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PROFESSOR: So today,
the general theme

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00:00:22,610 --> 00:00:24,400
is going to be to
try to understand

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00:00:24,400 --> 00:00:26,910
how bacteria find food.

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00:00:26,910 --> 00:00:30,350
And in particular, how do they
know which direction to swim?

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00:00:30,350 --> 00:00:31,880
How do they swim
given that they're

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00:00:31,880 --> 00:00:34,230
operating in this low
Reynolds number regime

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that you read about
maybe last night?

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00:00:37,770 --> 00:00:41,920
What did you guys think of the
life in low Reynolds number?

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00:00:41,920 --> 00:00:45,452
Yeah, it's kind of fun.

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00:00:45,452 --> 00:00:47,160
You very much get the
impression that you

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00:00:47,160 --> 00:00:51,460
are kind of there in that
lecture hall with him.

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00:00:51,460 --> 00:00:55,111
They do say that some
essential arm waving was not

20
00:00:55,111 --> 00:00:56,110
reproduced or so, right?

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00:00:56,110 --> 00:01:00,189
But you kind of imagine his
arms waving as he talks,

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00:01:00,189 --> 00:01:01,980
even though you don't
get to hear him talk.

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00:01:04,720 --> 00:01:06,340
So that's fun.

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00:01:06,340 --> 00:01:08,260
And it's a surprising
number of typos

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00:01:08,260 --> 00:01:11,347
given that-- every time I
read that, I'm like, somehow

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00:01:11,347 --> 00:01:13,430
this is still the best
version that I've been able

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00:01:13,430 --> 00:01:16,390
find of that talk.

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00:01:16,390 --> 00:01:20,130
But I think it's just
a neat example of how

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00:01:20,130 --> 00:01:22,710
clearly a talented physicist
can kind of from a naive

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00:01:22,710 --> 00:01:26,270
perspective say interesting
things about the way

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00:01:26,270 --> 00:01:28,660
that life has to work.

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00:01:28,660 --> 00:01:32,707
Now today, so we will
be talking indeed

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00:01:32,707 --> 00:01:34,040
about how bacteria are swimming.

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00:01:34,040 --> 00:01:36,160
But we want to
start by just making

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00:01:36,160 --> 00:01:40,060
sure we understand something
about how diffusion works,

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00:01:40,060 --> 00:01:42,640
what life is like in this
low Reynolds number regime.

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00:01:42,640 --> 00:01:46,970
And for these sorts of problems,
it's really very, very valuable

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00:01:46,970 --> 00:01:49,230
if you can just apply some
simple dimensional analysis

39
00:01:49,230 --> 00:01:49,730
ideas.

40
00:01:49,730 --> 00:01:53,410
And so we're going to practice
that in a few different cases,

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00:01:53,410 --> 00:01:57,490
that dimension analysis can
often tell you a great deal.

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00:01:57,490 --> 00:02:01,387
In many cases, you can
discover new physical laws.

43
00:02:01,387 --> 00:02:03,220
You can figure out how
things have to scale.

44
00:02:03,220 --> 00:02:06,026
And in some cases, the
scalings are rather surprising.

45
00:02:09,130 --> 00:02:11,400
And given some of
the constraints

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00:02:11,400 --> 00:02:15,240
that microscopic life is facing
in this low Reynolds number

47
00:02:15,240 --> 00:02:18,810
regime where diffusion is very
important, viscous forces are

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00:02:18,810 --> 00:02:21,130
dominating over these
inertial forces,

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00:02:21,130 --> 00:02:24,060
it really constrains what
bacteria, for example,

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00:02:24,060 --> 00:02:24,700
are able to do.

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00:02:24,700 --> 00:02:27,080
And it really
constrains how they

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00:02:27,080 --> 00:02:31,630
have to go about solving
the challenges of life.

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00:02:31,630 --> 00:02:34,940
So what I want to do
is start by thinking

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00:02:34,940 --> 00:02:39,590
about just a very kind of simple
situation, which is we have in,

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00:02:39,590 --> 00:02:44,310
we'll say water, a
non-permeable membrane that

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00:02:44,310 --> 00:02:47,420
is separating two compartments.

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00:02:47,420 --> 00:02:52,670
Whereas over here, we have maybe
salt or something on the right.

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00:02:52,670 --> 00:02:55,130
So this is a salty solution.

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00:03:01,370 --> 00:03:03,575
And this is pure water.

60
00:03:08,590 --> 00:03:12,590
Now, this is a
non-permeable membrane.

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00:03:12,590 --> 00:03:15,506
So it doesn't allow
the salt to cross.

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00:03:15,506 --> 00:03:16,880
But the question
is, what happens

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00:03:16,880 --> 00:03:21,770
if I puncture, and I put a
little hole in the membrane

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00:03:21,770 --> 00:03:25,660
of some radius a, let's say?

65
00:03:25,660 --> 00:03:32,840
Now, the question is, what will
be the flow rate, the net flow,

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00:03:32,840 --> 00:03:36,996
of salt, in this case,
from the right to the left?

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00:03:48,080 --> 00:03:53,190
All right, so first of all,
can somebody kind of answer

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00:03:53,190 --> 00:03:55,670
at least for a couple-- what
are possible things that

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00:03:55,670 --> 00:03:58,890
might be relevant if we're
going to try to figure this out?

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00:03:58,890 --> 00:04:01,772
What quantities will we
need to know or measure?

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00:04:01,772 --> 00:04:02,990
AUDIENCE: Concentration.

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00:04:02,990 --> 00:04:04,906
PROFESSOR: Concentration
seems relevant, yeah.

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00:04:09,430 --> 00:04:11,850
And in this case, there's
only one concentration.

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00:04:11,850 --> 00:04:14,170
Because well, this
conservation is

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00:04:14,170 --> 00:04:17,510
0 here, and some
concentration c over here.

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00:04:17,510 --> 00:04:19,134
What else might we need to know?

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00:04:19,134 --> 00:04:20,649
AUDIENCE: Pore size.

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00:04:20,649 --> 00:04:22,390
PROFESSOR: The
pore size, perfect.

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00:04:22,390 --> 00:04:25,848
We'll use a as the
radius of the pore.

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00:04:31,700 --> 00:04:33,246
And is that going to do it?

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00:04:37,070 --> 00:04:38,947
AUDIENCE: You need a
time for something.

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00:04:38,947 --> 00:04:40,030
PROFESSOR: We need a time.

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00:04:40,030 --> 00:04:45,150
Because what we want is
some number of molecules

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00:04:45,150 --> 00:04:48,010
per unit time that is going
to be crossing from the right

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00:04:48,010 --> 00:04:48,510
to the left.

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00:04:48,510 --> 00:04:51,701
So we need something
that has a time.

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00:04:51,701 --> 00:04:52,200
Yes.

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00:04:52,200 --> 00:04:53,445
AUDIENCE: The diffusivity.

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00:04:53,445 --> 00:04:55,060
PROFESSOR: All right,
the diffusivity.

90
00:04:55,060 --> 00:04:59,390
So D, so we'll say D, which
is the diffusion coefficient.

91
00:05:04,550 --> 00:05:07,594
How many S's?

92
00:05:07,594 --> 00:05:15,723
All right, now other things
that might be relevant?

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00:05:20,240 --> 00:05:21,140
AUDIENCE: Temperature

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00:05:21,140 --> 00:05:22,266
PROFESSOR: Temperature, OK.

95
00:05:22,266 --> 00:05:24,598
AUDIENCE: I guess it's probably
going to be incorporated

96
00:05:24,598 --> 00:05:25,450
into the [INAUDIBLE]

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00:05:25,450 --> 00:05:28,710
PROFESSOR: Exactly, right,
so indeed the flow rate

98
00:05:28,710 --> 00:05:31,050
will be dependent
upon the temperature.

99
00:05:31,050 --> 00:05:34,728
But the way that that manifests
is by doing what to D?

100
00:05:37,360 --> 00:05:39,790
In general, if we go up
to higher temperature,

101
00:05:39,790 --> 00:05:41,512
how will D change?

102
00:05:41,512 --> 00:05:42,892
AUDIENCE: Should get bigger.

103
00:05:42,892 --> 00:05:44,350
PROFESSOR: Should
get bigger, yeah,

104
00:05:44,350 --> 00:05:48,700
so indeed, D will be typically
given by this Einstein

105
00:05:48,700 --> 00:05:49,840
relation of kT over gamma.

106
00:05:49,840 --> 00:05:52,160
We'll talk more about
this in a little bit.

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00:05:52,160 --> 00:05:54,537
But for now, let's just
say that D is something

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00:05:54,537 --> 00:05:55,370
that we've measured.

109
00:05:58,650 --> 00:06:02,824
Now, other things that may
enter into the expression?

110
00:06:07,060 --> 00:06:08,850
All right, so this
is pretty simple.

111
00:06:08,850 --> 00:06:10,200
We have three things.

112
00:06:10,200 --> 00:06:16,510
Now, the question is, if we
double the size of the pore,

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00:06:16,510 --> 00:06:22,770
i.e. if we double a, how
does the flow rate change?

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00:06:22,770 --> 00:06:25,690
So we'll go ahead and do a
vote, since we like to do this.

115
00:06:35,617 --> 00:06:36,700
All right, so we double a.

116
00:06:36,700 --> 00:06:42,282
The flow goes up by what factor?

117
00:06:42,282 --> 00:06:44,164
All right, maybe not all.

118
00:06:58,050 --> 00:07:01,000
All right, and if you
think it's something else,

119
00:07:01,000 --> 00:07:05,500
you can splay your cards out.

120
00:07:05,500 --> 00:07:08,540
Now, the idea here is that
using dimensional analysis,

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00:07:08,540 --> 00:07:10,730
we should be able
to figure this out.

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00:07:10,730 --> 00:07:16,208
Now, I'll give you just a minute
to try to think through this.

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00:07:30,186 --> 00:07:31,665
AUDIENCE: [INAUDIBLE]

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00:07:31,665 --> 00:07:33,850
PROFESSOR: All right,
so you don't even

125
00:07:33,850 --> 00:07:35,568
need to know this in principle.

126
00:07:35,568 --> 00:07:37,692
I'm going to give you
something else that may help.

127
00:07:55,881 --> 00:07:56,964
AUDIENCE: Will it be by x?

128
00:07:59,920 --> 00:08:05,410
PROFESSOR: Yeah, so this is
the mean squared distance

129
00:08:05,410 --> 00:08:06,825
that something will diffuse.

130
00:08:10,020 --> 00:08:11,456
So I wasn't
explaining, because I

131
00:08:11,456 --> 00:08:12,830
was sort of-- a
hint in the sense

132
00:08:12,830 --> 00:08:14,760
that in case you might
remember something

133
00:08:14,760 --> 00:08:17,023
like an equation like-- Yes.

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00:08:17,023 --> 00:08:20,410
AUDIENCE: Are we doing this
at the very beginning of--

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00:08:20,410 --> 00:08:25,446
PROFESSOR: Yes,
right, OK, so there's

136
00:08:25,446 --> 00:08:27,320
going to be some question
of timescales here.

137
00:08:27,320 --> 00:08:31,180
OK, so it's really that
after you puncture the hole,

138
00:08:31,180 --> 00:08:34,570
very rapidly it's
going to assume

139
00:08:34,570 --> 00:08:37,020
some kind of steady state.

140
00:08:37,020 --> 00:08:40,610
And then the flow rate will be
constant for some rather long

141
00:08:40,610 --> 00:08:41,270
time.

142
00:08:41,270 --> 00:08:44,960
But then eventually the
concentration on this half

143
00:08:44,960 --> 00:08:46,170
will decrease.

144
00:08:46,170 --> 00:08:48,120
So we're thinking
about the flow rate

145
00:08:48,120 --> 00:08:50,200
after you've reached
that steady state,

146
00:08:50,200 --> 00:08:55,010
but before you've
equilibrated across the two

147
00:08:55,010 --> 00:08:57,119
sides of the tube.

148
00:09:07,280 --> 00:09:09,690
All right, do you
need more time?

149
00:09:09,690 --> 00:09:12,280
All right, let's go ahead
and see where we are.

150
00:09:12,280 --> 00:09:16,261
Ready-- three, two, one.

151
00:09:19,560 --> 00:09:24,720
All right, so we have a
pretty good agreement.

152
00:09:24,720 --> 00:09:31,167
It seems that it's D.
But didn't I tell you

153
00:09:31,167 --> 00:09:32,417
how to do the problem already?

154
00:09:42,470 --> 00:09:47,250
OK, so it's not--

155
00:09:47,250 --> 00:09:48,972
AUDIENCE: So the
naive thing, I guess,

156
00:09:48,972 --> 00:09:52,806
would be C. Because--
oh, sorry, never mind.

157
00:09:55,310 --> 00:09:58,644
Except for-- [INAUDIBLE]

158
00:09:58,644 --> 00:10:00,060
PROFESSOR: All
right, so I'm going

159
00:10:00,060 --> 00:10:04,120
to give you another--
collaborate with a neighbor.

160
00:10:04,120 --> 00:10:08,540
And I'll tell you that
it's not D to nudge you

161
00:10:08,540 --> 00:10:11,455
in the right direction.

162
00:10:11,455 --> 00:10:13,965
So spend a minute.

163
00:10:13,965 --> 00:10:16,790
[INTERPOSING VOICES]

164
00:11:17,830 --> 00:11:20,105
PROFESSOR: All right, let's
go ahead and reconvene.

165
00:11:23,637 --> 00:11:25,710
All right, I just want
to see where we are now.

166
00:11:25,710 --> 00:11:29,350
So if you still believe it's
D despite my telling you

167
00:11:29,350 --> 00:11:31,370
that I don't think it's
D, you can still vote D,

168
00:11:31,370 --> 00:11:32,260
and then we can argue about it.

169
00:11:32,260 --> 00:11:33,135
And that's fine, too.

170
00:11:33,135 --> 00:11:35,400
All right, ready--
three, two, one.

171
00:11:39,400 --> 00:11:43,650
OK, so now there's maybe a
disagreement between A and C

172
00:11:43,650 --> 00:11:44,420
now.

173
00:11:44,420 --> 00:11:47,418
All right, well,
nobody liked my root 2?

174
00:11:50,110 --> 00:11:52,540
Maybe you saw me hesitate a
little bit before I wrote it.

175
00:11:52,540 --> 00:11:55,670
I was like, oh, let
me throw in a root.

176
00:11:55,670 --> 00:12:03,750
So indeed the flow rate
does not scale as the area.

177
00:12:03,750 --> 00:12:07,600
It only scales with a linear
dimension of this hole.

178
00:12:07,600 --> 00:12:11,264
This is weird, which is
why everybody says D.

179
00:12:11,264 --> 00:12:12,930
And even after you've
done this problem,

180
00:12:12,930 --> 00:12:15,320
you still think it's
D in your intuition.

181
00:12:15,320 --> 00:12:17,780
But you have to like pound
that intuition away from you.

182
00:12:17,780 --> 00:12:20,230
We'll try to understand
in multiple different ways

183
00:12:20,230 --> 00:12:21,262
why this might be.

184
00:12:21,262 --> 00:12:22,720
But what I want to
do first is just

185
00:12:22,720 --> 00:12:25,460
make sure that we understand
from dimensional analysis

186
00:12:25,460 --> 00:12:28,970
why this has to be the case.

187
00:12:28,970 --> 00:12:31,730
Now, the first thing you need
to know for dimensional analysis

188
00:12:31,730 --> 00:12:34,520
is the units of your answer.

189
00:12:34,520 --> 00:12:36,120
We're looking for
something that is

190
00:12:36,120 --> 00:12:40,100
going to be units, some
number of molecules over time.

191
00:12:40,100 --> 00:12:42,080
Number-- there's no
units attached to that.

192
00:12:42,080 --> 00:12:45,400
So the answer is supposed
to be just 1 over time.

193
00:12:45,400 --> 00:12:47,210
Indeed, this is why
we needed something

194
00:12:47,210 --> 00:12:49,390
that had some unit
of time somewhere

195
00:12:49,390 --> 00:12:52,660
in order to get an answer
that could possibly do this.

196
00:12:52,660 --> 00:12:55,370
And indeed, we know that
the diffusion coefficient

197
00:12:55,370 --> 00:12:59,130
has units of, for example,
microns squared per second.

198
00:12:59,130 --> 00:13:01,860
So it's a length
squared over time.

199
00:13:01,860 --> 00:13:04,942
Now, the one barrier
that people have

200
00:13:04,942 --> 00:13:06,650
to use dimensional
analysis is because we

201
00:13:06,650 --> 00:13:09,760
have trouble remembering
what the units of things are.

202
00:13:09,760 --> 00:13:12,030
So if you don't
remember something

203
00:13:12,030 --> 00:13:13,780
like the units of a
diffusion coefficient,

204
00:13:13,780 --> 00:13:18,010
then what you have to do is
try to find some equation

205
00:13:18,010 --> 00:13:22,100
in your brain where
that symbol is used.

206
00:13:22,100 --> 00:13:26,980
And of course it should be
the correct use of the symbol.

207
00:13:26,980 --> 00:13:29,590
So here this is
why I'm saying it.

208
00:13:29,590 --> 00:13:31,160
This is essentially
the definition

209
00:13:31,160 --> 00:13:34,020
of this linear
diffusion coefficient.

210
00:13:34,020 --> 00:13:37,200
And from that,
you get the units.

211
00:13:37,200 --> 00:13:39,460
This is just very, very common.

212
00:13:39,460 --> 00:13:43,150
Something like viscosity,
impossible to remember what

213
00:13:43,150 --> 00:13:44,390
the units of that thing are.

214
00:13:44,390 --> 00:13:46,550
But then you just have
equations in your brain

215
00:13:46,550 --> 00:13:48,800
where the viscosity enters,
and you can figure it out.

216
00:13:53,190 --> 00:13:54,980
All right, so then
if you want to use

217
00:13:54,980 --> 00:13:57,425
this dimensional analysis
study, just be very clear.

218
00:13:57,425 --> 00:13:59,050
Write down what all
the dimensions are.

219
00:13:59,050 --> 00:14:02,690
And then the answer just
comes screaming out at you

220
00:14:02,690 --> 00:14:04,030
in general.

221
00:14:04,030 --> 00:14:07,100
Now, we see that we want
something 1 over time.

222
00:14:07,100 --> 00:14:08,790
There's only one
of these symbols

223
00:14:08,790 --> 00:14:10,050
that has a time in there.

224
00:14:10,050 --> 00:14:12,050
So we know exactly how
the diffusion coefficient

225
00:14:12,050 --> 00:14:13,710
has to enter.

226
00:14:13,710 --> 00:14:17,220
We know that the
flow is going to have

227
00:14:17,220 --> 00:14:20,580
to be something
where we know that D

228
00:14:20,580 --> 00:14:22,339
has to enter in there linearly.

229
00:14:22,339 --> 00:14:23,880
Because if it was
any other way, then

230
00:14:23,880 --> 00:14:25,910
the time would have
some weird units.

231
00:14:25,910 --> 00:14:27,875
It would be time
squared or square root

232
00:14:27,875 --> 00:14:30,020
of time or something.

233
00:14:30,020 --> 00:14:36,480
Now, it's true that we
could in principle dream up

234
00:14:36,480 --> 00:14:44,300
something crazy between
concentration and the pore size

235
00:14:44,300 --> 00:14:47,110
in the sense that
if we wanted to,

236
00:14:47,110 --> 00:14:58,140
we could do c times-- we could
always add a c times a cubed

237
00:14:58,140 --> 00:14:59,310
or something like that.

238
00:14:59,310 --> 00:15:01,790
But of course this is diffusion.

239
00:15:01,790 --> 00:15:06,100
What that means is that these
are non-interacting diffusing

240
00:15:06,100 --> 00:15:06,660
molecules.

241
00:15:06,660 --> 00:15:09,690
That means that if we double
the number of the molecules

242
00:15:09,690 --> 00:15:11,190
over on the right
hand side, we have

243
00:15:11,190 --> 00:15:15,580
to double the flow rate in
this diffusion equation.

244
00:15:15,580 --> 00:15:17,266
And for diffusion,
these molecules

245
00:15:17,266 --> 00:15:18,390
are not interacting at all.

246
00:15:18,390 --> 00:15:19,820
They don't see each other.

247
00:15:19,820 --> 00:15:21,945
And that means if you just
double the concentration

248
00:15:21,945 --> 00:15:25,360
on the right, you have
to double the flow rate.

249
00:15:25,360 --> 00:15:30,180
So that means we know that it
has to show up as a single c.

250
00:15:30,180 --> 00:15:33,620
And all we're left with is
going to be one unit of length

251
00:15:33,620 --> 00:15:37,390
that we have to get rid of.

252
00:15:37,390 --> 00:15:43,540
Now, this is not, of course,
a rigorous definition.

253
00:15:43,540 --> 00:15:47,200
And does this have to be
actually an equal sign?

254
00:15:47,200 --> 00:15:49,219
No, so what we can
actually say is that it

255
00:15:49,219 --> 00:15:50,510
has to be proportional to this.

256
00:15:56,680 --> 00:15:59,050
But actually just from
dimensional analysis

257
00:15:59,050 --> 00:16:00,640
and this idea of,
say, superposition

258
00:16:00,640 --> 00:16:03,050
of the concentrations,
you can get

259
00:16:03,050 --> 00:16:05,830
something that's really kind
of a deep statement about how

260
00:16:05,830 --> 00:16:08,110
diffusion is going to operate.

261
00:16:08,110 --> 00:16:09,531
Yes.

262
00:16:09,531 --> 00:16:11,530
AUDIENCE: It seems to me
that we can break that.

263
00:16:11,530 --> 00:16:13,093
There's this degeneracy.

264
00:16:13,093 --> 00:16:15,820
We can't quite sort out the
powers of c and a, right?

265
00:16:15,820 --> 00:16:18,280
PROFESSOR: Right, so that's
what I was saying, that just

266
00:16:18,280 --> 00:16:23,130
from the units alone, we could
always add a c times a cubed,

267
00:16:23,130 --> 00:16:25,550
or to some power.

268
00:16:25,550 --> 00:16:27,790
And that's why I'm
saying that you

269
00:16:27,790 --> 00:16:31,540
have to be able to invoke
this idea that if you

270
00:16:31,540 --> 00:16:34,084
double the concentration
of the salt on the right,

271
00:16:34,084 --> 00:16:35,250
you'll double the flow rate.

272
00:16:35,250 --> 00:16:38,060
Because these are
non-interacting molecules.

273
00:16:38,060 --> 00:16:40,586
Because they're just
diffusing around in solution.

274
00:16:40,586 --> 00:16:43,574
AUDIENCE: I was going
to say that if we simply

275
00:16:43,574 --> 00:16:49,052
chose the measure of flow in
90 units of number of atoms,

276
00:16:49,052 --> 00:16:51,991
but in units of
mass per unit time,

277
00:16:51,991 --> 00:16:54,032
and concentration in units
of mass per unit time,

278
00:16:54,032 --> 00:17:00,008
then we would clearly see
what power of c it would be.

279
00:17:00,008 --> 00:17:02,498
And it would have to be 1.

280
00:17:05,510 --> 00:17:07,220
PROFESSOR: It's an
interesting-- I think

281
00:17:07,220 --> 00:17:08,511
I'm going to disagree with you.

282
00:17:08,511 --> 00:17:15,170
But I want to make sure
that I-- so you want to say,

283
00:17:15,170 --> 00:17:16,410
OK, you're entering.

284
00:17:16,410 --> 00:17:21,399
Yeah, the diffusion
coefficient is about what

285
00:17:21,399 --> 00:17:22,710
the molecule is doing.

286
00:17:22,710 --> 00:17:26,400
So I'm not sure how we're
going to-- if we measure

287
00:17:26,400 --> 00:17:29,130
concentration in units
of, you're saying,

288
00:17:29,130 --> 00:17:33,560
mgs per mil of some protein, or
so, then you're really saying

289
00:17:33,560 --> 00:17:34,890
that there's some mass density.

290
00:17:38,020 --> 00:17:44,480
And I guess you can always
define diffusion coefficient,

291
00:17:44,480 --> 00:17:48,310
even from the standpoint
of-- because we don't even

292
00:17:48,310 --> 00:17:49,570
need numbers.

293
00:17:49,570 --> 00:17:53,750
We can measure some
fluorescence or something

294
00:17:53,750 --> 00:17:54,850
of how it diffuses across.

295
00:17:54,850 --> 00:17:56,995
So that doesn't require numbers.

296
00:17:56,995 --> 00:17:58,370
Yeah, I'm trying
to think if this

297
00:17:58,370 --> 00:18:00,600
is going to end up being
different from the argument.

298
00:18:03,710 --> 00:18:06,368
I'd have to--

299
00:18:06,368 --> 00:18:08,422
AUDIENCE: If we put in
the number of molecules,

300
00:18:08,422 --> 00:18:10,880
then it's going to have to be
number of molecules per time.

301
00:18:10,880 --> 00:18:13,370
And then c has to be linear.

302
00:18:13,370 --> 00:18:18,780
Because c is number of
molecules per something, right?

303
00:18:18,780 --> 00:18:24,770
PROFESSOR: OK, yeah, you could,
say, treat molecules as a unit.

304
00:18:24,770 --> 00:18:28,457
And then that also ends up
requiring that it's-- yeah, no,

305
00:18:28,457 --> 00:18:29,290
I think that's fair.

306
00:18:32,360 --> 00:18:34,750
But I'd have to think about
this, how it would play out

307
00:18:34,750 --> 00:18:37,333
if you try to measure things in
terms of mass per unit volume.

308
00:18:40,990 --> 00:18:44,139
Of course, you should be able
to calculate this explicitly,

309
00:18:44,139 --> 00:18:46,680
and then also to figure out what
the proportionality constant

310
00:18:46,680 --> 00:18:47,179
is.

311
00:18:47,179 --> 00:18:50,130
But indeed, what you find
is that it scales linearly

312
00:18:50,130 --> 00:18:52,429
with the radius instead
of as the area, which is,

313
00:18:52,429 --> 00:18:53,470
I think, very surprising.

314
00:18:53,470 --> 00:18:55,320
Yes.

315
00:18:55,320 --> 00:18:59,422
AUDIENCE: We can also say D
scales with a squared times

316
00:18:59,422 --> 00:18:59,922
flow.

317
00:19:20,290 --> 00:19:23,160
PROFESSOR: Oh, you're saying
that maybe the concentration is

318
00:19:23,160 --> 00:19:25,760
not relevant.

319
00:19:25,760 --> 00:19:30,080
AUDIENCE: If we look at it
that way, then the answer is D.

320
00:19:30,080 --> 00:19:33,730
PROFESSOR: OK,
what you're saying

321
00:19:33,730 --> 00:19:37,899
is that maybe somehow
the flow is just equal

322
00:19:37,899 --> 00:19:38,690
to or proportional.

323
00:19:38,690 --> 00:19:41,530
It goes as D times a
squared in the sense

324
00:19:41,530 --> 00:19:42,610
that you would say that--

325
00:19:42,610 --> 00:19:43,800
AUDIENCE: Over.

326
00:19:43,800 --> 00:19:44,300
Over.

327
00:19:49,530 --> 00:19:54,698
PROFESSOR: Right, I think that
this non-interacting particle

328
00:19:54,698 --> 00:19:56,656
business really does tell
you that the flow has

329
00:19:56,656 --> 00:19:58,960
to be proportional
to the concentration.

330
00:19:58,960 --> 00:20:03,970
And yeah, I think that
is sufficient already.

331
00:20:03,970 --> 00:20:06,410
I agree that just from
units, you could do this.

332
00:20:06,410 --> 00:20:12,180
But I think, well, it's not
true, and it's nonsensical.

333
00:20:12,180 --> 00:20:14,226
And it's not proportional to c.

334
00:20:18,680 --> 00:20:21,160
Let's move on.

335
00:20:21,160 --> 00:20:24,447
But I think it's
important that it's

336
00:20:24,447 --> 00:20:26,280
possible to say something
that's a deep kind

337
00:20:26,280 --> 00:20:31,050
of physical principle just
based on analysis of the units.

338
00:20:31,050 --> 00:20:32,580
And then, of
course, can somebody

339
00:20:32,580 --> 00:20:35,310
offer the intuitive explanation
for why this might be?

340
00:20:35,310 --> 00:20:39,168
Why is it that it goes as
a instead of a squared?

341
00:20:44,090 --> 00:20:47,830
And this is a little bit weird
given the fact that we all

342
00:20:47,830 --> 00:20:49,895
check our intuition,
we all say D.

343
00:20:49,895 --> 00:20:51,770
But the thing is, once
you know an answer,

344
00:20:51,770 --> 00:20:54,240
you should still try to figure
out what the intuition might

345
00:20:54,240 --> 00:21:10,180
have been that you missed when
you originally said D. Yes.

346
00:21:10,180 --> 00:21:12,620
AUDIENCE: OK, this might
not make any sense.

347
00:21:12,620 --> 00:21:16,583
So say you would first think
it goes like a squared,

348
00:21:16,583 --> 00:21:18,124
because the area
goes like a squared.

349
00:21:18,124 --> 00:21:21,575
But then maybe the fact that
the perimeter grows like a,

350
00:21:21,575 --> 00:21:24,533
and the perimeter
adds drag, maybe that

351
00:21:24,533 --> 00:21:28,491
would subtract from
or cancel out any--

352
00:21:28,491 --> 00:21:30,800
not necessarily make sense?

353
00:21:30,800 --> 00:21:37,030
PROFESSOR: Yeah, well, OK,
so I don't think that's

354
00:21:37,030 --> 00:21:39,087
the intuition that I would use.

355
00:21:39,087 --> 00:21:41,420
But then we have to figure
out, OK, what might be wrong?

356
00:21:41,420 --> 00:21:43,520
Well first, I don't think
you can subtract an a.

357
00:21:43,520 --> 00:21:48,840
But also we haven't invoked
any actual drag in the sense

358
00:21:48,840 --> 00:21:52,230
that there's no sense that the
perimeter should be-- right?

359
00:21:57,992 --> 00:21:59,450
All right, so this
is so mysterious

360
00:21:59,450 --> 00:22:02,910
that we are-- that's fine.

361
00:22:02,910 --> 00:22:06,130
So the way that I like
to think about this

362
00:22:06,130 --> 00:22:11,732
is that it's not that
we're shooting bullets

363
00:22:11,732 --> 00:22:12,440
at this membrane.

364
00:22:12,440 --> 00:22:15,100
If we were shooting
bullets, or if these

365
00:22:15,100 --> 00:22:16,820
were raindrops
coming, then indeed

366
00:22:16,820 --> 00:22:19,470
it would scale as the area.

367
00:22:19,470 --> 00:22:22,060
But because it's
diffusion, diffusion

368
00:22:22,060 --> 00:22:27,500
operates on gradients
of concentration.

369
00:22:27,500 --> 00:22:29,380
And what's happening
here is that you

370
00:22:29,380 --> 00:22:34,780
get a local depletion of the
concentration at the pore.

371
00:22:34,780 --> 00:22:37,699
So it's not the case that
you have concentration

372
00:22:37,699 --> 00:22:39,740
c all the way until you
get to the pore, and then

373
00:22:39,740 --> 00:22:42,190
all of a sudden you have
a linear gradient of c

374
00:22:42,190 --> 00:22:44,350
across that little pore,
but rather that you

375
00:22:44,350 --> 00:22:46,590
have a depletion of
the concentration

376
00:22:46,590 --> 00:22:49,510
as you approach the pore.

377
00:22:49,510 --> 00:22:52,070
And so as you grow
the pore, you're

378
00:22:52,070 --> 00:22:53,450
somehow competing with yourself.

379
00:22:56,540 --> 00:23:00,110
Another way to
think about this is

380
00:23:00,110 --> 00:23:02,151
to think about what happens
if we have two pores.

381
00:23:06,850 --> 00:23:08,560
For example, let's
imagine that we

382
00:23:08,560 --> 00:23:10,320
have this poor here of size a.

383
00:23:10,320 --> 00:23:15,430
But now I'm going to add a
second pore over here far away,

384
00:23:15,430 --> 00:23:15,960
same size.

385
00:23:18,680 --> 00:23:27,004
What should this do to the
total flow rate, A, B, C, D, E?

386
00:23:27,004 --> 00:23:29,170
I'll give you 10 seconds
to come up with an opinion.

387
00:23:31,720 --> 00:23:34,128
So I'm adding a
second pore far away.

388
00:23:37,720 --> 00:23:39,830
Ready?

389
00:23:39,830 --> 00:23:47,470
Three, two, one-- all right,
so we're all agreeing.

390
00:23:47,470 --> 00:23:51,850
And this should be 2x, again.

391
00:23:51,850 --> 00:23:55,820
This is for two pores now.

392
00:23:55,820 --> 00:23:57,240
We also get 2x.

393
00:23:57,240 --> 00:24:01,630
This is because if the two pores
are far away from each other,

394
00:24:01,630 --> 00:24:04,370
as relative to the
size of the pores,

395
00:24:04,370 --> 00:24:07,610
then these are really
non-interacting in the sense

396
00:24:07,610 --> 00:24:11,182
that there's none of
this depletion effect

397
00:24:11,182 --> 00:24:12,390
that I was telling you about.

398
00:24:12,390 --> 00:24:15,950
Whereas as we bring
these pores together,

399
00:24:15,950 --> 00:24:19,010
then they start interfering
with each other.

400
00:24:19,010 --> 00:24:22,580
Because they each involve
some local depletion

401
00:24:22,580 --> 00:24:23,560
of the concentration.

402
00:24:23,560 --> 00:24:27,290
And once that kind of depletion
area starts to overlap,

403
00:24:27,290 --> 00:24:29,540
then the point is that
there is some molecule that

404
00:24:29,540 --> 00:24:30,640
was diffusing.

405
00:24:30,640 --> 00:24:34,700
And if you have the pores
right next to each other,

406
00:24:34,700 --> 00:24:37,010
you're not going to
get twice the net flow.

407
00:24:37,010 --> 00:24:38,660
Because that molecule,
it might have

408
00:24:38,660 --> 00:24:43,854
diffused through the other
pore even if it-- if one pore

409
00:24:43,854 --> 00:24:45,645
were not there, then
it might have diffused

410
00:24:45,645 --> 00:24:46,603
through the other pore.

411
00:24:50,296 --> 00:24:51,670
AUDIENCE: Can you
not think of it

412
00:24:51,670 --> 00:24:55,630
as having to do with the fact
that under a random walk,

413
00:24:55,630 --> 00:24:58,105
the displacement goes like
square root of the time?

414
00:24:58,105 --> 00:25:05,990
So if you increase the
number of molecules

415
00:25:05,990 --> 00:25:08,715
going through by the
square of the perimeter,

416
00:25:08,715 --> 00:25:12,060
then actually be like
total molecule displacement

417
00:25:12,060 --> 00:25:17,300
through [INAUDIBLE]
will only go linearly

418
00:25:17,300 --> 00:25:20,390
in that perimeter, which
would account for the fact

419
00:25:20,390 --> 00:25:22,680
that molecule is
coming back across.

420
00:25:22,680 --> 00:25:24,170
PROFESSOR: Right, so it's true
that the molecules coming back

421
00:25:24,170 --> 00:25:25,753
across you always
have to worry about.

422
00:25:25,753 --> 00:25:34,740
But I don't see this
argument actually necessarily

423
00:25:34,740 --> 00:25:35,793
in my work.

424
00:25:35,793 --> 00:25:37,793
AUDIENCE: Maybe in the
case of, like [INAUDIBLE]

425
00:25:37,793 --> 00:25:38,740
which doubled in size.

426
00:25:38,740 --> 00:25:40,240
PROFESSOR: Oh, I see.

427
00:25:40,240 --> 00:25:44,065
Yeah, I think that
it may be connected,

428
00:25:44,065 --> 00:25:47,220
although I'm not sure if
I'm sold on the argument.

429
00:25:54,170 --> 00:25:56,255
Yeah, I think it's a
little bit dangerous.

430
00:25:59,300 --> 00:25:59,950
Yeah.

431
00:25:59,950 --> 00:26:01,516
AUDIENCE: So you
mentioned you appeal

432
00:26:01,516 --> 00:26:03,240
to some sort of
depletion with that.

433
00:26:03,240 --> 00:26:07,220
But if you just
punctured a hole in the--

434
00:26:07,220 --> 00:26:09,530
PROFESSOR: OK, and
this is why there

435
00:26:09,530 --> 00:26:11,210
is this question of timescale.

436
00:26:11,210 --> 00:26:12,970
Immediately after you
puncture the hole,

437
00:26:12,970 --> 00:26:16,110
then the flow rate
will actually be

438
00:26:16,110 --> 00:26:18,494
higher than it will be
at that steady state.

439
00:26:18,494 --> 00:26:21,830
AUDIENCE: So it's
going to be a squared.

440
00:26:21,830 --> 00:26:25,790
PROFESSOR: Yeah, right, I think
that the flow right after you

441
00:26:25,790 --> 00:26:29,060
puncture it indeed will
scale as a squared.

442
00:26:29,060 --> 00:26:30,840
Because what you're
doing is you're

443
00:26:30,840 --> 00:26:34,020
taking the number of
molecules across this area.

444
00:26:34,020 --> 00:26:36,730
And half of them are
going to come through.

445
00:26:36,730 --> 00:26:39,225
And there's just more
molecules in that area

446
00:26:39,225 --> 00:26:40,350
when you first puncture it.

447
00:26:40,350 --> 00:26:45,532
But that's going to equilibrate
in, whatever, a microsecond.

448
00:26:45,532 --> 00:26:46,490
That'll be really fast.

449
00:26:46,490 --> 00:26:49,650
Whereas in the steady
state flow rate,

450
00:26:49,650 --> 00:26:52,020
that could go in
principle forever

451
00:26:52,020 --> 00:26:53,445
if it's a large reservoir.

452
00:26:57,310 --> 00:27:00,210
OK, so I want to move on.

453
00:27:00,210 --> 00:27:03,640
But I want to make
sure that we understand

454
00:27:03,640 --> 00:27:06,150
how this basic
principle allows us

455
00:27:06,150 --> 00:27:07,930
to say some interesting things.

456
00:27:07,930 --> 00:27:10,380
For example, there's
a related problem,

457
00:27:10,380 --> 00:27:15,468
which is, how much food
could a cell possibly eat?

458
00:27:19,750 --> 00:27:23,410
And can somebody
say why there might

459
00:27:23,410 --> 00:27:25,800
be an upper bound on this?

460
00:27:34,572 --> 00:27:35,988
AUDIENCE: There's
a point at which

461
00:27:35,988 --> 00:27:38,225
you get too many nutrients
inside of the cell.

462
00:27:38,225 --> 00:27:39,219
PROFESSOR: I'm sorry?

463
00:27:39,219 --> 00:27:40,875
AUDIENCE: There's
a point at which

464
00:27:40,875 --> 00:27:43,700
you get too many nutrients
inside of the cell.

465
00:27:43,700 --> 00:27:46,930
PROFESSOR: Oh no, this
is a super hungry cell.

466
00:27:46,930 --> 00:27:53,240
It can in principle be able to
use every glucose molecule that

467
00:27:53,240 --> 00:27:54,292
came in.

468
00:27:54,292 --> 00:27:56,690
AUDIENCE: No, but I'm
saying that the osmosis

469
00:27:56,690 --> 00:28:00,170
of the water coming in
would pop the membrane.

470
00:28:00,170 --> 00:28:03,150
PROFESSOR: Oh, OK, we're
talking about osmotic effects.

471
00:28:03,150 --> 00:28:08,630
OK, yeah, but we can
just say that it's

472
00:28:08,630 --> 00:28:10,500
got a super tough cell wall.

473
00:28:10,500 --> 00:28:14,486
So it can take,
we'll say, infinite.

474
00:28:14,486 --> 00:28:16,360
I guess what I'm saying
is there's something.

475
00:28:16,360 --> 00:28:19,840
Even if I invoke such a
thing, there still is a limit.

476
00:28:19,840 --> 00:28:22,770
That's really what
I'm trying to say.

477
00:28:22,770 --> 00:28:25,356
AUDIENCE: The concentration
of the glucose inside the cell

478
00:28:25,356 --> 00:28:27,900
is a bit high.

479
00:28:27,900 --> 00:28:30,150
PROFESSOR: Well, the glucose
is going to get imported.

480
00:28:30,150 --> 00:28:32,150
And then it's going to
be immediately converted

481
00:28:32,150 --> 00:28:35,060
into useful things.

482
00:28:35,060 --> 00:28:36,580
But then still there's a limit.

483
00:28:36,580 --> 00:28:37,513
There's a maximum.

484
00:28:42,190 --> 00:28:45,168
This is the maximum kind of
possible nutrient uptake.

485
00:28:48,380 --> 00:28:50,870
Yes.

486
00:28:50,870 --> 00:28:53,120
AUDIENCE: If the diffusion
is somewhat slow,

487
00:28:53,120 --> 00:28:56,110
or maybe there's
[INAUDIBLE] around the cell.

488
00:28:56,110 --> 00:28:59,240
PROFESSOR: OK,
right, so maybe it

489
00:28:59,240 --> 00:29:02,370
could be somehow
diffusion limited.

490
00:29:02,370 --> 00:29:05,120
And what's the best that
a cell could possibly

491
00:29:05,120 --> 00:29:07,390
do in this regard?

492
00:29:07,390 --> 00:29:09,920
So here's a cell.

493
00:29:09,920 --> 00:29:14,810
And now maybe we will,
again, have it be radius a.

494
00:29:14,810 --> 00:29:16,200
So this is the cell of radius a.

495
00:29:20,630 --> 00:29:23,066
The best they can be,
how would you describe

496
00:29:23,066 --> 00:29:24,190
that relative to diffusion?

497
00:29:34,769 --> 00:29:38,203
AUDIENCE: Can you
be more specific?

498
00:29:38,203 --> 00:29:40,167
AUDIENCE: Everything
that [INAUDIBLE] membrane

499
00:29:40,167 --> 00:29:41,360
could be absorbed.

500
00:29:41,360 --> 00:29:42,360
PROFESSOR: That's right.

501
00:29:42,360 --> 00:29:45,630
So that's what I was-- so the
best that it could possibly be

502
00:29:45,630 --> 00:29:47,995
is a perfect absorber.

503
00:29:47,995 --> 00:29:49,370
What we mean by
that is something

504
00:29:49,370 --> 00:29:54,153
where the moment that a molecule
touches it, it gets imported.

505
00:29:59,210 --> 00:30:01,510
Now of course, a cell is
not a perfect absorber.

506
00:30:01,510 --> 00:30:06,110
But what's fascinating
is that even if it were,

507
00:30:06,110 --> 00:30:08,860
then it does not
mean that it's going

508
00:30:08,860 --> 00:30:12,286
to be consuming, say,
glucose at an infinite rate.

509
00:30:16,102 --> 00:30:17,920
And particularly,
we can ask, well,

510
00:30:17,920 --> 00:30:19,720
if we have a situation
like this where

511
00:30:19,720 --> 00:30:24,460
we have a cell, a sphere of
size a, it's a perfect absorber,

512
00:30:24,460 --> 00:30:29,012
how will its uptake rate
scale with its size?

513
00:30:36,290 --> 00:30:39,210
In particular, if
we double a, what

514
00:30:39,210 --> 00:30:41,400
is it going to do for the flow?

515
00:30:41,400 --> 00:30:44,212
All right, we're going to have
10 seconds to think about this.

516
00:30:50,150 --> 00:30:52,590
If we imagine a cell as a
perfect absorber, the moment

517
00:30:52,590 --> 00:30:56,710
that this nutrient touches it,
it gets imported and chewed up.

518
00:30:56,710 --> 00:31:00,270
The question is, how does
that maximum uptake rate

519
00:31:00,270 --> 00:31:05,909
change if I double the size of
the cell, double the radius?

520
00:31:05,909 --> 00:31:06,450
Are we ready?

521
00:31:09,390 --> 00:31:10,515
OK, let's see where we are.

522
00:31:10,515 --> 00:31:11,015
Ready?

523
00:31:11,015 --> 00:31:14,700
Three, two, one.

524
00:31:14,700 --> 00:31:20,820
OK, so now it's a bit more of
a mixture of A's, C's, D's,

525
00:31:20,820 --> 00:31:22,350
although it is a majority.

526
00:31:22,350 --> 00:31:28,900
We're getting a majority
C. And again, this

527
00:31:28,900 --> 00:31:32,250
is, cell doubles size.

528
00:31:32,250 --> 00:31:37,020
And indeed, from the standpoint
of dimensional analysis,

529
00:31:37,020 --> 00:31:42,230
this is the same problem as
that pore that we just did.

530
00:31:42,230 --> 00:31:45,436
Because again, it's diffusion.

531
00:31:45,436 --> 00:31:46,569
We're interested in uptake.

532
00:31:46,569 --> 00:31:48,110
Now, it's not flow
across a membrane.

533
00:31:48,110 --> 00:31:50,370
But this is uptake
rate by the cell.

534
00:31:50,370 --> 00:31:53,100
It's going to be numbers
of molecules per unit time.

535
00:31:53,100 --> 00:31:54,880
Again, there's diffusion.

536
00:31:54,880 --> 00:31:58,440
The concentration
is still relevant.

537
00:31:58,440 --> 00:32:01,880
And so all the same
arguments apply.

538
00:32:01,880 --> 00:32:10,900
And indeed, this is, again,
proportional to the radius a.

539
00:32:10,900 --> 00:32:17,590
And remember, we're following
the convention of Purcell

540
00:32:17,590 --> 00:32:18,610
and calling this a.

541
00:32:18,610 --> 00:32:24,700
Because Reynolds numbers is
an r, and then problems arise.

542
00:32:24,700 --> 00:32:28,930
Are there any questions about
why the argument is the same

543
00:32:28,930 --> 00:32:29,935
here as it was there?

544
00:32:34,910 --> 00:32:38,230
OK, now, there's something
interesting about this though,

545
00:32:38,230 --> 00:32:46,350
which is that you can imagine
the fact that this only scales

546
00:32:46,350 --> 00:32:50,240
as the radius of the cell
tells you that there's

547
00:32:50,240 --> 00:32:52,490
some sense in
which we're wasting

548
00:32:52,490 --> 00:32:56,040
an awful lot of
that surface area.

549
00:32:56,040 --> 00:32:59,040
Because we have this
large surface area.

550
00:32:59,040 --> 00:33:02,380
But it's not as good
as you would have

551
00:33:02,380 --> 00:33:03,560
thought it would have been.

552
00:33:03,560 --> 00:33:06,220
And what that means
is in principle,

553
00:33:06,220 --> 00:33:10,040
you could have a
smaller fraction

554
00:33:10,040 --> 00:33:18,020
of the area covered by a bunch
of little pores or transporters

555
00:33:18,020 --> 00:33:20,010
that are kind of analogous
to this situation

556
00:33:20,010 --> 00:33:23,050
that we had before with the
salt crossing the membrane.

557
00:33:23,050 --> 00:33:25,810
And indeed, if you
just imagine that these

558
00:33:25,810 --> 00:33:31,220
are each little absorbing
patches that are perhaps

559
00:33:31,220 --> 00:33:35,640
one nanometer in
size, small, there's

560
00:33:35,640 --> 00:33:37,400
an amusing calculation
you can do,

561
00:33:37,400 --> 00:33:42,930
which is you can ask, well, how
is it that the-- this is now

562
00:33:42,930 --> 00:33:43,650
uptake rate.

563
00:33:47,780 --> 00:33:50,170
We want to know,
how is it that that

564
00:33:50,170 --> 00:33:55,110
is going to change as a function
of the fraction of the surface

565
00:33:55,110 --> 00:33:59,160
area that we have covered
with these transporters,

566
00:33:59,160 --> 00:34:03,130
with these little perfectly
absorbing patches?

567
00:34:03,130 --> 00:34:09,090
So we imagine the cell is
1 to 10 microns in size.

568
00:34:09,090 --> 00:34:11,670
Now we have a bunch
of these little pores

569
00:34:11,670 --> 00:34:13,130
that we're going to go on.

570
00:34:13,130 --> 00:34:18,409
And of course, this is
going to saturate somewhere.

571
00:34:18,409 --> 00:34:20,000
And it's going to saturate.

572
00:34:20,000 --> 00:34:22,650
And as it turns out, this
goes as 4 pi D, I think.

573
00:34:22,650 --> 00:34:24,679
Yeah, so the uptake
rate is actually

574
00:34:24,679 --> 00:34:31,679
equal to 4 pi cDa for a sphere.

575
00:34:31,679 --> 00:34:34,580
So this is the part that we
could not have figured out

576
00:34:34,580 --> 00:34:36,710
from dimensional analysis.

577
00:34:36,710 --> 00:34:39,340
So indeed, the uptake rate,
once we get up to-- this

578
00:34:39,340 --> 00:34:40,714
is a fraction that's covered.

579
00:34:40,714 --> 00:34:46,286
Once that's 1, indeed
it's going to be 4 pi cDa.

580
00:34:50,510 --> 00:34:56,130
Now, the question is, how
does it behave over here?

581
00:34:56,130 --> 00:34:58,310
Now, you might have
thought that it would just

582
00:34:58,310 --> 00:35:02,350
be some line like that.

583
00:35:02,350 --> 00:35:08,870
It's just the more of these
transporters that you have,

584
00:35:08,870 --> 00:35:13,985
you just get a linear increase
in your total uptake rate.

585
00:35:13,985 --> 00:35:16,620
But it turns out
that that's not true.

586
00:35:16,620 --> 00:35:19,290
Because even at just
having, say, 1%,

587
00:35:19,290 --> 00:35:25,500
roughly, of the surface
covered by this transporter,

588
00:35:25,500 --> 00:35:32,441
you can get something like 50%
of the total maximal uptake

589
00:35:32,441 --> 00:35:32,940
rate.

590
00:35:32,940 --> 00:35:35,060
And that's because
these guys are not

591
00:35:35,060 --> 00:35:37,605
competing with each other so
much, because they're far away.

592
00:35:37,605 --> 00:35:46,195
So this curve really looks
something like-- well, OK?

593
00:35:52,244 --> 00:35:54,160
Now, this is, I think,
especially interesting.

594
00:35:54,160 --> 00:35:57,820
Because a cell is not trying
to import just one thing.

595
00:35:57,820 --> 00:36:00,910
In principle, it's trying to
import many different things.

596
00:36:00,910 --> 00:36:03,290
So you need carbon sources.

597
00:36:03,290 --> 00:36:06,190
You need all sorts
of trace metals.

598
00:36:06,190 --> 00:36:07,247
You need nitrogen.

599
00:36:07,247 --> 00:36:09,330
So in principle, there are
many, many, many things

600
00:36:09,330 --> 00:36:11,660
that a cell is trying to
get from the environment.

601
00:36:11,660 --> 00:36:14,512
And by simply having
different transporters

602
00:36:14,512 --> 00:36:16,220
specific to different
things-- all right,

603
00:36:16,220 --> 00:36:19,870
so here are some squares
that uptake something else,

604
00:36:19,870 --> 00:36:25,530
and little, I don't know,
triangles, whatever.

605
00:36:25,530 --> 00:36:28,480
So you can have many, many, many
of these different transporters

606
00:36:28,480 --> 00:36:29,980
on the cell surface.

607
00:36:29,980 --> 00:36:32,460
And they're each
uptaking their own thing.

608
00:36:32,460 --> 00:36:34,420
And kind of
surprisingly, you can

609
00:36:34,420 --> 00:36:39,530
get near optimal uptake rates
of many, many different things.

610
00:36:43,700 --> 00:36:47,210
I feel that I've lost you.

611
00:36:47,210 --> 00:36:49,310
Somebody ask a question.

612
00:36:49,310 --> 00:36:50,735
Yes.

613
00:36:50,735 --> 00:36:52,318
AUDIENCE: So this
[INAUDIBLE], is

614
00:36:52,318 --> 00:36:55,490
it really proportional to the
actual surface area covered?

615
00:36:55,490 --> 00:36:57,690
PROFESSOR: Right, so
this initial thing,

616
00:36:57,690 --> 00:37:01,130
this increases linearly
for small areas.

617
00:37:01,130 --> 00:37:02,860
Because just like
what we said before,

618
00:37:02,860 --> 00:37:06,700
if you had these two little
holes in the membrane that

619
00:37:06,700 --> 00:37:09,890
are far away from each other,
then they don't compete.

620
00:37:09,890 --> 00:37:13,019
What that means is that you do
get an initial linear increase

621
00:37:13,019 --> 00:37:13,519
here.

622
00:37:13,519 --> 00:37:15,475
AUDIENCE: But I guess
my question is sort of,

623
00:37:15,475 --> 00:37:18,409
how can we get that 50% of 1%?

624
00:37:18,409 --> 00:37:20,202
Is there a central
calculation that you can

625
00:37:20,202 --> 00:37:21,832
do to get that, or [INAUDIBLE]?

626
00:37:24,780 --> 00:37:28,020
PROFESSOR: Oh no, we can do it.

627
00:37:28,020 --> 00:37:34,870
Because actually, I think that
the most straightforward way

628
00:37:34,870 --> 00:37:39,650
to do this is to ask,
what is the absorbing rate

629
00:37:39,650 --> 00:37:45,500
of a little patch, let's
say a circular patch?

630
00:37:45,500 --> 00:37:48,905
And we already figured
out that when--

631
00:37:48,905 --> 00:37:50,560
so these are very
related problems,

632
00:37:50,560 --> 00:37:52,070
as you can kind of see.

633
00:37:52,070 --> 00:37:54,860
I just don't remember
whether it's 2 pi

634
00:37:54,860 --> 00:37:57,130
or if it's 4 pi for
that particular problem.

635
00:38:02,030 --> 00:38:12,590
Because this is some uptake
of a patch, we'll say.

636
00:38:12,590 --> 00:38:20,460
It's going to be either
1, 2 or 4 pi Dca.

637
00:38:20,460 --> 00:38:26,370
But there's some uptake
rate from each patch.

638
00:38:26,370 --> 00:38:28,726
But then that means just
that the uptake total

639
00:38:28,726 --> 00:38:31,465
for small numbers
is indeed just going

640
00:38:31,465 --> 00:38:33,780
to be equal to the
number of patches

641
00:38:33,780 --> 00:38:37,440
or transporters times the
uptake rate of each one.

642
00:38:37,440 --> 00:38:39,910
Because they're just not
competing against each other.

643
00:38:39,910 --> 00:38:42,990
And the striking thing is that
because of this difference

644
00:38:42,990 --> 00:38:45,670
in scaling between the
radius and the area,

645
00:38:45,670 --> 00:38:50,040
this can give you a
lot of uptake rate

646
00:38:50,040 --> 00:38:53,950
while only occupying a
small fraction of the area.

647
00:38:53,950 --> 00:39:01,620
This is a fraction
of area covered.

648
00:39:06,550 --> 00:39:09,620
Do you see why that
ends up being the case?

649
00:39:09,620 --> 00:39:10,552
Yes.

650
00:39:10,552 --> 00:39:16,150
AUDIENCE: So that's also saying
that [INAUDIBLE] pretty soon.

651
00:39:16,150 --> 00:39:19,830
PROFESSOR: Well, it depends
on what you mean by soon.

652
00:39:19,830 --> 00:39:23,450
I mean, it's soon on the
scale of the amount of area.

653
00:39:23,450 --> 00:39:27,101
But it could be that you can
have 1,000 of them or so,

654
00:39:27,101 --> 00:39:27,600
right?

655
00:39:27,600 --> 00:39:31,330
So the way I would
think about it

656
00:39:31,330 --> 00:39:34,487
is that you don't
have to cover up

657
00:39:34,487 --> 00:39:36,070
very much of the
cell surface in order

658
00:39:36,070 --> 00:39:38,864
to get really good
uptake kinetics.

659
00:39:48,690 --> 00:39:52,660
Now, these are some, say,
fundamental limits to the way

660
00:39:52,660 --> 00:39:54,830
that bacteria, for
example, and other cells

661
00:39:54,830 --> 00:39:57,810
can uptake food and
other nutrients.

662
00:39:57,810 --> 00:39:59,900
But from the
standpoint of trying

663
00:39:59,900 --> 00:40:02,110
to find where the
nutrients are better,

664
00:40:02,110 --> 00:40:04,000
what you need to
be able to do is

665
00:40:04,000 --> 00:40:06,041
you need to be able to
measure the concentration.

666
00:40:08,190 --> 00:40:12,290
Now, the basic way that you
could, for example, measure

667
00:40:12,290 --> 00:40:16,280
a concentration of something,
the best that you could do,

668
00:40:16,280 --> 00:40:19,170
is if you're a
perfect absorber, you

669
00:40:19,170 --> 00:40:22,180
uptake molecules at this rate.

670
00:40:22,180 --> 00:40:25,720
And then you count the
number of molecules

671
00:40:25,720 --> 00:40:29,790
that you uptake
over some time t.

672
00:40:29,790 --> 00:40:34,170
Indeed, the number that
you uptake over some time t

673
00:40:34,170 --> 00:40:36,995
will just be given by this.

674
00:40:44,710 --> 00:40:47,405
What will be the distribution
of these numbers?

675
00:40:55,480 --> 00:40:58,590
I should just have somewhere
up on the side always

676
00:40:58,590 --> 00:41:00,410
a bunch of probability
distributions

677
00:41:00,410 --> 00:41:04,172
that we can vote on.

678
00:41:04,172 --> 00:41:06,380
But think about it for a
moment, and then we'll vote.

679
00:41:16,240 --> 00:41:18,230
All right, ready?

680
00:41:18,230 --> 00:41:22,630
Three, two, one.

681
00:41:22,630 --> 00:41:24,140
What is going to
be the probability

682
00:41:24,140 --> 00:41:28,290
distribution of the number
of these things absorbed

683
00:41:28,290 --> 00:41:29,057
over some time?

684
00:41:29,057 --> 00:41:30,890
Right, it's going to
be distributed Poisson.

685
00:41:35,500 --> 00:41:39,290
What is going to be the
distribution of times

686
00:41:39,290 --> 00:41:44,180
between successive
absorption events?

687
00:41:44,180 --> 00:41:44,680
Ready?

688
00:41:44,680 --> 00:41:46,784
Three, two, one.

689
00:41:50,170 --> 00:41:55,307
All right, yeah, that is going
to end up being exponential.

690
00:41:55,307 --> 00:41:56,390
And this is all confusing.

691
00:41:56,390 --> 00:41:57,740
So this is a Poisson process.

692
00:41:57,740 --> 00:42:01,510
These things are occurring at
random times, at some rate.

693
00:42:01,510 --> 00:42:04,284
And if you have such a
process that's random,

694
00:42:04,284 --> 00:42:06,200
and you ask, how many
events over some time t,

695
00:42:06,200 --> 00:42:10,230
that is the definition
of a Poisson.

696
00:42:10,230 --> 00:42:15,050
If this is a large number, then
the probability distribution

697
00:42:15,050 --> 00:42:16,751
is going to look
like something else.

698
00:42:16,751 --> 00:42:18,000
What is it going to look like?

699
00:42:18,000 --> 00:42:19,555
Ready, three, two, one.

700
00:42:23,990 --> 00:42:26,410
Most people are
trying to find the D,

701
00:42:26,410 --> 00:42:30,480
the central limit theorem there.

702
00:42:30,480 --> 00:42:35,120
What is going to determine the
resolution of that measurement

703
00:42:35,120 --> 00:42:36,150
that we make?

704
00:42:41,950 --> 00:42:44,920
So in some ways, what you
would like to know is, well,

705
00:42:44,920 --> 00:42:47,600
what's the standard
deviation in your measurement

706
00:42:47,600 --> 00:42:55,590
of the concentration divided
by the concentration?

707
00:42:55,590 --> 00:42:57,565
So this is maybe
a fractional error

708
00:42:57,565 --> 00:42:59,273
in your measurement
of the concentration.

709
00:43:04,840 --> 00:43:08,960
Do you guys
understand why this is

710
00:43:08,960 --> 00:43:15,590
the error in the measurement?

711
00:43:22,710 --> 00:43:25,720
Well, this actually will go
as the standard deviation

712
00:43:25,720 --> 00:43:29,150
in our measurement of the
number divided by the number

713
00:43:29,150 --> 00:43:30,136
that we measure.

714
00:43:32,841 --> 00:43:35,216
What's going to be the standard
deviation of this number?

715
00:43:38,565 --> 00:43:39,642
AUDIENCE: The scale--

716
00:43:39,642 --> 00:43:40,600
PROFESSOR: What's that?

717
00:43:40,600 --> 00:43:41,770
AUDIENCE: [INAUDIBLE]

718
00:43:41,770 --> 00:43:44,110
PROFESSOR: The mean,
yeah, so it's close to--

719
00:43:44,110 --> 00:43:45,820
AUDIENCE: [INAUDIBLE]

720
00:43:45,820 --> 00:43:49,780
PROFESSOR: Yeah, it's the
square root of the mean number.

721
00:43:49,780 --> 00:43:52,120
What that means is that
this thing is going to be

722
00:43:52,120 --> 00:43:56,510
the square root of a 4 pi cDaT.

723
00:43:56,510 --> 00:43:58,660
And this is just 4 pi cDaT.

724
00:44:18,720 --> 00:44:20,950
The number that
you measure here,

725
00:44:20,950 --> 00:44:22,950
the number that you uptake
over some time period

726
00:44:22,950 --> 00:44:24,908
t, that's going to be
distributed as a Poisson.

727
00:44:24,908 --> 00:44:26,340
And in a Poisson,
the variance is

728
00:44:26,340 --> 00:44:29,380
equal to the mean, which means
the standard deviation, which

729
00:44:29,380 --> 00:44:32,510
is the square root
of the variance.

730
00:44:32,510 --> 00:44:33,960
It's going to be this.

731
00:44:33,960 --> 00:44:34,758
Yes.

732
00:44:34,758 --> 00:44:36,924
AUDIENCE: I'm not understanding
the first step where

733
00:44:36,924 --> 00:44:39,080
you changed the c [INAUDIBLE].

734
00:44:39,080 --> 00:44:45,170
PROFESSOR: Right, so from
the standpoint of the cell,

735
00:44:45,170 --> 00:44:48,320
what the cell is doing is
it's counting the number

736
00:44:48,320 --> 00:44:49,820
of these things that have come.

737
00:44:49,820 --> 00:44:56,350
And it's true that from
that number of, say,

738
00:44:56,350 --> 00:44:59,770
molecules that are absorbed,
to get a concentration of units

739
00:44:59,770 --> 00:45:04,400
of number per cubic
micron and whatnot,

740
00:45:04,400 --> 00:45:06,630
the cell would need to
know kind of how big it

741
00:45:06,630 --> 00:45:09,140
is, what the diffusion
coefficient is, and so forth.

742
00:45:09,140 --> 00:45:11,770
But in terms of the
fractional error,

743
00:45:11,770 --> 00:45:15,700
those all end up disappearing.

744
00:45:15,700 --> 00:45:17,800
Because the fractional
error is telling us,

745
00:45:17,800 --> 00:45:20,500
well, if this time I measure it,
and I get 10, and next time I

746
00:45:20,500 --> 00:45:24,270
measure it I get 20,
then this is telling us

747
00:45:24,270 --> 00:45:25,710
about how much error.

748
00:45:25,710 --> 00:45:27,870
This is indeed just
the fractional error

749
00:45:27,870 --> 00:45:29,810
in our measurement
of the concentration,

750
00:45:29,810 --> 00:45:32,747
and irrespective
of whether the cell

751
00:45:32,747 --> 00:45:35,960
is able to estimate
the concentration

752
00:45:35,960 --> 00:45:38,010
units of a physical quantity.

753
00:45:38,010 --> 00:45:46,000
Because that sort of error
would scale together.

754
00:45:46,000 --> 00:45:48,517
I'm worried that I'm
not helping you, though.

755
00:45:48,517 --> 00:45:50,767
AUDIENCE: So you're kind of
saying because c and n are

756
00:45:50,767 --> 00:45:53,175
linearly related, any
proportionality factors

757
00:45:53,175 --> 00:45:54,369
would cancel out.

758
00:45:54,369 --> 00:45:55,035
PROFESSOR: Yeah.

759
00:45:57,974 --> 00:45:59,432
AUDIENCE: Would I
be right to think

760
00:45:59,432 --> 00:46:04,736
that since n is distributed
as a Poisson, so is c?

761
00:46:04,736 --> 00:46:07,990
PROFESSOR: Yeah,
our estimate of c

762
00:46:07,990 --> 00:46:10,960
would be-- because
ultimately, what the cell

763
00:46:10,960 --> 00:46:12,970
would do is it would
measure some number.

764
00:46:12,970 --> 00:46:14,595
And then it would
say, all right, well,

765
00:46:14,595 --> 00:46:16,270
I multiply by all these things.

766
00:46:16,270 --> 00:46:18,102
And then that
results in an error

767
00:46:18,102 --> 00:46:20,560
in c that is distributed in
the same way as the error in n.

768
00:46:20,560 --> 00:46:23,153
But they're kind of
proportional to each other.

769
00:46:23,153 --> 00:46:24,528
AUDIENCE: What
has me confused is

770
00:46:24,528 --> 00:46:28,610
I'm thinking that c is
Poisson distributed,

771
00:46:28,610 --> 00:46:33,010
and then our original ratio
that we want, sigma of c over c,

772
00:46:33,010 --> 00:46:41,446
should be 1 over square root of
c, because c is Poisson itself.

773
00:46:41,446 --> 00:46:43,340
But then that's different
from what we got.

774
00:46:43,340 --> 00:46:46,162
PROFESSOR: Sure, OK, I
think we have to be careful.

775
00:46:46,162 --> 00:46:48,850
So it's not that the c is
actually Poisson distributed.

776
00:46:48,850 --> 00:46:52,460
Because it's really that you
take a Poisson distribution

777
00:46:52,460 --> 00:46:55,160
and multiply it by something
to get an estimate of c.

778
00:46:55,160 --> 00:46:57,050
But if you take a
Poisson distribution

779
00:46:57,050 --> 00:47:00,459
and you multiply
the numbers by 10,

780
00:47:00,459 --> 00:47:02,250
then it's no longer a
Poisson distribution.

781
00:47:02,250 --> 00:47:05,130
Because the mean goes up
by 10, but the variance

782
00:47:05,130 --> 00:47:06,500
goes up by 100.

783
00:47:06,500 --> 00:47:08,500
So then the variance no
longer equals the mean,

784
00:47:08,500 --> 00:47:09,750
so it's not a Poisson anymore.

785
00:47:09,750 --> 00:47:13,130
So you can't just multiply a
number times a Poisson to get--

786
00:47:13,130 --> 00:47:14,630
AUDIENCE: All right, thanks.

787
00:47:17,187 --> 00:47:18,770
PROFESSOR: And this
expression kind of

788
00:47:18,770 --> 00:47:21,900
makes sense in that the
longer that the cell measures,

789
00:47:21,900 --> 00:47:24,975
the less error there is.

790
00:47:24,975 --> 00:47:26,560
If there's more
diffusion or higher

791
00:47:26,560 --> 00:47:29,790
concentration-- less error.

792
00:47:29,790 --> 00:47:32,820
And this is really in
some ways some estimate

793
00:47:32,820 --> 00:47:34,510
of the best the cell could do.

794
00:47:34,510 --> 00:47:38,490
And surprisingly, cells
can get close to this.

795
00:47:38,490 --> 00:47:42,790
Now, this is a measurement
of the concentration

796
00:47:42,790 --> 00:47:46,390
for a perfectly absorbing
sphere as a cell.

797
00:47:46,390 --> 00:47:50,960
Now, you might ask, well,
how would our error change

798
00:47:50,960 --> 00:47:53,190
if instead of
absorbing the molecules

799
00:47:53,190 --> 00:47:56,670
we simply had a detector?

800
00:47:56,670 --> 00:47:59,280
So they just asked, OK, each
time that a molecule bounces

801
00:47:59,280 --> 00:48:03,960
against my cell, I count one?

802
00:48:03,960 --> 00:48:05,870
Do you understand
the difference?

803
00:48:05,870 --> 00:48:07,335
So this is for a
perfect absorber.

804
00:48:13,940 --> 00:48:18,255
So what about a perfect monitor?

805
00:48:22,380 --> 00:48:25,550
And the question there
is, does a perfect monitor

806
00:48:25,550 --> 00:48:28,141
do better or worse than
a perfect detector?

807
00:48:31,830 --> 00:48:39,030
So we'll say, better,
worse, or no change.

808
00:48:39,030 --> 00:48:41,390
And it's of course
not possible for you

809
00:48:41,390 --> 00:48:42,730
to actually do this calculation.

810
00:48:42,730 --> 00:48:45,480
But it's useful to
imagine the situation

811
00:48:45,480 --> 00:48:54,040
and make your best guess--
better, worse, and same.

812
00:48:54,040 --> 00:48:56,080
I'll give you 5, 10
seconds to think about it.

813
00:48:56,080 --> 00:48:59,280
Because it's interesting
to imagine the situation.

814
00:49:14,620 --> 00:49:17,686
All right, let's see where we
are-- ready, three, two, one.

815
00:49:21,310 --> 00:49:25,425
Oh, we have a fair
range of answers.

816
00:49:25,425 --> 00:49:27,160
All right, I think
everybody agrees

817
00:49:27,160 --> 00:49:28,480
it's going to do something.

818
00:49:28,480 --> 00:49:30,340
Because it would be
kind of a coincidence

819
00:49:30,340 --> 00:49:31,860
if it didn't change things.

820
00:49:31,860 --> 00:49:35,810
Can I get somebody
to volunteer what

821
00:49:35,810 --> 00:49:37,260
their neighbor is thinking?

822
00:49:37,260 --> 00:49:39,468
I know you didn't actually
talk to your neighbor yet.

823
00:49:39,468 --> 00:49:41,012
But you can still invoke that.

824
00:49:47,500 --> 00:49:48,370
Yes.

825
00:49:48,370 --> 00:49:50,350
AUDIENCE: So I don't
know about this so much.

826
00:49:50,350 --> 00:49:53,132
But the first thing
I was thinking

827
00:49:53,132 --> 00:49:56,958
is that maybe this monitor
might be encountered on

828
00:49:56,958 --> 00:50:00,241
at a different rate
proportional to its area.

829
00:50:03,670 --> 00:50:07,364
It seemed like it wouldn't
change the concentration

830
00:50:07,364 --> 00:50:09,438
gradient, because it
wouldn't be subtracting any.

831
00:50:09,438 --> 00:50:11,188
And so then things
would bounce off of it,

832
00:50:11,188 --> 00:50:13,914
and that would be
proportional to its area.

833
00:50:13,914 --> 00:50:15,330
PROFESSOR: OK,
that's interesting.

834
00:50:15,330 --> 00:50:18,240
So you're arguing for
a different scaling.

835
00:50:18,240 --> 00:50:22,030
And then it would actually be
either-- depending on the area,

836
00:50:22,030 --> 00:50:23,440
then it would be different.

837
00:50:23,440 --> 00:50:25,544
AUDIENCE: Yeah, so that
seemed weird to me,

838
00:50:25,544 --> 00:50:28,002
which is why I wanted to just
check where other people were

839
00:50:28,002 --> 00:50:29,005
on that.

840
00:50:29,005 --> 00:50:30,380
PROFESSOR: Yeah,
no, that's fair.

841
00:50:30,380 --> 00:50:31,210
That's fair.

842
00:50:31,210 --> 00:50:32,970
In the end, the
scaling is the same.

843
00:50:32,970 --> 00:50:36,880
Although this is very
complicated, confusing.

844
00:50:36,880 --> 00:50:39,661
Can I hear somebody else
argue for one or the other?

845
00:50:39,661 --> 00:50:41,160
Because he actually
argued for both.

846
00:50:44,000 --> 00:50:48,764
AUDIENCE: I just thought if it's
not taking up the molecules,

847
00:50:48,764 --> 00:50:52,033
then it could bounce against
the same molecule several times.

848
00:50:52,033 --> 00:50:54,800
PROFESSOR: Yeah, and
therefore, so which

849
00:50:54,800 --> 00:50:56,110
one are you arguing for?

850
00:50:56,110 --> 00:50:56,949
AUDIENCE: For worse.

851
00:50:56,949 --> 00:50:58,240
PROFESSOR: Yeah, for worse, OK.

852
00:50:58,240 --> 00:51:00,517
Yeah, and indeed, it
does end up being worse.

853
00:51:00,517 --> 00:51:02,100
And it's actually
significantly worse.

854
00:51:02,100 --> 00:51:06,060
It's something like
10 times worse.

855
00:51:06,060 --> 00:51:08,660
And I think the intuitive
explanation is indeed

856
00:51:08,660 --> 00:51:11,530
what you just said, that a
perfect monitor, that's great.

857
00:51:11,530 --> 00:51:13,707
Except for the fact that
it's counting every time

858
00:51:13,707 --> 00:51:14,540
that something hits.

859
00:51:14,540 --> 00:51:17,640
But then it doesn't know if it
already counted that molecule

860
00:51:17,640 --> 00:51:18,230
or not.

861
00:51:18,230 --> 00:51:20,510
So then there's
extra uncertainty

862
00:51:20,510 --> 00:51:23,180
that results from that.

863
00:51:23,180 --> 00:51:25,810
Of course, ultimately, you have
to go and do the calculation

864
00:51:25,810 --> 00:51:27,935
if you want to be convinced
of something like this.

865
00:51:27,935 --> 00:51:31,200
But indeed, the perfect
absorber is the best

866
00:51:31,200 --> 00:51:32,430
that you can possibly do.

867
00:51:37,330 --> 00:51:41,260
AUDIENCE: Unless it tagged
the molecule [INAUDIBLE].

868
00:51:41,260 --> 00:51:42,270
PROFESSOR: That's right.

869
00:51:42,270 --> 00:51:44,150
Right, so that's equivalent
to being a perfect absorber,

870
00:51:44,150 --> 00:51:44,650
right?

871
00:51:44,650 --> 00:51:46,721
AUDIENCE: Oh, OK.

872
00:51:46,721 --> 00:51:48,729
Well, no, because then-- oh.

873
00:51:48,729 --> 00:51:51,270
PROFESSOR: Yeah, because these
are non-interacting molecules.

874
00:51:51,270 --> 00:51:53,270
So if you tag it green,
then all that you

875
00:51:53,270 --> 00:51:54,820
do is when the green
molecule comes back, you say,

876
00:51:54,820 --> 00:51:55,950
I'm going to ignore that.

877
00:51:55,950 --> 00:51:59,480
But then that's equivalent to
if you had just absorbed it.

878
00:51:59,480 --> 00:52:01,330
AUDIENCE: But there's
no local depletion.

879
00:52:01,330 --> 00:52:03,740
PROFESSOR: Yeah, there's
no local depletion.

880
00:52:03,740 --> 00:52:06,620
But because these are all
non-interacting particles,

881
00:52:06,620 --> 00:52:09,610
then it can't make a difference.

882
00:52:09,610 --> 00:52:12,290
And I agree that
this is confusing.

883
00:52:12,290 --> 00:52:15,450
But I'm pretty confident
what I said is true.

884
00:52:15,450 --> 00:52:17,200
Because since they're
all non-interacting,

885
00:52:17,200 --> 00:52:19,610
there's just no more
information there.

886
00:52:19,610 --> 00:52:20,345
Yeah.

887
00:52:20,345 --> 00:52:25,200
AUDIENCE: So what
exactly is the error?

888
00:52:25,200 --> 00:52:28,146
PROFESSOR: Right,
OK, this is asking,

889
00:52:28,146 --> 00:52:30,520
over some period of time, I
count the number of molecules

890
00:52:30,520 --> 00:52:34,720
that hit me or that I absorb,
and then I say, OK, well, I

891
00:52:34,720 --> 00:52:37,215
think that I counted 20.

892
00:52:37,215 --> 00:52:38,590
And the question
is really, well,

893
00:52:38,590 --> 00:52:44,100
if I did this again, how
close to 20 would I get?

894
00:52:44,100 --> 00:52:49,820
Would I get, again, 20 or 21, or
would I next time get 10 or 50?

895
00:52:49,820 --> 00:52:51,510
And that's the real question.

896
00:52:51,510 --> 00:52:54,640
How repeatable are
my measurements?

897
00:52:54,640 --> 00:52:58,900
So if I plot a histogram
of a bunch of numbers

898
00:52:58,900 --> 00:53:07,930
that I get over this time-- this
is the frequency that I observe

899
00:53:07,930 --> 00:53:11,090
a particular number of molecules
over some period of time t.

900
00:53:11,090 --> 00:53:12,410
I get some histogram.

901
00:53:12,410 --> 00:53:14,170
And this is telling us about
the width of that histogram

902
00:53:14,170 --> 00:53:15,044
relative to the mean.

903
00:53:21,810 --> 00:53:23,450
Now let's imagine
that you're the cell,

904
00:53:23,450 --> 00:53:25,900
and you calculated
the concentration

905
00:53:25,900 --> 00:53:27,840
in a really wonderful way.

906
00:53:27,840 --> 00:53:30,670
Now, the question
is, do you know where

907
00:53:30,670 --> 00:53:33,700
you should go to get more food?

908
00:53:33,700 --> 00:53:34,200
No, right?

909
00:53:34,200 --> 00:53:36,158
So you know the concentration
at your location.

910
00:53:36,158 --> 00:53:40,450
But what you need to know for
that purpose is a gradient.

911
00:53:40,450 --> 00:53:42,920
And I'm trying to
remember, in this paper,

912
00:53:42,920 --> 00:53:44,980
they maybe are
not very explicit.

913
00:53:44,980 --> 00:53:48,320
Why is it that a
bacterial cell might

914
00:53:48,320 --> 00:53:51,400
do this biased random walk
that's described in the paper

915
00:53:51,400 --> 00:53:53,970
instead of just
measure the gradient?

916
00:53:56,590 --> 00:53:57,574
Yeah.

917
00:53:57,574 --> 00:53:58,560
AUDIENCE: Because
cell size is so small.

918
00:53:58,560 --> 00:54:00,450
PROFESSOR: Yeah, so
bacteria are just small,

919
00:54:00,450 --> 00:54:03,380
I think this is the
argument that we often give.

920
00:54:03,380 --> 00:54:07,530
And that's just saying that if
you want to measure a gradient,

921
00:54:07,530 --> 00:54:10,930
then what you would do is, if
you have a sphere like this,

922
00:54:10,930 --> 00:54:13,590
you might say, OK,
I'm going to count

923
00:54:13,590 --> 00:54:16,007
the number of the molecules
that hit me over here,

924
00:54:16,007 --> 00:54:18,340
and I'm going to compare that
to the number of molecules

925
00:54:18,340 --> 00:54:20,870
that hit me over there.

926
00:54:20,870 --> 00:54:23,692
So you get two measurements
of concentration.

927
00:54:23,692 --> 00:54:26,650
But then in order to get a
gradient, what's relevant

928
00:54:26,650 --> 00:54:30,630
is you have to look
at this distance.

929
00:54:30,630 --> 00:54:33,740
And of course if you
have a bigger cell, then

930
00:54:33,740 --> 00:54:35,810
a given gradient shows
up as a larger difference

931
00:54:35,810 --> 00:54:37,970
in concentration, or
a large difference

932
00:54:37,970 --> 00:54:40,430
in the number of molecules
that are going to hit the cell.

933
00:54:40,430 --> 00:54:44,560
And does that scale linearly
or quadratically with a?

934
00:54:48,898 --> 00:54:49,870
AUDIENCE: Linearly?

935
00:54:49,870 --> 00:54:53,540
PROFESSOR: Linearly--
difference in concentration

936
00:54:53,540 --> 00:54:58,660
is going to be equal to the
size of the cell, maybe 2a,

937
00:54:58,660 --> 00:55:01,810
times dc/dx.

938
00:55:01,810 --> 00:55:04,450
So there's some gradient
that's out there.

939
00:55:04,450 --> 00:55:08,680
And then so we get a linear
amplification of our signal

940
00:55:08,680 --> 00:55:10,640
based on the size of the cell.

941
00:55:10,640 --> 00:55:12,860
So if you look at
many eukaryotic cells,

942
00:55:12,860 --> 00:55:15,930
for example Dicty and
others, they indeed

943
00:55:15,930 --> 00:55:19,210
often do this direct
gradient measurement.

944
00:55:19,210 --> 00:55:23,226
And then from that,
they decide where to go.

945
00:55:23,226 --> 00:55:24,600
Bacteria, on the
other hand, they

946
00:55:24,600 --> 00:55:28,370
do this other thing that's
described in the paper where

947
00:55:28,370 --> 00:55:29,800
they have a biased random walk.

948
00:55:29,800 --> 00:55:32,258
We're going to be talking much
more about this and the gene

949
00:55:32,258 --> 00:55:34,510
network that allows cells
to do it on Tuesday.

950
00:55:34,510 --> 00:55:36,380
But the basic idea
is that the bacteria,

951
00:55:36,380 --> 00:55:39,480
they're swimming in
some random direction.

952
00:55:39,480 --> 00:55:44,800
And then instead of directly
measuring the concentration

953
00:55:44,800 --> 00:55:48,860
as dc/dx, which you might
do as a eukaryotic cell,

954
00:55:48,860 --> 00:55:51,950
instead what they measure is
the change in concentration

955
00:55:51,950 --> 00:55:53,500
with respect to time.

956
00:55:53,500 --> 00:55:54,580
Because they are moving.

957
00:55:54,580 --> 00:55:55,880
So they ask, are
things getting better?

958
00:55:55,880 --> 00:55:57,060
Are they getting worse?

959
00:55:57,060 --> 00:55:59,350
If things are getting
better, you keep on going.

960
00:55:59,350 --> 00:56:00,891
If things are getting
worse, then you

961
00:56:00,891 --> 00:56:04,120
randomize your direction,
and you try again.

962
00:56:04,120 --> 00:56:07,460
And this implements
a biased random walk.

963
00:56:07,460 --> 00:56:10,070
We'll talk about the
decision making circuit

964
00:56:10,070 --> 00:56:12,730
in the cell that allows
it to do that on Tuesday.

965
00:56:12,730 --> 00:56:17,260
But that's the
basic notion there.

966
00:56:17,260 --> 00:56:19,350
And can someone
remind us, how long

967
00:56:19,350 --> 00:56:22,636
is a typical run for E. coli?

968
00:56:22,636 --> 00:56:25,620
AUDIENCE: [INAUDIBLE]

969
00:56:25,620 --> 00:56:28,810
PROFESSOR: Yeah, it's
maybe 30 microns.

970
00:56:28,810 --> 00:56:42,420
So there's this idea
that a cell might

971
00:56:42,420 --> 00:56:45,170
go like this for 30 microns.

972
00:56:45,170 --> 00:56:47,740
And it takes about a
second for it to do that.

973
00:56:47,740 --> 00:56:50,810
But then that's a run.

974
00:56:50,810 --> 00:56:53,730
But then it'll engage in
some random tumble that

975
00:56:53,730 --> 00:56:56,840
leads it to go some
other direction, and then

976
00:56:56,840 --> 00:56:59,730
another tumble, and then it kind
of goes a different direction,

977
00:56:59,730 --> 00:57:02,210
and so forth.

978
00:57:02,210 --> 00:57:04,345
So these are runs and tumbles.

979
00:57:10,650 --> 00:57:12,760
Now, the nice thing
here is that you

980
00:57:12,760 --> 00:57:16,850
can see that to measure this
concentration over here,

981
00:57:16,850 --> 00:57:19,880
the change of concentration
across the cell

982
00:57:19,880 --> 00:57:22,860
went as a, which might only be
1 or 2 microns, whereas here

983
00:57:22,860 --> 00:57:25,910
now the E. coli can measure
a change in a concentration

984
00:57:25,910 --> 00:57:28,170
over tens of microns.

985
00:57:28,170 --> 00:57:31,330
So it's in some ways like a
longer antenna that allows it

986
00:57:31,330 --> 00:57:32,650
to pick up the signal better.

987
00:57:36,910 --> 00:57:40,580
There's a comment
about what happens here

988
00:57:40,580 --> 00:57:45,790
in the sense that if you
imagine you have a cell,

989
00:57:45,790 --> 00:57:51,290
it's a few microns in size,
flagella is pushing it,

990
00:57:51,290 --> 00:57:56,140
it's going at 30
microns per second.

991
00:57:56,140 --> 00:57:59,420
The question is,
if at some time it

992
00:57:59,420 --> 00:58:05,537
stops spinning its flagellum,
how far will it coast?

993
00:58:16,540 --> 00:58:18,400
And this is after
stopping swimming.

994
00:58:26,300 --> 00:58:32,653
OK, so we'll do an approximate,
so 10 micron maybe.

995
00:58:51,040 --> 00:58:55,146
How many microns does a cell
coast after it stops swimming?

996
00:58:55,146 --> 00:58:57,020
AUDIENCE: This is
a bacterial cell?

997
00:58:57,020 --> 00:58:58,720
PROFESSOR: This is
a bacterial cell.

998
00:58:58,720 --> 00:59:04,060
This is E. coli, and this is,
we'll say, 3 microns in size.

999
00:59:09,830 --> 00:59:15,290
Ready-- three, two, one.

1000
00:59:15,290 --> 00:59:18,790
All right, OK, so everybody
says-- and is this actually the

1001
00:59:18,790 --> 00:59:19,619
right answer?

1002
00:59:19,619 --> 00:59:20,160
AUDIENCE: No.

1003
00:59:20,160 --> 00:59:22,450
PROFESSOR: No, it's not.

1004
00:59:22,450 --> 00:59:24,710
So this is the
closest, it's true.

1005
00:59:24,710 --> 00:59:26,450
But none of these
things are true.

1006
00:59:26,450 --> 00:59:28,546
So how far does it coast?

1007
00:59:28,546 --> 00:59:30,300
[INTERPOSING VOICES]

1008
00:59:30,300 --> 00:59:35,240
PROFESSOR: Right, yeah, so
it's some ridiculous-- I mean,

1009
00:59:35,240 --> 00:59:38,450
and this is really,
really weird.

1010
00:59:38,450 --> 00:59:41,660
This guy-- 30 microns a second.

1011
00:59:41,660 --> 00:59:44,980
So this is 10 times its
body length every second.

1012
00:59:44,980 --> 00:59:47,710
It's really shooting along.

1013
00:59:47,710 --> 00:59:51,610
But the moment that it stops
swimming, it stops moving.

1014
00:59:51,610 --> 00:59:54,090
So it's none of these things.

1015
00:59:54,090 --> 00:59:57,350
It was 1 angstrom or
less than an angstrom?

1016
00:59:57,350 --> 00:59:58,770
AUDIENCE: 0.1.

1017
00:59:58,770 --> 01:00:04,180
PROFESSOR: OK, 0.1
angstroms, which this is

1018
01:00:04,180 --> 01:00:05,688
10 to the minus five microns.

1019
01:00:11,430 --> 01:00:13,800
OK, so it's orders
of magnitude smaller

1020
01:00:13,800 --> 01:00:15,590
than you would ever imagine.

1021
01:00:15,590 --> 01:00:19,600
And that's because our intuition
comes from kind of our length

1022
01:00:19,600 --> 01:00:24,090
scales where if I were
swimming this fast,

1023
01:00:24,090 --> 01:00:27,070
I would keep on drifting
after I stopped swimming.

1024
01:00:27,070 --> 01:00:30,410
But that's not the world
that bacteria live in.

1025
01:00:30,410 --> 01:00:32,846
And that's because of
the low Reynolds number.

1026
01:00:38,830 --> 01:00:41,990
The Reynolds number is
much, much less than 1.

1027
01:00:41,990 --> 01:00:43,900
So this thing is the
dimensionless number

1028
01:00:43,900 --> 01:00:47,410
that quantifies the relative
importance of inertial forces

1029
01:00:47,410 --> 01:00:48,400
to viscous forces.

1030
01:00:51,200 --> 01:00:54,070
Now there was some
discussion of the Reynolds

1031
01:00:54,070 --> 01:00:56,830
number in your reading.

1032
01:00:56,830 --> 01:01:00,780
I maybe won't get into
too much detail about it.

1033
01:01:00,780 --> 01:01:02,890
But what I'll say
is that if anybody

1034
01:01:02,890 --> 01:01:05,370
wants to talk about a way of
thinking about the Reynolds

1035
01:01:05,370 --> 01:01:07,310
number, maybe you can
ask me after class,

1036
01:01:07,310 --> 01:01:12,670
and I can explain to you why
this weird expression can

1037
01:01:12,670 --> 01:01:18,160
be thought of as the ratio of
the inertial to viscous forces.

1038
01:01:18,160 --> 01:01:23,410
But I want to think
a little bit more

1039
01:01:23,410 --> 01:01:26,070
about the strategy that
these cells are following,

1040
01:01:26,070 --> 01:01:33,780
which is, why is it that a
run is of the length scale

1041
01:01:33,780 --> 01:01:35,300
that it is?

1042
01:01:35,300 --> 01:01:39,710
In particular, why
not go further?

1043
01:01:39,710 --> 01:01:44,110
We already said that its
ability to measure concentration

1044
01:01:44,110 --> 01:01:47,570
is somehow-- if you did
a direct measurement,

1045
01:01:47,570 --> 01:01:49,670
it would scale with
the size of the cell.

1046
01:01:49,670 --> 01:01:53,300
In this case, it sort of scales
like the length of the run.

1047
01:01:53,300 --> 01:01:56,680
So why not run for
10 times longer,

1048
01:01:56,680 --> 01:01:59,930
100 times longer, so you can
get a really good measurement

1049
01:01:59,930 --> 01:02:02,430
of a gradient?

1050
01:02:02,430 --> 01:02:04,430
AUDIENCE: Because if
you're going the wrong way,

1051
01:02:04,430 --> 01:02:06,384
then you'd go 10 times longer.

1052
01:02:06,384 --> 01:02:07,800
PROFESSOR: OK, so
it could be just

1053
01:02:07,800 --> 01:02:08,780
if you're going the wrong way.

1054
01:02:08,780 --> 01:02:10,154
Although then we
could maybe even

1055
01:02:10,154 --> 01:02:12,462
be just more sensitive in that.

1056
01:02:12,462 --> 01:02:14,170
Because what's striking
is that even when

1057
01:02:14,170 --> 01:02:17,460
it's going in the right
direction, it still tumbles.

1058
01:02:17,460 --> 01:02:20,170
The difference between
the run lengths

1059
01:02:20,170 --> 01:02:23,120
and the correct and incorrect
directions are kind of modest,

1060
01:02:23,120 --> 01:02:23,900
I guess.

1061
01:02:23,900 --> 01:02:26,290
So maybe we could come up
with a different strategy

1062
01:02:26,290 --> 01:02:27,748
where we say, all
right, well, if I

1063
01:02:27,748 --> 01:02:30,160
feel like things are getting
worse, I'll tumble quickly.

1064
01:02:30,160 --> 01:02:31,910
But if I feel like
things are going great,

1065
01:02:31,910 --> 01:02:35,667
maybe then I'll go in
the same direction,

1066
01:02:35,667 --> 01:02:37,500
and I'll measure that
gradient really finely

1067
01:02:37,500 --> 01:02:39,522
so I can get precisely
the right direction.

1068
01:02:39,522 --> 01:02:41,680
So I guess the
question is, is there

1069
01:02:41,680 --> 01:02:44,690
anything that's going
to limit how useful it

1070
01:02:44,690 --> 01:02:51,680
is to go swimming, even in a
good direction, in principle?

1071
01:02:51,680 --> 01:02:52,648
Yeah.

1072
01:02:52,648 --> 01:02:57,488
AUDIENCE: So it's so small that
collision with actual molecules

1073
01:02:57,488 --> 01:02:59,908
can randomize its
direction as it goes along,

1074
01:02:59,908 --> 01:03:02,640
and it sort of loses memory.

1075
01:03:02,640 --> 01:03:05,730
PROFESSOR: And this is a weird
thing, that not only does

1076
01:03:05,730 --> 01:03:10,180
diffusion act kind of in a
linear sense on the center

1077
01:03:10,180 --> 01:03:12,540
of mass position of
an object, but it also

1078
01:03:12,540 --> 01:03:17,180
acts rotationally, which means
that the cell and other objects

1079
01:03:17,180 --> 01:03:22,910
in liquid lose their
orientational order.

1080
01:03:22,910 --> 01:03:24,730
What that means is that
if this cell starts

1081
01:03:24,730 --> 01:03:26,484
swimming in one
direction, it's actually

1082
01:03:26,484 --> 01:03:27,900
the case that after
a few seconds,

1083
01:03:27,900 --> 01:03:30,200
it forgets where it was going.

1084
01:03:30,200 --> 01:03:33,080
So you can't actually
collect information

1085
01:03:33,080 --> 01:03:34,670
about the gradient
in that direction,

1086
01:03:34,670 --> 01:03:36,920
because you're actually going
in a different direction

1087
01:03:36,920 --> 01:03:38,940
from what you were
a few seconds ago.

1088
01:03:38,940 --> 01:03:41,900
This is actually a
pretty tough limitation

1089
01:03:41,900 --> 01:03:43,265
that these cells are facing.

1090
01:03:43,265 --> 01:03:45,390
On the one hand, they're
small, so it's really hard

1091
01:03:45,390 --> 01:03:46,360
to measure gradients.

1092
01:03:46,360 --> 01:03:48,580
On the other hand,
every few seconds,

1093
01:03:48,580 --> 01:03:50,520
they get turned around.

1094
01:03:50,520 --> 01:03:52,330
And if you imagine
trying to navigate

1095
01:03:52,330 --> 01:03:55,460
in that kind of situation,
it would be pretty hard.

1096
01:03:55,460 --> 01:03:59,440
Now, I want to make sure that
we understand how to figure out

1097
01:03:59,440 --> 01:04:04,815
what the timescale is that this
orientational order is lost in.

1098
01:04:04,815 --> 01:04:06,940
And again, we're going to
use dimensional analysis,

1099
01:04:06,940 --> 01:04:08,520
because I love it.

1100
01:04:08,520 --> 01:04:11,230
And I can tell that you
guys are big fans as well.

1101
01:04:14,930 --> 01:04:21,749
OK, so I will tell you
something that hopefully--

1102
01:04:21,749 --> 01:04:23,040
how much do I want to tell you?

1103
01:04:32,080 --> 01:04:35,580
What I'll tell you is
this, that the diffusion

1104
01:04:35,580 --> 01:04:42,220
coefficient for
angular fluctuations

1105
01:04:42,220 --> 01:04:46,380
is defined just the way it
is for diffusion coefficient

1106
01:04:46,380 --> 01:04:47,956
for the center of mass.

1107
01:04:51,000 --> 01:04:56,680
Given that, the question is,
if I have a sphere-- again,

1108
01:04:56,680 --> 01:05:06,560
size a-- now I want to know,
how does the typical correlation

1109
01:05:06,560 --> 01:05:17,900
time, the time to randomize,
randomize orientation,

1110
01:05:17,900 --> 01:05:20,610
how does it scale
with the radius?

1111
01:05:30,650 --> 01:05:34,180
So it could be that
this timescale tau

1112
01:05:34,180 --> 01:05:35,452
could go a to the 0.

1113
01:05:46,600 --> 01:05:48,121
And you should be
starting to think.

1114
01:06:36,346 --> 01:06:38,268
All right, do you
need more time?

1115
01:07:25,120 --> 01:07:27,480
All right, why don't we go
and see where the group is.

1116
01:07:30,030 --> 01:07:32,570
And it's OK if you've
not figured it out.

1117
01:07:32,570 --> 01:07:34,960
Make your best guess.

1118
01:07:34,960 --> 01:07:39,840
Ready-- three, two, one.

1119
01:07:39,840 --> 01:07:47,020
OK, so I'd say it might
be a slight majority B.

1120
01:07:47,020 --> 01:07:51,860
But then actually there are
fair numbers spread around.

1121
01:07:51,860 --> 01:07:53,960
Yeah, there's A, B,
C's, D's, and E's.

1122
01:07:57,645 --> 01:08:00,750
All right, maybe I'll
give you a minute

1123
01:08:00,750 --> 01:08:01,810
to talk to your neighbor.

1124
01:08:01,810 --> 01:08:06,030
And I'm just going to give
you an extra hint, which

1125
01:08:06,030 --> 01:08:09,310
is it's good to look at
the units of the diffusion

1126
01:08:09,310 --> 01:08:13,167
coefficient, which is why
I wrote this equation.

1127
01:08:18,229 --> 01:08:20,350
Yes, no?

1128
01:08:20,350 --> 01:08:21,147
You did that.

1129
01:08:21,147 --> 01:08:24,712
You're like, that's what we did.

1130
01:08:24,712 --> 01:08:26,170
Well, I'll give
you another minute.

1131
01:08:26,170 --> 01:08:29,290
Because I think that it's
important to work through this

1132
01:08:29,290 --> 01:08:31,568
at least some way mentally.

1133
01:08:31,568 --> 01:08:34,600
[INTERPOSING VOICES]

1134
01:09:57,154 --> 01:09:59,570
PROFESSOR: All right, why don't
we go ahead and reconvene.

1135
01:09:59,570 --> 01:10:04,910
I know that you may not have had
time to fully converge on this.

1136
01:10:04,910 --> 01:10:09,560
But it turns out that this
actually goes as a cubed.

1137
01:10:09,560 --> 01:10:11,480
AUDIENCE: Oh, no one
would have guessed that.

1138
01:10:11,480 --> 01:10:13,379
Come on.

1139
01:10:13,379 --> 01:10:16,340
PROFESSOR: He's so upset.

1140
01:10:16,340 --> 01:10:18,260
It's the first sphere, yeah.

1141
01:10:18,260 --> 01:10:19,610
It's a sphere.

1142
01:10:19,610 --> 01:10:22,296
AUDIENCE: So it's not a ball.

1143
01:10:22,296 --> 01:10:25,120
[LAUGHING]

1144
01:10:25,120 --> 01:10:27,783
PROFESSOR: I don't
know what to say.

1145
01:10:27,783 --> 01:10:28,908
AUDIENCE: Where's the mass?

1146
01:10:28,908 --> 01:10:31,830
Is the mass on the surface,
or is the mass in the middle?

1147
01:10:31,830 --> 01:10:34,270
Does it make a difference?

1148
01:10:34,270 --> 01:10:35,570
PROFESSOR: No, no, no.

1149
01:10:39,830 --> 01:10:42,480
We're in the low
Reynolds number regime.

1150
01:10:42,480 --> 01:10:46,460
So what that means
is that this is all

1151
01:10:46,460 --> 01:10:48,570
dominated by the interactions
between the object

1152
01:10:48,570 --> 01:10:50,030
and the fluid.

1153
01:10:50,030 --> 01:10:55,380
So it actually doesn't
enter into-- yeah,

1154
01:10:55,380 --> 01:10:58,630
mass is irrelevant.

1155
01:10:58,630 --> 01:11:00,230
So it doesn't scale.

1156
01:11:00,230 --> 01:11:02,430
That would've been a
fine question as well.

1157
01:11:02,430 --> 01:11:07,600
So a cubed, all
right, this is weird.

1158
01:11:07,600 --> 01:11:09,842
Now, of course you
can think about things

1159
01:11:09,842 --> 01:11:10,800
in many different ways.

1160
01:11:10,800 --> 01:11:13,240
But the thing that
you can see maybe

1161
01:11:13,240 --> 01:11:21,236
is that before, for the sort of
linear diffusion, so D linear

1162
01:11:21,236 --> 01:11:22,610
with respect to
time, you can see

1163
01:11:22,610 --> 01:11:25,240
that the difference in units
between these two diffusion

1164
01:11:25,240 --> 01:11:30,410
coefficients is a
factor of-- there's

1165
01:11:30,410 --> 01:11:34,210
a length squared difference
between those two.

1166
01:11:34,210 --> 01:11:38,070
And it's that length
squared that turns

1167
01:11:38,070 --> 01:11:42,720
this linear dependence of a
for normal linear diffusion

1168
01:11:42,720 --> 01:11:45,520
up to an a cubed.

1169
01:11:45,520 --> 01:11:47,550
Now, of course there
are other things

1170
01:11:47,550 --> 01:11:49,290
we can write down
that may be useful.

1171
01:11:49,290 --> 01:11:52,530
So remember this
D linear, Einstein

1172
01:11:52,530 --> 01:11:56,960
told us this is kT
over gamma linear.

1173
01:11:56,960 --> 01:11:58,770
So this is if we try to move it.

1174
01:11:58,770 --> 01:12:01,460
This tells us how
hard we have to push

1175
01:12:01,460 --> 01:12:02,770
to get something to move.

1176
01:12:02,770 --> 01:12:08,600
So this is if you apply force,
then gamma linear tells you

1177
01:12:08,600 --> 01:12:15,140
how fast it will move in this
low Reynolds number regime.

1178
01:12:15,140 --> 01:12:18,220
Now, there's a similar set of
things that we can say here.

1179
01:12:18,220 --> 01:12:20,675
I'm going to erase this.

1180
01:12:20,675 --> 01:12:22,050
So we can still
say it's actually

1181
01:12:22,050 --> 01:12:23,810
Einstein is just always right.

1182
01:12:23,810 --> 01:12:27,320
So the D rotational
again is going to be kT.

1183
01:12:27,320 --> 01:12:29,860
Now this is a gamma rotational.

1184
01:12:29,860 --> 01:12:33,059
But instead of thinking about
forces, now this is torques.

1185
01:12:33,059 --> 01:12:35,350
So what's relevant is that
if you apply-- oh, I already

1186
01:12:35,350 --> 01:12:38,470
used a tau.

1187
01:12:38,470 --> 01:12:40,640
OK, I'll just say, torque.

1188
01:12:40,640 --> 01:12:44,540
So if you apply torque, on,
say, a sphere, or this object,

1189
01:12:44,540 --> 01:12:48,860
then this will cause
an angular velocity,

1190
01:12:48,860 --> 01:12:52,663
so gamma rotational times
some omega around it.

1191
01:12:57,460 --> 01:13:06,740
Now, for concreteness,
if you have a sphere,

1192
01:13:06,740 --> 01:13:13,050
Stokes' drag tells us
this is 6 pi eta a times

1193
01:13:13,050 --> 01:13:16,170
v. So this thing tells
us how hard we have

1194
01:13:16,170 --> 01:13:18,670
to push an object in
order to make it move

1195
01:13:18,670 --> 01:13:20,046
some velocity if it's a sphere.

1196
01:13:26,210 --> 01:13:30,360
Whereas the gamma
rotational here,

1197
01:13:30,360 --> 01:13:37,430
this ends up being 8
pi eta a cubed omega.

1198
01:13:37,430 --> 01:13:39,925
So the gamma rotational
ends up being that.

1199
01:13:44,370 --> 01:13:50,000
So you can see that in terms
of rotational diffusion,

1200
01:13:50,000 --> 01:13:56,340
rotational order, objects
scale-- what do I want to say?

1201
01:13:56,340 --> 01:14:00,090
The scaling is slower than
you would expect somehow.

1202
01:14:00,090 --> 01:14:02,410
Well, the scaling here is
larger than you would expect,

1203
01:14:02,410 --> 01:14:04,826
whereas the scaling here is
smaller than you would expect.

1204
01:14:09,190 --> 01:14:10,219
Yes.

1205
01:14:10,219 --> 01:14:11,760
AUDIENCE: Would this
be any different

1206
01:14:11,760 --> 01:14:13,530
for a bar-shaped bacteria?

1207
01:14:13,530 --> 01:14:15,600
PROFESSOR: Yes, it's
always different,

1208
01:14:15,600 --> 01:14:17,790
but not by very much.

1209
01:14:17,790 --> 01:14:21,280
So for example, this
thing, the Stokes' drag,

1210
01:14:21,280 --> 01:14:23,190
is only true for a sphere.

1211
01:14:23,190 --> 01:14:25,980
But this linear
dependence on a is

1212
01:14:25,980 --> 01:14:27,230
true for any of these objects.

1213
01:14:27,230 --> 01:14:29,590
So it's really that a is
some typical length scale.

1214
01:14:29,590 --> 01:14:31,560
And in particular, it's
actually the larger

1215
01:14:31,560 --> 01:14:33,890
of the two length scales.

1216
01:14:33,890 --> 01:14:38,457
So for example, if you take
a long kind of cylinder,

1217
01:14:38,457 --> 01:14:49,902
pipe, now you can ask, well,
now it's not isotropic anymore.

1218
01:14:49,902 --> 01:14:52,110
So there's going to actually
be two different gammas.

1219
01:14:52,110 --> 01:14:55,520
There's going to be a gamma
that we might call parallel

1220
01:14:55,520 --> 01:14:57,540
if you want to push
it in this direction,

1221
01:14:57,540 --> 01:14:59,146
and a gamma
perpendicular if you want

1222
01:14:59,146 --> 01:15:01,570
to push it in this direction.

1223
01:15:01,570 --> 01:15:05,130
What's fascinating is that if
this is something that's, say,

1224
01:15:05,130 --> 01:15:09,480
a nanometer in size
here but a micron here,

1225
01:15:09,480 --> 01:15:12,280
so an aspect ratio
of 1,000 to 1.

1226
01:15:12,280 --> 01:15:15,400
Still the difference between
gamma parallel and gamma

1227
01:15:15,400 --> 01:15:19,700
perpendicular is still
just a factor of two or so.

1228
01:15:19,700 --> 01:15:22,070
Because in both cases, it's
dominated by the longest

1229
01:15:22,070 --> 01:15:23,360
linear dimension.

1230
01:15:23,360 --> 01:15:30,210
So what's relevant is
this big thing, length.

1231
01:15:30,210 --> 01:15:34,750
So this is, again, an aspect
of the surprising nature

1232
01:15:34,750 --> 01:15:36,890
of the low Reynolds
number regime.

1233
01:15:36,890 --> 01:15:42,070
This is not true if you were a
skydiver or something, right?

1234
01:15:42,070 --> 01:15:45,120
But the reason is that this
longest dimension kind of

1235
01:15:45,120 --> 01:15:48,470
tells you how much water you're
going to be pulling with you.

1236
01:15:48,470 --> 01:15:50,950
What's relevant is not
the cross sectional area

1237
01:15:50,950 --> 01:15:52,140
that you're kind of pushing.

1238
01:15:52,140 --> 01:15:54,510
But rather it's
telling you how much

1239
01:15:54,510 --> 01:15:58,138
fluid are you kind of disturbing
along the whole length.

1240
01:16:04,536 --> 01:16:07,430
Are there any questions
about what I mean by this?

1241
01:16:11,960 --> 01:16:12,659
Yes.

1242
01:16:12,659 --> 01:16:16,491
AUDIENCE: I was wondering if
[INAUDIBLE] because the flow

1243
01:16:16,491 --> 01:16:19,844
is one different time.

1244
01:16:19,844 --> 01:16:22,240
So you're asking
about the time--

1245
01:16:22,240 --> 01:16:24,910
PROFESSOR: OK, so you're
wondering how I actually

1246
01:16:24,910 --> 01:16:26,246
get to a time from this.

1247
01:16:26,246 --> 01:16:28,120
Yeah, OK, sorry, I should
have finished this.

1248
01:16:28,120 --> 01:16:31,180
So let me just show
you what I meant.

1249
01:16:31,180 --> 01:16:32,980
So one way to
think about this is

1250
01:16:32,980 --> 01:16:37,950
we know that this
is where we start.

1251
01:16:37,950 --> 01:16:40,560
And what you could say
is, well, the typical time

1252
01:16:40,560 --> 01:16:43,290
to randomize, that's when
we've gone to enough time

1253
01:16:43,290 --> 01:16:47,420
so that change of theta squared
is, say, of order radian

1254
01:16:47,420 --> 01:16:48,390
squared.

1255
01:16:48,390 --> 01:16:49,910
So it's enough
that that's sort of

1256
01:16:49,910 --> 01:16:51,867
a characteristic reorientation.

1257
01:16:51,867 --> 01:16:53,450
So you just say, OK,
well, we're going

1258
01:16:53,450 --> 01:16:55,060
to set this equal to around 1.

1259
01:16:55,060 --> 01:17:01,095
So 2Dr times this reorientation
time tau will be around 1.

1260
01:17:03,710 --> 01:17:07,070
So that gives us that
the reorientation time

1261
01:17:07,070 --> 01:17:11,600
is going to be around-- so we
have 1/2, and then D rotation.

1262
01:17:11,600 --> 01:17:21,480
Now we have 8 pi
eta a cubed over kT.

1263
01:17:21,480 --> 01:17:26,880
Is that kT, yeah?

1264
01:17:26,880 --> 01:17:29,836
And actually in units-- so I
grew up in the single molecule

1265
01:17:29,836 --> 01:17:31,960
world where everything was
in units of piconewtons,

1266
01:17:31,960 --> 01:17:34,290
nanometers, and seconds.

1267
01:17:34,290 --> 01:17:36,490
And in those units,
we could actually

1268
01:17:36,490 --> 01:17:39,890
write, OK, we have a 4 pi.

1269
01:17:39,890 --> 01:17:43,160
kT is actually 4.1
piconewton nanometers.

1270
01:17:43,160 --> 01:17:45,160
So it's around 4 in these units.

1271
01:17:45,160 --> 01:17:49,520
And then eta is actually 10
to the minus 9 in those units.

1272
01:17:49,520 --> 01:17:54,200
a, for a bacteria,
is around a micron.

1273
01:17:54,200 --> 01:17:55,100
So it's 1,000.

1274
01:17:55,100 --> 01:17:57,490
But this is piconewton
nanometer second units.

1275
01:17:57,490 --> 01:18:01,680
So we have to do
10 to the 3 cubed.

1276
01:18:01,680 --> 01:18:05,240
Because a micron is 10
to the 3 nanometers.

1277
01:18:05,240 --> 01:18:07,930
And conveniently, 10 to the 3
to the third, 10 to the minus 9,

1278
01:18:07,930 --> 01:18:10,250
we can just cross these out.

1279
01:18:10,250 --> 01:18:11,975
We can cross the 4's out.

1280
01:18:11,975 --> 01:18:13,912
All right, so we get pi seconds.

1281
01:18:17,360 --> 01:18:20,870
Should we take the pi seriously?

1282
01:18:20,870 --> 01:18:22,310
No.

1283
01:18:22,310 --> 01:18:26,482
But the point is that it's
of order 1 to 10 seconds.

1284
01:18:26,482 --> 01:18:28,690
This is what you get from
such a back of the envelope

1285
01:18:28,690 --> 01:18:29,330
calculation.

1286
01:18:29,330 --> 01:18:31,121
But this is saying that
from the standpoint

1287
01:18:31,121 --> 01:18:33,714
of the bacterial
cell, it starts going.

1288
01:18:33,714 --> 01:18:35,880
And it forgets which direction
it started going over

1289
01:18:35,880 --> 01:18:38,700
the timescale of a few seconds.

1290
01:18:38,700 --> 01:18:40,332
So it's not worth
it for the cell

1291
01:18:40,332 --> 01:18:42,290
to actually have a run
that's longer than that.

1292
01:18:42,290 --> 01:18:44,495
Because it will have
lost its way anyways.

1293
01:18:50,070 --> 01:18:54,740
All right, what we're
going to do on Tuesday

1294
01:18:54,740 --> 01:18:56,760
is we're going to try
to understand something

1295
01:18:56,760 --> 01:19:00,440
about how bacteria use this
combination of runs and tumbles

1296
01:19:00,440 --> 01:19:03,040
in order to perform
these exquisitely

1297
01:19:03,040 --> 01:19:06,807
sensitive kind of travels
to find food sources.

1298
01:19:06,807 --> 01:19:09,390
If you have any questions about
anything that we talked about,

1299
01:19:09,390 --> 01:19:11,540
please come on up.