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PROFESSOR: Today what
we're going to do

9
00:00:21,910 --> 00:00:25,078
is we're going to talk about
some of the big ideas in class.

10
00:00:34,042 --> 00:00:36,775
Today what we're going to do
is talk about pattern formation

11
00:00:36,775 --> 00:00:39,220
in biology, some of the
different mechanisms

12
00:00:39,220 --> 00:00:43,010
that are employed in development
and other contexts in order

13
00:00:43,010 --> 00:00:46,130
for organisms to figure
out where to put things.

14
00:00:46,130 --> 00:00:49,110
Now before I get
into that too much,

15
00:00:49,110 --> 00:00:51,330
I wanted to make couple
administrative type

16
00:00:51,330 --> 00:00:52,830
announcements.

17
00:00:52,830 --> 00:00:55,090
So first, we have
graded the exams

18
00:00:55,090 --> 00:00:58,440
and we'll hand them back
at the end of class today.

19
00:00:58,440 --> 00:01:00,850
If I try to forget then
please somebody remind me.

20
00:01:00,850 --> 00:01:02,391
What we're going to
do is we're going

21
00:01:02,391 --> 00:01:05,736
to start by just talking
about diffusion a bit more.

22
00:01:05,736 --> 00:01:07,110
I think that
there's a fair range

23
00:01:07,110 --> 00:01:10,490
of different maybe experiences
thinking about diffusion

24
00:01:10,490 --> 00:01:11,770
and what it can do for you.

25
00:01:11,770 --> 00:01:14,090
So we'll work on our
intuition a little bit.

26
00:01:14,090 --> 00:01:16,370
And then I'll say
something about the reading

27
00:01:16,370 --> 00:01:18,810
that you did from
[INAUDIBLE] book

28
00:01:18,810 --> 00:01:22,010
about robust mechanisms
for pattern formation

29
00:01:22,010 --> 00:01:23,650
in the context of development.

30
00:01:23,650 --> 00:01:25,943
So simple diffusion
with degradation leads

31
00:01:25,943 --> 00:01:28,650
to these exponential
profiles that are not

32
00:01:28,650 --> 00:01:31,470
robust to changes in the
concentration or production

33
00:01:31,470 --> 00:01:34,690
rate of the morphogen.
Whereas if you have self

34
00:01:34,690 --> 00:01:39,940
enhanced degradation then
this leads to power law

35
00:01:39,940 --> 00:01:44,350
fall off it's more robust
to the concentration

36
00:01:44,350 --> 00:01:47,660
of this morphogen.

37
00:01:47,660 --> 00:01:50,640
Then what we're going to do
is transition to these turning

38
00:01:50,640 --> 00:01:52,640
patterns, reaction
to fusion systems

39
00:01:52,640 --> 00:01:55,620
where you can have a
really surprising effect,

40
00:01:55,620 --> 00:01:59,670
whereby in a well mixed
situation these chemicals

41
00:01:59,670 --> 00:02:02,220
or proteins or reactants,
they might just

42
00:02:02,220 --> 00:02:04,800
reach a stable state.

43
00:02:04,800 --> 00:02:07,100
However, once you add
diffusion, then you

44
00:02:07,100 --> 00:02:09,120
can start getting
pattern formation, which

45
00:02:09,120 --> 00:02:12,690
is a very funny thing because
diffusion is normally something

46
00:02:12,690 --> 00:02:14,485
that's smooths out profiles.

47
00:02:14,485 --> 00:02:15,860
Normally we think
about diffusion

48
00:02:15,860 --> 00:02:18,260
as being something
that removes patterns

49
00:02:18,260 --> 00:02:20,210
and indeed that's
generally what happens.

50
00:02:20,210 --> 00:02:23,360
But following from
this class of work

51
00:02:23,360 --> 00:02:25,970
from Alan Turing at
the end of his life,

52
00:02:25,970 --> 00:02:28,430
he showed that it's
actually in principle

53
00:02:28,430 --> 00:02:31,780
possible to have the emergence
of patterns from diffusion.

54
00:02:31,780 --> 00:02:34,720
And we'll talk about some of
the ways that could possibly

55
00:02:34,720 --> 00:02:35,220
happen.

56
00:02:38,790 --> 00:02:42,510
So Turing patterns are already
a surprising case where

57
00:02:42,510 --> 00:02:44,140
diffusion leads to patterns.

58
00:02:44,140 --> 00:02:45,860
There's another
interesting phenomenon

59
00:02:45,860 --> 00:02:48,850
where if you add another
source of noise that-- again,

60
00:02:48,850 --> 00:02:51,940
you think noise, demographic
noise, the random creation

61
00:02:51,940 --> 00:02:56,260
destruction of individuals
or chemicals or proteins.

62
00:02:56,260 --> 00:02:58,245
Normally, we think this
should also kind of be

63
00:02:58,245 --> 00:02:59,980
a force for removing patterns.

64
00:02:59,980 --> 00:03:03,520
But there's been
a number of pieces

65
00:03:03,520 --> 00:03:06,920
of work over the last decade
showing that in some cases,

66
00:03:06,920 --> 00:03:08,880
demographic noise
can actually lead

67
00:03:08,880 --> 00:03:11,210
to patterns either
in space or in time.

68
00:03:11,210 --> 00:03:12,990
We're going to talk
more about this

69
00:03:12,990 --> 00:03:15,380
in the context of these
so-called noise induced

70
00:03:15,380 --> 00:03:16,810
predator-prey oscillations.

71
00:03:16,810 --> 00:03:18,890
But since we're
talking about patterns

72
00:03:18,890 --> 00:03:22,785
here I will tell you something
about how such noise can

73
00:03:22,785 --> 00:03:24,830
enhance the
formations of patterns

74
00:03:24,830 --> 00:03:27,340
in the context of
the Turing mechanism.

75
00:03:27,340 --> 00:03:31,020
And then we'll end by talking
some about recent work of how

76
00:03:31,020 --> 00:03:36,250
E. coli find the center of their
cell when they want to divide.

77
00:03:36,250 --> 00:03:39,609
So you can imagine this
is not an obvious thing

78
00:03:39,609 --> 00:03:40,900
to figure out how you might do.

79
00:03:40,900 --> 00:03:44,751
And we'll talk about
the so-called Min system

80
00:03:44,751 --> 00:03:46,750
that E. coli use to find
the center of the cells

81
00:03:46,750 --> 00:03:48,395
so they know where
to septate, where

82
00:03:48,395 --> 00:03:49,915
to cut off and make two cells.

83
00:03:55,000 --> 00:03:56,750
So I just want to
start by making sure

84
00:03:56,750 --> 00:03:58,456
that when we talk
about diffusion,

85
00:03:58,456 --> 00:04:00,080
we're all talking
about the same thing.

86
00:04:05,060 --> 00:04:07,072
Now, hopefully for
those of you that

87
00:04:07,072 --> 00:04:09,405
have not been thinking about
diffusion so much recently,

88
00:04:09,405 --> 00:04:12,420
you did read the notes that
Alexander Van Oudenaarden

89
00:04:12,420 --> 00:04:16,550
put together for Systems
Biology a few years ago.

90
00:04:16,550 --> 00:04:20,950
I think it's useful, but
in addition to the math,

91
00:04:20,950 --> 00:04:24,300
as maybe many of you
know I'm a huge fan

92
00:04:24,300 --> 00:04:26,760
of graphical
representations of ideas.

93
00:04:26,760 --> 00:04:31,940
So I just want to
make sure that we all

94
00:04:31,940 --> 00:04:35,900
agree on some of the basic
ideas of how, for example, we

95
00:04:35,900 --> 00:04:40,240
get flux in the
context of diffusion.

96
00:04:40,240 --> 00:04:42,350
So imagine that we
have a one dimension

97
00:04:42,350 --> 00:04:49,270
system where there's some
chemical with concentration c.

98
00:04:49,270 --> 00:04:55,720
We can imagine, for example, it
starts out looking like this.

99
00:04:55,720 --> 00:04:58,690
This is a concentration profile
as a function of position

100
00:04:58,690 --> 00:05:05,880
x where we have maybe
a box of length l.

101
00:05:10,200 --> 00:05:12,800
Concentration of some chemicals
of function of position

102
00:05:12,800 --> 00:05:17,715
x in some space.

103
00:05:17,715 --> 00:05:20,340
Now I just want to make
sure that we're all

104
00:05:20,340 --> 00:05:21,590
thinking about the same thing.

105
00:05:21,590 --> 00:05:23,340
So it starts out some
concentration of c1,

106
00:05:23,340 --> 00:05:25,600
over might be some c2.

107
00:05:25,600 --> 00:05:26,670
Linearly here.

108
00:05:26,670 --> 00:05:29,350
I just want to make
sure we all agree

109
00:05:29,350 --> 00:05:36,060
on the magnitude of the flux
at some different points.

110
00:05:36,060 --> 00:05:47,650
So, in particular, we can think
about A, B, this is capital C.

111
00:05:47,650 --> 00:05:50,360
So where is the magnitude
of the flux largest?

112
00:06:02,840 --> 00:06:08,130
A, B, C. D corresponds
to them being the same.

113
00:06:08,130 --> 00:06:10,170
E as don't know.

114
00:06:12,889 --> 00:06:14,680
Are there any questions
about the question?

115
00:06:14,680 --> 00:06:15,180
Yes.

116
00:06:15,180 --> 00:06:17,530
AUDIENCE: On the y-axis,
you're plotting concentration?

117
00:06:17,530 --> 00:06:19,320
PROFESSOR: Yes, this
y-axis is the concentration

118
00:06:19,320 --> 00:06:20,570
is the function of position x.

119
00:06:24,700 --> 00:06:29,614
All right, I'll give another
five seconds to think about it.

120
00:06:36,175 --> 00:06:37,550
All right, do you
need more time?

121
00:06:40,461 --> 00:06:40,960
Let's vote.

122
00:06:40,960 --> 00:06:44,770
Ready three, two, one.

123
00:06:44,770 --> 00:06:46,530
OK, great.

124
00:06:46,530 --> 00:06:48,800
So I'd say a vast
majority here are agreeing

125
00:06:48,800 --> 00:06:54,250
that it's going to D. The
flux is going to the same

126
00:06:54,250 --> 00:06:56,080
everywhere.

127
00:06:56,080 --> 00:06:58,110
Now the flux-- can
somebody remind us?

128
00:06:58,110 --> 00:07:01,995
We'll say the flux of J.
And what is it going to be?

129
00:07:08,050 --> 00:07:15,720
So there's a minus D times the
change in c with respect to x.

130
00:07:15,720 --> 00:07:18,170
This was derived in the notes.

131
00:07:18,170 --> 00:07:21,990
So this is highlighting
that what leads to a flux

132
00:07:21,990 --> 00:07:26,880
is the change in a concentration
with respect to position.

133
00:07:26,880 --> 00:07:29,790
And so it doesn't matter that
the concentration is higher

134
00:07:29,790 --> 00:07:32,310
here than here, we
have the same flux.

135
00:07:32,310 --> 00:07:34,610
And the flux is in
which direction?

136
00:07:34,610 --> 00:07:36,170
Left, right, up, down?

137
00:07:36,170 --> 00:07:37,880
Three, two, one.

138
00:07:37,880 --> 00:07:40,170
Left, so the flux is here.

139
00:07:47,430 --> 00:07:52,400
Minus sign-- like always, it's
hard to remember from equations

140
00:07:52,400 --> 00:07:54,230
whether they're plus
minus signs but you

141
00:07:54,230 --> 00:07:56,680
should be able to just
remember you can draw something

142
00:07:56,680 --> 00:08:00,900
like this and say OK,
this is a positive DCDX,

143
00:08:00,900 --> 00:08:04,495
so flux is going to be in the
negative direction, right?

144
00:08:11,770 --> 00:08:16,260
Now, what will be the change in
the concentration with respect

145
00:08:16,260 --> 00:08:19,940
to time at this point right now?

146
00:08:22,560 --> 00:08:25,110
So this is change in the
conservation at point

147
00:08:25,110 --> 00:08:28,100
b with respect to time.

148
00:08:28,100 --> 00:08:33,390
Is it greater than
0, equal to 0,

149
00:08:33,390 --> 00:08:37,110
less than 0, or can't determine?

150
00:08:49,240 --> 00:08:51,830
Does everybody understand the
question I'm trying to ask?

151
00:08:51,830 --> 00:08:53,360
Change in the
concentration at this

152
00:08:53,360 --> 00:08:58,112
point with respect to time,
at this time that's wrong.

153
00:08:58,112 --> 00:08:59,400
I'll give you 5 seconds.

154
00:09:08,100 --> 00:09:09,050
Ready?

155
00:09:09,050 --> 00:09:13,890
Three, two, one.

156
00:09:13,890 --> 00:09:15,580
OK.

157
00:09:15,580 --> 00:09:22,370
And so this, again, it's
pretty good but not everyone.

158
00:09:22,370 --> 00:09:23,420
So it is going to be b.

159
00:09:23,420 --> 00:09:27,301
Can somebody explain
why it's b here?

160
00:09:27,301 --> 00:09:27,800
Yes.

161
00:09:27,800 --> 00:09:29,892
AUDIENCE: There's as
much flux coming in

162
00:09:29,892 --> 00:09:31,520
as coming out on the other side?

163
00:09:31,520 --> 00:09:32,520
PROFESSOR: That's right.

164
00:09:32,520 --> 00:09:36,689
So it's true that there's a net
flux here coming to the left.

165
00:09:36,689 --> 00:09:38,980
But from the sample, if we
want to know whether there's

166
00:09:38,980 --> 00:09:42,270
a change in the
concentration at that point,

167
00:09:42,270 --> 00:09:45,150
we need to know what's the
net number of particles moving

168
00:09:45,150 --> 00:09:47,990
to the left verses the net
particle number there that

169
00:09:47,990 --> 00:09:51,730
are coming in from the right.

170
00:09:51,730 --> 00:09:56,740
Now of course, on this left
face-- if I draw a little box

171
00:09:56,740 --> 00:10:02,092
here-- are all the particles
crossing this position?

172
00:10:02,092 --> 00:10:03,425
Are they all moving to the left?

173
00:10:09,161 --> 00:10:09,660
No.

174
00:10:12,760 --> 00:10:16,450
So the idea is that all the
particle motion is random.

175
00:10:16,450 --> 00:10:18,600
And indeed, there's
only a slight access

176
00:10:18,600 --> 00:10:20,975
of particles moving to the
left as compared to the right.

177
00:10:20,975 --> 00:10:23,584
In the sense that, if you look
at the concentration there,

178
00:10:23,584 --> 00:10:25,000
the concentration
is rather large,

179
00:10:25,000 --> 00:10:26,970
it's only a little
bit larger to right

180
00:10:26,970 --> 00:10:28,450
than to the left of this plane.

181
00:10:28,450 --> 00:10:30,480
Which means that there's only
a slight excess of particles

182
00:10:30,480 --> 00:10:32,313
defusing to the left
as coming to the right,

183
00:10:32,313 --> 00:10:35,862
but that leads to a net
flux of particles crossing

184
00:10:35,862 --> 00:10:36,820
this plane to the left.

185
00:10:44,970 --> 00:10:50,280
So where will the
concentration be changing?

186
00:10:50,280 --> 00:10:52,206
At this point here
will the concentration

187
00:10:52,206 --> 00:10:53,580
be changing with
respect to time?

188
00:10:53,580 --> 00:10:56,120
Ready three, two, one.

189
00:10:56,120 --> 00:10:57,490
No.

190
00:10:57,490 --> 00:11:03,980
So well does it change at all
ever anywhere in this example?

191
00:11:03,980 --> 00:11:05,730
Is this a steady state
profile, yes or no?

192
00:11:05,730 --> 00:11:07,329
Ready three, two, one.

193
00:11:07,329 --> 00:11:07,870
AUDIENCE: No.

194
00:11:07,870 --> 00:11:09,430
PROFESSOR: No.

195
00:11:09,430 --> 00:11:11,695
What should be the
steady state profile?

196
00:11:11,695 --> 00:11:12,320
AUDIENCE: Flat.

197
00:11:12,320 --> 00:11:13,700
PROFESSOR: Flat, OK.

198
00:11:13,700 --> 00:11:16,330
So how does that come about
if the concentration is not

199
00:11:16,330 --> 00:11:18,210
changing any?

200
00:11:18,210 --> 00:11:19,190
Yeah.

201
00:11:19,190 --> 00:11:21,260
AUDIENCE: It'll only
change at the edges.

202
00:11:21,260 --> 00:11:23,343
PROFESSOR: It's only
changing the edges initially.

203
00:11:25,454 --> 00:11:27,720
Now, it's important
to note that it's not

204
00:11:27,720 --> 00:11:31,230
that the concentration
profile is not going

205
00:11:31,230 --> 00:11:34,680
to start looking like this.

206
00:11:34,680 --> 00:11:37,633
So that will not actually
be how it-- eventually it's

207
00:11:37,633 --> 00:11:38,800
going to be flat.

208
00:11:38,800 --> 00:11:42,020
But it's going to
smooth out on the edges.

209
00:11:44,794 --> 00:11:46,710
Do you guys understand
what I'm trying to say?

210
00:11:46,710 --> 00:11:49,080
It's not that it's a line
that kind of goes like this,

211
00:11:49,080 --> 00:11:51,520
but instead we do end
up getting curvature.

212
00:11:54,670 --> 00:11:56,430
One more of these,
and then we'll

213
00:11:56,430 --> 00:11:59,540
consider ourselves to
be expert diffusionists.

214
00:12:01,477 --> 00:12:03,185
All right, so let's
say the concentration

215
00:12:03,185 --> 00:12:05,525
as a function of position
looks something like this.

216
00:12:12,310 --> 00:12:18,210
What I want to know is,
where is DCDT maximal?

217
00:12:26,931 --> 00:12:28,180
We have some different points.

218
00:12:28,180 --> 00:12:42,920
We have A, B, C,
D. Five seconds.

219
00:12:50,770 --> 00:12:54,040
Ready three, two, one.

220
00:12:59,370 --> 00:13:01,920
All right, so we finally
got a lot of disagreement.

221
00:13:01,920 --> 00:13:03,860
I like it.

222
00:13:03,860 --> 00:13:05,550
Turn to your
neighbor, you should

223
00:13:05,550 --> 00:13:08,626
be able to find somebody
that disagrees with you.

224
00:13:08,626 --> 00:13:09,750
What is it that determines?

225
00:13:09,750 --> 00:13:12,240
AUDIENCE: [ALL DISCUSSING]

226
00:13:58,675 --> 00:14:00,550
PROFESSOR: All right,
did you guys all agree?

227
00:14:08,730 --> 00:14:10,262
You all agree on it?

228
00:14:13,220 --> 00:14:15,420
OK let's go ahead and reconvene.

229
00:14:15,420 --> 00:14:18,150
And let me see, maybe
they were forming

230
00:14:18,150 --> 00:14:19,610
the domains of some sort.

231
00:14:19,610 --> 00:14:21,330
All right, ready.

232
00:14:21,330 --> 00:14:22,430
Wait, he says, no, no.

233
00:14:22,430 --> 00:14:26,570
All right, OK, one question.

234
00:14:26,570 --> 00:14:31,784
AUDIENCE: Is this
instantaneous in some sense?

235
00:14:31,784 --> 00:14:32,450
PROFESSOR: Yeah.

236
00:14:32,450 --> 00:14:34,609
AUDIENCE: Exactly at the inset.

237
00:14:34,609 --> 00:14:36,650
PROFESSOR: All right, so
I guess what I would say

238
00:14:36,650 --> 00:14:39,830
is that this is
concentration profile

239
00:14:39,830 --> 00:14:43,790
at some time as a function
of position in time.

240
00:14:43,790 --> 00:14:47,010
If you want to know the
concentration profile

241
00:14:47,010 --> 00:14:48,900
some time delta t later, right?

242
00:14:51,970 --> 00:14:54,190
t plus delta t.

243
00:14:54,190 --> 00:14:56,860
Well then, it's going to
be the concentration we

244
00:14:56,860 --> 00:14:59,530
had before then a little bit.

245
00:14:59,530 --> 00:15:05,460
Plus delta t times the
derivative with respect

246
00:15:05,460 --> 00:15:08,130
to time.

247
00:15:08,130 --> 00:15:12,780
Right, so this is the
concentrate file at this time.

248
00:15:12,780 --> 00:15:15,370
And I want to how
much is it going

249
00:15:15,370 --> 00:15:16,910
to change in the next delta t?

250
00:15:22,982 --> 00:15:24,690
And I'm assuming, of
course, that I'm not

251
00:15:24,690 --> 00:15:26,822
adding any particles or
taking any particles away.

252
00:15:26,822 --> 00:15:28,240
So just do the diffusion.

253
00:15:28,240 --> 00:15:29,656
All right, let's
see where we are.

254
00:15:29,656 --> 00:15:33,120
Ready three, two, one.

255
00:15:33,120 --> 00:15:34,080
OK.

256
00:15:34,080 --> 00:15:37,540
All right so it's we got
domain, definitely some domains.

257
00:15:37,540 --> 00:15:38,040
All right.

258
00:15:40,640 --> 00:15:42,676
Well, it's going
to end up being A,

259
00:15:42,676 --> 00:15:46,730
as I think the majority of
the group is now saying.

260
00:15:46,730 --> 00:15:50,625
And why is it a?

261
00:15:50,625 --> 00:15:52,500
AUDIENCE: Because of
the second derivative.

262
00:15:52,500 --> 00:15:54,000
PROFESSOR: Second
derivative, right?

263
00:15:54,000 --> 00:15:56,550
So just what we
said before is that,

264
00:15:56,550 --> 00:15:58,955
if we wanted to know the
change of concentration

265
00:15:58,955 --> 00:16:01,900
with respect of time, at
this point that was zero.

266
00:16:01,900 --> 00:16:08,930
Because the flux particle was
leading that point and the flux

267
00:16:08,930 --> 00:16:09,930
coming in were the same.

268
00:16:15,147 --> 00:16:17,230
And we can also think about
the second derivative.

269
00:16:17,230 --> 00:16:19,640
So this indeed just
from fix, like I said,

270
00:16:19,640 --> 00:16:26,550
was first law/ we got a D, so
this gives second derivative c

271
00:16:26,550 --> 00:16:28,532
with respect to x squared.

272
00:16:28,532 --> 00:16:29,115
So that's a 2.

273
00:16:31,645 --> 00:16:35,180
What this is saying is that,
to determine how rapidly

274
00:16:35,180 --> 00:16:39,360
this concentration is going to
change at a particular point

275
00:16:39,360 --> 00:16:41,132
with respect of
time, we need to know

276
00:16:41,132 --> 00:16:43,090
the curvature of the
concentration with respect

277
00:16:43,090 --> 00:16:45,530
to position.

278
00:16:45,530 --> 00:16:47,950
So many people
were saying c here.

279
00:16:47,950 --> 00:16:51,660
And actually, this is
where the concentration

280
00:16:51,660 --> 00:16:56,420
will remain exactly constant
over this next delta t.

281
00:16:56,420 --> 00:16:57,861
And of course, if
we want to know

282
00:16:57,861 --> 00:17:00,360
about what the concentration
is going to do for a long time,

283
00:17:00,360 --> 00:17:01,950
then we have to do
something more subtle.

284
00:17:01,950 --> 00:17:03,700
But if we want to how
the concentration is

285
00:17:03,700 --> 00:17:08,730
changing right now, then we
just look at the curvature.

286
00:17:08,730 --> 00:17:11,139
And indeed, the curvature
here is maximal.

287
00:17:14,599 --> 00:17:18,131
So concentration
DCDT at this point,

288
00:17:18,131 --> 00:17:19,839
is it going to be
greater or less than 0?

289
00:17:19,839 --> 00:17:22,770
Ready three, two, one.

290
00:17:22,770 --> 00:17:24,770
PROFESSOR: Less than 0, right?

291
00:17:24,770 --> 00:17:28,413
Now, it's true that only the
curvature at a particular point

292
00:17:28,413 --> 00:17:31,760
and matter is to know how the
concentrations going to change

293
00:17:31,760 --> 00:17:33,191
over this next delta t.

294
00:17:33,191 --> 00:17:34,690
But if you want to
know about what's

295
00:17:34,690 --> 00:17:36,150
going to happen over
longer periods of time,

296
00:17:36,150 --> 00:17:38,566
than you have to worry about
more of the global structure.

297
00:17:38,566 --> 00:17:42,650
And what's true is that this
local maximum may quickly

298
00:17:42,650 --> 00:17:44,374
go away.

299
00:17:44,374 --> 00:17:46,790
So you don't have to wait very
long before this thing gets

300
00:17:46,790 --> 00:17:53,590
smooth and then DCDT will--
the magnitude will go to 0.

301
00:17:53,590 --> 00:18:00,810
And sometimes this big hump
will stick around longer,

302
00:18:00,810 --> 00:18:02,940
but because of the
curvature, this

303
00:18:02,940 --> 00:18:06,830
has the maximal
magnitude of DCDT.

304
00:18:10,972 --> 00:18:12,680
And it makes sense
that the concentration

305
00:18:12,680 --> 00:18:15,680
is going to go down because the
net flux here is to the right

306
00:18:15,680 --> 00:18:16,635
and to the left.

307
00:18:16,635 --> 00:18:18,510
And so there's a net
flux, particles leaving,

308
00:18:18,510 --> 00:18:20,010
concentration is
going to come down.

309
00:18:23,590 --> 00:18:28,730
Now are there any questions
about what we've said?

310
00:18:28,730 --> 00:18:31,870
If you find this
discussion confusing,

311
00:18:31,870 --> 00:18:35,560
I would strongly encourage you
to get together with a friend

312
00:18:35,560 --> 00:18:38,485
and just draw some
random lines, curves,

313
00:18:38,485 --> 00:18:39,860
and just make sure
you understand

314
00:18:39,860 --> 00:18:42,197
where diffusion is going to
pull things and so forth.

315
00:18:44,940 --> 00:18:45,505
No questions?

316
00:18:55,030 --> 00:18:56,780
All right, I just want
to say a few things

317
00:18:56,780 --> 00:19:01,120
about the so-called French
flag model in development.

318
00:19:01,120 --> 00:19:05,709
This is perhaps the most famous.

319
00:19:05,709 --> 00:19:07,875
Does everybody know what
the French flag looks like?

320
00:19:11,400 --> 00:19:12,860
What's that?

321
00:19:12,860 --> 00:19:13,470
All right.

322
00:19:13,470 --> 00:19:16,430
There are three stripes
and some orientation.

323
00:19:16,430 --> 00:19:19,310
No, but the idea here is
just that a simple way

324
00:19:19,310 --> 00:19:23,610
to specify a structure
across one axis

325
00:19:23,610 --> 00:19:28,280
is to have a defusing chemical
or generally a protein.

326
00:19:28,280 --> 00:19:31,040
And then based on the
concentration of that protein

327
00:19:31,040 --> 00:19:34,430
or morphogen, you can just
have the tissue read out

328
00:19:34,430 --> 00:19:37,500
what it's supposed to be,
or what it's supposed to do.

329
00:19:37,500 --> 00:19:40,720
So the idea here
is that, you have

330
00:19:40,720 --> 00:19:45,620
some morphogen that starts
maybe on one end of an embryo

331
00:19:45,620 --> 00:19:48,340
and then diffuses.

332
00:19:48,340 --> 00:19:50,790
So we end up with
some curve that

333
00:19:50,790 --> 00:19:55,090
tells us about the
morphogen concentration

334
00:19:55,090 --> 00:19:57,730
as a function of position.

335
00:19:57,730 --> 00:19:59,450
This is a functional position.

336
00:19:59,450 --> 00:20:03,620
So this is along the embryo.

337
00:20:03,620 --> 00:20:06,040
Now the general challenge in
the context of development

338
00:20:06,040 --> 00:20:10,900
is asking-- we
started this embryo

339
00:20:10,900 --> 00:20:13,130
with many different
cells, they're

340
00:20:13,130 --> 00:20:15,870
all genetically identical,
and they all start out maybe

341
00:20:15,870 --> 00:20:18,050
without any positional
information.

342
00:20:18,050 --> 00:20:19,360
They don't know where they are.

343
00:20:19,360 --> 00:20:20,735
But what that
means is they don't

344
00:20:20,735 --> 00:20:25,000
know they need to develop into
a head or a tail or something

345
00:20:25,000 --> 00:20:26,630
in between.

346
00:20:26,630 --> 00:20:31,920
Now what often happens is that
you can have a deposition often

347
00:20:31,920 --> 00:20:35,100
by the mother so you have a
maternal deposition of, say,

348
00:20:35,100 --> 00:20:39,170
RNA that leads to expression
of some morphogen on one end

349
00:20:39,170 --> 00:20:40,134
and then it diffuses.

350
00:20:40,134 --> 00:20:41,550
And based on the
concentration you

351
00:20:41,550 --> 00:20:46,120
can say what part of the body
you should be developing into.

352
00:20:46,120 --> 00:20:49,410
So the idea is you just
have these thresholds that

353
00:20:49,410 --> 00:20:51,570
might be some M1, M2.

354
00:20:55,140 --> 00:20:59,870
All right, now all
the cells over here

355
00:20:59,870 --> 00:21:03,030
know that they should
maybe develop into a head.

356
00:21:03,030 --> 00:21:06,830
Over here, they're going to
develop into the mid body.

357
00:21:06,830 --> 00:21:10,420
And over down here it could
be the low body, for example.

358
00:21:10,420 --> 00:21:14,270
So just from the
concentration that a cell

359
00:21:14,270 --> 00:21:15,840
feels at a particular location.

360
00:21:15,840 --> 00:21:18,260
So if you had a cell
right here, you just

361
00:21:18,260 --> 00:21:21,540
say OK the concentration of
morphogens between M1 and M2

362
00:21:21,540 --> 00:21:26,770
and then from that you know
what l you're supposed be.

363
00:21:26,770 --> 00:21:35,500
Now if you have no
degradation within the body.

364
00:21:35,500 --> 00:21:43,950
Let's just imagine you
say, that at one end is 0.

365
00:21:43,950 --> 00:21:48,380
We're told morphogen
concentration is M0.

366
00:21:48,380 --> 00:21:51,810
And let's say that the other
end, we know the morphogen

367
00:21:51,810 --> 00:21:53,560
concentration is going to be 0.

368
00:21:53,560 --> 00:21:58,520
Let's say there's a lot
of degradation at an end.

369
00:21:58,520 --> 00:22:00,480
If there's no
degradation in between,

370
00:22:00,480 --> 00:22:04,620
what should the profile
look like given these?

371
00:22:04,620 --> 00:22:05,580
So, no.

372
00:22:08,990 --> 00:22:09,490
All right.

373
00:22:09,490 --> 00:22:13,960
So just diffusion with
these boundary conditions.

374
00:22:13,960 --> 00:22:17,030
I'm going to draw some
options and hopefully you

375
00:22:17,030 --> 00:22:19,970
can close your eyes and
imagine what it should be.

376
00:22:19,970 --> 00:22:26,350
All right, we know that starts
out M0 over L is equal to 0.

377
00:22:42,220 --> 00:22:45,930
Is it closer to A, B, C, or D?

378
00:22:56,450 --> 00:22:59,030
Do you understand
what I'm asking?

379
00:22:59,030 --> 00:22:59,800
Yes.

380
00:22:59,800 --> 00:23:01,360
AUDIENCE: That is
a steady state.

381
00:23:01,360 --> 00:23:04,620
PROFESSOR: This is the steady
state concentration profile

382
00:23:04,620 --> 00:23:09,390
of the morphogen given there's
no degradation in the interior.

383
00:23:09,390 --> 00:23:12,270
Ready three, two, one.

384
00:23:15,410 --> 00:23:16,020
All right.

385
00:23:16,020 --> 00:23:19,100
So at least a majority are
agreeing it's going to be B.

386
00:23:19,100 --> 00:23:23,940
Although, there's a significant
minority that's saying A.

387
00:23:23,940 --> 00:23:27,030
This is why I'm bring
this up because, we're

388
00:23:27,030 --> 00:23:29,100
so used to seeing
exponential profiles

389
00:23:29,100 --> 00:23:31,610
and think that you should
always be getting them.

390
00:23:31,610 --> 00:23:34,860
And I want to be clear why
it's an exponential profile.

391
00:23:34,860 --> 00:23:36,940
That's the thing we start
with, and for example

392
00:23:36,940 --> 00:23:39,670
this model in [INAUDIBLE] book.

393
00:23:39,670 --> 00:23:43,140
Noori's book, you get
exponential profiles

394
00:23:43,140 --> 00:23:47,140
as a result of first
order degradation.

395
00:23:47,140 --> 00:23:51,390
Whereas if you don't have any
degradation in the interior,

396
00:23:51,390 --> 00:23:58,840
then we say, OK, DCD-- well,
do we want to use C or M now?

397
00:23:58,840 --> 00:24:00,610
We'll use M, just
because now we're

398
00:24:00,610 --> 00:24:03,130
thinking very particularly
for morphogen.

399
00:24:03,130 --> 00:24:04,585
All so DMDT.

400
00:24:11,840 --> 00:24:15,290
We want to know the
steady state profile.

401
00:24:15,290 --> 00:24:16,535
We set this thing equal to 0.

402
00:24:21,110 --> 00:24:22,920
Integrate twice.

403
00:24:22,920 --> 00:24:25,790
Right, so we're going to get--
morphogen profile should just

404
00:24:25,790 --> 00:24:28,914
be some Ax plus B, OK?

405
00:24:35,310 --> 00:24:37,200
My line's kind of
crappy, I'm sorry.

406
00:24:41,510 --> 00:24:43,780
However, when we have
first order degradation,

407
00:24:43,780 --> 00:24:46,100
then we get these
exponential profiles.

408
00:24:49,350 --> 00:24:52,750
Are there any questions about--?

409
00:24:52,750 --> 00:24:56,710
AUDIENCE: Is it possible
to have [INAUDIBLE]?

410
00:24:56,710 --> 00:24:59,420
PROFESSOR: Yeah, right.

411
00:24:59,420 --> 00:25:01,659
What does this require?

412
00:25:01,659 --> 00:25:05,603
AUDIENCE: The morphogen
is generated into halves.

413
00:25:05,603 --> 00:25:07,876
X equals 0 and then--

414
00:25:07,876 --> 00:25:09,000
PROFESSOR: Degraded it at--

415
00:25:09,000 --> 00:25:09,900
AUDIENCE: Disappearing.

416
00:25:09,900 --> 00:25:10,566
PROFESSOR: Yeah.

417
00:25:10,566 --> 00:25:11,820
Right.

418
00:25:11,820 --> 00:25:16,430
That's what I'm saying, is that
when I said that at the end

419
00:25:16,430 --> 00:25:19,940
it was a 0, this means that we
have degradation at this point.

420
00:25:19,940 --> 00:25:24,470
Any time that the morphogen
reaches the end of the embryos,

421
00:25:24,470 --> 00:25:25,400
it gets degraded.

422
00:25:28,100 --> 00:25:31,420
So it's not degradation inside
if it's only degradation here.

423
00:25:31,420 --> 00:25:34,100
And this of course a situation
where you have constant flux

424
00:25:34,100 --> 00:25:36,860
throughout the embryo,
constant rate of production

425
00:25:36,860 --> 00:25:40,100
of the morphogen, constant
rate of degradation over here.

426
00:25:44,690 --> 00:25:47,490
But when you have this
first order degradation--

427
00:25:47,490 --> 00:25:51,430
if we're told that
instead-- well, maybe.

428
00:26:05,300 --> 00:26:12,000
So, if in addition
to having diffusion,

429
00:26:12,000 --> 00:26:15,180
you also have this
first order degradation.

430
00:26:15,180 --> 00:26:17,910
Now remember M is a function
of both x and t, right?

431
00:26:17,910 --> 00:26:23,890
And this is the situation where
we get exponential profiles.

432
00:26:23,890 --> 00:26:27,500
Now in Murray's book,
he often discusses

433
00:26:27,500 --> 00:26:32,310
this where the
boundary condition

434
00:26:32,310 --> 00:26:37,540
is that you have some
concentration M0 at one end.

435
00:26:37,540 --> 00:26:44,002
And then we get an
exponential profile here.

436
00:26:44,002 --> 00:26:45,710
What was the characteristic
length scale?

437
00:26:52,320 --> 00:26:54,930
This is something we should
be able to figure out.

438
00:27:03,590 --> 00:27:07,160
You should, of course, be
able to solve the equations,

439
00:27:07,160 --> 00:27:10,590
but you should be
able to figure it out

440
00:27:10,590 --> 00:27:13,670
from our favorite approach
of dimensional analysis.

441
00:27:13,670 --> 00:27:17,040
What's the units of things here?

442
00:27:17,040 --> 00:27:20,740
So units of D is what?

443
00:27:20,740 --> 00:27:21,910
Length squared over time.

444
00:27:28,600 --> 00:27:29,500
Units of alpha?

445
00:27:34,930 --> 00:27:38,220
So this is a morphogen
concentration over a time.

446
00:27:38,220 --> 00:27:40,812
This is a morphogen
concentration.

447
00:27:40,812 --> 00:27:42,395
So this thing has
to be one over time.

448
00:27:46,110 --> 00:27:47,620
So the length
scale is then going

449
00:27:47,620 --> 00:27:58,960
to have to go as the
square root of D over L.

450
00:27:58,960 --> 00:28:01,790
And indeed, if you solve it,
it's exactly equal to that,

451
00:28:01,790 --> 00:28:04,199
but this dimensional
analysis just told us

452
00:28:04,199 --> 00:28:05,490
that it had to scale like that.

453
00:28:08,350 --> 00:28:10,780
So this is telling us about
a characteristic length

454
00:28:10,780 --> 00:28:13,930
of which this morphogen
profile is going to fall off.

455
00:28:13,930 --> 00:28:20,100
This is the
characteristic length L.

456
00:28:20,100 --> 00:28:22,860
Incidentally, you'll
see many cases

457
00:28:22,860 --> 00:28:24,800
of things that look like this.

458
00:28:24,800 --> 00:28:28,060
So if there's first
order rate that's

459
00:28:28,060 --> 00:28:32,940
something is either degraded
or is uptaken or whatnot, then

460
00:28:32,940 --> 00:28:37,450
together with diffusion, you'll
get an exponential profile

461
00:28:37,450 --> 00:28:39,964
with a [INAUDIBLE] length scale
that's given by this ratio.

462
00:28:39,964 --> 00:28:41,880
And you'll see this in
many different contexts

463
00:28:41,880 --> 00:28:46,890
of, for example, nutrients
going into a biofilm.

464
00:28:46,890 --> 00:28:49,780
If you have cells
picking up nutrients

465
00:28:49,780 --> 00:28:53,340
then it's not that
it's being degraded,

466
00:28:53,340 --> 00:28:55,162
but it's being imported.

467
00:28:55,162 --> 00:28:56,620
Then you'll have
a similar process.

468
00:29:02,800 --> 00:29:07,310
This is a nice, fine situation
except for what was the problem

469
00:29:07,310 --> 00:29:08,520
that [INAUDIBLE] pointed out?

470
00:29:11,490 --> 00:29:12,187
It's not robust.

471
00:29:12,187 --> 00:29:13,395
It's not robust against what?

472
00:29:18,200 --> 00:29:20,500
Initial condition.

473
00:29:20,500 --> 00:29:24,630
Yes, but I think we have to be
a little more specific on that.

474
00:29:24,630 --> 00:29:26,420
Right, it's not robust
to changes in M0.

475
00:29:36,240 --> 00:29:39,760
Incidentally, in
most context, I think

476
00:29:39,760 --> 00:29:41,300
for mathematical
simplicity, it's

477
00:29:41,300 --> 00:29:45,820
useful to just have a banner
condition where M0 is constant.

478
00:29:45,820 --> 00:29:50,560
But in general, is
that what the mother is

479
00:29:50,560 --> 00:29:53,070
going to be fixing necessarily?

480
00:29:53,070 --> 00:29:54,820
What would the mother
typically be fixing?

481
00:29:58,370 --> 00:29:59,625
What was that?

482
00:29:59,625 --> 00:30:00,960
AUDIENCE: Production rate.

483
00:30:00,960 --> 00:30:02,824
PROFESSOR: Production rate.

484
00:30:02,824 --> 00:30:04,490
How do we figure out
what the production

485
00:30:04,490 --> 00:30:05,700
rate is in this situation?

486
00:30:10,430 --> 00:30:13,990
That's right, the flux ware.

487
00:30:13,990 --> 00:30:15,020
That's right.

488
00:30:15,020 --> 00:30:18,055
The production rate is
the flux right at or just

489
00:30:18,055 --> 00:30:21,310
to the right of 0.

490
00:30:21,310 --> 00:30:23,680
Incidentally, where's
the flux maximal here?

491
00:30:23,680 --> 00:30:27,560
Is it maximal at A, B, or C?

492
00:30:27,560 --> 00:30:29,660
Ready three, two, one.

493
00:30:29,660 --> 00:30:30,690
You can do it verbally.

494
00:30:30,690 --> 00:30:31,270
Fun.

495
00:30:31,270 --> 00:30:33,160
A. Right.

496
00:30:33,160 --> 00:30:36,720
And that's because the
profile steepest here and then

497
00:30:36,720 --> 00:30:39,510
shallows out.

498
00:30:39,510 --> 00:30:41,450
So flux is decreasing here.

499
00:30:41,450 --> 00:30:46,110
And that's because some of the
morphogen is being degraded.

500
00:30:46,110 --> 00:30:49,990
So indeed, we can figure
out the production rate.

501
00:30:53,820 --> 00:30:54,360
Right.

502
00:30:54,360 --> 00:31:02,950
This is equal to the flux,
which is equal to minus

503
00:31:02,950 --> 00:31:06,290
DCDX at that location.

504
00:31:06,290 --> 00:31:08,825
Right, so the
morphogen profile M

505
00:31:08,825 --> 00:31:13,380
is a function of x is
some M0e to the minus x

506
00:31:13,380 --> 00:31:16,160
over L, where L is given by
this character of lengthscale,

507
00:31:16,160 --> 00:31:18,080
right?

508
00:31:18,080 --> 00:31:22,290
So if we ask what's DMDX?

509
00:31:22,290 --> 00:31:26,330
Well, it's going to
be-- we get a minus M0

510
00:31:26,330 --> 00:31:35,690
over L. Evaluated at x equal
to 0 just makes that go away,

511
00:31:35,690 --> 00:31:38,640
so we end up with this.

512
00:31:38,640 --> 00:31:42,030
So you figure out
what flux is, right?

513
00:31:42,030 --> 00:31:45,310
Yeah, so, it's just an extra
couple terms, it's fine.

514
00:31:45,310 --> 00:31:47,450
But that would be what
would typically be constant

515
00:31:47,450 --> 00:31:51,800
because there might be
some RNA deposit there

516
00:31:51,800 --> 00:31:56,110
or if it's actually
transcription off of the genes

517
00:31:56,110 --> 00:32:00,230
at in the cells in
this location and then

518
00:32:00,230 --> 00:32:03,538
it's based on whether there's
one or two copies and so forth.

519
00:32:09,990 --> 00:32:14,630
So this is all fine, except
that this thing is not robust.

520
00:32:14,630 --> 00:32:17,647
It changes in M0 or
this production rate.

521
00:32:17,647 --> 00:32:19,730
Whereas in many cases,
what we seek experimentally

522
00:32:19,730 --> 00:32:23,670
is that if we, for example,
have the production rate,

523
00:32:23,670 --> 00:32:26,410
then the profile
that we see out here

524
00:32:26,410 --> 00:32:29,511
is somehow remarkably similar.

525
00:32:29,511 --> 00:32:30,010
Yes?

526
00:32:30,010 --> 00:32:31,978
AUDIENCE: [INAUDIBLE].

527
00:32:35,172 --> 00:32:35,880
PROFESSOR: Right.

528
00:32:35,880 --> 00:32:38,940
Yeah, OK.

529
00:32:38,940 --> 00:32:41,980
So I'd say that in
many cases, it's

530
00:32:41,980 --> 00:32:46,500
not that it's directly
this M squared business.

531
00:32:46,500 --> 00:32:50,230
But there are a number of
cases in Drosophila, where

532
00:32:50,230 --> 00:32:53,060
they actually do see this.

533
00:32:53,060 --> 00:33:00,310
In many cases, it was a self
enhanced degradation but not

534
00:33:00,310 --> 00:33:03,690
that it was directly, it was
via binding to a receptor.

535
00:33:03,690 --> 00:33:07,180
So this was seen in
patch lists together.

536
00:33:07,180 --> 00:33:15,060
I want to remember which
were the models that-- I'm

537
00:33:15,060 --> 00:33:17,310
worried I'm always doing it
get so it's like patchless

538
00:33:17,310 --> 00:33:20,780
and frizzled, or-- are
there any Drosophila people

539
00:33:20,780 --> 00:33:24,650
here that came up with
these crazy names?

540
00:33:31,830 --> 00:33:33,700
All right, yeah, so
I always got confused

541
00:33:33,700 --> 00:33:35,857
which are the
morphogens and which

542
00:33:35,857 --> 00:33:37,440
are the receptors
in these situations.

543
00:33:37,440 --> 00:33:41,320
But I'd say that it
is something that's

544
00:33:41,320 --> 00:33:46,790
observed in number of
different contexts,

545
00:33:46,790 --> 00:33:49,050
at different stages of
development, both in Drosophila

546
00:33:49,050 --> 00:33:53,160
and-- they see this in the
wing as well as in the body.

547
00:33:53,160 --> 00:33:58,480
And so in different stages
I think it is signature.

548
00:33:58,480 --> 00:34:01,520
But I'm very much not a
developmental biologist.

549
00:34:11,500 --> 00:34:16,050
All right so the proposal that
[INAUDIBLE] suggests as a way

550
00:34:16,050 --> 00:34:23,540
to make this thing more robust
is to somehow change this term.

551
00:34:23,540 --> 00:34:30,429
Now, the goal in all this was
to have some degradation rate.

552
00:34:30,429 --> 00:34:33,570
And he writes it as F is
a function concentration

553
00:34:33,570 --> 00:34:35,710
M. In principle,
this thing could

554
00:34:35,710 --> 00:34:38,370
be a function of the position
x, but he doesn't want that.

555
00:34:38,370 --> 00:34:40,036
And why was it that
he didn't want that?

556
00:34:44,642 --> 00:34:45,142
Yeah.

557
00:34:45,142 --> 00:34:48,723
AUDIENCE: Because we
already know [INAUDIBLE].

558
00:34:48,723 --> 00:34:49,639
PROFESSOR: Yes, right.

559
00:34:49,639 --> 00:34:51,420
So the whole goal
is that we want

560
00:34:51,420 --> 00:34:53,170
to start with some
situation where

561
00:34:53,170 --> 00:34:56,469
we don't know where we are
as a function of position x.

562
00:34:56,469 --> 00:34:59,504
And then from that get
some spatial information.

563
00:34:59,504 --> 00:35:00,920
So we really want
something that's

564
00:35:00,920 --> 00:35:03,210
not a function of x in there.

565
00:35:03,210 --> 00:35:05,947
Because then we've
already solved the problem

566
00:35:05,947 --> 00:35:07,155
that we're trying to address.

567
00:35:18,050 --> 00:35:20,060
And so one proposal
is indeed just

568
00:35:20,060 --> 00:35:21,660
to have something
that-- and this is

569
00:35:21,660 --> 00:35:24,230
very much a phenomenological
approach-- which

570
00:35:24,230 --> 00:35:28,250
is to have this thing be
self enhanced degradation.

571
00:35:32,080 --> 00:35:44,837
So the idea is that here if we
have the morphogen profile that

572
00:35:44,837 --> 00:35:45,920
looks something like this.

573
00:35:51,310 --> 00:35:52,810
There are multiple
ways you can have

574
00:35:52,810 --> 00:35:54,290
something that looks like this.

575
00:35:54,290 --> 00:35:59,040
So if along the way to
degradation, if the morphogen

576
00:35:59,040 --> 00:36:02,930
actually binds the receptor, but
then activates the degradation.

577
00:36:02,930 --> 00:36:07,590
And this is seen, for
example, in Drosophila.

578
00:36:07,590 --> 00:36:10,110
This is Hedgehog and Patched.

579
00:36:10,110 --> 00:36:15,380
So R is-- in this case,
you have the morphogen

580
00:36:15,380 --> 00:36:17,110
is diffusing outside the cells.

581
00:36:17,110 --> 00:36:20,830
It binds to the receptor,
and then the receptor

582
00:36:20,830 --> 00:36:24,860
activates imported
degradation of the morphogen.

583
00:36:24,860 --> 00:36:27,950
So it's not that the
morphogen is directly

584
00:36:27,950 --> 00:36:31,980
leading to its own degradation,
but it's via some network here.

585
00:36:31,980 --> 00:36:34,695
And there was another
one that they talk about,

586
00:36:34,695 --> 00:36:40,710
which is that, when the
degradation is mediated by 2.

587
00:36:40,710 --> 00:36:42,640
These look like this,
and this is in context.

588
00:36:42,640 --> 00:36:46,260
And this is again Drosophila,
but you could probably

589
00:36:46,260 --> 00:36:47,650
guess what this thing does.

590
00:36:51,410 --> 00:36:53,460
And so it is a wing
development as well.

591
00:36:58,646 --> 00:37:01,020
If you write down an actual
model of one of these things,

592
00:37:01,020 --> 00:37:02,260
and you could play
with it and see

593
00:37:02,260 --> 00:37:03,730
where you're going to get
something that's going to look

594
00:37:03,730 --> 00:37:05,700
like an M squared type term.

595
00:37:05,700 --> 00:37:08,714
But the nice thing about this
phenomenological approach

596
00:37:08,714 --> 00:37:11,005
is it kind of gives you a
sense of the kinds of effects

597
00:37:11,005 --> 00:37:13,239
that you should be looking
for that will enhance

598
00:37:13,239 --> 00:37:14,488
the robustness of the pattern.

599
00:37:17,240 --> 00:37:21,690
Now the key thing in this
whole discussion to remember

600
00:37:21,690 --> 00:37:27,970
is that if you have a
profile here, if you change,

601
00:37:27,970 --> 00:37:29,495
it's just somehow
translationally

602
00:37:29,495 --> 00:37:31,260
invarion in the sense
that since there's

603
00:37:31,260 --> 00:37:33,850
no information about
where you are here,

604
00:37:33,850 --> 00:37:35,930
these patterns can
kind of slide over.

605
00:37:35,930 --> 00:37:38,860
So if you change the
boundary condition M0,

606
00:37:38,860 --> 00:37:41,340
that's just equivalent to
sliding this over some distance

607
00:37:41,340 --> 00:37:44,120
and you could figure out
where everything has to go.

608
00:37:48,270 --> 00:37:50,990
And that's true not just
for the exponential profile,

609
00:37:50,990 --> 00:37:52,580
but for any of the profiles.

610
00:37:52,580 --> 00:37:55,460
And that's just because there's
no information about what's

611
00:37:55,460 --> 00:37:57,910
going on once you're
inside the embryo.

612
00:38:00,730 --> 00:38:03,620
Can somebody give the
intuitive explanation

613
00:38:03,620 --> 00:38:06,490
for why some sort of self
enhanced degradation like this

614
00:38:06,490 --> 00:38:09,245
might make the
pattern more robust?

615
00:38:15,566 --> 00:38:19,030
AUDIENCE: Decrease
faster [INAUDIBLE].

616
00:38:19,030 --> 00:38:21,200
PROFESSOR: Decrease
faster when M is big.

617
00:38:21,200 --> 00:38:21,950
That's right.

618
00:38:21,950 --> 00:38:26,947
And so the idea is that
if you change the M0 here,

619
00:38:26,947 --> 00:38:28,780
for example, you double
it, then with a self

620
00:38:28,780 --> 00:38:30,863
enhanced degradation you
kind of quickly come back

621
00:38:30,863 --> 00:38:31,892
to where you were.

622
00:38:31,892 --> 00:38:36,340
So there's some sense that if
the pattern quickly comes down,

623
00:38:36,340 --> 00:38:40,280
then you can get it
being more robust.

624
00:38:40,280 --> 00:38:47,230
In particular, this
profile for large x

625
00:38:47,230 --> 00:38:50,210
in the proper regime,
this thing falls off

626
00:38:50,210 --> 00:38:56,540
as some A a over x squared.

627
00:38:56,540 --> 00:38:58,950
And indeed, one of the
problems in Murray's book

628
00:38:58,950 --> 00:39:02,070
is that if this is just
degrading as M to some power n,

629
00:39:02,070 --> 00:39:04,940
then you can calculate
what happens here.

630
00:39:04,940 --> 00:39:07,980
But this is some sort
of power law fall off.

631
00:39:16,010 --> 00:39:20,290
So of course, it's falling
off rapidly here, but not

632
00:39:20,290 --> 00:39:23,130
as rapidly as if it
were an exponential.

633
00:39:23,130 --> 00:39:25,350
And this is just like in
the case of the power law

634
00:39:25,350 --> 00:39:27,700
distributions of networks
that we were talking about

635
00:39:27,700 --> 00:39:31,110
before, that for the
exponential fall off,

636
00:39:31,110 --> 00:39:32,620
then it's just
very, very unlikely

637
00:39:32,620 --> 00:39:35,310
that you'll find some
node with many edges.

638
00:39:35,310 --> 00:39:39,790
Similarly here, you quickly
fall to zero here as compared

639
00:39:39,790 --> 00:39:41,040
to a power law distribution.

640
00:39:41,040 --> 00:39:46,380
So that means that you can
specify patterns out further

641
00:39:46,380 --> 00:39:47,070
in some sense.

642
00:39:54,120 --> 00:39:56,340
Are there any questions
about this basic idea

643
00:39:56,340 --> 00:39:57,510
and where it comes from?

644
00:40:06,161 --> 00:40:07,910
I think the discussion
in [INAUDIBLE] book

645
00:40:07,910 --> 00:40:09,820
is actually reasonable.

646
00:40:09,820 --> 00:40:14,960
So I don't really want to
spend too much time on it.

647
00:40:14,960 --> 00:40:19,100
In the context of
the eldar model

648
00:40:19,100 --> 00:40:24,014
that was from the last
section of that chapter.

649
00:40:24,014 --> 00:40:25,430
One thing I just
want to highlight

650
00:40:25,430 --> 00:40:31,360
is that it's useful to be
able to look at these models,

651
00:40:31,360 --> 00:40:33,580
and from the equations
that you see-- for example,

652
00:40:33,580 --> 00:40:35,440
the supplemental
material of a paper--

653
00:40:35,440 --> 00:40:39,160
make sure you are very clear
about what assumptions they've

654
00:40:39,160 --> 00:40:42,630
made in getting to this model.

655
00:40:42,630 --> 00:40:49,615
And so maybe we'll just spend a
few minutes talking about this.

656
00:40:52,530 --> 00:40:56,270
So this is again a
Drosphil embryo two,

657
00:40:56,270 --> 00:41:02,970
two and a half hours
after fertilization

658
00:41:02,970 --> 00:41:09,710
where at that stage, you
really have the embryo that

659
00:41:09,710 --> 00:41:13,020
looks like a-- we're trying
to understand the patterning

660
00:41:13,020 --> 00:41:17,060
around the front back.

661
00:41:17,060 --> 00:41:19,810
So we think about diffusion
along in one dimension,

662
00:41:19,810 --> 00:41:23,180
but it's really around
this radial direction.

663
00:41:23,180 --> 00:41:28,290
And in particular, where this is
trying to understand patterning

664
00:41:28,290 --> 00:41:29,330
in the dorsal region.

665
00:41:32,690 --> 00:41:36,810
And what we have is
a situation where we

666
00:41:36,810 --> 00:41:40,610
have this protease inhibitor.

667
00:41:40,610 --> 00:41:43,090
And so the protease
distributed uniformly.

668
00:41:43,090 --> 00:41:46,210
Inhibitor starts out
out there, and then we

669
00:41:46,210 --> 00:41:48,820
have the morphogen that
somehow starts out in here.

670
00:41:48,820 --> 00:41:50,650
And we want to end
up with a situation

671
00:41:50,650 --> 00:41:55,090
where the morphogen profile in
here is kind of well defined.

672
00:41:55,090 --> 00:41:56,590
So where you just
have the morphogen

673
00:41:56,590 --> 00:41:59,401
at the center of
this dorsal region.

674
00:42:02,840 --> 00:42:05,490
And then we have diffusion
of these inhibitors

675
00:42:05,490 --> 00:42:07,710
coming in and then the
morphogen coming out.

676
00:42:07,710 --> 00:42:13,040
And the question is, how
can we make sense of this?

677
00:42:13,040 --> 00:42:19,220
And if you look at the equations
that they write down-- So

678
00:42:19,220 --> 00:42:20,770
we have diffusion
of the inhibitor.

679
00:42:33,350 --> 00:42:36,950
This is the complex
of the inhibitor

680
00:42:36,950 --> 00:42:40,660
and the morphogen-- complex c.

681
00:43:10,610 --> 00:43:13,380
And so this is a case
where they wrote down

682
00:43:13,380 --> 00:43:15,950
what they basically knew
about the biologists system.

683
00:43:15,950 --> 00:43:19,130
And then they asked, well,
what range of parameters

684
00:43:19,130 --> 00:43:26,690
would lead to a robust pattern
of the morphogen in here,

685
00:43:26,690 --> 00:43:30,340
if you vary things such as the
total concentration inhibitor,

686
00:43:30,340 --> 00:43:32,130
the morphogen or the protease?

687
00:43:36,400 --> 00:43:39,540
So we have diffusion of each
of these things, the inhibitor,

688
00:43:39,540 --> 00:43:42,690
the morphogen complex.

689
00:43:42,690 --> 00:43:44,670
We're not going to worry
about the protease,

690
00:43:44,670 --> 00:43:46,645
because this thing is
uniformly distributed.

691
00:43:46,645 --> 00:43:49,020
You can argue about whether
that's the right thing to do,

692
00:43:49,020 --> 00:43:51,350
but that's based on the model.

693
00:43:51,350 --> 00:43:54,920
So we have diffusion
of each of these guys.

694
00:43:54,920 --> 00:43:56,745
Can somebody say
what's going on here?

695
00:44:07,870 --> 00:44:10,304
What are these things
trying to capture?

696
00:44:10,304 --> 00:44:10,804
John?

697
00:44:10,804 --> 00:44:12,414
AUDIENCE: [INAUDIBLE].

698
00:44:12,414 --> 00:44:13,830
PROFESSOR: So this
is just saying,

699
00:44:13,830 --> 00:44:17,412
OK, the complex is formed
as a result of binding

700
00:44:17,412 --> 00:44:18,870
of the inhibitor
and the morphogen.

701
00:44:18,870 --> 00:44:21,080
It's proportional to
concentration those two things,

702
00:44:21,080 --> 00:44:24,080
at that particular position
at the particular time.

703
00:44:24,080 --> 00:44:28,820
And just remember that any
time that you create something,

704
00:44:28,820 --> 00:44:31,130
if you have binding at
two things together,

705
00:44:31,130 --> 00:44:34,875
then you have to consider
these things going away.

706
00:44:34,875 --> 00:44:36,850
And it's not they're
being degraded,

707
00:44:36,850 --> 00:44:39,482
it's just that they're
forming this complex.

708
00:44:39,482 --> 00:44:40,940
So in these equations
you just have

709
00:44:40,940 --> 00:44:42,523
to be very careful
about keeping track

710
00:44:42,523 --> 00:44:44,624
of which things are
really going away

711
00:44:44,624 --> 00:44:46,290
and which things are
just changing form.

712
00:44:46,290 --> 00:44:46,772
Yeah?

713
00:44:46,772 --> 00:44:47,688
AUDIENCE: [INAUDIBLE].

714
00:44:53,580 --> 00:44:55,930
PROFESSOR: Right.

715
00:44:55,930 --> 00:45:00,540
You're saying why that
there's not a minus term?

716
00:45:00,540 --> 00:45:02,242
So this is an assumption.

717
00:45:02,242 --> 00:45:03,700
And I think this
is a kind of thing

718
00:45:03,700 --> 00:45:06,300
that you have to be clear
about in any of these,

719
00:45:06,300 --> 00:45:09,440
because that's always going
to occur at some rate.

720
00:45:09,440 --> 00:45:12,830
And the assumption is
that this is dominant.

721
00:45:16,154 --> 00:45:18,320
And then just be clear in
words, what is this thing?

722
00:45:27,554 --> 00:45:30,202
AUDIENCE: [INAUDIBLE].

723
00:45:30,202 --> 00:45:30,910
PROFESSOR: Right.

724
00:45:30,910 --> 00:45:32,900
The protease.

725
00:45:32,900 --> 00:45:34,949
And you're saying
cleaving inhibitor.

726
00:45:34,949 --> 00:45:36,490
AUDIENCE: Yeah,
cleaving the compacts

727
00:45:36,490 --> 00:45:39,372
by degrading the [INAUDIBLE].

728
00:45:39,372 --> 00:45:40,080
PROFESSOR: Right.

729
00:45:40,080 --> 00:45:43,340
So at a rate proportional to
the Proteus concentration--

730
00:45:43,340 --> 00:45:47,519
the complex concentration--
this complex goes away.

731
00:45:47,519 --> 00:45:49,810
Now the question is we have
to figure out where it went

732
00:45:49,810 --> 00:45:52,180
or what happened, right?

733
00:45:52,180 --> 00:45:54,210
Well we have a
minus alpha c here

734
00:45:54,210 --> 00:45:56,750
but then we have a
plus alpha c here.

735
00:45:56,750 --> 00:46:00,570
But this isn't the
morphogen right.

736
00:46:00,570 --> 00:46:05,020
This is at the rate
that the complex somehow

737
00:46:05,020 --> 00:46:07,180
is being disappeared
by the protease,

738
00:46:07,180 --> 00:46:11,340
the morphogen is appearing.

739
00:46:11,340 --> 00:46:15,620
But then, we don't have
the same term here.

740
00:46:15,620 --> 00:46:19,270
And that's how you know that
what the protease is doing

741
00:46:19,270 --> 00:46:26,670
is it's degrading the inhibitor
as part of the complex.

742
00:46:26,670 --> 00:46:28,070
So by looking at
these equations,

743
00:46:28,070 --> 00:46:30,710
you can actually figure out
what are the assumptions that

744
00:46:30,710 --> 00:46:32,805
have been made of
what the biology is

745
00:46:32,805 --> 00:46:33,960
that's being captured here.

746
00:46:33,960 --> 00:46:36,209
But in many cases, there are
assumptions of what's big

747
00:46:36,209 --> 00:46:38,547
and what's small.

748
00:46:38,547 --> 00:46:39,880
And then what's this thing here?

749
00:46:50,050 --> 00:46:52,647
Somebody else that has not
yet-- do these equations

750
00:46:52,647 --> 00:46:53,230
look familiar?

751
00:46:55,920 --> 00:46:56,981
Yes, please.

752
00:46:56,981 --> 00:47:00,750
AUDIENCE: It's the
protease [INAUDIBLE].

753
00:47:00,750 --> 00:47:01,750
PROFESSOR: That's right.

754
00:47:01,750 --> 00:47:05,380
It's the protease degrading the
inhibitor when the inhibitor is

755
00:47:05,380 --> 00:47:09,840
not down to the
morphogen. Now of course,

756
00:47:09,840 --> 00:47:13,540
what the authors did next
is they searched numerically

757
00:47:13,540 --> 00:47:16,000
across parameter space
over [INAUDIBLE] magnitude,

758
00:47:16,000 --> 00:47:18,570
and they found that for
some parameter regime

759
00:47:18,570 --> 00:47:22,960
there was a robust pattern
of the morphogen developed

760
00:47:22,960 --> 00:47:24,879
in here.

761
00:47:24,879 --> 00:47:26,920
There's no reason for you
to necessarily remember

762
00:47:26,920 --> 00:47:29,620
which thing it was.

763
00:47:29,620 --> 00:47:35,070
But what they found is
that if this term went away

764
00:47:35,070 --> 00:47:40,340
and this term went
away, then you

765
00:47:40,340 --> 00:47:46,460
end up getting a robust
profile of the morphogen

766
00:47:46,460 --> 00:47:49,430
against changes in the
overall morphogen inhibitor

767
00:47:49,430 --> 00:47:50,760
concentration.

768
00:47:50,760 --> 00:47:52,860
Now this is pretty
weird, I would say.

769
00:47:55,192 --> 00:47:57,150
I don't know, do you
think-- is this one weird?

770
00:48:01,830 --> 00:48:03,910
Weirdness is in the
eye of the beholder.

771
00:48:03,910 --> 00:48:06,520
But I would say it's
not so shocking.

772
00:48:11,650 --> 00:48:15,465
Can somebody say verbally
what this would correspond to?

773
00:48:15,465 --> 00:48:17,671
AUDIENCE: The protease
doesn't degrade the inhibitor

774
00:48:17,671 --> 00:48:18,170
on its own.

775
00:48:18,170 --> 00:48:19,800
PROFESSOR: That's right,
the protease doesn't degrade

776
00:48:19,800 --> 00:48:21,310
the inhibitor on
its own, only when

777
00:48:21,310 --> 00:48:23,560
it's part of the complex
with the morphogen.

778
00:48:23,560 --> 00:48:25,780
And independently there was
experimental data already

779
00:48:25,780 --> 00:48:26,821
indicating that was true.

780
00:48:26,821 --> 00:48:28,780
So you could have if
you want to just written

781
00:48:28,780 --> 00:48:31,285
the model without that,
because there was already

782
00:48:31,285 --> 00:48:32,120
evidence for that.

783
00:48:32,120 --> 00:48:39,190
But as I said, this is
a pretty weird thing.

784
00:48:39,190 --> 00:48:43,180
And so, this would be saying
that, for some reason,

785
00:48:43,180 --> 00:48:49,940
diffusion of the morphogen
on its own is very small.

786
00:48:49,940 --> 00:48:52,280
In particular, small compared
diffusion of the morphogen

787
00:48:52,280 --> 00:48:53,195
when it's a complex.

788
00:48:58,190 --> 00:49:01,499
Now can somebody say why I
might think that's weird?

789
00:49:01,499 --> 00:49:03,040
You don't have to
believe it's weird.

790
00:49:08,752 --> 00:49:10,755
AUDIENCE: You'd think
bigger is better.

791
00:49:10,755 --> 00:49:12,210
PROFESSOR: That's right.

792
00:49:12,210 --> 00:49:13,102
Exactly.

793
00:49:13,102 --> 00:49:15,310
This is saying, well, when
it's complex, it's bigger,

794
00:49:15,310 --> 00:49:16,800
it's somehow diffusing faster.

795
00:49:16,800 --> 00:49:22,750
So this would not happen do
the simple Stokes type drag.

796
00:49:22,750 --> 00:49:24,940
This would have to be biology.

797
00:49:28,776 --> 00:49:31,150
Somebody's biology is allowed
to do anything that doesn't

798
00:49:31,150 --> 00:49:32,550
violate the laws of physics.

799
00:49:32,550 --> 00:49:35,720
Indeed, later there was
experimental evidence

800
00:49:35,720 --> 00:49:39,015
of something like this
was actually happening.

801
00:49:39,015 --> 00:49:40,670
And you could
imagine having either

802
00:49:40,670 --> 00:49:44,530
from sort of active transport
type dynamics for the complex,

803
00:49:44,530 --> 00:49:52,380
or from binding type
dynamics of the morphogen.

804
00:49:52,380 --> 00:49:54,650
I think the key thing to
take from this example

805
00:49:54,650 --> 00:49:58,030
is just that, in
general, we would

806
00:49:58,030 --> 00:50:01,420
like biological function--
biology would like

807
00:50:01,420 --> 00:50:05,150
biological function to be
robust to things that are often

808
00:50:05,150 --> 00:50:06,840
fluctuating or varying.

809
00:50:06,840 --> 00:50:10,755
And that's one way to try to
make guesses about what might

810
00:50:10,755 --> 00:50:11,880
be happening in the system.

811
00:50:11,880 --> 00:50:14,560
This sort of
computational exercise

812
00:50:14,560 --> 00:50:18,200
is not all a proof that this
has to be what's happening.

813
00:50:18,200 --> 00:50:19,700
They wrote down a
particular model--

814
00:50:19,700 --> 00:50:22,074
it could have been that there
are other terms they're not

815
00:50:22,074 --> 00:50:23,010
aware of and so forth.

816
00:50:23,010 --> 00:50:25,010
But it's at least a way
of generating hypotheses

817
00:50:25,010 --> 00:50:28,690
that you can go and test.

818
00:50:28,690 --> 00:50:30,840
And quite generally,
I'd also say

819
00:50:30,840 --> 00:50:37,030
that its essential for all
of us as consumers of models

820
00:50:37,030 --> 00:50:40,190
to be able to look at
some of these equations

821
00:50:40,190 --> 00:50:42,440
and figure out what it
is that they've assumed.

822
00:50:49,620 --> 00:50:53,970
I want to move on to
this idea of pattern

823
00:50:53,970 --> 00:50:57,150
formation via the
reaction diffusion

824
00:50:57,150 --> 00:50:59,480
or Turing-type patterns.

825
00:50:59,480 --> 00:51:01,400
And I think it's
really important

826
00:51:01,400 --> 00:51:05,950
to start by just acknowledging
that this is really

827
00:51:05,950 --> 00:51:09,230
a surprising finding, that
it's even possible for this

828
00:51:09,230 --> 00:51:10,430
to happen.

829
00:51:10,430 --> 00:51:12,860
Because this is
a situation where

830
00:51:12,860 --> 00:51:17,490
you have a couple
chemicals-- or proteins,

831
00:51:17,490 --> 00:51:23,910
or reacting elements--
and if you just mix them,

832
00:51:23,910 --> 00:51:26,730
if you have them in
some well mixed tube,

833
00:51:26,730 --> 00:51:30,270
they reach some steady state.

834
00:51:30,270 --> 00:51:32,250
Whereas somehow if
you allow diffusion,

835
00:51:32,250 --> 00:51:36,590
then you can get these patterns.

836
00:51:36,590 --> 00:51:38,667
And I would say based on
my intuition at least,

837
00:51:38,667 --> 00:51:40,875
I would not have thought
that this would be possible.

838
00:51:43,590 --> 00:51:47,520
And this really just is because
if you can just imagine,

839
00:51:47,520 --> 00:51:49,140
you start with some profile.

840
00:51:49,140 --> 00:51:51,810
Now there's some structure here.

841
00:51:51,810 --> 00:51:53,700
But diffusion's going
to actually cause

842
00:51:53,700 --> 00:51:56,080
this to come down,
and this to come up.

843
00:51:56,080 --> 00:52:00,720
So diffusion acts to remove
these spatial patterns.

844
00:52:00,720 --> 00:52:03,870
But somehow in
some prime regimes,

845
00:52:03,870 --> 00:52:07,280
if you have coupled
reactions that

846
00:52:07,280 --> 00:52:09,140
are activating and
inhibiting each other,

847
00:52:09,140 --> 00:52:12,900
then you can actually get
these spatial patterns.

848
00:52:12,900 --> 00:52:16,910
And as you read about,
there's some idea

849
00:52:16,910 --> 00:52:19,730
that you need to have
a local activation

850
00:52:19,730 --> 00:52:21,550
and a global inhibition.

851
00:52:21,550 --> 00:52:25,266
But ultimately the
mathematical-- you always

852
00:52:25,266 --> 00:52:27,640
have to go and think about
more carefully about the math,

853
00:52:27,640 --> 00:52:30,060
and would be indicated
just by those words.

854
00:52:30,060 --> 00:52:33,820
This is at least a way to guide
your thinking a little bit.

855
00:52:33,820 --> 00:52:38,010
But ultimately what
matters are around this

856
00:52:38,010 --> 00:52:40,330
stable what would be
the stable fixed point,

857
00:52:40,330 --> 00:52:43,490
you have to actually look at
these derivatives and so forth.

858
00:52:43,490 --> 00:52:47,312
And the derivatives can
be subtle I would say.

859
00:52:47,312 --> 00:52:48,770
Just because you
think of something

860
00:52:48,770 --> 00:52:51,070
as activator
inhibitor doesn't mean

861
00:52:51,070 --> 00:52:54,300
that it's going to have that
role around the fix point

862
00:52:54,300 --> 00:52:55,200
that you're studying.

863
00:52:55,200 --> 00:52:56,770
So this is just a caution.

864
00:53:04,030 --> 00:53:05,630
What I want to do
is just give you,

865
00:53:05,630 --> 00:53:08,410
for example, one example
of a simple model

866
00:53:08,410 --> 00:53:11,850
that does experience
these Turing patterns.

867
00:53:11,850 --> 00:53:13,610
And in an appropriate regime.

868
00:53:13,610 --> 00:53:18,254
And this Levin-Segel model
of pattern formation.

869
00:53:25,900 --> 00:53:29,120
And this was published in
1976, and it was actually

870
00:53:29,120 --> 00:53:33,520
meant as a model of
predator-prey interactions

871
00:53:33,520 --> 00:53:35,990
in ecology.

872
00:53:35,990 --> 00:53:38,570
We'll talk
significantly more about

873
00:53:38,570 --> 00:53:40,530
predator-prey interactions
in a few weeks.

874
00:53:40,530 --> 00:53:45,900
But I just want to write
down what they said.

875
00:53:45,900 --> 00:53:48,710
And this is a simplified
model of their stuff.

876
00:53:48,710 --> 00:53:55,780
So it's actually trying to think
about some plankton herbivore

877
00:53:55,780 --> 00:53:58,110
before interactions.

878
00:53:58,110 --> 00:54:02,120
And this is derivative
with respect to time.

879
00:54:02,120 --> 00:54:05,410
We're going to follow the
nomenclature of a paper

880
00:54:05,410 --> 00:54:08,350
by Butler and Goldenfeld
a few years ago,

881
00:54:08,350 --> 00:54:09,850
because they're the
ones who thought

882
00:54:09,850 --> 00:54:11,974
about the demographic noise
enhancing the patterns.

883
00:54:30,090 --> 00:54:53,160
PROFESSOR: All right, so we have
[? phi ?] and [? phi. ?] First

884
00:54:53,160 --> 00:54:54,420
of all, who's eating whom?

885
00:55:07,460 --> 00:55:09,560
All right, we're going
to say it verbally.

886
00:55:09,560 --> 00:55:11,710
Who is the predator?

887
00:55:11,710 --> 00:55:14,730
Ready three, two, one.

888
00:55:14,730 --> 00:55:15,620
AUDIENCE: Herbivore.

889
00:55:15,620 --> 00:55:16,495
PROFESSOR: Herbivore.

890
00:55:19,590 --> 00:55:22,210
Herbivores are not
what they used to be.

891
00:55:22,210 --> 00:55:26,570
So indeed, the herbivore
benefits from the presence

892
00:55:26,570 --> 00:55:29,140
of the plankton.

893
00:55:29,140 --> 00:55:31,456
So nobody cares about
plankton, I guess.

894
00:55:34,210 --> 00:55:38,740
So this corresponds to the
predator-prey type interaction.

895
00:55:38,740 --> 00:55:41,100
So there's some death rate.

896
00:55:41,100 --> 00:55:45,100
So let's see, the
herbivore or the predator.

897
00:55:45,100 --> 00:55:47,460
And then what you see is
there's these two growth

898
00:55:47,460 --> 00:55:49,640
terms for the plankton.

899
00:55:49,640 --> 00:55:51,400
So there's this term,
which would be kind

900
00:55:51,400 --> 00:55:54,779
of simple, exponential growth.

901
00:55:54,779 --> 00:55:56,320
And then this term,
which is actually

902
00:55:56,320 --> 00:55:58,850
some sort of super
exponential growth.

903
00:55:58,850 --> 00:56:05,690
There's some sense in which the
plankton benefit each other.

904
00:56:05,690 --> 00:56:08,320
And this was originally
introduced because of something

905
00:56:08,320 --> 00:56:10,200
called predator satiation.

906
00:56:10,200 --> 00:56:14,150
But it's just a general
reflection of the fact

907
00:56:14,150 --> 00:56:16,650
that in many cases, individuals
benefit from the presence

908
00:56:16,650 --> 00:56:19,679
of other individuals.

909
00:56:19,679 --> 00:56:21,845
And in particular, this is
known as the ally effect,

910
00:56:21,845 --> 00:56:24,960
and we'll spend time talking
about this in a few weeks

911
00:56:24,960 --> 00:56:27,550
as well in the context of
populations and ecology.

912
00:56:27,550 --> 00:56:32,360
For now, I just want to use this
as an example of a model that

913
00:56:32,360 --> 00:56:35,435
gives you these Turing patterns.

914
00:56:41,367 --> 00:56:43,450
So if you want to, you can
go ahead and ask, well,

915
00:56:43,450 --> 00:56:46,600
what happens if we just
have a well mixed situation?

916
00:56:46,600 --> 00:56:50,030
If everything is not a
function of x, but it's still

917
00:56:50,030 --> 00:56:51,370
can to be a function of time.

918
00:56:51,370 --> 00:56:53,030
You can solve these
equations, and you

919
00:56:53,030 --> 00:56:56,150
can find what the steady
states are equal to.

920
00:56:56,150 --> 00:57:00,110
So there is indeed a steady
state stable coexistence

921
00:57:00,110 --> 00:57:04,260
of the predator and the prey
in a well mixed situation.

922
00:57:09,950 --> 00:57:12,070
However, if you then
go and you analyze

923
00:57:12,070 --> 00:57:16,240
the stability of different
spatial modes, what you'll find

924
00:57:16,240 --> 00:57:20,570
is that in some situations,
particular wavelength or wave

925
00:57:20,570 --> 00:57:23,650
vectors become unstable.

926
00:57:23,650 --> 00:57:26,050
And it's just over some
range of wavelengths,

927
00:57:26,050 --> 00:57:29,340
and that corresponds to the wave
length of the Turing patterns

928
00:57:29,340 --> 00:57:30,290
that you'll see.

929
00:57:30,290 --> 00:57:32,640
If you're curious about
these things in more depth,

930
00:57:32,640 --> 00:57:37,410
I encourage you to attend Mehran
Kardar's class Statistical

931
00:57:37,410 --> 00:57:40,790
Physics in Biology, he is
an expert on these topics

932
00:57:40,790 --> 00:57:44,115
and I think is a
wonderful clear lecturer.

933
00:57:49,410 --> 00:57:55,400
Now there's going to be some
condition for these things

934
00:57:55,400 --> 00:57:56,890
leading to Turing patterns.

935
00:58:00,500 --> 00:58:05,420
Now from your reading, do you
think it's going to be mu--?

936
00:58:19,130 --> 00:58:31,620
So this is Turing patterns
require-- I'll give you

937
00:58:31,620 --> 00:58:33,130
30 seconds to think about what.

938
00:58:48,620 --> 00:58:50,260
Do you need more time?

939
00:58:50,260 --> 00:58:51,360
I'm not sure where we are.

940
00:58:54,050 --> 00:58:55,871
Maybe another 15
seconds just to--

941
00:59:17,000 --> 00:59:18,450
Let's see where we are.

942
00:59:18,450 --> 00:59:21,710
Ready three, two, one.

943
00:59:24,650 --> 00:59:27,460
OK, we're all over the place.

944
00:59:27,460 --> 00:59:30,940
I think we're uniformly
distributed between A, B,

945
00:59:30,940 --> 00:59:34,490
and C. So there's
an idea that you're

946
00:59:34,490 --> 00:59:37,670
supposed to have
so-called local activation

947
00:59:37,670 --> 00:59:38,715
and global inhibition.

948
00:59:48,040 --> 00:59:51,240
Why don't we turn to a neighbor
and see if you can figure out

949
00:59:51,240 --> 00:59:52,734
what's going on.

950
01:01:21,010 --> 01:01:23,420
OK, why don't we
reconvene, I'm curious

951
01:01:23,420 --> 01:01:25,440
where you're thinking is.

952
01:01:25,440 --> 01:01:26,500
Let's go ahead and vote.

953
01:01:26,500 --> 01:01:29,310
Ready three, two, one.

954
01:01:32,460 --> 01:01:36,440
OK, so we have many, many C's.

955
01:01:36,440 --> 01:01:39,105
The idea that some mu has
to be much less the nu.

956
01:01:42,500 --> 01:01:44,070
Yes.

957
01:01:44,070 --> 01:01:44,980
I want to make sure.

958
01:01:49,210 --> 01:01:54,370
All right, mu is telling us
about the diffusion coefficient

959
01:01:54,370 --> 01:01:57,620
of the prey.

960
01:01:57,620 --> 01:02:02,580
Nu is the diffusion
coefficient of the predator.

961
01:02:02,580 --> 01:02:07,990
So what you want is to
have local activation.

962
01:02:07,990 --> 01:02:11,740
And the thing that's activating
itself is the plankton.

963
01:02:11,740 --> 01:02:14,740
So you can see that that's like,
for example, psi squared term.

964
01:02:18,710 --> 01:02:21,450
And from the stand point of a
spatial situation, you can say,

965
01:02:21,450 --> 01:02:23,720
all right, well, let's
say we have some region

966
01:02:23,720 --> 01:02:25,580
with a lot of this prey.

967
01:02:25,580 --> 01:02:28,340
Well, it's able to activate
itself, so it kind of comes up.

968
01:02:28,340 --> 01:02:33,610
And as it does that, it is
creating more of the predator,

969
01:02:33,610 --> 01:02:34,970
phi.

970
01:02:34,970 --> 01:02:40,024
But because the predator has a
larger diffusion coefficient--

971
01:02:40,024 --> 01:02:41,690
well, it's obviously
going to grow here.

972
01:02:41,690 --> 01:02:44,740
But it will also diffuse away.

973
01:02:44,740 --> 01:02:46,990
And that's this
global inhibition

974
01:02:46,990 --> 01:02:49,920
of neighboring regions.

975
01:02:49,920 --> 01:02:52,060
Now of course, these
are just words,

976
01:02:52,060 --> 01:02:55,170
you have to take it a little
bit of a grain of salt.

977
01:02:55,170 --> 01:02:59,300
But if you can actually do
this calculation analytically,

978
01:02:59,300 --> 01:03:02,890
and derive in this
model the condition

979
01:03:02,890 --> 01:03:07,400
for Turing patterns to emerge
as a function of everything.

980
01:03:07,400 --> 01:03:12,261
And for example, if
you have b 1/2, p 1.

981
01:03:16,560 --> 01:03:21,400
So for unity type parameters,
then the requirement

982
01:03:21,400 --> 01:03:25,050
to get Turing patterns
is that nu over mu

983
01:03:25,050 --> 01:03:32,555
is the greater than 27.8.

984
01:03:32,555 --> 01:03:34,180
The key thing though
is that this thing

985
01:03:34,180 --> 01:03:35,280
is much larger than 1.

986
01:03:37,800 --> 01:03:41,190
So a general thing that emerges
in these Turing patterns

987
01:03:41,190 --> 01:03:45,850
is that you need this
so-called inhibiting type

988
01:03:45,850 --> 01:03:48,400
partner to have a much larger
effective diffusion coefficient

989
01:03:48,400 --> 01:03:51,900
than the activating partner.

990
01:03:51,900 --> 01:03:55,930
Now the problem with this
is that this thing is indeed

991
01:03:55,930 --> 01:04:03,840
much larger than 1, which limits
the overall usefulness of this

992
01:04:03,840 --> 01:04:04,670
is a mechanism.

993
01:04:04,670 --> 01:04:11,700
Because if it really is simple
diffusion, how much bigger

994
01:04:11,700 --> 01:04:15,644
would the-- let's say, low
Reynolds number regime,

995
01:04:15,644 --> 01:04:17,560
these are just molecules
experience diffusion.

996
01:04:24,650 --> 01:04:30,320
How much bigger does the--
this is so confusing.

997
01:04:30,320 --> 01:04:33,550
So this is big and
this is little, right?

998
01:04:33,550 --> 01:04:43,270
So how much bigger
does the activator--

999
01:04:43,270 --> 01:04:45,530
they're too many things
to keep track of here.

1000
01:04:45,530 --> 01:04:47,529
It's because I don't have
anything written down.

1001
01:04:47,529 --> 01:04:49,870
So that's the activator.

1002
01:04:49,870 --> 01:04:54,780
So this one's the activator,
and this one's the inhibitor.

1003
01:04:58,920 --> 01:05:01,150
And the inhibitors has
to diffuse much more

1004
01:05:01,150 --> 01:05:03,340
than the activator.

1005
01:05:03,340 --> 01:05:06,106
So the inhibitor has to be much
smaller than the activator.

1006
01:05:06,106 --> 01:05:08,230
OK good, so how much bigger
does the activator have

1007
01:05:08,230 --> 01:05:11,640
to be than the inhibitor
for this to be true?

1008
01:05:11,640 --> 01:05:16,137
Does it scale as this 28
or 28 squared or 28 cubed?

1009
01:05:21,870 --> 01:05:23,210
In terms of radius, how much?

1010
01:05:27,180 --> 01:05:29,730
Scales linearly, remember?

1011
01:05:29,730 --> 01:05:33,150
Einstein equation, you guys had
fun thinking about on the exam.

1012
01:05:33,150 --> 01:05:34,950
KT over gamma.

1013
01:05:34,950 --> 01:05:37,060
This would require, for
example, the activator

1014
01:05:37,060 --> 01:05:40,650
to be 30 times bigger
in terms of radius.

1015
01:05:40,650 --> 01:05:42,720
And the inhibitor,
in order to have this

1016
01:05:42,720 --> 01:05:46,870
emerge just as a result of
diffusion, simple diffusion.

1017
01:05:46,870 --> 01:05:50,650
And that's just
not a typical thing

1018
01:05:50,650 --> 01:05:53,305
for just a range
of protein sizes.

1019
01:05:53,305 --> 01:05:55,180
But you could imagine
putting up various ways

1020
01:05:55,180 --> 01:05:56,570
to make this happen.

1021
01:05:56,570 --> 01:06:00,530
But there is a very nice
development that I mentioned,

1022
01:06:00,530 --> 01:06:04,910
which is that there's another
surprising aspect in these

1023
01:06:04,910 --> 01:06:07,150
problems, which is
that in many cases,

1024
01:06:07,150 --> 01:06:10,410
demographic noise can
lead to patterns-- --

1025
01:06:10,410 --> 01:06:12,480
or maybe you might call
them quasi-patterns.

1026
01:06:12,480 --> 01:06:15,070
But things that, for all
intents and purposes, look

1027
01:06:15,070 --> 01:06:18,870
like some of this
pseudo-periodic.

1028
01:06:23,160 --> 01:06:27,780
So there's a pseudo-Turing
pattern, maybe.

1029
01:06:27,780 --> 01:06:32,340
And indeed, if you
actually do this simulation

1030
01:06:32,340 --> 01:06:36,370
with those parameters
where you do

1031
01:06:36,370 --> 01:06:40,160
the explicit demographic
fluctuations-- i.e.

1032
01:06:40,160 --> 01:06:42,580
you take into account
that this corresponds

1033
01:06:42,580 --> 01:06:45,165
to a prey giving birth.

1034
01:06:45,165 --> 01:06:47,040
This corresponds to a
predator eating a prey,

1035
01:06:47,040 --> 01:06:48,700
random events just
like what we've

1036
01:06:48,700 --> 01:06:52,860
done in the context of
a Gillespie simulation.

1037
01:06:52,860 --> 01:06:58,415
Then what you find is that
with demographic fluctuations

1038
01:06:58,415 --> 01:07:04,730
or demographic noise, then
the condition here nu over mu

1039
01:07:04,730 --> 01:07:08,370
has to be only 2.48.

1040
01:07:08,370 --> 01:07:15,900
And indeed in Butler
and Goldenfeld,

1041
01:07:15,900 --> 01:07:21,140
they derive these things using
field theoretic approaches.

1042
01:07:21,140 --> 01:07:25,781
There was a 2011 paper
and also 2008 or 2009.

1043
01:07:25,781 --> 01:07:27,280
The nice thing here
is that what you

1044
01:07:27,280 --> 01:07:32,020
see is that in the presence of
these demographic noise that

1045
01:07:32,020 --> 01:07:36,910
will be there, the difference
in diffusivity that are required

1046
01:07:36,910 --> 01:07:39,630
is not nearly as large
as when you're thinking

1047
01:07:39,630 --> 01:07:42,495
about the mean field equations.

1048
01:07:52,774 --> 01:07:54,190
Are there any
questions about this

1049
01:07:54,190 --> 01:07:57,717
before we talk about
center fighting in E. coli?

1050
01:07:57,717 --> 01:07:58,262
Yeah.

1051
01:07:58,262 --> 01:07:59,178
AUDIENCE: [INAUDIBLE].

1052
01:08:05,974 --> 01:08:07,390
PROFESSOR: OK, now
you're thinking

1053
01:08:07,390 --> 01:08:08,890
about the actual
plankton-herbivore.

1054
01:08:13,230 --> 01:08:16,800
I think there are
many things to say.

1055
01:08:16,800 --> 01:08:19,680
One of them would just be that
maybe the patterns that you

1056
01:08:19,680 --> 01:08:22,090
see between actual plankton
and actual herbivores

1057
01:08:22,090 --> 01:08:23,660
is not due to the
Turing mechanism.

1058
01:08:30,720 --> 01:08:32,950
For the Turing
mechanism to be at play

1059
01:08:32,950 --> 01:08:37,550
to generate spatial
patterns, it requires

1060
01:08:37,550 --> 01:08:42,281
that this inhibitor do
something equivalent

1061
01:08:42,281 --> 01:08:43,364
to diffusion more rapidly.

1062
01:08:43,364 --> 01:08:46,549
It has to have to move
away more rapidly.

1063
01:08:46,549 --> 01:08:47,830
AUDIENCE: [INAUDIBLE].

1064
01:08:47,830 --> 01:08:50,029
PROFESSOR: Right.

1065
01:08:50,029 --> 01:08:53,060
It could be on the plankton only
move around on the currents,

1066
01:08:53,060 --> 01:08:55,644
whereas the herbivore
actually has direct motions.

1067
01:08:55,644 --> 01:08:58,060
I have to say, I don't actually
know enough about plankton

1068
01:08:58,060 --> 01:09:01,766
to know whether this is at
all-- That base statement I thin

1069
01:09:01,766 --> 01:09:05,729
is true, but I don't know
how the numbers work out.

1070
01:09:05,729 --> 01:09:06,229
Yeah.

1071
01:09:06,229 --> 01:09:07,145
AUDIENCE: [INAUDIBLE].

1072
01:09:10,670 --> 01:09:12,560
PROFESSOR: Yes.

1073
01:09:12,560 --> 01:09:16,290
Both in this case and the simple
predator-prey oscillations.

1074
01:09:18,554 --> 01:09:20,970
One thing we're going to see
the predator-prey populations

1075
01:09:20,970 --> 01:09:23,630
later is that you can also
get this sort of oscillations

1076
01:09:23,630 --> 01:09:24,630
in time.

1077
01:09:24,630 --> 01:09:28,319
So this could be in time or this
could be a function of space.

1078
01:09:28,319 --> 01:09:30,149
But in both cases,
it's really that you

1079
01:09:30,149 --> 01:09:34,090
have a resonant
enhancement in some areas.

1080
01:09:34,090 --> 01:09:36,250
So that the demographic
fluctuation somehow

1081
01:09:36,250 --> 01:09:40,840
excite all frequencies
or all wavelengths.

1082
01:09:40,840 --> 01:09:43,870
But then what happens
is that, in this case,

1083
01:09:43,870 --> 01:09:49,975
if you look at the eigenvalues--
OK, I want to be close to 0,

1084
01:09:49,975 --> 01:09:50,474
though.

1085
01:09:54,040 --> 01:09:56,290
If you look at how
the modes decay--

1086
01:09:56,290 --> 01:10:00,260
so the eigenvalue's function
of a wave number-- what you see

1087
01:10:00,260 --> 01:10:02,900
is that you end up getting
things that look like this.

1088
01:10:02,900 --> 01:10:05,680
So whereas a Turing
pattern is when

1089
01:10:05,680 --> 01:10:08,420
particular wave
length or wave modes

1090
01:10:08,420 --> 01:10:12,180
actually becomes
unstable, and then you

1091
01:10:12,180 --> 01:10:13,850
get mean field type patterns.

1092
01:10:13,850 --> 01:10:15,840
And in this situation
what happens

1093
01:10:15,840 --> 01:10:20,110
is that you're just close here,
so then you excite all the wave

1094
01:10:20,110 --> 01:10:21,390
numbers or wavelengths.

1095
01:10:21,390 --> 01:10:23,740
And then some of them take
a long time to go away,

1096
01:10:23,740 --> 01:10:26,360
so then they build up and then
that's the resulting patterns

1097
01:10:26,360 --> 01:10:27,821
that you see.

1098
01:10:33,190 --> 01:10:35,440
So what I want to do is
just for the last 10 minutes

1099
01:10:35,440 --> 01:10:37,520
to talk about this Min system.

1100
01:10:37,520 --> 01:10:39,520
Because it's I think
it's a beautiful example

1101
01:10:39,520 --> 01:10:43,780
of a combination of Systems
Biology questions together

1102
01:10:43,780 --> 01:10:46,440
with reaction diffusion
systems, and also

1103
01:10:46,440 --> 01:10:48,240
some beautiful in
vitro experiments

1104
01:10:48,240 --> 01:10:53,130
that I've quite appreciated.

1105
01:10:53,130 --> 01:10:55,660
So the question is--
imagine yourself,

1106
01:10:55,660 --> 01:10:58,080
you've gone to all this work
right, so now you're nice

1107
01:10:58,080 --> 01:10:59,070
and long.

1108
01:10:59,070 --> 01:11:00,960
So you'd like to
divide into two cells.

1109
01:11:00,960 --> 01:11:03,168
And the question is, how do
you know where to divide?

1110
01:11:06,350 --> 01:11:10,890
So you imagine that
you're some E. coli cell.

1111
01:11:10,890 --> 01:11:16,320
You're maybe six microns
long, everything is great.

1112
01:11:16,320 --> 01:11:19,650
And the question is,
you want to divide?

1113
01:11:19,650 --> 01:11:22,260
Where would you like to divide
if you're an E. coli cell?

1114
01:11:22,260 --> 01:11:22,760
Middle.

1115
01:11:22,760 --> 01:11:25,890
OK, so what you'd
like to do is go here.

1116
01:11:25,890 --> 01:11:30,300
And indeed, what happens is that
there's the so-called z-ring.

1117
01:11:30,300 --> 01:11:32,420
There's a a
pseudo-polymer protein

1118
01:11:32,420 --> 01:11:33,946
that forms a ring
around here, then

1119
01:11:33,946 --> 01:11:38,120
it constricts itself and
pinches off the membrane.

1120
01:11:38,120 --> 01:11:40,250
And that's how
you get two cells.

1121
01:11:40,250 --> 01:11:43,220
So there's this
formation of this z-ring

1122
01:11:43,220 --> 01:11:47,480
that constricts and that's
this cell division event.

1123
01:11:47,480 --> 01:11:50,960
The question is how do
you know where to put it?

1124
01:11:50,960 --> 01:11:54,360
And that's what this
Min system is for.

1125
01:11:54,360 --> 01:11:56,900
And it's, of course,
like everything

1126
01:11:56,900 --> 01:12:01,460
in bacterial genetics, it
was identify by mutations.

1127
01:12:05,030 --> 01:12:11,790
So the Min system it's called
that because these mutants

1128
01:12:11,790 --> 01:12:13,370
formed so-called mini cells.

1129
01:12:18,150 --> 01:12:21,560
If you, instead of
dividing the center where

1130
01:12:21,560 --> 01:12:24,520
you might have two copies
of the genome here.

1131
01:12:24,520 --> 01:12:26,770
If instead you divide
over here, then what

1132
01:12:26,770 --> 01:12:28,810
your going to end up
with is a very long cell

1133
01:12:28,810 --> 01:12:31,220
with two copies of the genome.

1134
01:12:31,220 --> 01:12:35,290
You're going to end up with
this mini cell without any DNA.

1135
01:12:35,290 --> 01:12:37,520
And those mini
cells, are they going

1136
01:12:37,520 --> 01:12:40,120
to do well for the long term?

1137
01:12:40,120 --> 01:12:41,060
AUDIENCE: No.

1138
01:12:41,060 --> 01:12:42,000
PROFESSOR: Right.

1139
01:12:42,000 --> 01:12:44,430
They actually can
survive for a little bi.

1140
01:12:44,430 --> 01:12:46,830
And people have argued
that maybe that could

1141
01:12:46,830 --> 01:12:49,897
be useful for synthetic
biology because they still

1142
01:12:49,897 --> 01:12:52,480
make proteins, but they're not
going to go take over the world

1143
01:12:52,480 --> 01:12:54,790
because they don't have any DNA.

1144
01:12:54,790 --> 01:12:57,140
But these are the
so-called mini cells

1145
01:12:57,140 --> 01:13:02,380
that happen if you have
mutations in The min system.

1146
01:13:02,380 --> 01:13:08,750
Now, I think the three players
are MinC, MinD, and MinE.

1147
01:13:16,660 --> 01:13:20,080
So I think that MinA and MinB
ended up not actually existing

1148
01:13:20,080 --> 01:13:22,970
or something, in the sense that
they identified the mutants,

1149
01:13:22,970 --> 01:13:25,290
but then they were
wrong about something.

1150
01:13:25,290 --> 01:13:27,720
So the three that
actually ended up

1151
01:13:27,720 --> 01:13:30,640
being involved in the actual
Min system are C, D and E.

1152
01:13:30,640 --> 01:13:35,606
So this MinC prevents
formation of the z-ring.

1153
01:13:43,550 --> 01:13:46,890
Now MinD and MinE are
kind of the amazing guys.

1154
01:13:46,890 --> 01:13:54,070
So this guy binds to the
membrane and recruits MinC.

1155
01:14:01,930 --> 01:14:09,430
Whereas MinE-- what it does is
it pulls MinD off the membrane.

1156
01:14:09,430 --> 01:14:15,800
So it binds to MinD and
ejects from the membrane.

1157
01:14:22,710 --> 01:14:28,020
And what has been seen by doing
imaging in live E. coli cells

1158
01:14:28,020 --> 01:14:31,040
with fluorescently
labeled MinD and MinE

1159
01:14:31,040 --> 01:14:35,540
is that there are remarkable
oscillations from pole to pole.

1160
01:14:35,540 --> 01:14:37,940
And indeed, that
doesn't require MinC,

1161
01:14:37,940 --> 01:14:39,740
so MinC is somehow
following the others.

1162
01:14:39,740 --> 01:14:43,420
What happens is that
MinD binds over here,

1163
01:14:43,420 --> 01:14:46,300
then MinE binds
and pushes it off.

1164
01:14:46,300 --> 01:14:48,110
And then the MinD
comes over here,

1165
01:14:48,110 --> 01:14:52,140
and then MinE pulls it off,
and it goes back and forth.

1166
01:14:52,140 --> 01:14:56,220
The period is minutes, maybe.

1167
01:14:58,920 --> 01:15:03,600
So it's a really remarkable
thing that happens in vivo.

1168
01:15:03,600 --> 01:15:07,060
And the idea of what's happening
is that if MinD comes here

1169
01:15:07,060 --> 01:15:09,240
and then it comes here,
on each of the edges,

1170
01:15:09,240 --> 01:15:11,670
then it doesn't hang
out in the middle.

1171
01:15:11,670 --> 01:15:16,080
And that means that it's only in
the middle where MinC can then

1172
01:15:16,080 --> 01:15:20,150
bind-- oh, sorry.

1173
01:15:20,150 --> 01:15:22,860
Because MinC is following
MinD, then only in the middle

1174
01:15:22,860 --> 01:15:25,750
can you form the z-ring, because
that's where MinC is not.

1175
01:15:25,750 --> 01:15:26,250
Right.

1176
01:15:30,450 --> 01:15:32,170
It's wonderful to do
things in live cells

1177
01:15:32,170 --> 01:15:34,866
because that's the native
context and so forth.

1178
01:15:34,866 --> 01:15:36,990
But I think that in some
cases, it's also wonderful

1179
01:15:36,990 --> 01:15:39,370
if you can take
purified components

1180
01:15:39,370 --> 01:15:42,980
and recapitulate interesting
behaviors in vitro.

1181
01:15:42,980 --> 01:15:45,131
Because then at
least you know what

1182
01:15:45,131 --> 01:15:51,360
is sufficient to generate
a particular kind

1183
01:15:51,360 --> 01:15:52,890
of dynamical behavior.

1184
01:15:52,890 --> 01:15:55,155
And because this system
had been proposes

1185
01:15:55,155 --> 01:15:59,040
as a model system for
reaction diffusion mechanism

1186
01:15:59,040 --> 01:16:00,360
to get this behavior.

1187
01:16:00,360 --> 01:16:03,800
And so there's this
paper by Martin Loose,

1188
01:16:03,800 --> 01:16:09,610
in Petra Schwille's
lab in-- where was it?

1189
01:16:09,610 --> 01:16:12,660
In Dresden.

1190
01:16:12,660 --> 01:16:15,990
And what they did is they took
a supported lipid bilayer-- so

1191
01:16:15,990 --> 01:16:19,240
a membrane on
glass-- and then they

1192
01:16:19,240 --> 01:16:23,110
added purified components
of MinD and MinE

1193
01:16:23,110 --> 01:16:24,810
that were fluorescently labeled.

1194
01:16:24,810 --> 01:16:28,580
And then they saw
amazing patterns,

1195
01:16:28,580 --> 01:16:32,959
these amazing reaction
diffusion waves traveling along.

1196
01:16:32,959 --> 01:16:34,750
Now on the first day
of lecture, I actually

1197
01:16:34,750 --> 01:16:38,780
showed you what some of
those things look like.

1198
01:16:38,780 --> 01:16:42,270
I didn't want to use
up the projector again,

1199
01:16:42,270 --> 01:16:44,580
but maybe I can
pull up this movie

1200
01:16:44,580 --> 01:16:47,730
because it is kind of fabulous.

1201
01:16:47,730 --> 01:16:50,590
So maybe you can't see
these things very well.

1202
01:16:50,590 --> 01:16:57,640
But this is a MinD and MinE
and an overlay of the two,

1203
01:16:57,640 --> 01:17:01,790
imaged on a two
dimensional membrane.

1204
01:17:01,790 --> 01:17:04,580
And the movies are just amazing.

1205
01:17:04,580 --> 01:17:07,160
When you see this, you think
that it's a simulation.

1206
01:17:07,160 --> 01:17:10,250
It's so incredible
watching these things go.

1207
01:17:10,250 --> 01:17:12,720
After class, you can come
up, I can show you the paper,

1208
01:17:12,720 --> 01:17:16,510
and you can look at the
movie in more depth.

1209
01:17:16,510 --> 01:17:19,330
But they're really
amazing patterns,

1210
01:17:19,330 --> 01:17:21,330
and it's patterns
that you would predict

1211
01:17:21,330 --> 01:17:23,845
from some sort of Turing type
reaction diffusion mechanism.

1212
01:17:28,560 --> 01:17:31,140
So there are few things
that are may be worth

1213
01:17:31,140 --> 01:17:32,500
saying in this business.

1214
01:17:32,500 --> 01:17:36,260
So these are fluorescently
labeled MinD, MinE.

1215
01:17:36,260 --> 01:17:39,036
They started out with
MinD that was uniform.

1216
01:17:39,036 --> 01:17:41,077
And then they added MinE,
and over the time scale

1217
01:17:41,077 --> 01:17:43,770
of about an hour they started
seeing these sorts of patterns.

1218
01:17:46,460 --> 01:17:51,150
And as maybe you expected
from the in vivo behavior,

1219
01:17:51,150 --> 01:17:59,430
what you see are these things
where MinD looks like this,

1220
01:17:59,430 --> 01:18:05,270
whereas MinE looks like that.

1221
01:18:05,270 --> 01:18:08,460
And then the wave
travels here to the left.

1222
01:18:08,460 --> 01:18:12,780
Because the MinE is ejecting
the MinD from the membrane,

1223
01:18:12,780 --> 01:18:17,060
causing this whole
thing to move.

1224
01:18:17,060 --> 01:18:19,180
So this is a situation
where you have

1225
01:18:19,180 --> 01:18:22,570
both of these proteins
in the liquid.

1226
01:18:22,570 --> 01:18:25,440
All right, so in buffer, as well
as on the membrane and they're

1227
01:18:25,440 --> 01:18:28,020
coming on and off and so forth.

1228
01:18:28,020 --> 01:18:30,060
And in this situation,
it's wonderful

1229
01:18:30,060 --> 01:18:32,360
because they can do all
sorts of things like control

1230
01:18:32,360 --> 01:18:34,560
the concentration
of MinE, and see

1231
01:18:34,560 --> 01:18:37,920
that the velocity of
these waves changes as you

1232
01:18:37,920 --> 01:18:39,480
change concentration in MinE.

1233
01:18:39,480 --> 01:18:40,860
So this is a really
experimentally trackable

1234
01:18:40,860 --> 01:18:41,430
system.

1235
01:18:41,430 --> 01:18:43,750
And then you can ask
what kind of model

1236
01:18:43,750 --> 01:18:45,250
would lead to that
sort of behavior?

1237
01:18:49,160 --> 01:18:51,790
Such waves, do you think
that it requires ATP?

1238
01:19:00,820 --> 01:19:03,540
Just for fun, we can vote
yes or no, it's all right.

1239
01:19:03,540 --> 01:19:05,486
Ready three, two, one.

1240
01:19:05,486 --> 01:19:06,134
AUDIENCE: Yes.

1241
01:19:06,134 --> 01:19:06,800
PROFESSOR: Yeah.

1242
01:19:06,800 --> 01:19:08,600
So indeed it does require ATP.

1243
01:19:14,680 --> 01:19:18,550
And that's a general
feature of these Turing type

1244
01:19:18,550 --> 01:19:22,277
patterns is that there is
a non-equilibrium structure

1245
01:19:22,277 --> 01:19:22,776
formation.

1246
01:19:28,840 --> 01:19:31,810
Now one nice thing you can
do in this sort of system

1247
01:19:31,810 --> 01:19:33,435
is you can ask, well,
what happens if I

1248
01:19:33,435 --> 01:19:36,310
photo bleach a particular area?

1249
01:19:36,310 --> 01:19:37,680
So I come in and I photo bleach.

1250
01:19:37,680 --> 01:19:43,860
So now the profile
looks like-- I locally

1251
01:19:43,860 --> 01:19:48,160
deplete the fluorescence here.

1252
01:19:48,160 --> 01:19:51,860
Question is, will this move
together with the traveling

1253
01:19:51,860 --> 01:19:54,280
wave, or does it stay fixed?

1254
01:19:56,960 --> 01:19:59,830
And in these mechanisms, do
you think should this locally

1255
01:19:59,830 --> 01:20:02,696
depleted area, should it move
or should it stay where it is?

1256
01:20:02,696 --> 01:20:03,987
Do you understand the question?

1257
01:20:06,930 --> 01:20:07,960
Move, yes/no.

1258
01:20:07,960 --> 01:20:10,044
Ready three, two, one.

1259
01:20:10,044 --> 01:20:10,960
AUDIENCE: [INAUDIBLE].

1260
01:20:14,320 --> 01:20:17,430
PROFESSOR: It
actually doesn't move.

1261
01:20:17,430 --> 01:20:19,336
Depleted area doesn't move.

1262
01:20:27,070 --> 01:20:29,650
And this is just a reflection.

1263
01:20:29,650 --> 01:20:33,032
These waves are not the result
of individual molecules moving

1264
01:20:33,032 --> 01:20:33,990
together with the wave.

1265
01:20:37,040 --> 01:20:41,130
A wave like what you see in
the ocean, or on a stream,

1266
01:20:41,130 --> 01:20:41,940
or what not.

1267
01:20:41,940 --> 01:20:48,657
So the motion is a result of
the individual molecules coming

1268
01:20:48,657 --> 01:20:50,490
and going and communicating
with each other,

1269
01:20:50,490 --> 01:20:53,270
but it's not a reflection
of actual molecules

1270
01:20:53,270 --> 01:20:55,210
having directed motion.

1271
01:20:55,210 --> 01:20:58,580
Because indeed,
there's no mechanism

1272
01:20:58,580 --> 01:21:02,070
in these models for
directed motion.

1273
01:21:02,070 --> 01:21:05,700
What you see is that
in all these models,

1274
01:21:05,700 --> 01:21:11,070
the only thing that is
a function of position

1275
01:21:11,070 --> 01:21:12,760
is diffusion terms.

1276
01:21:12,760 --> 01:21:18,070
So it's all random diffusion,
but then you get global motion.