

815
00:40:08,240 --> 00:40:10,360
And it depends on
the determinant.

816
00:40:10,360 --> 00:40:12,866
So the trace should be negative.

817
00:40:12,866 --> 00:40:13,990
And is this trace negative?

818
00:40:17,180 --> 00:40:17,980
Yes.

819
00:40:17,980 --> 00:40:22,827
Yes because beta-- does anybody
remember what beta was again.

820
00:40:22,827 --> 00:40:24,035
AUDIENCE: Ratio of lifetimes.

821
00:40:24,035 --> 00:40:25,150
PROFESSOR: Ratio of lifetimes.

822
00:40:25,150 --> 00:40:26,108
Lifetimes are positive.

823
00:40:26,108 --> 00:40:27,100
So beta is positive.

824
00:40:27,100 --> 00:40:35,210
All right, so the trace is
equal to minus 1 minus beta.

825
00:40:35,210 --> 00:40:37,080
This is, indeed, less than 0.

826
00:40:37,080 --> 00:40:39,410
So this is consistent
for stability.

827
00:40:39,410 --> 00:40:41,760
Does prove that it's stable?

828
00:40:41,760 --> 00:40:43,980
No.

829
00:40:43,980 --> 00:40:49,070
But we also need to know
about the determinant of a,

830
00:40:49,070 --> 00:40:53,960
which is going to be
beta, this times this,

831
00:40:53,960 --> 00:40:55,710
minus this times this.

832
00:40:55,710 --> 00:40:59,910
So that's minus.

833
00:40:59,910 --> 00:41:02,530
And this is a beta
times what x was.

834
00:41:02,530 --> 00:41:04,160
So this gives us--
we can write this

835
00:41:04,160 --> 00:41:08,015
all down just so that it's clear
that it has to be positive.

836
00:41:12,900 --> 00:41:14,480
So beta is positive.

837
00:41:14,480 --> 00:41:16,830
Positive, positive,
positive, positive, positive.

838
00:41:16,830 --> 00:41:19,240
Everything's positive.

839
00:41:19,240 --> 00:41:23,730
So this thing has to
be greater than 0.

840
00:41:23,730 --> 00:41:27,630
So what does this mean about
the stability of Ethics Point?

841
00:41:27,630 --> 00:41:28,130
Stable.

842
00:41:30,960 --> 00:41:31,970
Fixed point stable.

843
00:41:31,970 --> 00:41:33,856
And what does that mean
about oscillations?

844
00:41:33,856 --> 00:41:35,864
It means there are
no oscillations.

845
00:41:35,864 --> 00:41:36,655
Fixed point stable.

846
00:41:39,580 --> 00:41:40,910
Therefore, no oscillations.

847
00:41:47,860 --> 00:41:51,950
So what this is saying is that
the original, kind of simple,

848
00:41:51,950 --> 00:41:54,260
equation we wrote down for
negative auto regulation,

849
00:41:54,260 --> 00:41:58,650
that thing was not allowed
to oscillate mathematically.

850
00:41:58,650 --> 00:42:01,612
But that doesn't mean that, if
you explicitly model the mRNA,

851
00:42:01,612 --> 00:42:02,570
it could go either way.

852
00:42:02,570 --> 00:42:07,040
But still, that's insufficient
to generate oscillations.

853
00:42:07,040 --> 00:42:10,989
However, maybe if you
included more steps,

854
00:42:10,989 --> 00:42:12,030
maybe it would oscillate.

855
00:42:12,030 --> 00:42:12,620
Question?

856
00:42:12,620 --> 00:42:14,762
AUDIENCE: So just to double
check-- when you said,

857
00:42:14,762 --> 00:42:17,210
no oscillations, you
mean stable oscillations?

858
00:42:17,210 --> 00:42:18,700
PROFESSOR: That's right, sorry.

859
00:42:18,700 --> 00:42:21,070
When I mean no oscillations,
what I mean are indeed,

860
00:42:21,070 --> 00:42:22,760
no limit cycle oscillations.

861
00:42:22,760 --> 00:42:25,250
AUDIENCE: This is
like a dampened--

862
00:42:25,250 --> 00:42:26,840
PROFESSOR: Yeah.

863
00:42:26,840 --> 00:42:30,870
Yeah, so we, actually,
have not solved

864
00:42:30,870 --> 00:42:32,280
exactly what it looks like.

865
00:42:32,280 --> 00:42:34,729
And I've drawn this is a
pretty oscillatory thing.

866
00:42:34,729 --> 00:42:36,520
But it might just look
like this, depending

867
00:42:36,520 --> 00:42:38,220
on the parameters and so forth.

868
00:42:38,220 --> 00:42:41,070
And indeed, we haven't
even proven that this thing

869
00:42:41,070 --> 00:42:43,770
has complex eigenvalues.

870
00:42:43,770 --> 00:42:46,190
But certainly, there are no
limit cycle oscillations.

871
00:42:46,190 --> 00:42:49,510
And I'd say it's really limit
cycle oscillations that people

872
00:42:49,510 --> 00:42:58,130
find most exciting as because
limit cycle oscillations have

873
00:42:58,130 --> 00:43:00,259
a characteristic amplitude.

874
00:43:00,259 --> 00:43:01,800
So it doesn't matter
where you start.

875
00:43:01,800 --> 00:43:04,316
The oscillations go
to some amplitude.

876
00:43:04,316 --> 00:43:06,190
And they have a
characteristic period, again,

877
00:43:06,190 --> 00:43:08,250
independent of your
starting condition.

878
00:43:10,940 --> 00:43:13,748
So a limit cycle
oscillation has a feeling

879
00:43:13,748 --> 00:43:15,581
similar to a stable
fixed point in the since

880
00:43:15,581 --> 00:43:16,460
that it doesn't matter
where you start.

881
00:43:16,460 --> 00:43:17,460
You always end up there.

882
00:43:17,460 --> 00:43:18,980
So they're the ones
that are really

883
00:43:18,980 --> 00:43:21,411
what you would
call mathematically

884
00:43:21,411 --> 00:43:22,160
nice oscillations.

885
00:43:26,530 --> 00:43:28,950
And when I say this,
I'm, in particular,

886
00:43:28,950 --> 00:43:32,660
comparing them to
neutrally stable orbits.

887
00:43:32,660 --> 00:43:39,060
So there are cases in
which, in two variables,

888
00:43:39,060 --> 00:43:40,640
you have a fixed point here.

889
00:43:40,640 --> 00:43:45,990
And at least in the case
of linear stability,

890
00:43:45,990 --> 00:43:49,220
if you have purely
imaginary eigenvalues,

891
00:43:49,220 --> 00:43:53,450
what that means is that
you have orbits that

892
00:43:53,450 --> 00:43:54,890
go around your fixed point.

893
00:43:57,990 --> 00:44:01,570
And we'll see some cases
that look like this later on.

894
00:44:01,570 --> 00:44:04,190
And this is, indeed, the
nature of the oscillations

895
00:44:04,190 --> 00:44:07,424
in the Lotka-Volterra model
for predator prey oscillations.

896
00:44:07,424 --> 00:44:09,340
They're not actually
limit cycle oscillations.

897
00:44:09,340 --> 00:44:13,190
They're of this kind that are
considered less interesting

898
00:44:13,190 --> 00:44:16,050
because they're less robust.

899
00:44:16,050 --> 00:44:18,900
Small changes in the model
can cause these things

900
00:44:18,900 --> 00:44:25,870
to either go away, to turn into
this kind of stable spiral,

901
00:44:25,870 --> 00:44:28,530
or to turn into limit
cycle oscillations.

902
00:44:28,530 --> 00:44:32,950
So we'll talk about this
more in a couple months.

903
00:44:32,950 --> 00:44:34,460
These are neutrally
stable orbits.

904
00:44:48,810 --> 00:44:51,710
OK, but what I
wanted to highlight,

905
00:44:51,710 --> 00:44:56,880
though, is that just because
the original, simple, protein

906
00:44:56,880 --> 00:44:59,832
only model didn't oscillate
and this protein mRNA together

907
00:44:59,832 --> 00:45:01,540
doesn't oscillate does
not mean that it's

908
00:45:01,540 --> 00:45:04,530
impossible to get oscillations
using negative auto

909
00:45:04,530 --> 00:45:07,060
regulation, either
experimentally

910
00:45:07,060 --> 00:45:10,080
or computationally.

911
00:45:10,080 --> 00:45:12,050
And the question
is, what might you

912
00:45:12,050 --> 00:45:13,440
need to do to get oscillations?

913
00:45:22,120 --> 00:45:24,570
AUDIENCE: So in the paper
they talk about leakage

914
00:45:24,570 --> 00:45:25,550
in the negative--

915
00:45:30,446 --> 00:45:31,590
PROFESSOR: OK, right.

916
00:45:31,590 --> 00:45:33,590
So in the paper, they
talk about various things,

917
00:45:33,590 --> 00:45:34,964
including things
such as leakage.

918
00:45:34,964 --> 00:45:37,390
It terms out that
leakage in an expression

919
00:45:37,390 --> 00:45:39,800
only inhibits
oscillations though.

920
00:45:39,800 --> 00:45:43,360
So in some sense, if you're
trying to get oscillations,

921
00:45:43,360 --> 00:45:45,880
leakage is a problem, actually.

922
00:45:45,880 --> 00:45:48,420
And that's why they use
this especially tight-- well

923
00:45:48,420 --> 00:45:50,030
we're going to talk about
that in a few minutes.

924
00:45:50,030 --> 00:45:52,321
They use an especially tight
version of these promoters

925
00:45:52,321 --> 00:45:55,920
to have low rates of
leakage in a synthesis.

926
00:45:55,920 --> 00:45:58,410
But what might you need in
order to get oscillations

927
00:45:58,410 --> 00:45:59,740
in negative autoregulation?

928
00:45:59,740 --> 00:46:02,000
Did you have-- have delay.

929
00:46:02,000 --> 00:46:02,740
Yes indeed.

930
00:46:02,740 --> 00:46:05,480
And that's something that they
mentioned in the Elowitz paper

931
00:46:05,480 --> 00:46:09,150
is if you add explicit delay.

932
00:46:09,150 --> 00:46:14,761
So for example, if instead of
having the repression depend

933
00:46:14,761 --> 00:46:16,260
on--OK, I already
erased everything.

934
00:46:16,260 --> 00:46:22,164
But instead of having
the protein, for example,

935
00:46:22,164 --> 00:46:24,330
being a function of the
mRNA now, maybe if you said,

936
00:46:24,330 --> 00:46:27,190
oh, it's a function of
the mRNA five minutes ago.

937
00:46:27,190 --> 00:46:29,860
And that's just because maybe it
takes time to make the protein.

938
00:46:29,860 --> 00:46:31,276
Or it takes time
for this or that.

939
00:46:31,276 --> 00:46:34,560
You could introduce an
explicit delay like that.

940
00:46:34,560 --> 00:46:37,470
Or you could even,
instead, have a model

941
00:46:37,470 --> 00:46:40,610
where you just have more steps.

942
00:46:40,610 --> 00:46:42,730
So what you do is you
say, oh, well yeah, sure.

943
00:46:42,730 --> 00:46:45,165
What happens is that,
first, the mRNA is made.

944
00:46:45,165 --> 00:46:46,540
But then, after
the mRNA is made,

945
00:46:46,540 --> 00:46:49,960
then you have to make
the peptide chain.

946
00:46:49,960 --> 00:46:53,010
Then, the that peptide
chain has to fold.

947
00:46:53,010 --> 00:46:56,050
And then, maybe, those
proteins have to multimerize.

948
00:46:56,050 --> 00:46:58,740
Indeed, if you right
down such a model

949
00:46:58,740 --> 00:47:00,310
then, for some
reasonable parameters,

950
00:47:00,310 --> 00:47:03,630
you can get oscillations just
with negative auto regulation.

951
00:47:03,630 --> 00:47:06,250
And indeed, I would say
that over the last 10 years,

952
00:47:06,250 --> 00:47:09,890
probably, the reigning
king of oscillations

953
00:47:09,890 --> 00:47:12,260
in the field of system
synthetic biology

954
00:47:12,260 --> 00:47:14,230
is Jeff Hasty at San Diego.

955
00:47:14,230 --> 00:47:17,880
And he's written a whole train
of beautiful papers exploring

956
00:47:17,880 --> 00:47:21,991
how you can make these
oscillators in simple G

957
00:47:21,991 --> 00:47:22,490
network.

958
00:47:22,490 --> 00:47:24,420
So he's been
focusing in E. coli.

959
00:47:24,420 --> 00:47:26,420
There's also been great
work in higher organisms

960
00:47:26,420 --> 00:47:27,320
in this regard.

961
00:47:27,320 --> 00:47:30,450
But let's say, Hasty's
work stands out

962
00:47:30,450 --> 00:47:32,770
in terms of really being
able to take these models

963
00:47:32,770 --> 00:47:35,015
and then implement them
in cells and, kind of,

964
00:47:35,015 --> 00:47:35,890
going back and forth.

965
00:47:35,890 --> 00:47:39,180
And he's shown that you can
generate oscillations just

966
00:47:39,180 --> 00:47:40,730
using negative auto
regulation if you

967
00:47:40,730 --> 00:47:45,120
have enough delays in that
negative feedback loop.

968
00:47:48,510 --> 00:47:51,654
Are there any questions
about where we are right now?

969
00:47:51,654 --> 00:47:53,320
I know that we're
supposed to be talking

970
00:47:53,320 --> 00:47:54,319
about the repressilator.

971
00:47:54,319 --> 00:47:57,283
But we first have to make sure
we understand the negative auto

972
00:47:57,283 --> 00:47:57,782
regulation.

973
00:48:00,580 --> 00:48:05,670
So everything that we've said,
so far, in terms of the models

974
00:48:05,670 --> 00:48:07,231
was all known.

975
00:48:07,231 --> 00:48:09,730
But what Michael wanted to do
is ask whether he could really

976
00:48:09,730 --> 00:48:12,710
construct an oscillator.

977
00:48:12,710 --> 00:48:17,570
And he did this using these
three mutual reppressors.

978
00:48:17,570 --> 00:48:21,100
We'll say x, y,
and z just for now.

979
00:48:21,100 --> 00:48:27,740
x represses y, represses
z, represses x.

980
00:48:27,740 --> 00:48:31,500
And has a nice
model of this system

981
00:48:31,500 --> 00:48:34,380
that helped him guide the
design of his circuits.

982
00:48:34,380 --> 00:48:36,920
So experiments--
as most of us who

983
00:48:36,920 --> 00:48:39,570
have done them know--
experiments are hard.

984
00:48:39,570 --> 00:48:43,370
So if you can do
a week of thinking

985
00:48:43,370 --> 00:48:46,890
before you do a year of
experimental biology,

986
00:48:46,890 --> 00:48:49,180
then you should do that.

987
00:48:49,180 --> 00:48:52,680
And what were the
lessons that he

988
00:48:52,680 --> 00:48:56,854
learned from the modeling
that guided his construction

989
00:48:56,854 --> 00:48:57,520
of this circuit?

990
00:49:01,484 --> 00:49:01,984
Yeah?

991
00:49:01,984 --> 00:49:03,412
AUDIENCE: Lifetime of mRNA.

992
00:49:03,412 --> 00:49:04,120
PROFESSOR: Right.

993
00:49:04,120 --> 00:49:05,619
So you want to have
similar lifetime

994
00:49:05,619 --> 00:49:08,300
of the mRNA and the protein.

995
00:49:08,300 --> 00:49:10,740
And this is, somehow,
similar to this idea

996
00:49:10,740 --> 00:49:13,875
that you need more delay
elements because if you

997
00:49:13,875 --> 00:49:17,030
have very different lifetimes,
then the more rapid process,

998
00:49:17,030 --> 00:49:19,920
somehow, doesn't count.

999
00:49:19,920 --> 00:49:23,020
It's very hard to increase the
lifetime of the mRNA that much

1000
00:49:23,020 --> 00:49:24,220
in bacteria.

1001
00:49:24,220 --> 00:49:26,500
So instead, what he
did is he decreased

1002
00:49:26,500 --> 00:49:29,960
the lifetime of the proteins
of the transcription factors.

1003
00:49:29,960 --> 00:49:31,130
In this case, x, y, and z.

1004
00:49:33,914 --> 00:49:36,080
And you mentioned the other
thing that he maybe did.

1005
00:49:36,080 --> 00:49:37,580
AUDIENCE: He
introduced the leakage,

1006
00:49:37,580 --> 00:49:39,510
but he didn't
mention that that was

1007
00:49:39,510 --> 00:49:40,510
PROFESSOR: That's right.

1008
00:49:40,510 --> 00:49:44,820
So I guess, he knew that leakage
was going to be a problem.

1009
00:49:44,820 --> 00:49:47,910
I.e, that you want
tight repression.

1010
00:49:47,910 --> 00:49:52,610
So he used these
synthetic promoters

1011
00:49:52,610 --> 00:49:55,910
that both had high
level of expression when

1012
00:49:55,910 --> 00:49:57,875
on but then very low
level of expression

1013
00:49:57,875 --> 00:49:58,750
when being repressed.

1014
00:50:07,800 --> 00:50:10,580
He made this thing.

1015
00:50:10,580 --> 00:50:15,190
And in particular, he
looked at it in a test tube.

1016
00:50:15,190 --> 00:50:18,720
He was able to use, in this
case, IPDG to synchronize them.

1017
00:50:18,720 --> 00:50:22,860
And he looked at the
fluorescence in the test tube.

1018
00:50:22,860 --> 00:50:27,020
So the fluorescence is reporting
on one of the proteins.

1019
00:50:27,020 --> 00:50:29,890
We can call it x if we'd like.

1020
00:50:29,890 --> 00:50:32,850
But fluorescence is kind
of telling about the state.

1021
00:50:32,850 --> 00:50:37,390
And if it starts out, say,
here, he saw a single cycle.

1022
00:50:37,390 --> 00:50:40,550
Damped oscillations, maybe.

1023
00:50:40,550 --> 00:50:44,870
So the question is,
why did this happen?

1024
00:50:54,920 --> 00:50:56,560
So why is it that,
in the test tube,

1025
00:50:56,560 --> 00:51:00,030
he didn't see something
that looked very nice?

1026
00:51:00,030 --> 00:51:02,660
Oscillations.

1027
00:51:02,660 --> 00:51:03,250
Noise.

1028
00:51:03,250 --> 00:51:05,210
And in particular,
what kind of noise?

1029
00:51:05,210 --> 00:51:06,040
Or what's going on?

1030
00:51:11,510 --> 00:51:13,351
Desynchronization, exactly.

1031
00:51:13,351 --> 00:51:15,100
So the idea is that,
even if you start out

1032
00:51:15,100 --> 00:51:17,980
with them all synchronized--
you give it IPDG pulls,

1033
00:51:17,980 --> 00:51:20,105
and they're synchronized
in some way--

1034
00:51:20,105 --> 00:51:21,980
it may be that, at the
beginning, all of them

1035
00:51:21,980 --> 00:51:23,980
are oscillating in
phase with each other.

1036
00:51:23,980 --> 00:51:27,420
But over time, random
noise, phase drift,

1037
00:51:27,420 --> 00:51:31,320
and the different oscillators
leads to some of them

1038
00:51:31,320 --> 00:51:33,320
come down and come back up.

1039
00:51:33,320 --> 00:51:34,644
And then, others are slower.

1040
00:51:34,644 --> 00:51:36,560
You start averaging all
these things together.

1041
00:51:36,560 --> 00:51:42,230
And it leads to damped
oscillations at the test tube

1042
00:51:42,230 --> 00:51:44,530
level within the bulk.

1043
00:51:44,530 --> 00:51:45,100
Yes?

1044
00:51:45,100 --> 00:51:46,555
AUDIENCE: So what do you
mean in the test tube?

1045
00:51:46,555 --> 00:51:48,500
Like, you just take all
these components and put it--

1046
00:51:48,500 --> 00:51:50,624
PROFESSOR: Sorry, when I
say test tube, what I mean

1047
00:51:50,624 --> 00:51:52,100
is that you have all the cells.

1048
00:51:52,100 --> 00:51:54,690
So they still are intact cells.

1049
00:51:54,690 --> 00:51:56,069
But it's just many cells.

1050
00:51:56,069 --> 00:51:58,110
So then, the signal that
you get the fluorescence

1051
00:51:58,110 --> 00:52:00,484
is some average over
all or sum overall.

1052
00:52:00,484 --> 00:52:02,400
The fluorescence you get
from all those cells.

1053
00:52:06,200 --> 00:52:07,850
So there's a sense
that this is really

1054
00:52:07,850 --> 00:52:11,172
what you expect given
the fact that they're

1055
00:52:11,172 --> 00:52:12,130
going to desynchronize.

1056
00:52:12,130 --> 00:52:14,171
Of course, the better the
oscillator in the sense

1057
00:52:14,171 --> 00:52:16,730
that the lower the
phase drift, then maybe

1058
00:52:16,730 --> 00:52:21,110
you can see a slower rate of
this kind of desynchronization.

1059
00:52:21,110 --> 00:52:23,730
But this is really what
you, kind of, expect.

1060
00:52:23,730 --> 00:52:24,230
All right.

1061
00:52:24,230 --> 00:52:26,760
So that's what, maybe,
led him to go and look

1062
00:52:26,760 --> 00:52:28,870
at the single cell
level where he put down

1063
00:52:28,870 --> 00:52:31,240
single cells on this
agar pad and just

1064
00:52:31,240 --> 00:52:34,090
imaged as the cells
oscillated and divided.

1065
00:52:37,170 --> 00:52:40,330
Now there are a
few features that

1066
00:52:40,330 --> 00:52:43,340
are important to
note from the data.

1067
00:52:43,340 --> 00:52:45,480
The first is that
they do oscillate.

1068
00:52:48,590 --> 00:52:51,030
That's a big deal
because this was, indeed,

1069
00:52:51,030 --> 00:52:53,320
the first demonstration
of being able to put

1070
00:52:53,320 --> 00:52:55,070
these random components
together like that

1071
00:52:55,070 --> 00:52:56,520
and generate oscillation.

1072
00:52:56,520 --> 00:52:59,200
But they didn't
oscillate very well.

1073
00:52:59,200 --> 00:53:01,850
So they said, oh, maybe 40%
of the cells oscillated.

1074
00:53:01,850 --> 00:53:07,470
And I have no idea what the
rest of the cells were doing.

1075
00:53:07,470 --> 00:53:09,720
But also, even the cells
that were oscillating, there

1076
00:53:09,720 --> 00:53:12,640
was a fair amount of
noise to the oscillation.

1077
00:53:12,640 --> 00:53:15,110
And the latter
half of this paper

1078
00:53:15,110 --> 00:53:17,620
has a fair amount of discussion
of why that might be.

1079
00:53:17,620 --> 00:53:22,410
And they allude to the ideas
that had been bouncing around

1080
00:53:22,410 --> 00:53:25,150
and from the theoretical
computational side

1081
00:53:25,150 --> 00:53:28,380
demonstrating that it may
be that the low numbers

1082
00:53:28,380 --> 00:53:31,140
of proteins, genes involved
here could introduce

1083
00:53:31,140 --> 00:53:34,900
stochastic noise into the
system and, thus, lead

1084
00:53:34,900 --> 00:53:38,060
to this kind of phase drift that
was observed experimentally.

1085
00:53:38,060 --> 00:53:39,970
I think that this
basic observation

1086
00:53:39,970 --> 00:53:42,900
that Michael had that
he got oscillations,

1087
00:53:42,900 --> 00:53:43,780
but they were noisy.

1088
00:53:43,780 --> 00:53:45,363
That is probably
what led him to start

1089
00:53:45,363 --> 00:53:48,570
thinking more and more about
the role of noise in G networks

1090
00:53:48,570 --> 00:53:51,410
and so forth and led,
later, to another hugely

1091
00:53:51,410 --> 00:53:54,790
influential paper that is
not going to be a required

1092
00:53:54,790 --> 00:53:56,420
reading in this
class but is listed

1093
00:53:56,420 --> 00:53:59,780
under the optional reading,
if you're interested.

1094
00:53:59,780 --> 00:54:03,360
But we'll really get into this
question of noise more a couple

1095
00:54:03,360 --> 00:54:04,752
weeks from now.

1096
00:54:08,520 --> 00:54:12,890
Were there any other questions
about the experimental side

1097
00:54:12,890 --> 00:54:15,560
of this paper?

1098
00:54:15,560 --> 00:54:19,666
I wanted to analyze maybe a
little bit of simple model

1099
00:54:19,666 --> 00:54:20,540
of the repressilator.

1100
00:54:26,782 --> 00:54:28,240
So the model that
they used to help

1101
00:54:28,240 --> 00:54:31,870
them design this experiment
involved all three proteins,

1102
00:54:31,870 --> 00:54:32,720
all three mRNAs.

1103
00:54:32,720 --> 00:54:35,220
And what that means is that,
when you go and you do a model,

1104
00:54:35,220 --> 00:54:37,830
you're going to end up
with a six by six matrix.

1105
00:54:37,830 --> 00:54:40,164
And I don't have boards
that are big enough.

1106
00:54:40,164 --> 00:54:41,580
So what I'm going
to do instead is

1107
00:54:41,580 --> 00:54:45,860
I'm going to analyze just
the protein only version

1108
00:54:45,860 --> 00:54:47,000
model of the repressilator.

1109
00:54:52,300 --> 00:54:52,800
All right.

1110
00:55:11,630 --> 00:55:15,080
So what we have here
is three proteins.

1111
00:55:15,080 --> 00:55:18,030
p1 2 3 p1 dot.

1112
00:55:18,030 --> 00:55:20,885
And we have degradation
of this protein.

1113
00:55:20,885 --> 00:55:23,540
And we're going to analyze
the symmetric version, just

1114
00:55:23,540 --> 00:55:25,430
like what Michael did.

1115
00:55:25,430 --> 00:55:28,070
So that means we're assuming
that all the proteins are

1116
00:55:28,070 --> 00:55:28,570
equivalent.

1117
00:55:28,570 --> 00:55:31,919
I'm sure that's not true because
these are different promoters

1118
00:55:31,919 --> 00:55:32,960
and different everything.

1119
00:55:32,960 --> 00:55:35,210
But this gives us the intuition.

1120
00:55:35,210 --> 00:55:35,920
So it's minus p1.

1121
00:55:35,920 --> 00:55:40,870
And this is protein 1 is
repressed by trajectory protein

1122
00:55:40,870 --> 00:55:41,370
3.

1123
00:55:47,050 --> 00:55:50,400
Protein 2 is going to be
repressed by protein 1.

1124
00:56:00,130 --> 00:56:05,360
And then protein 3 is going
to be repressed by protein 2.

1125
00:56:15,790 --> 00:56:18,105
So this is what you would
call the protein only model

1126
00:56:18,105 --> 00:56:18,980
of the repressilator.

1127
00:56:21,610 --> 00:56:23,630
Now just as before,
the fixed points

1128
00:56:23,630 --> 00:56:28,480
are when the pi
dots are equal to 0.

1129
00:56:28,480 --> 00:56:33,370
And we get the same
equation that we, basically,

1130
00:56:33,370 --> 00:56:37,160
had before where the
equilibrium or the fixed point,

1131
00:56:37,160 --> 00:56:43,030
again, is going to be given by
something that looks like this.

1132
00:56:43,030 --> 00:56:47,020
So it's the same requirement
that we had before.

1133
00:56:51,470 --> 00:56:55,500
Now the question is, how
can we get the stability

1134
00:56:55,500 --> 00:56:56,840
of that internal fixed point?

1135
00:56:56,840 --> 00:57:00,280
It's worth mentioning here that
now we have three proteins.

1136
00:57:00,280 --> 00:57:04,360
So the trajectories are in
this three dimensional space.

1137
00:57:04,360 --> 00:57:08,260
So from a mathematical
standpoint,

1138
00:57:08,260 --> 00:57:11,010
determining the stability of
that internal fixed point is

1139
00:57:11,010 --> 00:57:13,410
actually not sufficient to
tell you that there has to be

1140
00:57:13,410 --> 00:57:15,830
oscillations or there cannot
be oscillations because these

1141
00:57:15,830 --> 00:57:18,121
trajectories are, in principal,
allowed to do all sorts

1142
00:57:18,121 --> 00:57:20,170
of crazy things in
three dimensions.

1143
00:57:20,170 --> 00:57:23,210
But it turns out
that it still ends up

1144
00:57:23,210 --> 00:57:28,465
being true here that when this
internal fixed point is stable,

1145
00:57:28,465 --> 00:57:29,590
you don't get oscillations.

1146
00:57:29,590 --> 00:57:32,200
And when it's unstable, you do.

1147
00:57:32,200 --> 00:57:33,740
But that, sort of,
didn't have to be

1148
00:57:33,740 --> 00:57:38,720
true from a
mathematical standpoint.

1149
00:57:41,430 --> 00:57:41,930
All right.

1150
00:57:41,930 --> 00:57:46,740
Now since this is now going
to be a 3 by 3 matrix,

1151
00:57:46,740 --> 00:57:52,630
we're going to have to
calculate those eigenvalues.

1152
00:57:52,630 --> 00:57:56,690
Now how many eigenvalues
are there going to be?

1153
00:57:56,690 --> 00:57:57,930
Three?

1154
00:57:57,930 --> 00:57:58,430
OK.

1155
00:58:01,140 --> 00:58:05,840
So this thing I've written
in the form of a matrix

1156
00:58:05,840 --> 00:58:07,630
to help us out a little bit.

1157
00:58:07,630 --> 00:58:10,610
But in particular, we're
going to get the same thing

1158
00:58:10,610 --> 00:58:14,410
that we had before,
which is the p1 tilde.

1159
00:58:14,410 --> 00:58:20,421
So these are deviations,
again, from the fixed point.

1160
00:58:20,421 --> 00:58:22,670
And we got this matrix that's
going to look like this.

1161
00:58:22,670 --> 00:58:24,510
Minus 1, again, 0.

1162
00:58:24,510 --> 00:58:29,234
It's the same x that we had
before conveniently still

1163
00:58:29,234 --> 00:58:29,775
on the board.

1164
00:58:39,610 --> 00:58:43,820
So this is just after we
take these derivatives.

1165
00:58:43,820 --> 00:58:52,710
And then, we have p1 tilde,
p2 tilde, and p3 tilde.

1166
00:58:52,710 --> 00:58:55,440
Now what we need to know
is, for this Jacobian,

1167
00:58:55,440 --> 00:58:58,920
what are going to
be the eigenvalues?

1168
00:58:58,920 --> 00:59:01,318
For this thing to be
stable, it requires what?

1169
00:59:08,300 --> 00:59:12,090
What's the requirement for
stability of that fixed point?

1170
00:59:12,090 --> 00:59:14,704
That p0?

1171
00:59:14,704 --> 00:59:17,010
AUDIENCE: [INAUDIBLE].

1172
00:59:17,010 --> 00:59:17,920
PROFESSOR: OK.

1173
00:59:17,920 --> 00:59:18,500
Right.

1174
00:59:18,500 --> 00:59:22,030
For two dimensions, this trace
and determinant condition

1175
00:59:22,030 --> 00:59:22,920
works.

1176
00:59:22,920 --> 00:59:25,680
It's important to say that that
only works for two dimensions,

1177
00:59:25,680 --> 00:59:28,035
actually, the rule about
traces and determinants.

1178
00:59:31,980 --> 00:59:33,889
So be careful.

1179
00:59:33,889 --> 00:59:35,430
So what's the more
general statement?

1180
00:59:35,430 --> 00:59:35,930
Yeah.

1181
00:59:35,930 --> 00:59:38,009
AUDIENCE: Negative eigenvalues.

1182
00:59:38,009 --> 00:59:38,800
PROFESSOR: Exactly.

1183
00:59:38,800 --> 00:59:41,695
So in order for that
fixed point to be stable,

1184
00:59:41,695 --> 00:59:44,360
it requires that
all the eigenvalues

1185
00:59:44,360 --> 00:59:48,094
have real parts less than 0.

1186
00:59:48,094 --> 00:59:50,510
So in order to determine the
stability of the fixed point,

1187
00:59:50,510 --> 00:59:55,100
we need to ask what are the
eigenvalues of this matrix.

1188
00:59:55,100 --> 00:59:58,710
And to get the
eigenvalues, what we do

1189
00:59:58,710 --> 01:00:01,480
is we calculate this
characteristic equation,

1190
01:00:01,480 --> 01:00:05,570
this thing that we learned about
in linear algebra and so forth.

1191
01:00:05,570 --> 01:00:08,670
What we do is we take-- all
right, this is the matrix A,

1192
01:00:08,670 --> 01:00:09,550
we'll say.

1193
01:00:09,550 --> 01:00:12,410
This is matrix A. And
what we want to do

1194
01:00:12,410 --> 01:00:22,100
is we want to ask
whether the determinant

1195
01:00:22,100 --> 01:00:30,540
of the matrix A minus some
eigenvalue times the identity

1196
01:00:30,540 --> 01:00:31,230
matrix.

1197
01:00:31,230 --> 01:00:33,600
We want this thing
to be equal to 0.

1198
01:00:33,600 --> 01:00:40,120
So this is how we determine
what the eigenvalues are.

1199
01:00:40,120 --> 01:00:43,510
And this is not as bad as it
could be for general three

1200
01:00:43,510 --> 01:00:47,180
by three matrices because a
lot of these things are 0.

1201
01:00:47,180 --> 01:00:50,090
So this thing is just
this is the determinant

1202
01:00:50,090 --> 01:00:51,430
of the following matrix.

1203
01:00:54,270 --> 01:00:57,970
So we have minus
1 minus lambda 0.

1204
01:00:57,970 --> 01:01:01,550
This thing x that's,
in principle, bad.

1205
01:01:01,550 --> 01:01:04,870
Minus 1 minus lambda 0.

1206
01:01:04,870 --> 01:01:09,070
Getting 0 x minus
1 minus lambda.

1207
01:01:09,070 --> 01:01:10,860
Now to take the
determinant three by three

1208
01:01:10,860 --> 01:01:13,250
matrix, remember, you can
say, well, this determinant

1209
01:01:13,250 --> 01:01:16,700
is going to be equal
to-- we have this term.

1210
01:01:16,700 --> 01:01:23,040
So this is a minus 1 plus a
lambda times the determinant

1211
01:01:23,040 --> 01:01:24,950
of this matrix.

1212
01:01:24,950 --> 01:01:27,030
And then, we just
have that's this.

1213
01:01:27,030 --> 01:01:29,030
The product of these minus
the product of these.

1214
01:01:29,030 --> 01:01:31,080
So this just gives
us this thing again.

1215
01:01:31,080 --> 01:01:36,710
So this is actually just
minus 1 plus lambda cubed.

1216
01:01:36,710 --> 01:01:37,729
Next term, this is 0.

1217
01:01:37,729 --> 01:01:38,270
That's great.

1218
01:01:38,270 --> 01:01:39,686
We don't need to
worry about that.

1219
01:01:39,686 --> 01:01:41,390
The next one, we get plus.

1220
01:01:41,390 --> 01:01:43,470
We have an x.

1221
01:01:43,470 --> 01:01:45,640
Determining here, we
get, again, x squared.

1222
01:01:45,640 --> 01:01:48,814
So this is just an x cubed.

1223
01:01:48,814 --> 01:01:49,980
We want the same equal to 0.

1224
01:01:49,980 --> 01:01:52,780
So we actually get a
very simple requirement

1225
01:01:52,780 --> 01:01:56,280
for the eigenvalues,
which is that 1

1226
01:01:56,280 --> 01:02:02,335
plus the eigenvalues cubed is
equal to this thing x cubed.

1227
01:02:02,335 --> 01:02:03,710
Now be careful
because, remember,

1228
01:02:03,710 --> 01:02:08,310
x is actually a negative number.

1229
01:02:08,310 --> 01:02:11,230
So watch out.

1230
01:02:11,230 --> 01:02:13,840
So I think that the best
way to get a sense of what

1231
01:02:13,840 --> 01:02:15,670
this thing is is to plot it.

1232
01:02:28,380 --> 01:02:29,940
Of course, it's a
little bit tempting

1233
01:02:29,940 --> 01:02:31,740
here to just say,
all right, well,

1234
01:02:31,740 --> 01:02:34,360
can we just say that 1
plus lambda is equal to x?

1235
01:02:40,210 --> 01:02:41,139
No.

1236
01:02:41,139 --> 01:02:42,430
So what's the matter with that?

1237
01:02:46,960 --> 01:02:51,790
I mean, it's, sort of,
true, maybe, possibly.

1238
01:02:51,790 --> 01:02:52,290
Right.

1239
01:02:52,290 --> 01:02:55,030
So the problem
here is that we're

1240
01:02:55,030 --> 01:02:58,740
supposed to be getting
three different eigenvalues.

1241
01:02:58,740 --> 01:03:00,220
Or at lease, it's
possible to get

1242
01:03:00,220 --> 01:03:02,520
three different eigenvalues.

1243
01:03:02,520 --> 01:03:05,040
So this is really specifying
the solution for 1

1244
01:03:05,040 --> 01:03:07,400
plus lambda on
the complex plane.

1245
01:03:07,400 --> 01:03:12,010
So the solution
for 1 plus lambda

1246
01:03:12,010 --> 01:03:14,350
we can get by
thinking about this

1247
01:03:14,350 --> 01:03:17,920
is the real part
of 1 plus lambda.

1248
01:03:17,920 --> 01:03:22,720
And this is the imaginary
part of 1 plus lambda.

1249
01:03:22,720 --> 01:03:34,015
And we know that one solution
is going to be out here at x.

1250
01:03:37,830 --> 01:03:40,090
This distance here is
the magnitude of x.

1251
01:03:44,870 --> 01:03:46,880
Now the others,
however, are going

1252
01:03:46,880 --> 01:03:52,349
to be around the complex plane
similar distances where we get

1253
01:03:52,349 --> 01:03:53,640
something that looks like this.

1254
01:03:57,620 --> 01:04:01,270
So these are, like,
30, 60, 90 triangle.

1255
01:04:01,270 --> 01:04:03,440
So this is 30 degrees
here because what

1256
01:04:03,440 --> 01:04:06,370
you see is that, for each of
these three solutions for 1

1257
01:04:06,370 --> 01:04:14,320
plus lambda, if you cube
them, you end up with x cubed.

1258
01:04:14,320 --> 01:04:18,060
So this guy, you square it.

1259
01:04:18,060 --> 01:04:18,560
Cube.

1260
01:04:18,560 --> 01:04:20,640
You end up back here.

1261
01:04:20,640 --> 01:04:24,050
This one, if you cube
it, you start out here,

1262
01:04:24,050 --> 01:04:26,530
squared, and then cubed
comes back out here.

1263
01:04:26,530 --> 01:04:29,896
Same thing and this
goes around somehow.

1264
01:04:29,896 --> 01:04:31,570
All right, so there
are three solutions

1265
01:04:31,570 --> 01:04:33,280
to this 1 plus lambda.

1266
01:04:33,280 --> 01:04:37,127
And there are these points here.

1267
01:04:37,127 --> 01:04:38,710
Now, of course, it's
not 1 plus lambda

1268
01:04:38,710 --> 01:04:40,293
that we actually
wanted to know about.

1269
01:04:40,293 --> 01:04:42,490
It was lambda.

1270
01:04:42,490 --> 01:04:46,730
But if we know what
1 plus lambda is,

1271
01:04:46,730 --> 01:04:48,850
then we can get what lambda is.

1272
01:04:48,850 --> 01:04:50,800
What do we have to do?

1273
01:04:50,800 --> 01:04:52,560
Right, we have to
slide it to the left.

1274
01:04:52,560 --> 01:04:54,910
So this is the real axis.

1275
01:04:54,910 --> 01:04:56,310
This is the imaginary axis.

1276
01:04:56,310 --> 01:04:58,830
1.

1277
01:04:58,830 --> 01:05:02,600
So we have to move
everything over 1.

1278
01:05:02,600 --> 01:05:04,530
Now remember, the
requirement for stability

1279
01:05:04,530 --> 01:05:07,540
was that all of the
eigenvalues have

1280
01:05:07,540 --> 01:05:09,071
real parts that were negative.

1281
01:05:09,071 --> 01:05:11,570
That means the requirement for
stability of that fixed point

1282
01:05:11,570 --> 01:05:13,350
is that all three of
these fixed points

1283
01:05:13,350 --> 01:05:16,470
are in the left
half of the plane.

1284
01:05:16,470 --> 01:05:19,420
So what you can see is
that, in this problem,

1285
01:05:19,420 --> 01:05:21,695
the whole question of
stability and whether we

1286
01:05:21,695 --> 01:05:25,220
get oscillations boils down
to how big this thing is.

1287
01:05:25,220 --> 01:05:26,310
What's this distance?

1288
01:05:26,310 --> 01:05:29,950
If this distance is more
than 1, then we subtract 1,

1289
01:05:29,950 --> 01:05:35,931
we don't get it into the
left part of the plane.

1290
01:05:35,931 --> 01:05:38,370
OK, I can't remember
which case I just gave.

1291
01:05:38,370 --> 01:05:42,119
But yeah, we need to know
whether this thing is

1292
01:05:42,119 --> 01:05:43,160
larger or smaller than 1.

1293
01:05:45,820 --> 01:05:48,795
And that has to do with
the magnitude of x.

1294
01:05:51,600 --> 01:05:56,535
So if the magnitude
of x-- do you

1295
01:05:56,535 --> 01:05:59,872
guys remember your geometry
for a 30, 60, 90 triangle?

1296
01:06:03,099 --> 01:06:06,550
All right, so if the
magnitude of x-- and this

1297
01:06:06,550 --> 01:06:08,760
is indeed the magnitude of x.

1298
01:06:08,760 --> 01:06:16,320
This short edge on a 30, 60, 90
is half the long edge, right?

1299
01:06:16,320 --> 01:06:21,220
So what we can say is that
this fixed point stable, state

1300
01:06:21,220 --> 01:06:30,610
fixed point, is if and only
if the magnitude of x is what?

1301
01:06:35,630 --> 01:06:37,840
Lesson two.

1302
01:06:37,840 --> 01:06:38,930
OK.

1303
01:06:38,930 --> 01:06:41,140
That's nice.

1304
01:06:41,140 --> 01:06:44,100
And if we want, we could plug
in-- just to ride this out.

1305
01:06:44,100 --> 01:06:48,320
This is n alpha p 0 n minus 1.

1306
01:07:05,100 --> 01:07:08,000
So it's useful, once you
get to something like this,

1307
01:07:08,000 --> 01:07:12,332
to try to just ask, for
various kind of values,

1308
01:07:12,332 --> 01:07:13,290
how does this play out?

1309
01:07:13,290 --> 01:07:18,062
What does the
requirement end up being?

1310
01:07:18,062 --> 01:07:19,770
And a useful limit is
to think about what

1311
01:07:19,770 --> 01:07:22,070
happens in the limit of
very strong expression?

1312
01:07:22,070 --> 01:07:24,110
So strong expression
corresponds to what?

1313
01:07:32,696 --> 01:07:34,127
AUDIENCE: Big alpha?

1314
01:07:34,127 --> 01:07:35,080
PROFESSOR: Big alpha.

1315
01:07:35,080 --> 01:07:35,780
Yes, perfect.

1316
01:07:38,750 --> 01:07:44,730
And it turns out, big
alpha is a little bit-- OK,

1317
01:07:44,730 --> 01:07:47,340
and remember we have to
remember what p0 was.

1318
01:07:47,340 --> 01:07:50,766
p0 was this p0 times 1 plus p0.

1319
01:07:54,877 --> 01:07:56,460
All right, so this
is the requirement.

1320
01:08:00,860 --> 01:08:04,840
And actually, if you
play with these equations

1321
01:08:04,840 --> 01:08:06,300
just a little bit,
what you'll find

1322
01:08:06,300 --> 01:08:09,920
is that, if alpha is
much larger than 1,

1323
01:08:09,920 --> 01:08:18,340
then this requirement
is that n is less than 2

1324
01:08:18,340 --> 01:08:19,359
or less than around 2.

1325
01:08:24,080 --> 01:08:26,479
This is saying,
on the flip side,

1326
01:08:26,479 --> 01:08:32,500
the fixed point is stable if
you don't have very strong

1327
01:08:32,500 --> 01:08:33,920
cooperativity and repression.

1328
01:08:33,920 --> 01:08:36,740
And the flip side is, if you
have strong cooperativity

1329
01:08:36,740 --> 01:08:39,830
of repression, then you
can get oscillations

1330
01:08:39,830 --> 01:08:42,029
because this interior fixed
point becomes unstable.

1331
01:08:44,910 --> 01:08:49,754
So this is also saying that
n greater than around 2

1332
01:08:49,754 --> 01:08:50,670
leads to oscillations.

1333
01:09:14,370 --> 01:09:16,590
And this maybe
makes sense because,

1334
01:09:16,590 --> 01:09:19,140
when you have strong
productivity in the repression

1335
01:09:19,140 --> 01:09:21,390
there, what that's
telling you is that it's

1336
01:09:21,390 --> 01:09:22,979
a switch like response.

1337
01:09:22,979 --> 01:09:28,039
And in that regime,
it maybe becomes more

1338
01:09:28,039 --> 01:09:29,580
like a simple Boolean
kind of network

1339
01:09:29,580 --> 01:09:31,663
where, if you just write
down the ones and zeroes,

1340
01:09:31,663 --> 01:09:34,494
you can convince yourself that
this thing maybe, in principle,

1341
01:09:34,494 --> 01:09:35,160
could oscillate.

1342
01:09:38,010 --> 01:09:41,540
Now if you look at the
Elowitz repressilator paper,

1343
01:09:41,540 --> 01:09:46,270
you'll see that he gives
some expression for what

1344
01:09:46,270 --> 01:09:48,516
this thing should be like.

1345
01:09:48,516 --> 01:09:51,710
And it looks vaguely similar.

1346
01:09:51,710 --> 01:09:56,540
Of course, there he's
including the mRNAs, as well.

1347
01:09:56,540 --> 01:09:59,420
But if you think that this
was painful to do in class,

1348
01:09:59,420 --> 01:10:01,460
then including the
mRNAs is more painful.

1349
01:10:07,680 --> 01:10:14,764
Are there any questions
about this idea?

1350
01:10:14,764 --> 01:10:15,264
Yeah.

1351
01:10:15,264 --> 01:10:17,847
AUDIENCE: So in the paper, did
they also only do the stability

1352
01:10:17,847 --> 01:10:20,104
analysis to determine the--

1353
01:10:20,104 --> 01:10:23,390
PROFESSOR: I think they
did simulations, as well.

1354
01:10:23,390 --> 01:10:27,330
So the nature of simulations is
that you can convince yourself

1355
01:10:27,330 --> 01:10:31,790
that their exist places that do
oscillate or don't oscillate.

1356
01:10:31,790 --> 01:10:33,800
Although, you'll notice
that they have a very,

1357
01:10:33,800 --> 01:10:36,850
kind of, enigmatic
sentence in here,

1358
01:10:36,850 --> 01:10:39,160
which is that it is
possible that, in addition

1359
01:10:39,160 --> 01:10:41,770
to simple oscillations, this
and more realistic models

1360
01:10:41,770 --> 01:10:45,220
may exhibit other complex
types of dynamic behavior.

1361
01:10:45,220 --> 01:10:47,640
And this is just
a way of saying,

1362
01:10:47,640 --> 01:10:49,810
well, you know, I don't know.

1363
01:10:49,810 --> 01:10:52,990
Maybe someday
because once you talk

1364
01:10:52,990 --> 01:10:55,870
about six dimensional
system, you never

1365
01:10:55,870 --> 01:10:58,750
know if you've explored
all of the parameter space.

1366
01:10:58,750 --> 01:11:00,170
I mean, even for
fixed parameters,

1367
01:11:00,170 --> 01:11:02,720
you don't know if you started
at all the right locations.

1368
01:11:02,720 --> 01:11:06,047
You can kind develop
some sense that, oh,

1369
01:11:06,047 --> 01:11:08,380
this thing seems to oscillate
or seems to not oscillate.

1370
01:11:08,380 --> 01:11:10,171
And it does correspond
to these conditions.

1371
01:11:10,171 --> 01:11:13,080
But you don't know.

1372
01:11:13,080 --> 01:11:15,270
I mean, it could be
that, in some regions,

1373
01:11:15,270 --> 01:11:16,590
you get chaos or other things.

1374
01:11:16,590 --> 01:11:17,340
Right?

1375
01:11:17,340 --> 01:11:20,650
So it's funny because I've
read this paper many times.

1376
01:11:20,650 --> 01:11:22,905
But it was only last night
when I was re-reading it

1377
01:11:22,905 --> 01:11:24,529
that I kind of thought
about that sense

1378
01:11:24,529 --> 01:11:28,600
like, yeah, I'm not sure either
what this model could possibly

1379
01:11:28,600 --> 01:11:29,361
do.

1380
01:11:29,361 --> 01:11:29,860
Yes?

1381
01:11:29,860 --> 01:11:31,348
AUDIENCE: In this
linear analysis

1382
01:11:31,348 --> 01:11:33,780
the three x's are the same.

1383
01:11:33,780 --> 01:11:34,780
PROFESSOR: That's right.

1384
01:11:34,780 --> 01:11:36,529
AUDIENCE: Because
they're non-dimensional?

1385
01:11:39,375 --> 01:11:40,250
PROFESSOR: All right.

1386
01:11:40,250 --> 01:11:42,370
So the reason that the
three X's are the same

1387
01:11:42,370 --> 01:11:46,250
is because we've
assumed that this really

1388
01:11:46,250 --> 01:11:52,067
is the symmetric version
of the repressilator

1389
01:11:52,067 --> 01:11:54,150
because we're assuming
that all of the alphas, all

1390
01:11:54,150 --> 01:11:55,608
the ends, all the
K's, everything's

1391
01:11:55,608 --> 01:11:57,090
the same across
all three of them.

1392
01:11:57,090 --> 01:11:58,650
So given that
symmetry, then you're

1393
01:11:58,650 --> 01:12:02,190
always going to end up with
a symmetric version of this.

1394
01:12:02,190 --> 01:12:05,690
So I think if it were
asymmetric and then you

1395
01:12:05,690 --> 01:12:07,880
made the non-dimensional
versions of things,

1396
01:12:07,880 --> 01:12:10,555
I think you still won't end
up getting the same X's just

1397
01:12:10,555 --> 01:12:12,680
because, if it's asymmetric,
then and something has

1398
01:12:12,680 --> 01:12:13,388
to be asymmetric.

1399
01:12:16,382 --> 01:12:16,882
Yes?

1400
01:12:16,882 --> 01:12:21,489
AUDIENCE: [INAUDIBLE] so
large alpha leads to--

1401
01:12:21,489 --> 01:12:22,280
PROFESSOR: Yes, OK.

1402
01:12:22,280 --> 01:12:24,430
We can go ahead and do this.

1403
01:12:37,020 --> 01:12:41,350
So for large alpha,
this fixed point

1404
01:12:41,350 --> 01:12:48,360
is going to be-- p0 is going
to be much larger than 1.

1405
01:12:48,360 --> 01:12:51,190
So this is about
p0 to the n plus 1.

1406
01:12:51,190 --> 01:12:53,615
We can neglect the
1 for large alpha.

1407
01:12:53,615 --> 01:12:54,990
And then and then
what we can say

1408
01:12:54,990 --> 01:13:00,580
is that, over here, for
example, if we multiply

1409
01:13:00,580 --> 01:13:07,520
both sides by-- p0 squared, p0
squared, so multiply it by one.

1410
01:13:07,520 --> 01:13:11,650
Then this down here is
definitely alpha squared.

1411
01:13:11,650 --> 01:13:13,350
And then, up here,
what we have is

1412
01:13:13,350 --> 01:13:17,190
p0 to the n plus 1, which
we decided was around

1413
01:13:17,190 --> 01:13:20,980
alpha for a strong alpha.

1414
01:13:20,980 --> 01:13:26,770
So that gives us alpha times
alpha divided by alpha squared.

1415
01:13:26,770 --> 01:13:30,410
So this actually all goes
away for large alpha.

1416
01:13:30,410 --> 01:13:33,630
So then, you're just
left with n less than 2.

1417
01:13:33,630 --> 01:13:34,130
Did that--

1418
01:13:34,130 --> 01:13:35,030
AUDIENCE: Sorry.

1419
01:13:35,030 --> 01:13:36,380
Where did that top right
equation come from?

1420
01:13:36,380 --> 01:13:38,280
PROFESSOR: OK, so this
equation here is this

1421
01:13:38,280 --> 01:13:42,140
is the solution for where
that fixed point is.

1422
01:13:42,140 --> 01:13:47,780
So in this space of the p0's,
if you set the equations

1423
01:13:47,780 --> 01:13:49,770
for p1, p2, p3, if you
set that equal to 0,

1424
01:13:49,770 --> 01:13:54,880
this is the expression always
for a large alpha, small.

1425
01:13:54,880 --> 01:13:59,180
So this is a need be
location of that fixed point.

1426
01:13:59,180 --> 01:14:01,020
And it's just, as
alpha is large,

1427
01:14:01,020 --> 01:14:04,240
then we get that
p0 to the n plus 1

1428
01:14:04,240 --> 01:14:05,920
is approximately equal to alpha.

1429
01:14:05,920 --> 01:14:08,326
And this is for alpha
much greater than 1.

1430
01:14:08,326 --> 01:14:11,400
And in that case, all of
these things just go away.

1431
01:14:11,400 --> 01:14:14,890
And you're just left
with n less than 2.

1432
01:14:14,890 --> 01:14:20,370
So for example, as alpha
goes down in magnitude,

1433
01:14:20,370 --> 01:14:23,090
then you end up getting a
requirement that oscillations

1434
01:14:23,090 --> 01:14:24,158
require a larger n.

1435
01:14:32,110 --> 01:14:33,640
We'll give you practice on this.

1436
01:14:33,640 --> 01:14:46,680
All right, so I think I wrote
another-- if I can find my--

1437
01:14:46,680 --> 01:14:48,430
you can ask for
alpha equal to 2.

1438
01:14:51,180 --> 01:14:54,090
What n required
for oscillations.

1439
01:15:05,589 --> 01:15:07,130
I'll let you start
playing with that.

1440
01:15:07,130 --> 01:15:09,912
And I will make sure
that I've given you

1441
01:15:09,912 --> 01:15:10,870
the right alpha to use.

1442
01:15:36,110 --> 01:15:37,910
So in this case,
what we're asking is,

1443
01:15:37,910 --> 01:15:41,740
instead of having really strong
maximal expression, if instead

1444
01:15:41,740 --> 01:15:44,465
expression is just not quite
as strong, then what we'll find

1445
01:15:44,465 --> 01:15:45,840
is that you actually
need to have

1446
01:15:45,840 --> 01:15:49,170
a more cooperative repression
in order to get oscillations.

1447
01:15:49,170 --> 01:15:51,500
And that's just because,
if alpha is equal to 2,

1448
01:15:51,500 --> 01:15:53,922
then we can, kind of, figure
out what p0 is equal to.

1449
01:15:57,370 --> 01:15:57,950
1.

1450
01:15:57,950 --> 01:15:58,860
Right.

1451
01:15:58,860 --> 01:16:00,190
Great.

1452
01:16:00,190 --> 01:16:03,570
So the fixed point is at one.

1453
01:16:03,570 --> 01:16:05,660
That's great because this
we can then figure out.

1454
01:16:05,660 --> 01:16:06,560
Right?

1455
01:16:06,560 --> 01:16:08,230
So this is 1 plus 1 square.

1456
01:16:08,230 --> 01:16:09,880
That's a four.

1457
01:16:09,880 --> 01:16:10,760
1.

1458
01:16:10,760 --> 01:16:11,260
2.

1459
01:16:13,800 --> 01:16:16,380
So this tells us
that, in this case,

1460
01:16:16,380 --> 01:16:19,710
we need to have very
cooperative repression.

1461
01:16:19,710 --> 01:16:22,660
We have to have an n
greater than around 4

1462
01:16:22,660 --> 01:16:26,980
in order to get oscillations
in this protein only model.

1463
01:16:31,330 --> 01:16:31,830
Yes?

1464
01:16:31,830 --> 01:16:33,538
AUDIENCE: It is kind
of strange that even

1465
01:16:33,538 --> 01:16:37,650
for a really big alpha you
still need n greater than sum.

1466
01:16:37,650 --> 01:16:39,064
PROFESSOR: Yeah, right, right.

1467
01:16:39,064 --> 01:16:40,480
So this is an
interesting question

1468
01:16:40,480 --> 01:16:45,130
that you might think that
for a very large expression

1469
01:16:45,130 --> 01:16:49,020
that you wouldn't need to have
cooperative repression at all.

1470
01:16:49,020 --> 01:16:49,750
Right?

1471
01:16:49,750 --> 01:16:54,880
And I can't say that I have any
wonderful intuition about this

1472
01:16:54,880 --> 01:16:58,440
because it, somehow,
has to do with just

1473
01:16:58,440 --> 01:17:01,507
the slopes of those curves
around that fixed point.

1474
01:17:01,507 --> 01:17:02,715
And it's in three dimensions.

1475
01:17:06,050 --> 01:17:08,900
But I think that this
highlights that it's

1476
01:17:08,900 --> 01:17:11,160
a priori if you
go and say, oh, I

1477
01:17:11,160 --> 01:17:14,905
want to construct
this repressilator,

1478
01:17:14,905 --> 01:17:17,640
it's maybe not even obvious that
you want it to be more or less.

1479
01:17:17,640 --> 01:17:19,640
I mean, you might not
even think about this idea

1480
01:17:19,640 --> 01:17:20,870
of cooperative repression.

1481
01:17:20,870 --> 01:17:25,240
You might be tempted to
think that any chain of three

1482
01:17:25,240 --> 01:17:27,330
proteins repressing each
other just, kind of,

1483
01:17:27,330 --> 01:17:28,314
has to oscillate.

1484
01:17:28,314 --> 01:17:29,980
I mean, there's a
little bit of a sense.

1485
01:17:29,980 --> 01:17:34,180
And that's the logic that
you get at if you just do

1486
01:17:34,180 --> 01:17:34,980
0's and 1's.

1487
01:17:34,980 --> 01:17:36,030
If you say, oh, here's x.

1488
01:17:36,030 --> 01:17:37,000
Here's y.

1489
01:17:37,000 --> 01:17:37,880
Here's z.

1490
01:17:37,880 --> 01:17:39,310
And they're
repressing each other.

1491
01:17:39,310 --> 01:17:40,062
Right?

1492
01:17:40,062 --> 01:17:45,470
And you say, oh, OK, well if
I start out at, say, 0 1 0

1493
01:17:45,470 --> 01:17:47,950
and you say, OK,
that's all fine.

1494
01:17:47,950 --> 01:17:49,400
But OK, so this is repressing.

1495
01:17:49,400 --> 01:17:51,608
And it's OK, but this guy
wasn't repressing this one.

1496
01:17:51,608 --> 01:17:54,616
So now we get a 1, 1, 0, maybe.

1497
01:17:54,616 --> 01:17:55,490
Then you say, oh, OK.

1498
01:17:55,490 --> 01:17:57,870
Well now this guy starts
repressing this one.

1499
01:17:57,870 --> 01:18:00,170
So now it gives us a 1 0 0.

1500
01:18:00,170 --> 01:18:05,255
And what you see is that, over
these two steps, the on protein

1501
01:18:05,255 --> 01:18:05,755
has shifted.

1502
01:18:05,755 --> 01:18:07,120
And indeed, that's
going to continue

1503
01:18:07,120 --> 01:18:08,250
going all the way around.

1504
01:18:08,250 --> 01:18:12,720
So from this Boolean
logic kind of perspective,

1505
01:18:12,720 --> 01:18:15,930
you might think that any
three proteins mutually

1506
01:18:15,930 --> 01:18:18,332
repressing each other
just has to oscillate.

1507
01:18:18,332 --> 01:18:20,290
And it's only by looking
at things a little bit

1508
01:18:20,290 --> 01:18:21,998
more carefully that
you say, oh, well, we

1509
01:18:21,998 --> 01:18:25,520
have to actually worry
about this that you really

1510
01:18:25,520 --> 01:18:30,110
have to think about you want
to choose some transcription

1511
01:18:30,110 --> 01:18:33,176
factors that are multimerizing
and cooperatively repressing

1512
01:18:33,176 --> 01:18:36,930
the next protein just to have
some reasonable shot at having

1513
01:18:36,930 --> 01:18:40,925
this thing actually oscillate.

1514
01:18:40,925 --> 01:18:44,010
AUDIENCE: So in this, we
might still be able-- I mean,

1515
01:18:44,010 --> 01:18:47,700
oscillations like this
might still [INAUDIBLE]

1516
01:18:47,700 --> 01:18:50,188
but just not like,
maybe, oscillations

1517
01:18:50,188 --> 01:18:51,938
around some stable fix
point or something.

1518
01:18:51,938 --> 01:18:56,577
Like, they're just not
limit cycle oscillations.

1519
01:18:56,577 --> 01:18:57,994
Do you think that
in a [INAUDIBLE]

1520
01:18:57,994 --> 01:19:00,285
there would probably still
be some kind of oscillations

1521
01:19:00,285 --> 01:19:00,960
somewhere.

1522
01:19:00,960 --> 01:19:04,346
Just not this beautiful
limit cycle kind.

1523
01:19:04,346 --> 01:19:05,720
PROFESSOR: Yeah,
my understanding

1524
01:19:05,720 --> 01:19:09,820
is that in, for example,
this protein only model

1525
01:19:09,820 --> 01:19:13,610
of the repressilator
that if you do not

1526
01:19:13,610 --> 01:19:16,280
have cooperative repression,
then it really just

1527
01:19:16,280 --> 01:19:17,904
goes to that stable fixed point.

1528
01:19:17,904 --> 01:19:19,570
Of course, you have
to worry about maybe

1529
01:19:19,570 --> 01:19:22,200
these noise-induced
oscillation ideas.

1530
01:19:22,200 --> 01:19:24,690
But at least within the
realm of the deterministic,

1531
01:19:24,690 --> 01:19:28,130
differential equations,
then the system

1532
01:19:28,130 --> 01:19:30,900
just goes to that internal fixed
point that's specified by this.

1533
01:19:34,181 --> 01:19:34,680
Question?

1534
01:19:34,680 --> 01:19:41,750
AUDIENCE: Can we think like
that the cooperation, sort of,

1535
01:19:41,750 --> 01:19:44,570
introduced delay?

1536
01:19:44,570 --> 01:19:47,130
PROFESSOR: That's an
interesting question.

1537
01:19:47,130 --> 01:19:50,290
Whether cooperativity, maybe,
is introducing a delay.

1538
01:19:50,290 --> 01:19:52,400
And that's because, after
the proteins are made,

1539
01:19:52,400 --> 01:19:55,290
maybe it takes some extra
time to dimer and so forth.

1540
01:19:55,290 --> 01:19:58,000
So that statement may be true.

1541
01:19:58,000 --> 01:19:59,950
But it's not relevant.

1542
01:19:59,950 --> 01:20:00,530
OK?

1543
01:20:00,530 --> 01:20:02,220
And I think this
is very important.

1544
01:20:02,220 --> 01:20:06,170
This model has certainly
not taken that into account.

1545
01:20:06,170 --> 01:20:09,200
So the mechanism that's here
is not what you're saying.

1546
01:20:09,200 --> 01:20:11,440
But it may be true that,
for any experimental system,

1547
01:20:11,440 --> 01:20:14,935
such delay from dimerization
is relevant and helps

1548
01:20:14,935 --> 01:20:15,810
you get oscillations.

1549
01:20:15,810 --> 01:20:16,310
Right?

1550
01:20:16,310 --> 01:20:19,180
But at least within the
realm of this model,

1551
01:20:19,180 --> 01:20:21,910
we have very much not included
any sort of delay associated

1552
01:20:21,910 --> 01:20:23,400
with dimerization or anything.

1553
01:20:23,400 --> 01:20:25,340
So that is very much
not the explanation

1554
01:20:25,340 --> 01:20:28,300
for why dimerization leads
to oscillations here.

1555
01:20:28,300 --> 01:20:30,700
And I think this
is a wider point

1556
01:20:30,700 --> 01:20:33,460
that it's very important
always to keep track

1557
01:20:33,460 --> 01:20:35,815
of which effects you've
included in any given analysis

1558
01:20:35,815 --> 01:20:37,900
and which ones are not.

1559
01:20:37,900 --> 01:20:39,970
And it's very, very common.

1560
01:20:39,970 --> 01:20:41,470
There are many
things that are true.

1561
01:20:41,470 --> 01:20:44,390
But they may not actually be
relevant for the discussion

1562
01:20:44,390 --> 01:20:44,985
at hand.

1563
01:20:44,985 --> 01:20:46,360
And I think, in
those situations,

1564
01:20:46,360 --> 01:20:51,810
it's easy to get mixed up
because it still is true,

1565
01:20:51,810 --> 01:20:54,220
even if it's not what's
driving the effect that is

1566
01:20:54,220 --> 01:20:57,530
being, in this case, analyzed.

1567
01:20:57,530 --> 01:20:58,280
We're out of time.

1568
01:20:58,280 --> 01:20:59,190
So we should quit.

1569
01:20:59,190 --> 01:21:02,750
On Tuesday, we'll start by
wrapping up the oscillation

1570
01:21:02,750 --> 01:21:06,610
discussion by talking about
other oscillator designs that

1571
01:21:06,610 --> 01:21:09,460
allow for robustness
and tunability.

1572
01:21:09,460 --> 01:21:11,010
OK?