DNA cloning and recombinant DNA

WEBVTT
Kind: captions
Language: en

00:00:00.492 --> 00:00:03.511
- [Voiceover] Let's talk a
little bit about DNA Cloning.

00:00:03.511 --> 00:00:05.819
Which is all about making identical copies

00:00:05.819 --> 00:00:07.822
of a piece of DNA.

00:00:07.822 --> 00:00:09.999
And usually it's a piece of DNA

00:00:09.999 --> 00:00:11.798
that codes for something we care about,

00:00:11.798 --> 00:00:14.918
it is a gene that will
express itself as a protein

00:00:14.918 --> 00:00:18.258
that we think is useful in some way.

00:00:18.258 --> 00:00:19.975
Now you might have also
heard the term cloning

00:00:19.975 --> 00:00:23.225
in terms of the Clone Wars in Star Wars

00:00:23.225 --> 00:00:25.209
or Dolly the sheep

00:00:25.209 --> 00:00:26.931
and that is a related idea.

00:00:26.931 --> 00:00:30.523
If you're cloning an animal
or an organism, like a sheep,

00:00:30.523 --> 00:00:32.865
well then you are creating an animal

00:00:32.865 --> 00:00:36.814
that has the exact genetic
material as the original animal.

00:00:36.814 --> 00:00:39.284
But when we talk about
cloning and DNA cloning

00:00:39.284 --> 00:00:41.853
we're talking about something
a little bit simpler.

00:00:41.853 --> 00:00:44.778
Although, as we'll see, it's
still quite fascinating.

00:00:44.778 --> 00:00:48.550
It's identical copies of a piece of DNA.

00:00:48.550 --> 00:00:50.824
So how do we do that?

00:00:50.824 --> 00:00:54.433
Well let's say that this is a
strand of DNA right over here

00:00:54.433 --> 00:00:55.945
and I'm just drawing it as a long,

00:00:55.945 --> 00:00:57.960
but this is a double-stranded,

00:00:57.960 --> 00:00:58.793
and I'll just write it down,

00:00:58.793 --> 00:01:00.253
this is double stranded.

00:01:00.253 --> 00:01:01.681
I don't want to have to take the trouble

00:01:01.681 --> 00:01:03.357
of keep drawing the multiple strands.

00:01:03.357 --> 00:01:04.656
Actually, let me just draw,

00:01:04.656 --> 00:01:07.371
let me just try to draw the two strands

00:01:07.371 --> 00:01:08.883
just so we remind ourselves.

00:01:08.883 --> 00:01:10.021
So there we go.

00:01:10.021 --> 00:01:12.767
This is the double-stranded DNA

00:01:12.767 --> 00:01:16.100
and let's say that this part of this DNA

00:01:17.676 --> 00:01:20.426
has a gene that we want to clone.

00:01:21.561 --> 00:01:24.649
We wanna make copies of
this right over here.

00:01:24.649 --> 00:01:26.066
So gene to clone.

00:01:27.592 --> 00:01:28.759
Gene to clone.

00:01:29.964 --> 00:01:31.557
Well, the first thing we wanna do

00:01:31.557 --> 00:01:34.045
is we wanna cut this gene out some how.

00:01:34.045 --> 00:01:35.393
And the way we do that

00:01:35.393 --> 00:01:37.523
is using restriction enzymes.

00:01:37.523 --> 00:01:39.424
And there's a bunch of
restriction enzymes,

00:01:39.424 --> 00:01:40.919
and I personally find it fascinating

00:01:40.919 --> 00:01:43.748
that we as a civilization
have gotten to the point

00:01:43.748 --> 00:01:46.657
that we can find and
identify these enzymes

00:01:46.657 --> 00:01:49.908
and we know at what points
of DNA that they can cut.

00:01:49.908 --> 00:01:51.615
They recognize specific sequences

00:01:51.615 --> 00:01:52.834
and then we can figure out

00:01:52.834 --> 00:01:55.776
well which restriction
enzyme should we use

00:01:55.776 --> 00:01:57.855
to cut out different pieces of DNA,

00:01:57.855 --> 00:02:00.879
but we have gotten to that
point as a civilization.

00:02:00.879 --> 00:02:02.862
So we use restriction enzymes.

00:02:02.862 --> 00:02:04.847
We might use one restriction enzyme,

00:02:04.847 --> 00:02:06.894
Let me use a different color here,

00:02:06.894 --> 00:02:10.073
that latches on right over here

00:02:10.073 --> 00:02:12.960
and identifies the genetic
sequence right over here

00:02:12.960 --> 00:02:15.918
and cuts right in the right place.

00:02:15.918 --> 00:02:18.616
So that might be a restriction
enzyme right over there

00:02:18.616 --> 00:02:21.103
and then you might use
another restriction enzyme

00:02:21.103 --> 00:02:24.467
that identifies with the
sequence at the other side

00:02:24.467 --> 00:02:26.288
that we wanna cut.

00:02:26.288 --> 00:02:28.238
So let me label these.

00:02:28.238 --> 00:02:31.116
These, those things right over there

00:02:31.116 --> 00:02:34.091
those are restriction enzymes.

00:02:34.091 --> 00:02:35.758
Restriction enzymes.

00:02:39.161 --> 00:02:40.640
And so now you would have,

00:02:40.640 --> 00:02:42.494
after you applied the restriction enzymes,

00:02:42.494 --> 00:02:45.549
you will have just that gene.

00:02:45.549 --> 00:02:48.378
You might have a little bit
left over on either side

00:02:48.378 --> 00:02:50.621
but essentially you have cut out the gene.

00:02:50.621 --> 00:02:53.188
You've used the restriction
enzymes to cut out your gene

00:02:53.188 --> 00:02:56.050
and then what you wanna
do is you wanna paste it

00:02:56.050 --> 00:02:57.871
into what we'll call a plasmid.

00:02:57.871 --> 00:03:01.538
And a plasmid is a piece
of genetic material

00:03:02.990 --> 00:03:05.070
that sits outside of chromosomes

00:03:05.070 --> 00:03:06.516
but it can reproduce along,

00:03:06.516 --> 00:03:08.873
or I guess we can say can replicate

00:03:08.873 --> 00:03:12.125
along with the machinery of the,

00:03:12.125 --> 00:03:14.107
the genetic machinery of the organism.

00:03:14.107 --> 00:03:15.196
Or it can even express itself

00:03:15.196 --> 00:03:17.520
just like the genes of the organism

00:03:17.520 --> 00:03:20.349
that are in the chromosomes,
express themselves.

00:03:20.349 --> 00:03:22.787
So then so this is where we cut,

00:03:22.787 --> 00:03:23.713
let me write this,

00:03:23.713 --> 00:03:25.296
we cut out the gene

00:03:26.607 --> 00:03:28.330
and then we wanna paste it

00:03:28.330 --> 00:03:31.497
then we wanna paste it into a plasmid.

00:03:32.345 --> 00:03:35.839
And plasmids tend to be circular DNA

00:03:35.839 --> 00:03:38.326
so we will paste it into a plasmid.

00:03:38.326 --> 00:03:39.757
And in order for them to fit

00:03:39.757 --> 00:03:43.852
there's oftentimes these
overhangs over here.

00:03:43.852 --> 00:03:45.467
So you might have an overhang over there,

00:03:45.467 --> 00:03:47.628
you might have an overhang over there.

00:03:47.628 --> 00:03:50.424
And so the plasmid that we're placing in

00:03:50.424 --> 00:03:54.033
might have complimentary base
pairs over the overhangs,

00:03:54.033 --> 00:03:56.081
which will allow it easier,

00:03:56.081 --> 00:03:59.770
it will become easier for
them to react with each other

00:03:59.770 --> 00:04:02.825
if they have these overhangs.

00:04:02.825 --> 00:04:05.963
So let me, we're pasting
it into the plasmid.

00:04:05.963 --> 00:04:06.922
And this is amazing

00:04:06.922 --> 00:04:08.368
because obviously DNA,

00:04:08.368 --> 00:04:10.254
this isn't stuff that we can, you know,

00:04:10.254 --> 00:04:11.457
manipulate with our hands

00:04:11.457 --> 00:04:14.723
the way that we would copy
and paste things with tape.

00:04:14.723 --> 00:04:16.397
You're making these solutions

00:04:16.397 --> 00:04:18.219
and you're applying the
restriction enzymes.

00:04:18.219 --> 00:04:20.851
The restriction enzymes are just in mass

00:04:20.851 --> 00:04:21.924
cutting these things.

00:04:21.924 --> 00:04:23.517
They're bumping in just the right way

00:04:23.517 --> 00:04:25.338
to cause this reaction to happen

00:04:25.338 --> 00:04:26.996
then you're taking those genes

00:04:26.996 --> 00:04:28.782
and you're putting them with the plasmids

00:04:28.782 --> 00:04:31.237
that happen to have the
right sequences at their ends

00:04:31.237 --> 00:04:32.926
so that they match up

00:04:32.926 --> 00:04:36.502
and then you also put in
a bunch of DNA ligase.

00:04:36.502 --> 00:04:37.419
DNA ligase,

00:04:40.053 --> 00:04:42.490
to connect the backbones

00:04:42.490 --> 00:04:44.066
right over here.

00:04:44.066 --> 00:04:47.562
And we also saw DNA ligase
when we studied replication.

00:04:47.562 --> 00:04:49.395
So that is DNA ligase,

00:04:51.170 --> 00:04:53.493
which you can think of
it as helping to do,

00:04:53.493 --> 00:04:55.510
helping to do the pasting.

00:04:55.510 --> 00:04:58.321
And so now we have this plasmid

00:04:58.321 --> 00:05:01.003
and we want to insert it into an organism

00:05:01.003 --> 00:05:03.358
that can make the copies for us.

00:05:03.358 --> 00:05:05.456
And an organism that's typically used,

00:05:05.456 --> 00:05:10.267
or a type of organism is bacteria
and E. coli in particular,

00:05:10.267 --> 00:05:11.584
and so what we could do is,

00:05:11.584 --> 00:05:15.501
we could, let's say that we have a bunch,

00:05:15.501 --> 00:05:19.418
let's say you have a vial right over here.

00:05:19.418 --> 00:05:22.897
You have a vial and it
has a solution in it

00:05:22.897 --> 00:05:25.705
with a bunch of E. Coli.

00:05:25.705 --> 00:05:26.550
A bunch of E. coli.

00:05:26.550 --> 00:05:28.153
And you actually wouldn't
be able to see it visually

00:05:28.153 --> 00:05:31.615
but there is E. coli in that solution.

00:05:31.615 --> 00:05:34.248
And then you would put your plasmids,

00:05:34.248 --> 00:05:36.296
which would be even harder to see,

00:05:36.296 --> 00:05:40.524
in that solution and
somehow we want the E. coli,

00:05:40.524 --> 00:05:43.302
we want the bacteria
to take up the plasmid.

00:05:43.302 --> 00:05:45.285
And the technique that's typically done

00:05:45.285 --> 00:05:47.684
is giving some type of
a shock to the system

00:05:47.684 --> 00:05:51.033
that makes the bacteria
take up the plasmids.

00:05:51.033 --> 00:05:53.196
And the typical shock is a heat shock.

00:05:53.196 --> 00:05:56.464
And this isn't fully understood

00:05:56.464 --> 00:05:58.694
how the heat shock works

00:05:58.694 --> 00:05:59.816
but it does

00:05:59.816 --> 00:06:02.318
and so people have been
using this for some time.

00:06:02.318 --> 00:06:04.887
So if you have a bacteria,

00:06:04.887 --> 00:06:07.781
you have a bacteria right over here,

00:06:07.781 --> 00:06:09.781
it has its existing DNA,

00:06:11.243 --> 00:06:15.273
so this is its existing genetic material

00:06:15.273 --> 00:06:17.907
right over there, let me label this.

00:06:17.907 --> 00:06:19.657
This is the bacteria.

00:06:20.735 --> 00:06:24.441
You put it in the presence of our plasmids

00:06:24.441 --> 00:06:27.121
so you put it in the
presence of our plasmid

00:06:27.121 --> 00:06:29.136
and you apply the heat shock

00:06:29.136 --> 00:06:33.086
and some of that bacteria is
going to take in the plasmid.

00:06:33.086 --> 00:06:35.865
It's going to take in the plasmid.

00:06:35.865 --> 00:06:39.615
And so just like that,
it's going to take it,

00:06:40.513 --> 00:06:42.254
it's going to take it in.

00:06:42.254 --> 00:06:43.928
And so what you then do

00:06:43.928 --> 00:06:47.958
is you place the solution
that has your bacteria,

00:06:47.958 --> 00:06:50.509
some of which will have
taken up the plasmid,

00:06:50.509 --> 00:06:51.372
and you put it

00:06:51.372 --> 00:06:54.378
and then you try to grow
the bacteria on a plate.

00:06:54.378 --> 00:06:55.428
So let me draw that.

00:06:55.428 --> 00:06:56.678
So let me draw,

00:06:57.691 --> 00:06:59.608
so here we have a plate

00:07:02.030 --> 00:07:04.792
to grow our bacteria on,

00:07:04.792 --> 00:07:08.548
and it has nutrients right over here

00:07:08.548 --> 00:07:11.377
that bacteria can grow on.

00:07:11.377 --> 00:07:13.587
It has nutrients.

00:07:13.587 --> 00:07:15.780
It has nutrients, and so you could say,

00:07:15.780 --> 00:07:16.837
okay well put this here

00:07:16.837 --> 00:07:19.585
and then a bunch of
bacteria will just grow.

00:07:19.585 --> 00:07:21.144
So you would see things like this,

00:07:21.144 --> 00:07:24.347
which would be many, many,
many cells of bacteria,

00:07:24.347 --> 00:07:25.825
there would be colonies of bacteria.

00:07:25.825 --> 00:07:26.980
You could just let them grow

00:07:26.980 --> 00:07:28.378
but there's a problem here.

00:07:28.378 --> 00:07:30.702
Because I mentioned some of the bacteria

00:07:30.702 --> 00:07:32.181
will take up the plasmids

00:07:32.181 --> 00:07:33.384
and some won't.

00:07:33.384 --> 00:07:34.570
And so you won't know,

00:07:34.570 --> 00:07:36.553
hey when this bacteria,

00:07:36.553 --> 00:07:39.804
when it keeps replicating
it might form one of these,

00:07:39.804 --> 00:07:41.982
it might form one of these colonies.

00:07:41.982 --> 00:07:43.803
So this is a colony that you like.

00:07:43.803 --> 00:07:45.672
So this one is a good colony,

00:07:45.672 --> 00:07:47.020
put a checkmark there.

00:07:47.020 --> 00:07:51.360
But maybe this colony is
formed by an initial bacteria

00:07:51.360 --> 00:07:54.563
or a set of bacteria that
did not take up the plasmid

00:07:54.563 --> 00:07:57.489
so it won't contain the
actual gene in question.

00:07:57.489 --> 00:07:59.260
So you don't want that one.

00:07:59.260 --> 00:08:03.097
So how do you select for the bacteria

00:08:03.097 --> 00:08:06.396
that actually took up the plasmid?

00:08:06.396 --> 00:08:07.874
Well, what you do is

00:08:07.874 --> 00:08:10.492
besides the gene that you care about

00:08:10.492 --> 00:08:12.621
that you want to make copies of,

00:08:12.621 --> 00:08:14.371
you also place a gene

00:08:16.630 --> 00:08:18.713
for antibiotic resistance

00:08:19.816 --> 00:08:21.490
in your plasmid.

00:08:21.490 --> 00:08:24.611
So now you have a gene for
antibiotic resistance here,

00:08:24.611 --> 00:08:26.820
and so only the bacteria,

00:08:26.820 --> 00:08:28.854
and I think it's amazing
that we as humanity

00:08:28.854 --> 00:08:30.625
are able to do these types of things,

00:08:30.625 --> 00:08:33.762
but now only the bacteria
that have taken up the plasmid

00:08:33.762 --> 00:08:35.923
will have that antibiotic resistance.

00:08:35.923 --> 00:08:38.540
And so what you do is in your nutrients

00:08:38.540 --> 00:08:41.904
you grew nutrients plus antibiotics,

00:08:41.904 --> 00:08:43.807
plus an antibiotic.

00:08:43.807 --> 00:08:47.756
Antibiotic, and so this one will survive

00:08:47.756 --> 00:08:49.447
'cause it has that resistance.

00:08:49.447 --> 00:08:52.632
It has that gene that allows
it to not be susceptible

00:08:52.632 --> 00:08:53.884
to the antibiotics.

00:08:53.884 --> 00:08:56.387
But these are not going to survive.

00:08:56.387 --> 00:08:57.671
They're not even going to happen.

00:08:57.671 --> 00:08:58.728
They're not even going to grow

00:08:58.728 --> 00:09:00.775
because there's antibiotics mixed in

00:09:00.775 --> 00:09:02.271
with those nutrients.

00:09:02.271 --> 00:09:03.994
And so this is a pretty cool thing.

00:09:03.994 --> 00:09:06.205
You started with the gene
that you cared about,

00:09:06.205 --> 00:09:09.455
you cut and pasted it into our plasmid.

00:09:10.497 --> 00:09:12.706
Let me write the labels down,

00:09:12.706 --> 00:09:16.289
into our plasmid that
also contained a gene

00:09:17.355 --> 00:09:19.938
that gave antibiotic resistance

00:09:20.996 --> 00:09:23.271
to any bacteria that takes up the plasmid.

00:09:23.271 --> 00:09:26.571
You put these plasmids in
the presence of the bacteria

00:09:26.571 --> 00:09:28.083
or you provide some type of a shock,

00:09:28.083 --> 00:09:30.910
maybe a heat shock, so that
some of the bacteria takes it up

00:09:30.910 --> 00:09:33.040
and then the bacteria starts reproducing.

00:09:33.040 --> 00:09:37.965
And as it reproduces it also
is reproducing the plasmids

00:09:37.965 --> 00:09:41.639
and because it has this
antibiotic resistance

00:09:41.639 --> 00:09:44.906
it is going to grow on this
nutrient antibiotic mixture

00:09:44.906 --> 00:09:47.685
and the other bacteria that
did not take up the plasmids

00:09:47.685 --> 00:09:49.278
are not going to grow.

00:09:49.278 --> 00:09:51.650
And so just like that you can take this,

00:09:51.650 --> 00:09:54.983
you can take this colony right over here,

00:09:54.983 --> 00:09:57.145
and put it into another solution

00:09:57.145 --> 00:09:58.511
or continue to grow it

00:09:58.511 --> 00:10:01.729
and you will have multiple
copies of that gene

00:10:01.729 --> 00:10:05.157
that are inside of that bacteria.

00:10:05.157 --> 00:10:06.182
Now the next question,

00:10:06.182 --> 00:10:09.188
and I'm over simplifying
things fairly dramatically

00:10:09.188 --> 00:10:11.171
is well you now have a bunch of bacteria

00:10:11.171 --> 00:10:13.042
that have a bunch of copies of that gene,

00:10:13.042 --> 00:10:14.731
how do you make use of it?

00:10:14.731 --> 00:10:16.746
Well, the bacteria themselves,

00:10:16.746 --> 00:10:19.820
let's say that gene is for
something you want to manufacture

00:10:19.820 --> 00:10:22.277
say insulin for diabetics,

00:10:22.277 --> 00:10:24.764
well you could actually use
that bacteria's machinery,

00:10:24.764 --> 00:10:27.787
we used its reproductive machinery

00:10:27.787 --> 00:10:30.419
to keep replicating the
genetic information,

00:10:30.419 --> 00:10:34.141
but you can also use its
productive machinery,

00:10:34.141 --> 00:10:35.297
I guess you could say,

00:10:35.297 --> 00:10:37.231
it's going to express its existing DNA

00:10:37.231 --> 00:10:40.189
but it can also express the
genes that are on the plasmid.

00:10:40.189 --> 00:10:43.781
And in fact that's what
would give the bacteria

00:10:43.781 --> 00:10:45.341
its antibiotic resistance

00:10:45.341 --> 00:10:47.552
but if this gene was say for insulin,

00:10:47.552 --> 00:10:51.719
well then the bacteria will
produce a bunch of insulin,

00:10:53.419 --> 00:10:55.402
a bunch of insulin molecules,

00:10:55.402 --> 00:10:57.564
which you might be able
to use in some way.

00:10:57.564 --> 00:10:59.124
And I'm not going to
go into all the details

00:10:59.124 --> 00:11:00.718
of how you will get the insulin out

00:11:00.718 --> 00:11:02.181
and how you could make use of it,

00:11:02.181 --> 00:11:03.886
but needless to say,

00:11:03.886 --> 00:11:06.873
it's pretty cool that we
can even get to this point.


Speed and precision of DNA replication

WEBVTT
Kind: captions
Language: en

00:00:00.254 --> 00:00:03.321
- [Voiceover] In the earlier
video on DNA replication,

00:00:03.321 --> 00:00:04.691
we go into some detail

00:00:04.691 --> 00:00:07.129
about leading strands and lagging strands

00:00:07.129 --> 00:00:08.639
and all of the different actors,

00:00:08.639 --> 00:00:11.286
all of these different enzymatic actors.

00:00:11.286 --> 00:00:13.283
But I left out what is probably

00:00:13.283 --> 00:00:16.673
the most mind-boggling
aspect of all of this,

00:00:16.673 --> 00:00:19.239
and that's the speed and the precision

00:00:19.239 --> 00:00:21.823
with which this is actually happening.

00:00:21.823 --> 00:00:23.036
As we talked about in that video,

00:00:23.036 --> 00:00:24.735
it feels pretty complex.

00:00:24.735 --> 00:00:28.446
You have this topoisomerase
that's unwinding things,

00:00:28.446 --> 00:00:30.675
the helicase is unzipping it.

00:00:30.675 --> 00:00:32.602
Then you have the
polymerase that can only go

00:00:32.602 --> 00:00:34.994
from the five prime to
three prime direction,

00:00:34.994 --> 00:00:36.585
and needs a little primer to get started,

00:00:36.585 --> 00:00:38.268
but then it starts adding the,

00:00:38.268 --> 00:00:40.311
it starts adding the nucleotides.

00:00:40.311 --> 00:00:41.705
On the lagging strand,

00:00:41.705 --> 00:00:42.959
you have to have the R,

00:00:42.959 --> 00:00:44.051
you get the RNA primer,

00:00:44.051 --> 00:00:46.233
but then it's going from,

00:00:46.233 --> 00:00:49.716
once again, from five
prime to three prime,

00:00:49.716 --> 00:00:51.888
so you have these Okazaki fragments.

00:00:51.888 --> 00:00:53.606
And all of this craziness
that's happening,

00:00:53.606 --> 00:00:55.301
and remember, these
things don't have brains.

00:00:55.301 --> 00:00:56.217
These aren't computers.

00:00:56.217 --> 00:00:57.879
They don't know exactly where to go.

00:00:57.879 --> 00:00:59.864
It's all because of the chemistry.

00:00:59.864 --> 00:01:01.815
They're all bumping into each other

00:01:01.815 --> 00:01:03.649
and reacting in just the right way

00:01:03.649 --> 00:01:06.070
to make this incredible thing happen.

00:01:06.070 --> 00:01:07.036
Now what I'm about to tell you

00:01:07.036 --> 00:01:09.779
is really going to boggle your mind.

00:01:09.779 --> 00:01:13.391
Because this is happening incredibly fast.

00:01:13.391 --> 00:01:16.072
DNA polymerase has been clocked,

00:01:16.072 --> 00:01:17.674
at least in E. coli,

00:01:17.674 --> 00:01:22.204
has clocked at approaching
1,000 base pairs per second.

00:01:22.204 --> 00:01:23.401
I think the number that I saw

00:01:23.401 --> 00:01:26.518
was 700-something base pairs per second.

00:01:26.518 --> 00:01:28.286
So polymerase, let me write this down.

00:01:28.286 --> 00:01:29.494
This is worth writing down,

00:01:29.494 --> 00:01:30.701
because it's mind-boggling.

00:01:30.701 --> 00:01:34.509
Gives you a sense of just how amazing

00:01:34.509 --> 00:01:37.783
what the machinery in your cells are.

00:01:37.783 --> 00:01:39.222
So it's been as high as,

00:01:39.222 --> 00:01:40.288
and it can change.

00:01:40.288 --> 00:01:41.768
It can speed up and slow down,

00:01:41.768 --> 00:01:43.600
and that's actually been observed.

00:01:43.600 --> 00:01:45.945
But polymerase as fast as,

00:01:45.945 --> 00:01:46.778
as fast as

00:01:48.158 --> 00:01:50.158
700-plus base pairs per,

00:01:52.803 --> 00:01:53.720
per second.

00:01:55.752 --> 00:01:56.937
So if this, on this diagram,

00:01:56.937 --> 00:01:58.240
it, man, it's just zipping,

00:01:58.240 --> 00:01:59.403
it's just zipping along,

00:01:59.403 --> 00:02:02.243
at least from our perceptual
frame of reference.

00:02:02.243 --> 00:02:04.843
A second seems like a very
short amount of time to us,

00:02:04.843 --> 00:02:06.121
but on a molecular scale,

00:02:06.121 --> 00:02:07.328
these things are just bouncing around

00:02:07.328 --> 00:02:09.534
and just getting this stuff done.

00:02:09.534 --> 00:02:11.066
Now the second thing that
you might be wondering,

00:02:11.066 --> 00:02:12.576
okay, this is happening fast,

00:02:12.576 --> 00:02:14.886
but surely it has a lot of errors.

00:02:14.886 --> 00:02:15.719
Well, the first thing you might say,

00:02:15.719 --> 00:02:16.585
well, if it had a lot of errors,

00:02:16.585 --> 00:02:18.527
that would really not be good for biology,

00:02:18.527 --> 00:02:21.214
because you always have,
you have DNA replicating

00:02:21.214 --> 00:02:22.730
all throughout our lives.

00:02:22.730 --> 00:02:24.727
And at some point you
just have so many errors

00:02:24.727 --> 00:02:26.573
that the cells wouldn't function any more.

00:02:26.573 --> 00:02:28.059
And so lucky for us

00:02:28.059 --> 00:02:31.049
that this is actually a
fairly precise process.

00:02:31.049 --> 00:02:33.795
Even in the first pass of the polymerase,

00:02:33.795 --> 00:02:35.582
you have one mistake,

00:02:35.582 --> 00:02:37.882
you have one mistake,
let me write this down,

00:02:37.882 --> 00:02:39.519
'cause it's amazing.

00:02:39.519 --> 00:02:41.352
One mistake for every,

00:02:44.223 --> 00:02:47.723
for every approximately 10 to the seventh.

00:02:48.727 --> 00:02:50.817
So this is 10 million,

00:02:50.817 --> 00:02:53.093
10 million in nucleotides.

00:02:53.093 --> 00:02:54.093
Nucleotides.

00:02:55.392 --> 00:02:57.458
And that might seem pretty accurate,

00:02:57.458 --> 00:02:58.410
but you gotta remember,

00:02:58.410 --> 00:03:01.893
we have billions of
nucleotides in our DNA.

00:03:01.893 --> 00:03:03.542
So this would still
introduce a lot of errors.

00:03:03.542 --> 00:03:05.980
But then there's
proofreading that goes back

00:03:05.980 --> 00:03:09.312
and makes sure that those
errors don't stick around.

00:03:09.312 --> 00:03:11.634
And so once all the
proofreading takes place,

00:03:11.634 --> 00:03:13.283
it actually becomes one mistake,

00:03:13.283 --> 00:03:15.116
one mistake for every

00:03:16.383 --> 00:03:19.669
approximately 10 to the ninth nucleotides.

00:03:19.669 --> 00:03:21.743
So approximately, you can,

00:03:21.743 --> 00:03:24.232
it would do this at an
incredibly fast pace,

00:03:24.232 --> 00:03:28.458
as fast as 700-plus approaching
1,000 base pairs per second.

00:03:28.458 --> 00:03:32.620
And you have one error
every billion nucleotides,

00:03:32.620 --> 00:03:36.237
especially after you go through
these proofreading steps.

00:03:36.237 --> 00:03:38.025
And so it's incredibly fast,

00:03:38.025 --> 00:03:40.324
and it's incredibly precise.

00:03:40.324 --> 00:03:42.204
So hopefully that gives
you a better appreciation

00:03:42.204 --> 00:03:43.677
for just the magic that's literally,

00:03:43.677 --> 00:03:44.635
I would look at your hand,

00:03:44.635 --> 00:03:45.734
or just think about,

00:03:45.734 --> 00:03:47.929
this is happening in all of the cells

00:03:47.929 --> 00:03:51.762
or most of the cells of
your body as we speak.


Gel electrophoresis

WEBVTT
Kind: captions
Language: en

00:00:00.832 --> 00:00:02.856
- [Voiceover] Let's say that
you have some vials here,

00:00:02.856 --> 00:00:04.115
and you know that in the solution

00:00:04.115 --> 00:00:07.371
you have fragments of
DNA in each of these,

00:00:07.371 --> 00:00:08.843
and what you're curious about, well,

00:00:08.843 --> 00:00:10.887
what about the DNA fragments in our,

00:00:10.887 --> 00:00:12.735
in this first vial?

00:00:12.735 --> 00:00:13.799
In vial number one.

00:00:13.799 --> 00:00:15.395
How long are those fragments?

00:00:15.395 --> 00:00:16.515
How many base pairs?

00:00:16.515 --> 00:00:18.223
How long are they?

00:00:18.223 --> 00:00:20.239
Well, you might say, well why don't I just

00:00:20.239 --> 00:00:21.639
take them out and count them?

00:00:21.639 --> 00:00:24.384
Except for the fact that
they're incredibly small

00:00:24.384 --> 00:00:26.596
and incredibly hard to handle.

00:00:26.596 --> 00:00:29.199
Even a fairly large fragment of DNA,

00:00:29.199 --> 00:00:30.655
let's say we're talking about something

00:00:30.655 --> 00:00:33.905
that's on the order of 5000 base pairs,

00:00:35.781 --> 00:00:37.095
well that's going to be approximately

00:00:37.095 --> 00:00:40.007
one to two micrometers long

00:00:40.007 --> 00:00:41.911
if you were to completely stretch it out.

00:00:41.911 --> 00:00:44.404
And we can't even start
to think about how thin

00:00:44.404 --> 00:00:46.756
the actual diameter is, if we just,

00:00:46.756 --> 00:00:49.893
but length-wise, the long
way, it's only going to be

00:00:49.893 --> 00:00:52.635
one to two micrometers
which is super duper small.

00:00:52.635 --> 00:00:55.631
This is one to two
thousandths of a millimeter.

00:00:55.631 --> 00:00:57.367
So that's not going to help us to somehow

00:00:57.367 --> 00:01:00.783
try to manipulate it
physically with our hands

00:01:00.783 --> 00:01:03.863
or with, you know, kind of rough tools.

00:01:03.863 --> 00:01:05.095
So how do we do that?

00:01:05.095 --> 00:01:06.467
And we could have other vials there.

00:01:06.467 --> 00:01:09.016
How do we see how long the DNA strands

00:01:09.016 --> 00:01:11.367
that are sitting in
those vials actually are?

00:01:11.367 --> 00:01:12.599
And the technique we're going to use,

00:01:12.599 --> 00:01:15.512
gel electrophoresis, it
actually could be used for

00:01:15.512 --> 00:01:17.277
DNA strands, it could be used for RNA,

00:01:17.277 --> 00:01:18.395
if could also be used for proteins,

00:01:18.395 --> 00:01:19.991
any of these macromolecules,

00:01:19.991 --> 00:01:23.771
to see how long are those fragments?

00:01:23.771 --> 00:01:25.955
And so let me write this down.

00:01:25.955 --> 00:01:27.622
Gel electrophoresis.

00:01:34.271 --> 00:01:36.959
And it's called gel electrophoresis

00:01:36.959 --> 00:01:38.696
because it involves a gel,

00:01:38.696 --> 00:01:41.029
it involves electric charge,

00:01:41.973 --> 00:01:44.967
and phoresis is just
referring to the fact that

00:01:44.967 --> 00:01:47.571
we are going to cause the DNA fragments

00:01:47.571 --> 00:01:51.435
to migrate through a gel
because of the charge.

00:01:51.435 --> 00:01:53.759
So phoresis is referring to the migration,

00:01:53.759 --> 00:01:57.035
or the movement of the actual DNA.

00:01:57.035 --> 00:01:58.772
So how do we do this?

00:01:58.772 --> 00:02:01.264
Well here is our set up, right over here.

00:02:01.264 --> 00:02:04.264
We have our gel, that's inside of a,

00:02:05.687 --> 00:02:08.770
that's embedded in a buffer solution.

00:02:10.363 --> 00:02:11.959
So this gel, the most typical one is

00:02:11.959 --> 00:02:14.675
agarose gel, that's a polysaccharide

00:02:14.675 --> 00:02:16.607
that we get from seaweed,

00:02:16.607 --> 00:02:18.623
and it's literally a gel.

00:02:18.623 --> 00:02:20.500
It's a gelatinous material.

00:02:20.500 --> 00:02:23.019
And what we're going to do is,

00:02:23.019 --> 00:02:26.492
is we're going to put,
we're gonna take samples,

00:02:26.492 --> 00:02:28.648
so we might take a little sample from

00:02:28.648 --> 00:02:29.823
this one right over here,

00:02:29.823 --> 00:02:32.287
and we'll put it in this well,

00:02:32.287 --> 00:02:33.269
right over here.

00:02:33.269 --> 00:02:36.095
And you can view these wells
as little divets in the gel.

00:02:36.095 --> 00:02:37.973
You could take a little sample from here

00:02:37.973 --> 00:02:40.140
and put it into this well.

00:02:41.051 --> 00:02:42.955
And then you could put a sample from here,

00:02:42.955 --> 00:02:46.063
and you could put it in that well.

00:02:46.063 --> 00:02:48.779
And it's going to be bathed
inside of this buffer,

00:02:48.779 --> 00:02:50.852
so you can see the buffer
I drew, this fluid,

00:02:50.852 --> 00:02:53.429
and that's really just
water with some salt in it.

00:02:53.429 --> 00:02:55.109
And the buffer is going to keep the pH

00:02:55.109 --> 00:02:57.599
from going too far out of bounds

00:02:57.599 --> 00:03:00.527
as we place a charge
across this entire thing,

00:03:00.527 --> 00:03:02.599
because if the pH gets too far in the

00:03:02.599 --> 00:03:05.651
basic or acidic side, it
might actually affect the DNA,

00:03:05.651 --> 00:03:08.256
or affect the charge on the DNA.

00:03:08.256 --> 00:03:09.600
Now what we're going to do is,

00:03:09.600 --> 00:03:13.099
we're gonna put a charge
across this whole setup.

00:03:13.099 --> 00:03:15.675
Where the side where the wells are,

00:03:15.675 --> 00:03:17.327
where we're gonna place the DNA,

00:03:17.327 --> 00:03:19.119
that's going to be where we're

00:03:19.119 --> 00:03:21.869
gonna put the negative electrode,

00:03:22.960 --> 00:03:25.340
so that's our negative electrode there.

00:03:25.340 --> 00:03:29.507
And the other end is going
to be our positive electrode.

00:03:31.920 --> 00:03:33.572
And we're going to use the fact

00:03:33.572 --> 00:03:38.192
that DNA has a negative
charge at the typical pHs,

00:03:38.192 --> 00:03:41.048
or the pHs that we are
going to be dealing with.

00:03:41.048 --> 00:03:44.380
Now we can go back into previous videos,

00:03:44.380 --> 00:03:46.060
and we can see it right over here,

00:03:46.060 --> 00:03:47.432
you see these negative charges

00:03:47.432 --> 00:03:49.599
on our phosphate backbone.

00:03:51.128 --> 00:03:53.312
And so what is going to happen?

00:03:53.312 --> 00:03:56.056
What is going to happen once we connect

00:03:56.056 --> 00:03:57.932
both of these to a power source,

00:03:57.932 --> 00:04:00.984
and then this side is negative
and this side is positive?

00:04:00.984 --> 00:04:03.924
Well the DNA is going to want to migrate.

00:04:03.924 --> 00:04:06.500
Now, let's think about what will happen.

00:04:06.500 --> 00:04:10.000
Will shorter things migrate further,

00:04:10.000 --> 00:04:12.044
or will longer things migrate further?

00:04:12.044 --> 00:04:14.228
Well you might say, well
longer things are going to

00:04:14.228 --> 00:04:15.684
have more negative charge,

00:04:15.684 --> 00:04:17.588
so maybe they go farther away,

00:04:17.588 --> 00:04:18.904
but then you also have to remember

00:04:18.904 --> 00:04:21.004
that they're also moving more mass.

00:04:21.004 --> 00:04:22.712
So their charge per mass
is gonna be the same

00:04:22.712 --> 00:04:24.700
regardless of length.

00:04:24.700 --> 00:04:27.136
And so what determines
how far something gets,

00:04:27.136 --> 00:04:30.132
how much it migrates over
a certain amount of time,

00:04:30.132 --> 00:04:31.924
is how small it is.

00:04:31.924 --> 00:04:34.136
Remember, we have this agarose gel,

00:04:34.136 --> 00:04:36.824
and people are still
studying the exact mechanism

00:04:36.824 --> 00:04:39.120
of how this DNA, or these macromolecules,

00:04:39.120 --> 00:04:41.612
actually migrate through
the polysaccharide,

00:04:41.612 --> 00:04:43.740
but if you imagine this polysaccharide

00:04:43.740 --> 00:04:46.456
is kind of this mesh,
this net, this sieve,

00:04:46.456 --> 00:04:48.276
well smaller things are gonna be able

00:04:48.276 --> 00:04:51.048
to go through the gaps easier
than the larger things.

00:04:51.048 --> 00:04:53.008
And so if you let some time pass,

00:04:53.008 --> 00:04:55.220
if you let some time pass,

00:04:55.220 --> 00:04:57.628
some of the DNA, let's say this DNA,

00:04:57.628 --> 00:04:59.128
gets around there.

00:05:00.848 --> 00:05:02.696
Let's say, and I'm just color,

00:05:02.696 --> 00:05:04.236
you actually wouldn't see these colors,

00:05:04.236 --> 00:05:07.176
let's say this DNA gets around that far,

00:05:07.176 --> 00:05:08.296
so it doesn't get as far.

00:05:08.296 --> 00:05:09.864
Let's say that this DNA doesn't migrate,

00:05:09.864 --> 00:05:11.852
let's say it has some
that migrates that far

00:05:11.852 --> 00:05:15.592
and let's say it has some
that migrates that far.

00:05:15.592 --> 00:05:17.804
And so if you just saw this,
you wait some amount of time,

00:05:17.804 --> 00:05:21.052
and you were come back and you
were to see this migration,

00:05:21.052 --> 00:05:24.135
you were to see this migration occur,

00:05:25.924 --> 00:05:27.100
and the longer you wait,

00:05:27.100 --> 00:05:29.424
the further these things are gonna get.

00:05:29.424 --> 00:05:30.320
In fact, if you wait too long

00:05:30.320 --> 00:05:32.896
they're gonna fall off all
the way over the other edge.

00:05:32.896 --> 00:05:35.164
Is, if you just saw this you'd say

00:05:35.164 --> 00:05:38.468
okay, well this strand right over here

00:05:38.468 --> 00:05:41.468
these must be smaller DNA molecules.

00:05:42.640 --> 00:05:44.124
They must be shorter.

00:05:44.124 --> 00:05:45.776
These must be a little bit longer,

00:05:45.776 --> 00:05:47.456
and these must be even longer than that.

00:05:47.456 --> 00:05:49.024
And this grouping right over here

00:05:49.024 --> 00:05:50.620
is going to be the longest of all.

00:05:50.620 --> 00:05:52.440
So this was a mixture
of some longer strands

00:05:52.440 --> 00:05:55.100
and still longer ones,
but not quite as long.

00:05:55.100 --> 00:05:58.096
And, for example, maybe there
are some really short strands,

00:05:58.096 --> 00:06:00.616
maybe there were some really short strands

00:06:00.616 --> 00:06:04.533
in that, what I'm drawing
as, that orange group

00:06:05.936 --> 00:06:07.980
right over here.

00:06:07.980 --> 00:06:10.080
So, what I just did right over here

00:06:10.080 --> 00:06:14.140
this could tell you the
relative length of these strands

00:06:14.140 --> 00:06:16.044
but how would you actually measure them?

00:06:16.044 --> 00:06:19.152
Well that's where you can go
find standardized solutions,

00:06:19.152 --> 00:06:21.402
which we call a DNA ladder.

00:06:22.456 --> 00:06:24.192
And so let's say you
go get the DNA ladder,

00:06:24.192 --> 00:06:25.564
I'm gonna draw it in pink,

00:06:25.564 --> 00:06:27.104
so you literally could buy this.

00:06:27.104 --> 00:06:29.036
You can buy it online.

00:06:29.036 --> 00:06:30.492
And the standard solution let's say

00:06:30.492 --> 00:06:31.892
it separates like this.

00:06:31.892 --> 00:06:33.936
So it separates, that goes there,

00:06:33.936 --> 00:06:36.148
let's say some of it goes like, there,

00:06:36.148 --> 00:06:39.228
and some of it goes like, there.

00:06:39.228 --> 00:06:41.328
Well you would be able to
know from the labeling,

00:06:41.328 --> 00:06:42.840
or whichever one you choose to buy,

00:06:42.840 --> 00:06:45.920
that this grouping here,
this all of the DNA

00:06:45.920 --> 00:06:48.753
that is 5000 base pairs let's say.

00:06:50.456 --> 00:06:54.824
Let's say this right over
here is 1500 base pairs.

00:06:54.824 --> 00:06:56.952
And let's say this over here is,

00:06:56.952 --> 00:07:01.516
let's say this over here
is 500 base pairs long.

00:07:01.516 --> 00:07:03.672
And so now you can use this DNA ladder,

00:07:03.672 --> 00:07:04.988
these standardized ones,

00:07:04.988 --> 00:07:08.460
to gauge how long, how
many base pairs these are.

00:07:08.460 --> 00:07:10.420
So you say okay, this blue one here,

00:07:10.420 --> 00:07:11.400
this is a bunch of DNA

00:07:11.400 --> 00:07:14.480
that's a little bit
longer than 500 base pairs

00:07:14.480 --> 00:07:16.748
but it's shorter than 1500 base pairs.

00:07:16.748 --> 00:07:18.176
You can see this green one here,

00:07:18.176 --> 00:07:21.116
well it's a little bit
longer than 1500 base pairs,

00:07:21.116 --> 00:07:22.908
it didn't migrate quite as fair as this

00:07:22.908 --> 00:07:25.820
big bundle of 1500 base pairs guy did.

00:07:25.820 --> 00:07:28.592
And so then you can get
a better approximation.

00:07:28.592 --> 00:07:30.244
And you can choose your ladder based on

00:07:30.244 --> 00:07:32.512
what you think you are
going to find there,

00:07:32.512 --> 00:07:36.320
what you're actually going to look for.

00:07:36.320 --> 00:07:38.336
Now the other thing to appreciate is,

00:07:38.336 --> 00:07:42.228
when you see, when you see
the DNA having migrated

00:07:42.228 --> 00:07:43.404
this far, you might say okay,

00:07:43.404 --> 00:07:45.672
is this one DNA strand,

00:07:45.672 --> 00:07:48.808
is that one DNA strand
that I'm looking at?

00:07:48.808 --> 00:07:51.244
And just going back to
the measurements, no.

00:07:51.244 --> 00:07:53.484
That is many, many, many, many DNAs

00:07:53.484 --> 00:07:54.576
that you're looking at.

00:07:54.576 --> 00:07:56.704
And this is, they're not
all stretched out like that.

00:07:56.704 --> 00:08:00.296
Remember, even something
that is 5000 base pairs long

00:08:00.296 --> 00:08:04.244
is only going to be one to two micrometers

00:08:04.244 --> 00:08:05.224
if you stretch it out.

00:08:05.224 --> 00:08:06.624
So, you wouldn't even be able to see it,

00:08:06.624 --> 00:08:07.996
it's a thousandth of a millimeter.

00:08:07.996 --> 00:08:09.228
You wouldn't you even be able to see it.

00:08:09.228 --> 00:08:11.189
So this is many, many, many molecules

00:08:11.189 --> 00:08:13.932
of DNA, is migrating that far.

00:08:13.932 --> 00:08:14.968
And they wouldn't even
have to be that small

00:08:14.968 --> 00:08:18.356
to be able to migrate through
that polysaccharide gel.

00:08:18.356 --> 00:08:20.093
Now the last thing you're
probably saying is okay,

00:08:20.093 --> 00:08:22.332
wait, but how am I even
seeing it over here?

00:08:22.332 --> 00:08:24.600
How do I actually see this DNA?

00:08:24.600 --> 00:08:26.868
Especially if they're these
super, super small molecules?

00:08:26.868 --> 00:08:29.248
And the answer is you
put some type of marker

00:08:29.248 --> 00:08:32.020
on the DNA, that will make them visible.

00:08:32.020 --> 00:08:33.504
Some type of dye, or something that

00:08:33.504 --> 00:08:34.960
might become fluorescent.

00:08:34.960 --> 00:08:36.500
And one of the typical things

00:08:36.500 --> 00:08:39.945
that people often use it ethidium bromide.

00:08:39.945 --> 00:08:43.808
And ethidium bromide is
called an intercalating agent,

00:08:43.808 --> 00:08:45.880
and it's a molecule,
you can see the ethidium

00:08:45.880 --> 00:08:50.047
right over here, these are
two DNA, two backbones of DNA,

00:08:51.256 --> 00:08:53.608
you can see the base pairs bonding here,

00:08:53.608 --> 00:08:55.540
and then this right over here

00:08:55.540 --> 00:08:58.623
that is ethidium that has fit itself,

00:08:59.909 --> 00:09:01.813
that's why we call it intercalating,

00:09:01.813 --> 00:09:04.752
it has fit itself in between
the rungs of the ladder.

00:09:04.752 --> 00:09:07.189
And when it does so, inside of DNA,

00:09:07.189 --> 00:09:09.706
it actually becomes
fluorescent when you apply

00:09:09.706 --> 00:09:10.956
UV light to it.

00:09:11.920 --> 00:09:14.369
So if you put this ethidium bromide

00:09:14.369 --> 00:09:17.452
into all of your DNA right over here,

00:09:18.317 --> 00:09:20.137
and then as it migrates,
and then if you were to

00:09:20.137 --> 00:09:22.461
turn on a UV light, it
would become fluorescent,

00:09:22.461 --> 00:09:25.570
and you would actually see these things.

00:09:25.570 --> 00:09:26.997
And so if you wanted to see what it

00:09:26.997 --> 00:09:28.341
actually would look like in real life,

00:09:28.341 --> 00:09:29.434
well this is what it would look like

00:09:29.434 --> 00:09:32.122
when you were to, if you were
to look at it straight on.

00:09:32.122 --> 00:09:33.549
Where this would have been a well,

00:09:33.549 --> 00:09:36.713
let me make it a little
bit easier to read.

00:09:36.713 --> 00:09:38.925
So right over here would
have been the well,

00:09:38.925 --> 00:09:40.885
where you would put the DNA ladder,

00:09:40.885 --> 00:09:43.097
and it would come up with
standardized measurements.

00:09:43.097 --> 00:09:45.847
Maybe that's our 5000 base pairs,

00:09:46.709 --> 00:09:50.797
this right over here
is our 1500 base pairs,

00:09:50.797 --> 00:09:54.857
and this right over here
is our 500 base pairs.

00:09:54.857 --> 00:09:56.453
And then let's say you had some solution

00:09:56.453 --> 00:09:58.973
of some other DNA, and
you wait a little while,

00:09:58.973 --> 00:10:01.241
and you see look, it
migrated not quite as far

00:10:01.241 --> 00:10:03.706
as a 500 base pair, so
it must be little bit,

00:10:03.706 --> 00:10:05.581
this must be a bundle
of things a little bit

00:10:05.581 --> 00:10:07.485
longer than 500 base pairs,

00:10:07.485 --> 00:10:11.125
but for sure a lot shorter
than 1500 base pairs.

00:10:11.125 --> 00:10:12.161
Now once again, doesn't have to have

00:10:12.161 --> 00:10:13.729
just one fragment length,
you could have had

00:10:13.729 --> 00:10:17.957
another group that was, maybe
right at 1500 base pairs.

00:10:17.957 --> 00:10:19.189
And you've probably seen this,

00:10:19.189 --> 00:10:21.401
whenever you see people
talking about genetic analysis,

00:10:21.401 --> 00:10:23.613
and things like this,
you're often seeing people

00:10:23.613 --> 00:10:25.952
look at one of these read-outs from

00:10:25.952 --> 00:10:27.875
gel electrophoresis.

00:10:27.875 --> 00:10:30.395
So now you know what's
actually going on here.

00:10:30.395 --> 00:10:32.411
This isn't a strand of DNA, this is a big,

00:10:32.411 --> 00:10:34.736
this is a bunch of DNA
that has been tagged

00:10:34.736 --> 00:10:35.912
with some type of a dye,

00:10:35.912 --> 00:10:40.671
or the ethidium bromide,
or something like that.

00:10:40.671 --> 00:10:42.323
And it's a bunch of those molecules

00:10:42.323 --> 00:10:44.787
and they've migrated based on the charge.

00:10:44.787 --> 00:10:46.411
They're trying to get away
from that negative charge

00:10:46.411 --> 00:10:47.923
to the positive charge.

00:10:47.923 --> 00:10:49.491
And the smaller molecules,
this is a bunch of

00:10:49.491 --> 00:10:53.159
small molecules, right over
here, are able to get further

00:10:53.159 --> 00:10:55.371
because they're able
to get through the mesh

00:10:55.371 --> 00:10:57.038
of the agarose gel.


Leading and lagging strands in DNA replication

WEBVTT
Kind: captions
Language: en

00:00:00.627 --> 00:00:03.326
- [Voiceover] Let's talk
a little bit in more depth

00:00:03.326 --> 00:00:05.805
about how DNA actually copies itself,

00:00:05.805 --> 00:00:06.845
how it actually replicates,

00:00:06.845 --> 00:00:08.123
and we're gonna talk about

00:00:08.123 --> 00:00:11.150
the actual actors in the process.

00:00:11.150 --> 00:00:13.166
Now, as I talk about
it, I'm gonna talk a lot

00:00:13.166 --> 00:00:17.704
about the 3' and 5' ends
of the DNA molecule,

00:00:17.704 --> 00:00:20.013
and if that is completely
unfamiliar to you,

00:00:20.013 --> 00:00:22.163
I encourage you to watch the video

00:00:22.163 --> 00:00:26.041
on the antiparallel structure of DNA.

00:00:26.041 --> 00:00:28.163
And I'll give a little bit
of a quick review here,

00:00:28.163 --> 00:00:31.355
just in case you saw it but
it was a little while ago.

00:00:31.355 --> 00:00:33.042
This is a zoom-in of DNA,

00:00:33.042 --> 00:00:35.497
it's actually the zoom-in from that video,

00:00:35.497 --> 00:00:37.934
and when we talk about the 5' and 3' ends,

00:00:37.934 --> 00:00:40.848
we're referring to what's
happening on the riboses

00:00:40.848 --> 00:00:44.930
that formed part of this
phosphate sugar backbone.

00:00:44.930 --> 00:00:47.051
So we have ribose right over here,

00:00:47.051 --> 00:00:48.639
five-carbon sugar,

00:00:48.639 --> 00:00:50.900
and we can number the carbons;

00:00:50.900 --> 00:00:54.876
this is the 1' carbon,
that's the 2' carbon,

00:00:54.876 --> 00:00:58.516
that's the 3' carbon,
that's the 4' carbon,

00:00:58.516 --> 00:01:00.641
and that's the 5' carbon.

00:01:00.641 --> 00:01:02.891
So this side of the ladder,

00:01:04.641 --> 00:01:05.693
you could say,

00:01:05.693 --> 00:01:07.642
it is going in the ...

00:01:07.642 --> 00:01:10.526
it is going, let me
draw a little line here,

00:01:10.526 --> 00:01:13.859
this is going in the 3' to 5' direction.

00:01:15.308 --> 00:01:19.498
So this end is 3' and then this end is 5'.

00:01:19.498 --> 00:01:20.886
It's going 3' to 5'.

00:01:20.886 --> 00:01:23.550
Notice three, this phosphate
connects to the 3',

00:01:23.550 --> 00:01:25.934
then we go to the 5'
connects to a phosphate,

00:01:25.934 --> 00:01:27.358
this connects to a 3',

00:01:27.358 --> 00:01:28.464
then it connects--

00:01:28.464 --> 00:01:31.463
then we go to the 5'
connects to a phosphate.

00:01:31.463 --> 00:01:34.003
Now on this end, as we
said it's antiparallel.

00:01:34.003 --> 00:01:36.676
It's parallel, but it's
oriented the other way.

00:01:36.676 --> 00:01:39.509
So this is the 3', this is the 5',

00:01:40.379 --> 00:01:43.515
this is the 3', this is the 5'.

00:01:43.515 --> 00:01:44.979
And so this is just
what we're talking about

00:01:44.979 --> 00:01:47.563
when we talk about the
antiparallel structure.

00:01:47.563 --> 00:01:49.418
These two backbones,

00:01:49.418 --> 00:01:51.449
these two strands are
parallel to each other,

00:01:51.449 --> 00:01:54.661
but they're oriented
in opposite directions.

00:01:54.661 --> 00:01:58.833
So this is the 3' end
and this is the 5' end.

00:01:58.833 --> 00:02:00.402
And this is gonna be really important

00:02:00.402 --> 00:02:02.070
for understanding replication,

00:02:02.070 --> 00:02:04.030
because the DNA polymerase,

00:02:04.030 --> 00:02:06.767
the things that's adding
more and more nucleotides

00:02:06.767 --> 00:02:09.327
to grow a DNA strand;

00:02:09.327 --> 00:02:12.827
it can only add nucleotides on the 3' end.

00:02:13.872 --> 00:02:16.828
So if we were talking
about this right over here,

00:02:16.828 --> 00:02:18.456
we would only be able to add …

00:02:18.456 --> 00:02:21.741
We would only be able
to add going that way.

00:02:21.741 --> 00:02:24.250
We wouldn't be able to add going …

00:02:24.250 --> 00:02:27.000
We wouldn't be able to add going that way.

00:02:27.000 --> 00:02:29.386
So one way to think about it
is you can only add nucleotides

00:02:29.386 --> 00:02:33.074
on the 3' end or you can only extend …

00:02:33.074 --> 00:02:36.741
You can only extend DNA
going from 5' to 3'.

00:02:38.793 --> 00:02:40.978
If you're only adding on the 3' end,

00:02:40.978 --> 00:02:45.038
then you're going from the
5' to the 3' direction.

00:02:45.038 --> 00:02:47.733
You can't go from the
3' to the 5' direction.

00:02:47.733 --> 00:02:52.410
You can't continue to add on
the 5' side using polymerase.

00:02:52.410 --> 00:02:55.146
So what am I talking
about with polymerase.

00:02:55.146 --> 00:02:57.046
Well let's look at this
diagram right over here

00:02:57.046 --> 00:02:58.985
that really gives us an overview

00:02:58.985 --> 00:03:01.023
of all of the different actors.

00:03:01.023 --> 00:03:03.512
So here is just our of our DNA strand,

00:03:03.512 --> 00:03:07.270
and it's, you can imagine
it's somewhat natural,

00:03:07.270 --> 00:03:09.769
in it's natural unreplicated form,

00:03:09.769 --> 00:03:13.591
and you could see we've labeled
here the 3' and the 5' ends,

00:03:13.591 --> 00:03:15.436
and you could follow
one of these backbones.

00:03:15.436 --> 00:03:19.204
This 3', if you follow
it all the way over here,

00:03:19.204 --> 00:03:22.914
it goes, this is the corresponding 5' end.

00:03:22.914 --> 00:03:26.741
So this and this are the same strand,

00:03:26.741 --> 00:03:30.402
and this one, if you follow it along,

00:03:30.402 --> 00:03:32.409
if you go all the way over
here, it's the same strand.

00:03:32.409 --> 00:03:33.654
So this is the 3' end,

00:03:33.654 --> 00:03:37.821
and 3' end of it and then
this is the 5' end of it.

00:03:38.954 --> 00:03:40.118
Now the first thing,

00:03:40.118 --> 00:03:41.795
and we've talked about
this in previous videos

00:03:41.795 --> 00:03:44.154
where we give an overview of replication,

00:03:44.154 --> 00:03:48.321
is the general idea is that
the two sides of our helix,

00:03:49.992 --> 00:03:54.031
the two DNA, the double-helix
needs to get split,

00:03:54.031 --> 00:03:56.271
and then we can build another,

00:03:56.271 --> 00:03:58.757
we can build another side of the ladder

00:03:58.757 --> 00:04:01.076
on each of those two split ends.

00:04:01.076 --> 00:04:04.092
You could really view this
as if this is a zipper,

00:04:04.092 --> 00:04:08.299
you unzip it and then you put
new zippers on either end.

00:04:08.299 --> 00:04:10.041
But there's a lot of--

00:04:10.041 --> 00:04:12.419
in reality, it is far more
complex than just saying

00:04:12.419 --> 00:04:15.300
"Oh, let's open the zipper
and put new zippers on it."

00:04:15.300 --> 00:04:18.296
It involves a whole bunch of
enzymes and all sorts of things

00:04:18.296 --> 00:04:20.151
and even in this diagram,

00:04:20.151 --> 00:04:21.845
we're not showing all
of the different actors,

00:04:21.845 --> 00:04:24.000
but we're showing you the primary actors,

00:04:24.000 --> 00:04:26.459
at least the ones that
you'll hear discussed

00:04:26.459 --> 00:04:30.249
when people talk about DNA replication.

00:04:30.249 --> 00:04:31.576
So the first thing that needs to happen,

00:04:31.576 --> 00:04:35.484
right over here, it's all
tightly, tightly wound.

00:04:35.484 --> 00:04:36.445
So let me write that,

00:04:36.445 --> 00:04:38.862
it is tightly, tightly wound.

00:04:40.302 --> 00:04:41.443
And it actually turns out,

00:04:41.443 --> 00:04:44.052
the more that we unwind it on one side,

00:04:44.052 --> 00:04:46.421
the more tightly wound
it gets on this side.

00:04:46.421 --> 00:04:48.903
So in order for us to unzip the zipper,

00:04:48.903 --> 00:04:50.584
we need to have an enzyme

00:04:50.584 --> 00:04:54.861
that helps us unwind
this tightly wound helix.

00:04:54.861 --> 00:04:58.676
And that enzyme is the topoisomerase.

00:04:58.676 --> 00:05:00.993
And the way that it actually works is

00:05:00.993 --> 00:05:04.776
it breaks up parts of the
back bones temporarily,

00:05:04.776 --> 00:05:07.982
so that it can unwind and
then they get back together,

00:05:07.982 --> 00:05:11.097
but the general high-level
idea is it unwinds it,

00:05:11.097 --> 00:05:12.762
so then the helicase enzyme,

00:05:12.762 --> 00:05:14.044
and the helicase really doesn't look like

00:05:14.044 --> 00:05:16.351
this little triangle
that's cutting things.

00:05:16.351 --> 00:05:18.057
These things are actually
far more fascinating

00:05:18.057 --> 00:05:19.427
if you were to actually see a--

00:05:19.427 --> 00:05:21.659
the molecular structure of helicase.

00:05:21.659 --> 00:05:23.437
But what helicase is doing

00:05:23.437 --> 00:05:26.874
is it's breaking those
hydrogen bonds between our …

00:05:26.874 --> 00:05:29.156
Between our nitrogenous bases,

00:05:29.156 --> 00:05:31.550
in this case it's an adenine
here, this is a thymine

00:05:31.550 --> 00:05:35.514
and it would break that
hydrogen bond between these two.

00:05:35.514 --> 00:05:37.514
So, first you unwind it,

00:05:39.511 --> 00:05:40.841
then the helicase,

00:05:40.841 --> 00:05:42.831
the topoisomerase unwinds it,

00:05:42.831 --> 00:05:45.555
then the helicase breaks them up,

00:05:45.555 --> 00:05:47.504
and then we actually think about

00:05:47.504 --> 00:05:49.744
these two strands differently,

00:05:49.744 --> 00:05:51.109
because as I mentioned,

00:05:51.109 --> 00:05:53.633
you can only add nucleotides

00:05:53.633 --> 00:05:56.466
going from the 5' to 3' direction.

00:05:57.693 --> 00:05:59.864
So this strand on the
bottom right over here

00:05:59.864 --> 00:06:02.399
which we will call our leading strand,

00:06:02.399 --> 00:06:04.236
this one actually has a
pretty straightforward,

00:06:04.236 --> 00:06:06.428
remember this is the
5' end right over here,

00:06:06.428 --> 00:06:07.829
so it can add,

00:06:07.829 --> 00:06:10.349
it can add going in that direction,

00:06:10.349 --> 00:06:13.527
it can add going in that
direction right over here.

00:06:13.527 --> 00:06:15.277
This is the 5' to 3',

00:06:16.496 --> 00:06:20.116
so what needs to happen here
is to start the process,

00:06:20.116 --> 00:06:21.949
you need an RNA primer

00:06:22.818 --> 00:06:26.234
and the character that puts an RNA primer,

00:06:26.234 --> 00:06:28.107
that is DNA primase.

00:06:28.107 --> 00:06:29.684
We'll talk a little bit more about

00:06:29.684 --> 00:06:31.967
these characters up here
in the lagging strand,

00:06:31.967 --> 00:06:33.328
but they'll add an RNA,

00:06:33.328 --> 00:06:35.650
let me do this in a color you can see,

00:06:35.650 --> 00:06:37.768
an RNA primer will be added here,

00:06:37.768 --> 00:06:39.598
and then once there's a primer,

00:06:39.598 --> 00:06:44.239
then DNA polymerase can just
start adding nucleotides,

00:06:44.239 --> 00:06:47.159
it can start adding
nucleotides at the 3' end.

00:06:47.159 --> 00:06:49.430
And the reason why the leading
strand has it pretty easy

00:06:49.430 --> 00:06:52.680
is this DNA polymerase right over here,

00:06:53.613 --> 00:06:55.692
this polymerase, and once again,

00:06:55.692 --> 00:06:59.107
they aren't these perfect
rectangles as on this diagram.

00:06:59.107 --> 00:07:01.326
They're actually much more
fascinating than that.

00:07:01.326 --> 00:07:02.712
You see the polymerase up there,

00:07:02.712 --> 00:07:07.079
you also see you one
over here, polymerase.

00:07:07.079 --> 00:07:08.477
This polymerase can just,

00:07:08.477 --> 00:07:11.601
you can kind of think of it
as following the opened zipper

00:07:11.601 --> 00:07:13.473
and then just keep adding,

00:07:13.473 --> 00:07:17.191
keep adding nucleotides at the 3' end.

00:07:17.191 --> 00:07:20.831
And so this one seems
pretty straightforward.

00:07:20.831 --> 00:07:23.232
Now, you might say wouldn't it be easy

00:07:23.232 --> 00:07:26.893
if we could just add
nucleotides at a 5' end,

00:07:26.893 --> 00:07:28.433
because then we could say

00:07:28.433 --> 00:07:31.128
well this is going from 3' to 5',

00:07:31.128 --> 00:07:33.399
well maybe that polymerase
or different polymerase

00:07:33.399 --> 00:07:35.563
could just keep adding
nucleotides like that,

00:07:35.563 --> 00:07:37.308
and then everything would be easy.

00:07:37.308 --> 00:07:40.011
Well, it turns out that
that is not the case.

00:07:40.011 --> 00:07:43.841
you cannot add nucleotides at the 5' end,

00:07:43.841 --> 00:07:46.620
and let me be clear,
this 3' right over here,

00:07:46.620 --> 00:07:48.401
this, I'm talking about this strand.

00:07:48.401 --> 00:07:50.295
This strand right over here,

00:07:50.295 --> 00:07:52.332
this, let me do this in another color,

00:07:52.332 --> 00:07:55.454
this strand right over here,

00:07:55.454 --> 00:07:58.856
this is the 3' end, this is the 5' end,

00:07:58.856 --> 00:07:59.689
and so you can't,

00:07:59.689 --> 00:08:03.343
you can't just keep adding
nucleotides just like that,

00:08:03.343 --> 00:08:06.343
and so how does biology handle this?

00:08:07.439 --> 00:08:10.659
Well it handles this by adding primers

00:08:10.659 --> 00:08:14.278
right as this opening
happens, it'll add primers,

00:08:14.278 --> 00:08:17.197
and this diagram shows the
primer is just one nucleotide

00:08:17.197 --> 00:08:19.570
but a primer is typically
several nucleotides,

00:08:19.570 --> 00:08:21.467
roughly 10 nucleotides.

00:08:21.467 --> 00:08:25.634
So it'll add roughly 10 RNA
nucleotides right over here,

00:08:27.151 --> 00:08:29.060
and that's done by the DNA primase.

00:08:29.060 --> 00:08:32.415
So the DNA primase is
going along the lagging,

00:08:32.415 --> 00:08:36.800
is going along this side,
I can say the top strand,

00:08:36.800 --> 00:08:38.382
and it's adding,

00:08:38.382 --> 00:08:41.280
it's adding the RNA primer,

00:08:41.280 --> 00:08:42.773
which won't be just one nucleotide,

00:08:42.773 --> 00:08:45.186
it tends to be several of them,

00:08:45.186 --> 00:08:48.008
and then once you have that RNA primer,

00:08:48.008 --> 00:08:52.175
then the polymerase can add
in the 5' to 3' direction,

00:08:53.349 --> 00:08:55.432
it can add on the 3' end.

00:08:56.408 --> 00:08:58.606
So then it can just start adding,

00:08:58.606 --> 00:09:00.918
it can just start adding DNA like that.

00:09:00.918 --> 00:09:02.286
And so you can imagine this process,

00:09:02.286 --> 00:09:06.766
it's kind of, you add the
primase, put some primer here,

00:09:06.766 --> 00:09:10.933
and then you start building
from the 5' to 3' direction.

00:09:12.465 --> 00:09:14.684
You start building just like that,

00:09:14.684 --> 00:09:16.818
and then you skip a little bit
and then that happens again.

00:09:16.818 --> 00:09:20.665
So you end up with all
these fragments of DNA

00:09:20.665 --> 00:09:24.492
and those fragments are
called Okazaki fragments.

00:09:24.492 --> 00:09:26.909
So, it's a Okazaki fragments,

00:09:28.251 --> 00:09:30.750
and so what you have happening
here on the lagging strand,

00:09:30.750 --> 00:09:31.621
you can think of it as,

00:09:31.621 --> 00:09:32.705
why is it called the lagging strand?

00:09:32.705 --> 00:09:34.996
Well you have to do it in this kind of …

00:09:34.996 --> 00:09:36.555
it feels like a sub-optimal way

00:09:36.555 --> 00:09:39.020
where you have to keep creating
these Okazaki fragments

00:09:39.020 --> 00:09:41.337
as you follow this opening,

00:09:41.337 --> 00:09:45.600
and so it lags, it's going
to be a slower process,

00:09:45.600 --> 00:09:49.074
but then all of these
strands can be put together

00:09:49.074 --> 00:09:50.824
using the DNA ligase.

00:09:52.195 --> 00:09:55.927
The DNA ligase; not only will
the strands be put together,

00:09:55.927 --> 00:09:59.148
but then you also have the
RNA being actually replaced

00:09:59.148 --> 00:10:02.102
with DNA and then when all is said done,

00:10:02.102 --> 00:10:05.623
you are going to have a strand
of DNA being replicated,

00:10:05.623 --> 00:10:08.293
or being created right up here.

00:10:08.293 --> 00:10:10.995
So when it's all done, you're
gonna have two double strands,

00:10:10.995 --> 00:10:12.904
one up here for on the lagging strand,

00:10:12.904 --> 00:10:16.321
and one down here on the leading strand.


Introduction to reaction quotient Qc

WEBVTT
Kind: captions
Language: en

00:00:00.732 --> 00:00:03.245
- [Voiceover] Today we
are going to be talking

00:00:03.245 --> 00:00:05.001
about the Reaction Quotient, Q.

00:00:05.001 --> 00:00:06.897
In this video, I'm going to go over,

00:00:06.897 --> 00:00:10.147
how you calculate Q and how you use it.

00:00:10.147 --> 00:00:12.912
We're gonna start with
and example reaction

00:00:12.912 --> 00:00:15.300
between sulfur dioxide,

00:00:15.300 --> 00:00:16.133
S02 gas,

00:00:18.494 --> 00:00:21.244
which will react with oxygen gas,

00:00:22.266 --> 00:00:24.198
and this is a reversible reaction

00:00:24.198 --> 00:00:25.031
that makes

00:00:26.060 --> 00:00:27.426
sulfur trioxide

00:00:27.426 --> 00:00:28.259
or SO3.

00:00:29.728 --> 00:00:32.303
We should make sure this
is a balanced reaction.

00:00:32.303 --> 00:00:36.268
We have two sulfur dioxdes
reacting with one O2

00:00:36.268 --> 00:00:37.209
to give

00:00:37.209 --> 00:00:38.042
2S03.

00:00:39.010 --> 00:00:41.990
At equilibrium we can calculate

00:00:41.990 --> 00:00:44.407
the equilibrium constant, Kc.

00:00:46.195 --> 00:00:47.695
So, at equilibrium

00:00:50.300 --> 00:00:51.908
we know the concentrations

00:00:51.908 --> 00:00:54.193
should be constant because the rate of the

00:00:54.193 --> 00:00:56.720
forward and backward
reactions are the same.

00:00:56.720 --> 00:01:00.167
And if we plug those concentrations in

00:01:00.167 --> 00:01:02.461
to this expression, we will get Kc.

00:01:02.461 --> 00:01:03.402
So,

00:01:03.402 --> 00:01:04.235
Kc is

00:01:05.360 --> 00:01:08.508
the product concentration raised

00:01:08.508 --> 00:01:11.733
to the second power so that's from

00:01:11.733 --> 00:01:15.017
this stoichiometric coefficient.

00:01:15.017 --> 00:01:17.092
And then our reactant concentrations,

00:01:17.092 --> 00:01:18.519
so S02

00:01:18.519 --> 00:01:19.352
squared

00:01:20.957 --> 00:01:23.290
and the concentration of O2.

00:01:24.952 --> 00:01:27.010
So we know at some temperature,

00:01:27.010 --> 00:01:29.731
if you plug in the
equilibrium concentrations,

00:01:29.731 --> 00:01:31.314
Kc is equal to 4.3.

00:01:32.513 --> 00:01:34.597
But what if we're interested in looking

00:01:34.597 --> 00:01:36.082
at the reaction and it's not

00:01:36.082 --> 00:01:38.607
at equilibrium yet or
maybe we just don't know

00:01:38.607 --> 00:01:39.954
if it's at equilibrium.

00:01:39.954 --> 00:01:41.037
In that case,

00:01:42.026 --> 00:01:43.472
when you're not sure it's

00:01:43.472 --> 00:01:46.207
at equilibrium or really at any point

00:01:46.207 --> 00:01:48.732
in your reaction or any time,

00:01:48.732 --> 00:01:51.716
we can calculate the Reaction Quotient, Q.

00:01:51.716 --> 00:01:52.549
So Qc

00:01:53.388 --> 00:01:54.305
is equal to

00:01:55.548 --> 00:01:58.213
the concentration of our product

00:01:58.213 --> 00:01:59.521
squared,

00:01:59.521 --> 00:02:01.469
so the concentration of the product

00:02:01.469 --> 00:02:02.685
raised to

00:02:02.685 --> 00:02:05.745
the stoichiometric coefficient

00:02:05.745 --> 00:02:08.224
times the reactant concentrations,

00:02:08.224 --> 00:02:11.431
also raised to their
stoichiometric coefficients.

00:02:11.431 --> 00:02:12.803
So S02 squared

00:02:12.803 --> 00:02:13.636
and O2.

00:02:15.191 --> 00:02:17.303
So you may be wondering at this point,

00:02:17.303 --> 00:02:18.596
what's the difference?

00:02:18.596 --> 00:02:20.434
The equation for Qc and Kc will

00:02:20.434 --> 00:02:22.466
always look exactly the same

00:02:22.466 --> 00:02:25.723
and the main difference
is when you use them.

00:02:25.723 --> 00:02:28.607
The equilibrium constant K you calculate

00:02:28.607 --> 00:02:32.024
only with the equilibrium concentrations.

00:02:33.464 --> 00:02:35.091
So the c means everything

00:02:35.091 --> 00:02:38.341
is in terms of the molar concentration.

00:02:39.301 --> 00:02:41.196
And for the reaction quotient, Q,

00:02:41.196 --> 00:02:43.197
again everything is in terms of molar

00:02:43.197 --> 00:02:45.235
concentration but we can calculate it

00:02:45.235 --> 00:02:47.319
with any concentrations and we don't

00:02:47.319 --> 00:02:49.486
have to be at equilibrium.

00:02:51.987 --> 00:02:54.235
Molar concentration...

00:02:54.235 --> 00:02:55.482
So let's calculate this

00:02:55.482 --> 00:02:58.479
for a set of example concentrations.

00:02:58.479 --> 00:03:00.722
At some point in our reaction we have

00:03:00.722 --> 00:03:02.418
the following concentrations.

00:03:02.418 --> 00:03:04.256
We have 0.10

00:03:04.256 --> 00:03:05.245
molar

00:03:05.245 --> 00:03:06.488
S02,

00:03:06.488 --> 00:03:07.321
0.30

00:03:08.236 --> 00:03:09.069
molar

00:03:09.957 --> 00:03:10.968
O2,

00:03:10.968 --> 00:03:11.801
and 3.5

00:03:12.667 --> 00:03:14.731
molar of our product.

00:03:14.731 --> 00:03:16.717
So if we plug these numbers into

00:03:16.717 --> 00:03:18.681
our expression for Qc,

00:03:18.681 --> 00:03:19.811
we get

00:03:19.811 --> 00:03:20.644
3.5 molar

00:03:21.968 --> 00:03:23.264
squared

00:03:23.264 --> 00:03:24.712
in the numerator

00:03:24.712 --> 00:03:26.045
and 0.10 squared

00:03:27.674 --> 00:03:28.507
times 0.3

00:03:30.691 --> 00:03:33.030
in the denominator.

00:03:33.030 --> 00:03:35.730
So if I plug this into my calculator,

00:03:35.730 --> 00:03:37.251
I get that Qc

00:03:37.251 --> 00:03:39.242
with this set of concentrations

00:03:39.242 --> 00:03:40.075
is 4,083.

00:03:42.242 --> 00:03:45.159
So now we know how to calculate Qc.

00:03:46.216 --> 00:03:48.073
So next we're going to talk about

00:03:48.073 --> 00:03:49.742
what it tells you.

00:03:49.742 --> 00:03:52.909
So there are three possible scenarios.

00:03:54.484 --> 00:03:56.036
So when Q

00:03:56.036 --> 00:03:57.203
is equal to K,

00:03:59.258 --> 00:04:02.175
that tells us we're at equilibrium.

00:04:03.990 --> 00:04:05.812
So, if at any point you're not sure

00:04:05.812 --> 00:04:08.140
if your concentrations are the

00:04:08.140 --> 00:04:09.759
equilibrium concentrations,

00:04:09.759 --> 00:04:10.927
you can calculate Q

00:04:10.927 --> 00:04:12.697
and check if it's equal to K.

00:04:12.697 --> 00:04:14.686
And in this case, it's not.

00:04:14.686 --> 00:04:16.548
So the other two possibilities are

00:04:16.548 --> 00:04:19.228
that Q is greater than K,

00:04:19.228 --> 00:04:20.744
which is the case here.

00:04:20.744 --> 00:04:22.744
Or Q can be less than K.

00:04:25.890 --> 00:04:29.557
So let's go through both
of those possibilities.

00:04:29.557 --> 00:04:32.605
We can draw all of the
possible values of Q

00:04:32.605 --> 00:04:34.146
on a number line

00:04:34.146 --> 00:04:35.146
or a Q line.

00:04:36.521 --> 00:04:38.305
So Q can have values anywhere

00:04:38.305 --> 00:04:39.745
from zero

00:04:39.745 --> 00:04:40.967
to infinity.

00:04:40.967 --> 00:04:43.234
When you have no product

00:04:43.234 --> 00:04:45.236
your numerator is zero

00:04:45.236 --> 00:04:46.713
and Q is equal to zero.

00:04:46.713 --> 00:04:47.736
So that tells us

00:04:47.736 --> 00:04:48.764
Q equals zero

00:04:48.764 --> 00:04:50.874
when you have all reactants

00:04:50.874 --> 00:04:52.207
and no products.

00:04:54.980 --> 00:04:57.730
And then if you have no reactants left

00:04:57.730 --> 00:04:59.710
and all products,

00:04:59.710 --> 00:05:02.298
we have zero in the denominator

00:05:02.298 --> 00:05:05.034
and that gives us a Q value of infinity.

00:05:05.034 --> 00:05:07.213
So that means at Q equals infinity,

00:05:07.213 --> 00:05:08.963
we have all products.

00:05:11.968 --> 00:05:14.764
Then we have a bunch of values in between.

00:05:14.764 --> 00:05:17.479
And I'm gonna just write
some intermediate values

00:05:17.479 --> 00:05:18.312
in here

00:05:19.179 --> 00:05:21.305
but the actual intermediate values here

00:05:21.305 --> 00:05:22.742
aren't super important.

00:05:22.742 --> 00:05:25.340
We're mostly gonna wanna
compare the relative

00:05:25.340 --> 00:05:27.235
values of our Q and K.

00:05:27.235 --> 00:05:29.568
So Q here is equal to 4,083,

00:05:32.231 --> 00:05:33.882
which I will place

00:05:33.882 --> 00:05:35.195
right around here.

00:05:35.195 --> 00:05:36.962
So that's Qc

00:05:36.962 --> 00:05:38.093
and our K

00:05:38.093 --> 00:05:39.232
in yellow

00:05:39.232 --> 00:05:40.725
is 4.3.

00:05:40.725 --> 00:05:44.509
So we'll place that right around here.

00:05:44.509 --> 00:05:46.041
So we can see that

00:05:46.041 --> 00:05:47.874
our Q is larger than K

00:05:48.972 --> 00:05:52.254
and it's closer to having all products.

00:05:52.254 --> 00:05:55.464
At the concentrations we have up here,

00:05:55.464 --> 00:05:57.874
we have way more products than we should

00:05:57.874 --> 00:05:59.471
at equilibrium.

00:05:59.471 --> 00:06:01.451
So our reaction is gonna try to adjust

00:06:01.451 --> 00:06:03.736
the concentrations to get to equilibrium.

00:06:03.736 --> 00:06:06.370
And what that means in
terms of our number line

00:06:06.370 --> 00:06:09.501
is that our concentrations are gonna shift

00:06:09.501 --> 00:06:12.001
so that Q can get closer to K.

00:06:13.524 --> 00:06:16.024
Since our shift is to the right,

00:06:16.024 --> 00:06:18.971
and it's moving towards all reactants,

00:06:18.971 --> 00:06:21.113
our reaction is going to favor reactants

00:06:21.113 --> 00:06:23.235
to get to equilibrium.

00:06:23.235 --> 00:06:25.780
So when Q is greater than K, like here,

00:06:25.780 --> 00:06:27.447
we're going to favor

00:06:28.682 --> 00:06:29.682
reactants...

00:06:33.606 --> 00:06:35.208
reactants.

00:06:35.208 --> 00:06:38.518
And then the last scenario
when Q is less than K

00:06:38.518 --> 00:06:40.596
our reaction will

00:06:40.596 --> 00:06:41.846
favor products.

00:06:43.957 --> 00:06:47.707
And we can show that
also on our number line.

00:06:49.922 --> 00:06:52.251
If we had a different
set of concentrations,

00:06:52.251 --> 00:06:54.251
where Q was less than K,

00:06:55.271 --> 00:06:57.014
which I will show

00:06:57.014 --> 00:06:58.847
using this color here.

00:07:00.456 --> 00:07:03.539
If we had, say, a Q value around here

00:07:05.262 --> 00:07:07.921
then our shift would be

00:07:07.921 --> 00:07:09.720
to the right

00:07:09.720 --> 00:07:12.484
towards making more products

00:07:12.484 --> 00:07:16.400
and therefore that would mean our reaction

00:07:16.400 --> 00:07:19.012
is gonna try to reach equilibrium

00:07:19.012 --> 00:07:21.261
by favoring the forward reaction.

00:07:21.261 --> 00:07:23.905
So that's how you calculate
Q and how you use it

00:07:23.905 --> 00:07:26.050
to see how the reaction concentrations

00:07:26.050 --> 00:07:27.737
will shift to get to equilibrium.

00:07:27.737 --> 00:07:29.476
In our next video, we'll go over

00:07:29.476 --> 00:07:31.617
an example problem using Q and trying

00:07:31.617 --> 00:07:34.073
to figure out how the
reactant concentrations

00:07:34.073 --> 00:07:35.873
will shift for another reaction.


Laplacian intuition

WEBVTT
Kind: captions
Language: en

00:00:00.744 --> 00:00:03.786
- [Voiceover] So here I'm
gonna talk about the Laplacian.

00:00:03.786 --> 00:00:04.619
Laplacian.

00:00:05.759 --> 00:00:07.942
And the Laplacian is a certain operator

00:00:07.942 --> 00:00:09.985
in the same way that the
divergence, or the gradient,

00:00:09.985 --> 00:00:13.050
or the curl, or even just
the derivative are operators.

00:00:13.050 --> 00:00:15.071
The things that take in
some kind of function

00:00:15.071 --> 00:00:16.650
and give you another function.

00:00:16.650 --> 00:00:19.506
So in this case, let's say we
have a multivariable function

00:00:19.506 --> 00:00:23.825
like F, that just takes in
a two-dimensional input,

00:00:23.825 --> 00:00:25.241
F of X Y.

00:00:25.241 --> 00:00:28.167
So you might imagine its graph
as being something like this,

00:00:28.167 --> 00:00:31.232
where the input space
is this X-Y plane here,

00:00:31.232 --> 00:00:34.286
so each of the points,
X Y, is a point here,

00:00:34.286 --> 00:00:35.609
and then the output is just given

00:00:35.609 --> 00:00:37.467
by the height of that graph.

00:00:37.467 --> 00:00:39.603
So the Laplacian of F is denoted

00:00:39.603 --> 00:00:41.600
with a right-side-up triangle,

00:00:41.600 --> 00:00:44.595
and it's gonna give you a
new scalar valued function

00:00:44.595 --> 00:00:45.431
of X and Y.

00:00:45.431 --> 00:00:47.034
So it's gonna give you a new function

00:00:47.034 --> 00:00:49.100
that takes in a two-dimensional input

00:00:49.100 --> 00:00:50.516
and just outputs a number.

00:00:50.516 --> 00:00:53.071
And it's kind of like a second derivative,

00:00:53.071 --> 00:00:54.510
because the way that it's defined

00:00:54.510 --> 00:00:57.018
is that you take the divergence

00:00:57.018 --> 00:00:58.268
of the gradient

00:01:00.781 --> 00:01:02.806
of your function F.

00:01:02.806 --> 00:01:04.083
So that's kind of how it's defined,

00:01:04.083 --> 00:01:05.824
the divergence of the gradient of F.

00:01:05.824 --> 00:01:08.252
And a more common notation
that you might see here

00:01:08.252 --> 00:01:11.476
is to take that upside-down
triangle, nabla,

00:01:11.476 --> 00:01:12.925
dot product,

00:01:12.925 --> 00:01:14.449
with nabla

00:01:14.449 --> 00:01:15.282
of F.

00:01:16.353 --> 00:01:18.791
So remember, if F is a
scalar valued function,

00:01:18.791 --> 00:01:21.369
then the gradient of F
gives you a vector field,

00:01:21.369 --> 00:01:22.715
a certain vector field.

00:01:22.715 --> 00:01:25.130
But the divergence of a vector field

00:01:25.130 --> 00:01:27.267
gives you another scalar valued function.

00:01:27.267 --> 00:01:29.797
So this is the sense in which
it's a second derivative.

00:01:29.797 --> 00:01:31.818
But let's see if we can kind
of understand intuitively

00:01:31.818 --> 00:01:33.466
what this should mean.

00:01:33.466 --> 00:01:35.324
'Cause the gradient, if you remember,

00:01:35.324 --> 00:01:37.576
gives you the slope of steepest descent.

00:01:37.576 --> 00:01:41.315
So it's a vector field
in the input space of X.

00:01:41.315 --> 00:01:43.613
And each one of the vectors
points in the direction

00:01:43.613 --> 00:01:45.773
that you should walk, such that this graph

00:01:45.773 --> 00:01:47.700
is kind of a hill on top of you,

00:01:47.700 --> 00:01:49.256
it tells you the direction you should go

00:01:49.256 --> 00:01:52.762
to increase your direction
the most rapidly.

00:01:52.762 --> 00:01:54.643
And if that seems unfamiliar,
it doesn't make sense,

00:01:54.643 --> 00:01:57.870
maybe go take a look at that
video on gradients and graphs

00:01:57.870 --> 00:01:59.798
and how they relate to each other.

00:01:59.798 --> 00:02:02.120
So with the specific graph
that I have pictured here,

00:02:02.120 --> 00:02:04.209
when you have kind of the top of a hill,

00:02:04.209 --> 00:02:05.812
all of the points around it,

00:02:05.812 --> 00:02:07.158
the direction that you should walk,

00:02:07.158 --> 00:02:09.248
is towards the top of that hill.

00:02:09.248 --> 00:02:10.920
Whereas when you have
kind of like the bottom,

00:02:10.920 --> 00:02:13.219
a little goalie here,

00:02:13.219 --> 00:02:14.658
all of the directions you should walk

00:02:14.658 --> 00:02:16.818
to increase the value of
the function most rapidly,

00:02:16.818 --> 00:02:19.102
are directly away from that value,

00:02:19.102 --> 00:02:21.912
which you might call a local minimum.

00:02:21.912 --> 00:02:24.188
So let's temporarily get rid of the graph,

00:02:24.188 --> 00:02:25.627
just so we can look at the gradient field

00:02:25.627 --> 00:02:27.020
here pretty clearly.

00:02:27.020 --> 00:02:28.739
And now let's think
about what the divergence

00:02:28.739 --> 00:02:30.457
is supposed to represent.

00:02:30.457 --> 00:02:33.127
So now the divergence, and
again if this feels unfamiliar,

00:02:33.127 --> 00:02:36.100
maybe go back and take a look
at the divergence videos,

00:02:36.100 --> 00:02:38.700
but that divergence has you
imagining that this vector field

00:02:38.700 --> 00:02:41.115
corresponds to some kind of fluid flow.

00:02:41.115 --> 00:02:43.623
So you imagine little
like water molecules,

00:02:43.623 --> 00:02:46.734
and at any given moment,
they're moving along the vector

00:02:46.734 --> 00:02:48.429
that they're attached to.

00:02:48.429 --> 00:02:50.217
So for example, if you
had a water molecule

00:02:50.217 --> 00:02:51.773
that started off kind of here,

00:02:51.773 --> 00:02:53.816
you would start by going along that vector

00:02:53.816 --> 00:02:55.604
and then kind of follow the ones near it,

00:02:55.604 --> 00:02:59.018
and it looks like it kind
of ends up in this spot.

00:02:59.018 --> 00:03:00.643
And a lot of the water molecules seem to

00:03:00.643 --> 00:03:02.292
kind of converge over there.

00:03:02.292 --> 00:03:05.115
Whereas over here, the water
molecules tend to go away

00:03:05.115 --> 00:03:06.681
when they're following those vectors

00:03:06.681 --> 00:03:08.469
away from this point.

00:03:08.469 --> 00:03:09.546
And when they go away like that,

00:03:09.546 --> 00:03:10.962
when you have a whole bunch of vectors

00:03:10.962 --> 00:03:12.379
kind of pointed away,

00:03:12.379 --> 00:03:14.376
that's an indication that
divergence is positive,

00:03:14.376 --> 00:03:16.001
because they're diverging away.

00:03:16.001 --> 00:03:18.230
So over here, divergence is positive.

00:03:18.230 --> 00:03:20.761
Whereas the opposite case,
where all of the water molecules

00:03:20.761 --> 00:03:22.549
are kind of coming in towards a point,

00:03:22.549 --> 00:03:24.569
that's where divergence is negative.

00:03:24.569 --> 00:03:26.172
And in another area, let's
say it was kinda like

00:03:26.172 --> 00:03:28.633
this center point, where you
have some water molecules

00:03:28.633 --> 00:03:30.189
that looks like they're coming in,

00:03:30.189 --> 00:03:31.744
but other ones are going out,

00:03:31.744 --> 00:03:33.857
and at least from this
picture, it doesn't seem like

00:03:33.857 --> 00:03:35.831
the ones going out are
doing so at a faster rate

00:03:35.831 --> 00:03:37.248
or slower than they are here.

00:03:37.248 --> 00:03:39.477
This would be roughly zero divergence.

00:03:39.477 --> 00:03:41.265
So now let's think
about what it might mean

00:03:41.265 --> 00:03:44.678
when you take the divergence
of the gradient field of F.

00:03:44.678 --> 00:03:46.977
So let me kind of clear
up the markings I made

00:03:46.977 --> 00:03:48.637
on top of it here.

00:03:48.637 --> 00:03:50.103
Points of high divergence,

00:03:50.103 --> 00:03:52.100
points where it diverges a lot here,

00:03:52.100 --> 00:03:54.167
why are those vectors pointing away?

00:03:54.167 --> 00:03:56.256
And if we pull up the graph again,

00:03:56.256 --> 00:03:57.464
the reason they're pointing away

00:03:57.464 --> 00:03:59.484
is 'cause the direction of steepest descent

00:03:59.484 --> 00:04:02.549
is kind of uphill everywhere;
you are in a valley.

00:04:02.549 --> 00:04:04.383
Whereas in the opposite circumstance,

00:04:04.383 --> 00:04:06.729
where divergence is highly negative,

00:04:06.729 --> 00:04:08.818
'cause points are converging towards it,

00:04:08.818 --> 00:04:10.188
why are they pointing towards it?

00:04:10.188 --> 00:04:11.674
Well this is a gradient field,

00:04:11.674 --> 00:04:13.439
so they're pointing towards that spot,

00:04:13.439 --> 00:04:16.133
because that's where anywhere around it,

00:04:16.133 --> 00:04:19.035
you should walk towards
that spot to go uphill.

00:04:19.035 --> 00:04:21.729
So in other words, the
divergence of the gradient

00:04:21.729 --> 00:04:24.260
is very high at points that are

00:04:24.260 --> 00:04:27.322
kind of like minima, at points
where everyone around them

00:04:27.322 --> 00:04:28.751
tends to be higher.

00:04:28.751 --> 00:04:31.236
But the divergence of the gradient is low

00:04:31.236 --> 00:04:33.674
at points that look more
like maximum points,

00:04:33.674 --> 00:04:35.253
where when you evaluate the function

00:04:35.253 --> 00:04:37.482
at all of the points
around that input point,

00:04:37.482 --> 00:04:38.805
they give something smaller.

00:04:38.805 --> 00:04:41.940
So this Laplacian operator
is kind of like a measure

00:04:41.940 --> 00:04:46.352
of how much of a minimum
point is this X Y.

00:04:46.352 --> 00:04:47.861
You will be very positive

00:04:47.861 --> 00:04:50.044
when F evaluated at that point

00:04:50.044 --> 00:04:51.831
tends to give a smaller value

00:04:51.831 --> 00:04:54.246
than F evaluated at
neighbors of that point.

00:04:54.246 --> 00:04:55.756
But it'll be very negative

00:04:55.756 --> 00:04:57.590
if when you evaluate F at that point

00:04:57.590 --> 00:04:59.819
it tends to be bigger than its neighbors.

00:04:59.819 --> 00:05:01.166
And this should feel kind of analogous

00:05:01.166 --> 00:05:03.999
to the second derivative
in ordinary calculus,

00:05:03.999 --> 00:05:05.138
when you have some kind of graph

00:05:05.138 --> 00:05:06.926
of just a single variable function,

00:05:06.926 --> 00:05:09.713
the second derivative of X

00:05:09.713 --> 00:05:11.733
will be low, it'll be negative

00:05:11.733 --> 00:05:14.217
at points where it kind of
looks like a local maximum.

00:05:14.217 --> 00:05:16.748
But over here, the second derivative of x

00:05:16.748 --> 00:05:19.209
would be positive at points that kind of

00:05:19.209 --> 00:05:20.626
look like a local minimum.

00:05:20.626 --> 00:05:22.530
SO in that way, the Laplacian is sort of

00:05:22.530 --> 00:05:25.014
an analog of the second derivative

00:05:25.014 --> 00:05:28.265
for scalar valued multivariable functions.

00:05:28.265 --> 00:05:30.671
And in the next video, I'll
go through an example of that.


Laplacian computation example

WEBVTT
Kind: captions
Language: en

00:00:00.398 --> 00:00:02.116
- [Voiceover] In the last
video, I started introducing

00:00:02.116 --> 00:00:04.903
the intuition for the Laplacian operator

00:00:04.903 --> 00:00:07.132
in the context of the
function with this graph

00:00:07.132 --> 00:00:09.384
and with the gradient
field pictured below it.

00:00:09.384 --> 00:00:11.009
And here, I'd like to go
through the computation

00:00:11.009 --> 00:00:12.681
involved in that.

00:00:12.681 --> 00:00:15.421
So the function that I
had there was defined,

00:00:15.421 --> 00:00:17.163
it's a two-variable function.

00:00:17.163 --> 00:00:20.112
And it's defined as f(x,y)

00:00:20.112 --> 00:00:21.969
is equal to three

00:00:21.969 --> 00:00:26.969
plus cos(x/2)

00:00:27.530 --> 00:00:32.530
multiplied by sin(y/2).

00:00:34.530 --> 00:00:36.459
And then the Laplacian which we define

00:00:36.459 --> 00:00:38.664
with this right side up triangle

00:00:38.664 --> 00:00:40.475
is an operator of f.

00:00:40.475 --> 00:00:42.960
And it's defined to be the divergence,

00:00:42.960 --> 00:00:44.702
so kind of this nabla dot

00:00:44.702 --> 00:00:48.904
times the gradient which
is just nabla of f.

00:00:48.904 --> 00:00:50.274
So two different things going on.

00:00:50.274 --> 00:00:51.877
It's kind of like a second derivative.

00:00:51.877 --> 00:00:53.246
And the first thing we need to do

00:00:53.246 --> 00:00:54.802
is take the gradient of f.

00:00:54.802 --> 00:00:55.708
And the way we do that,

00:00:55.708 --> 00:00:56.822
we kind of imagine expanding

00:00:56.822 --> 00:00:58.634
this upside down triangle

00:00:58.634 --> 00:01:01.675
as a vector full of partial
differential operators:

00:01:01.675 --> 00:01:03.742
partial partial x

00:01:03.742 --> 00:01:08.363
and partial partial y.

00:01:08.363 --> 00:01:09.965
And with a gradient,
you just kind of imagine

00:01:09.965 --> 00:01:11.753
multiplying that by the function.

00:01:11.753 --> 00:01:13.053
So if you imagine multiplying that

00:01:13.053 --> 00:01:16.095
by the function, what it looks like

00:01:16.095 --> 00:01:18.788
is just a vector full
of partial derivatives.

00:01:18.788 --> 00:01:20.391
So you're taking the partial of f

00:01:20.391 --> 00:01:24.222
with respect to x and the partial of f

00:01:24.222 --> 00:01:25.220
with respect to y.

00:01:25.220 --> 00:01:26.590
Those are the two different components

00:01:26.590 --> 00:01:29.482
of this vector-valued
function that is the gradient.

00:01:29.482 --> 00:01:31.415
And in our specific example,

00:01:31.415 --> 00:01:33.134
when we take the partial derivative of f

00:01:33.134 --> 00:01:35.154
with respect to x

00:01:35.154 --> 00:01:37.151
what we get, is we look over here.

00:01:37.151 --> 00:01:38.590
Three just looks like a constant,

00:01:38.590 --> 00:01:39.728
so nothing happens.

00:01:39.728 --> 00:01:41.377
Cos(x) halves.

00:01:41.377 --> 00:01:43.165
The derivative of that with respect to x,

00:01:43.165 --> 00:01:45.649
we kind of take out that 1/2.

00:01:45.649 --> 00:01:49.809
So 1/2, and the derivative of cos is -sin.

00:01:49.809 --> 00:01:54.736
So that's -sin(x/2).

00:01:56.441 --> 00:01:59.668
And sin(y/2), well, y just
looks like a constant.

00:01:59.668 --> 00:02:02.548
So sin(y/2) is just some other constant.

00:02:02.548 --> 00:02:05.636
So in our derivative, we just
keep that constant in there,

00:02:05.636 --> 00:02:10.112
that sin(y/2).

00:02:11.441 --> 00:02:12.718
And then, for the second component,

00:02:12.718 --> 00:02:14.367
the partial derivative of f

00:02:14.367 --> 00:02:15.574
with respect to y.

00:02:15.574 --> 00:02:16.712
Three still looks like a constant

00:02:16.712 --> 00:02:17.989
cause it is a constant.

00:02:17.989 --> 00:02:20.264
Now, cos(x/2) looks like a constant

00:02:20.264 --> 00:02:21.844
because as far as y is concerned,

00:02:21.844 --> 00:02:22.865
x is a constant.

00:02:22.865 --> 00:02:24.746
So cos(x) is a constant.

00:02:24.746 --> 00:02:28.833
But then, the sin(y)
has a derivative of cos.

00:02:28.833 --> 00:02:30.435
And we also take out that 1/2.

00:02:30.435 --> 00:02:32.757
So you take out that 1/2

00:02:32.757 --> 00:02:34.707
when you take the derivative of the inside

00:02:34.707 --> 00:02:36.774
and then the derivative of the outside

00:02:36.774 --> 00:02:40.071
is cosine of whatever was in there.

00:02:40.071 --> 00:02:43.136
So in this case, y/2.

00:02:43.136 --> 00:02:45.296
And we're multiplying it
by that original constant,

00:02:45.296 --> 00:02:47.362
cos(x/2).

00:02:47.362 --> 00:02:50.613
So, still we have our cos(x/2)

00:02:50.613 --> 00:02:55.489
since it was a constant times
a certain variable thing, x/2.

00:02:55.489 --> 00:02:57.440
So that's the gradient.

00:02:57.440 --> 00:02:58.717
And then, the next step here

00:02:58.717 --> 00:03:01.271
is to take the divergence of that.

00:03:01.271 --> 00:03:02.687
So with the divergence,
we're going to imagine

00:03:02.687 --> 00:03:05.543
taking that del operator
and dot producting

00:03:05.543 --> 00:03:06.565
with this guy.

00:03:06.565 --> 00:03:08.910
So if I scroll down to
give some room here,

00:03:08.910 --> 00:03:11.255
we're going to take that,

00:03:11.255 --> 00:03:13.067
that vector that's kind
of the same vector,

00:03:13.067 --> 00:03:14.205
the partial partial x.

00:03:14.205 --> 00:03:16.387
And I say vector, but vector-ish thing,

00:03:16.387 --> 00:03:19.104
partial partial y.

00:03:19.104 --> 00:03:19.940
And now we're going to take

00:03:19.940 --> 00:03:21.519
the dot product with this entire guy.

00:03:21.519 --> 00:03:24.146
So I'll go ahead and just copy it over.

00:03:26.901 --> 00:03:29.803
Just kind of copy it over here.

00:03:29.803 --> 00:03:31.382
And let's see.

00:03:31.382 --> 00:03:35.393
So, I need a little bit
more room to evaluate this.

00:03:35.393 --> 00:03:37.646
So here, when you imagine
taking the dot product,

00:03:37.646 --> 00:03:40.107
you kind of multiply these
top components together.

00:03:40.107 --> 00:03:41.384
So we're taking the partial derivative

00:03:41.384 --> 00:03:44.008
with respect to x of this whole guy.

00:03:44.008 --> 00:03:46.214
And when you do that, well you get,

00:03:46.214 --> 00:03:48.373
you still have that 1/2

00:03:48.373 --> 00:03:51.624
and then the derivative of -sin(x/2).

00:03:51.624 --> 00:03:53.760
So that 1/2 gets pulled out

00:03:53.760 --> 00:03:54.643
when you're kind of taking

00:03:54.643 --> 00:03:55.850
the derivative of the inside.

00:03:55.850 --> 00:03:59.891
And the derivative of -sin is -cos.

00:03:59.891 --> 00:04:02.723
So -cos of that stuff on the inside,

00:04:02.723 --> 00:04:06.113
that x/2.

00:04:06.113 --> 00:04:07.617
And of course, we still
multiply it by this.

00:04:07.617 --> 00:04:09.754
This looks like a constant, the sin(y/2).

00:04:09.754 --> 00:04:11.713
And we multiply by that,

00:04:11.713 --> 00:04:15.713
sin(y/2).

00:04:16.953 --> 00:04:18.275
And then we add that

00:04:18.275 --> 00:04:19.483
because it's kind of like a dot product.

00:04:19.483 --> 00:04:20.876
You add that to what it looks like

00:04:20.876 --> 00:04:23.105
when you multiply these
next two components.

00:04:23.105 --> 00:04:24.220
So we're going to add.

00:04:24.220 --> 00:04:25.961
And you have that 1/2.

00:04:25.961 --> 00:04:29.978
And then cos(y/2), when
we differentiate that

00:04:29.978 --> 00:04:31.371
you also pull out the 1/2.

00:04:31.371 --> 00:04:34.436
So again, you have that pulled-out 1/2.

00:04:34.436 --> 00:04:37.153
And the derivative of cos is -sin.

00:04:37.153 --> 00:04:40.520
So now we're taking -sin of,

00:04:40.520 --> 00:04:44.653
and then that stuff on the inside, y/2.

00:04:44.653 --> 00:04:46.882
And we continue multiplying
by the constant.

00:04:46.882 --> 00:04:50.017
As far as y is concerned,
cos(x/2) is a constant.

00:04:50.017 --> 00:04:55.006
So we multiply it by that, cos(x/2).

00:04:56.095 --> 00:04:58.278
And then that, so that is the divergence

00:04:58.278 --> 00:04:59.671
of that gradient field.

00:04:59.671 --> 00:05:01.343
So the divergence of the gradient

00:05:01.343 --> 00:05:05.523
of our original function
gives us the Laplacian.

00:05:05.523 --> 00:05:07.264
And in fact, we could
simplify this further

00:05:07.264 --> 00:05:08.286
because both of these terms

00:05:08.286 --> 00:05:09.516
kind of look identical.

00:05:09.516 --> 00:05:10.747
But the main point of this video

00:05:10.747 --> 00:05:12.721
is kind of how you go through that process

00:05:12.721 --> 00:05:14.718
where you imagine taking the gradient

00:05:14.718 --> 00:05:17.063
of your function and then
the divergence of that.

00:05:17.063 --> 00:05:19.547
And that's what the Laplacian is.

00:05:19.547 --> 00:05:20.529
See you next video.


Explicit Laplacian formula

WEBVTT
Kind: captions
Language: en

00:00:00.325 --> 00:00:02.277
- [Voiceover] So let's say you
have yourselves some kind of

00:00:02.277 --> 00:00:03.299
multivariable function,

00:00:03.299 --> 00:00:05.983
and this time its got some
very high dimensional inputs.

00:00:05.983 --> 00:00:09.983
So x1, x2 on and on and
on up to sub x sub n,

00:00:11.154 --> 00:00:12.781
for some large number n.

00:00:12.781 --> 00:00:14.979
In the last couple videos I told you about

00:00:14.979 --> 00:00:16.565
the Laplacian operator,

00:00:16.565 --> 00:00:19.695
which is a way of taking in
your scalar valued function f

00:00:19.695 --> 00:00:21.563
and it gives you a new
scalar valued function

00:00:21.563 --> 00:00:23.474
that's kind of like a
second derivative thing

00:00:23.474 --> 00:00:26.647
because it takes the
divergence of the gradient

00:00:26.647 --> 00:00:27.991
of your function f.

00:00:27.991 --> 00:00:30.190
So the gradient of f
gives you a vector field

00:00:30.190 --> 00:00:33.199
and the divergence of that
gives you a scalar field.

00:00:33.199 --> 00:00:35.229
And what I want to show
you here is another formula

00:00:35.229 --> 00:00:37.954
that you might commonly
see for this Laplacian.

00:00:37.954 --> 00:00:40.023
So first let's kind of
abstractly write out

00:00:40.023 --> 00:00:42.259
what the gradient of f will look like.

00:00:42.259 --> 00:00:44.577
So we start by taking this del operator,

00:00:44.577 --> 00:00:47.308
which is going to be a vector
full of partial differential

00:00:47.308 --> 00:00:48.141
operators.

00:00:48.141 --> 00:00:49.867
Partial with respect to x1,

00:00:49.867 --> 00:00:52.117
partial with respect to x2,

00:00:53.362 --> 00:00:55.553
and kind of on and on and on up to

00:00:55.553 --> 00:00:59.900
partial with respect to
that last input variable.

00:00:59.900 --> 00:01:03.031
And then you kind of just
imagine multiplying it

00:01:03.031 --> 00:01:04.533
by your function,

00:01:04.533 --> 00:01:06.239
so what you end up getting

00:01:06.239 --> 00:01:08.882
is all the different
partial derivatives of f.

00:01:08.882 --> 00:01:12.665
It's partial of f with
respect to the first variable,

00:01:12.665 --> 00:01:14.453
and then kind of on and on and on

00:01:14.453 --> 00:01:18.354
up until you get the
partial derivative of f

00:01:18.354 --> 00:01:20.634
with respect to that last variable,

00:01:20.634 --> 00:01:21.467
x sub n.

00:01:24.693 --> 00:01:26.077
And the divergence of that

00:01:26.077 --> 00:01:29.125
and just to save myself
some writing I'm gonna say

00:01:29.125 --> 00:01:31.198
you take that nabla operator,

00:01:31.198 --> 00:01:33.641
and then you imagine
taking the dot product

00:01:33.641 --> 00:01:35.310
between that whole operator

00:01:35.310 --> 00:01:38.033
and this gradient vector
that you have here,

00:01:38.033 --> 00:01:39.817
what you end up getting is

00:01:39.817 --> 00:01:42.545
well, you start by multiplying
the first components

00:01:42.545 --> 00:01:44.532
which involves taking
the partial derivative

00:01:44.532 --> 00:01:45.872
with respect to x1,

00:01:45.872 --> 00:01:48.839
that first variable of the
partial derivative of f

00:01:48.839 --> 00:01:50.708
with respect to that same variable.

00:01:50.708 --> 00:01:54.730
So it looks like the second
partial derivative of f

00:01:54.730 --> 00:01:57.730
with respect to that first variable.

00:01:59.028 --> 00:02:01.592
So the second partial derivative
of f with respect to x1,

00:02:01.592 --> 00:02:03.097
that first variable.

00:02:03.097 --> 00:02:05.215
Then you imagine kind of adding

00:02:05.215 --> 00:02:08.342
what the product of these
next two items will be

00:02:08.342 --> 00:02:10.497
and for very similar reasons
that's gonna look like the

00:02:10.497 --> 00:02:13.952
second partial derivative
of f with respect

00:02:13.952 --> 00:02:15.580
to that second variable,

00:02:15.580 --> 00:02:17.163
partial x2 squared.

00:02:18.304 --> 00:02:19.522
And you do this to all of them

00:02:19.522 --> 00:02:21.963
and you're adding them all
up until you find yourself

00:02:21.963 --> 00:02:23.505
doing it to the last one.

00:02:23.505 --> 00:02:25.545
So you've got plus and
a whole bunch things

00:02:25.545 --> 00:02:29.367
and you'll be taking the
second partial derivative of f

00:02:29.367 --> 00:02:31.766
with respect to that last variable,

00:02:31.766 --> 00:02:33.599
partial of x sub n.

00:02:34.649 --> 00:02:37.491
This is another format in which
you might see the Laplacian,

00:02:37.491 --> 00:02:40.295
and often times it's
written kind of compactly,

00:02:40.295 --> 00:02:43.426
so people will say the
Laplacian of your function f,

00:02:43.426 --> 00:02:45.946
is equal to, using sigma notation,

00:02:45.946 --> 00:02:49.696
you'd say the sum from
i is equal to 1 up to,

00:02:51.998 --> 00:02:53.464
you know, 1, 2, 3 up to n.

00:02:53.464 --> 00:02:55.495
So the sum from that up to n,

00:02:55.495 --> 00:02:57.608
of your second partial derivatives.

00:02:57.608 --> 00:03:01.358
Partial squared of f
with that i-th variable.

00:03:03.922 --> 00:03:06.437
So if you were thinking in
terms of three variables

00:03:06.437 --> 00:03:09.448
often x1, x2, x3 we instead write x, y, z,

00:03:09.448 --> 00:03:12.578
but is common to more
generally just say x sub i.

00:03:12.578 --> 00:03:15.706
So this here is kind of
the alternate formula

00:03:15.706 --> 00:03:17.780
that you might see for the Laplacian.

00:03:17.780 --> 00:03:19.565
Personally I always like to think about it

00:03:19.565 --> 00:03:23.224
as taking the divergence
of the gradient of f,

00:03:23.224 --> 00:03:25.134
because you're thinking
about the gradient field,

00:03:25.134 --> 00:03:27.369
and the divergence of
that kind of corresponds

00:03:27.369 --> 00:03:30.250
to maxima and minima of
your original function,

00:03:30.250 --> 00:03:31.554
which is what I talked about

00:03:31.554 --> 00:03:33.872
in the initial intuition
of Laplacian video.

00:03:33.872 --> 00:03:36.535
But this formula is probably
a little more straightforward

00:03:36.535 --> 00:03:39.049
when it comes to actual computations and,

00:03:39.049 --> 00:03:43.398
oh wait sorry I forgot a
square there, didn't I?

00:03:43.398 --> 00:03:46.696
Partial x squared, so this
is the second derivative.

00:03:46.696 --> 00:03:50.074
So summing all these
second partial derivatives.

00:03:50.074 --> 00:03:51.704
And you can probably see this a kind of

00:03:51.704 --> 00:03:53.863
a more straightforward way
to compute a given example

00:03:53.863 --> 00:03:55.490
that you might come across,

00:03:55.490 --> 00:03:57.685
and it also makes it
clearer how the Laplacian

00:03:57.685 --> 00:04:01.590
is kind of an extension of the
idea of a second derivative.

00:04:01.590 --> 00:04:02.574
See you next video.