OK. Let's get started. We're going to complete our
discussion about mitosis and meiosis today.
And then get into a topic called Mendelian Genetics,
understanding genetic principles. And we'll give you as examples the
work of the famous Austrian monk Gregor Mendel who sort of set down
the rules that we still follow today regarding simple genetic inheritance.
So I mentioned to you last time, and I think you all know that your
genes are carried on chromosomes. And chromosomes come in different
sizes and shapes. You have 46 total chromosomes in
all of your cells. Those chromosomes come from your
parents. You get one set from one parent and another set from another
parent, so you have 22 pairs plus the two sex chromosomes.
And the chromosomes are made of DNA, and along those DNA sequences are
your genes scattered along the length of the chromosomes.
And there are about 30,000 genes in total.
Now, I've also told you that the genome, the whole sequence of your
DNA has been determined over the last couple of years.
We now know that in great detail. We know every single nucleotide of
the three times ten to the ninth base pairs of DNA,
and we can actually go to a website and look at it.
You can do this. It's publicly available,
sponsored by the government, but it's a little like hacking into
the database of humanity. So if you go to this website that I
just showed you, you'll find access to the genomes of
many organisms. This is an evolutionary tree which
shows the relationship of these organisms from an evolutionary
perspective. There are nine mammalian genomes that are present
within this database. If you clink on that link to
homosapiens, that's us, assuming this is functioning.
I had this working. I'm not sure why it's not working.
I'm getting a good signal here. Oh well. Had this worked,
you would have seen an alignment, a diagram of all the human
chromosomes, one through 22, plus X plus Y. You could then click
on chromosome whatever you're interested, let's say chromosome one.
It would show a diagram of chromosome one from end to end.
And you could then click on the sequence of chromosome one.
All two point four times ten to the eighth base pairs of chromosome one.
It would show you all of the genes on chromosomes one.
You would know exactly where they were. And you could then click on a
particular gene and look at its sequence. And it would actually
give you information about that gene, as much information as is known.
So, we know the nucleotide sequence of all the genes.
And the question is, is that sequence that's in that
database applicable to all of you? Is that the sequence of your DNA?
What do you say? Why do you say no? Right. So it's somebody's DNA.
Actually, the genome sequence that's present in a publicly
available database is a collection of a small number of people's DNA,
but it's a representation of the human genome. And you all have
subtle differences. In between genes quite a few,
and within your genes some. And we call those differences within genes
allelic differences, you have different alleles,
and that's what makes you all different, which alleles you carry,
and importantly which combinations of alleles of genes that you carry.
OK. I apologize for that not working. I don't know why it didn't.
All right. Well, last time we left off talking about
the fact that you inherit your chromosomes through the process
of fertilization. Fusion of sperm and egg produces a
diploid cell from two haploid cells. This diploid cell has two copies of
all the chromosomes, two copies of all the genes,
and then through the process of development there is a great deal of
mitosis. We talked about mitosis in some detail last time.
The process of chromosome duplication followed by chromosome
separation, chromatid separation really, which allows daughter cells,
during this process of mitosis to faithfully inherit all of the DNA.
And this process continues through development, and it also continues
in adults. It's not over when the baby is born. There's a lot of
mitosis that takes place in many of your organs, and your intestines for
example, your blood. Not in all of your organs.
In the brain, for example, there is relatively little mitosis.
Once the brain forms, cells don't divide anymore.
It's true of the cardiac muscle as well in the heart.
But many tissues do continue to divide. So this is happening inside
of you as we speak. Now, what I want to turn to for
today, though, is to talk about this process.
How do you go back to the beginning? How do you generate haploid cells
from diploid cells? And this occurs through a related
process called meiosis. Now, meiosis takes place at
different times, depending on whether you're male or
female. As I mentioned last time in males meiosis takes place
in puberty. So diploid germ cells are set aside
in embryogenesis, but they don't undergo this process
of meiosis until males reach puberty at which point it happens very
abundantly, as I said, about a million or a couple of
million meiotic cells produced per hour. In females meiosis actually
begins in embryogenesis, quite amazing. The cells are set
aside and they start to undergo meiosis. And they don't make it all
the way through. You don't make haploid germ cells
in embryogenesis. They get stuck partway through.
Then in puberty, during ovulation a subset of those
cells continue through the subsequent stages of meiosis.
Then they get stuck again. And only at fertilization does meiosis
get completed, a haploid cell is produced which is
then fused with the sperm cell, a diploid cell is generated and
development proceeds. So what is meiosis and how does it
compare to mitosis?
Well, they're related in the sense that both of them follow S phase.
If you remember during S phase chromosomes get duplicated.
And during chromosome duplication two chromatids are produced.
You're not going to start singing,
are you? No. Just kidding. When a chromosome gets produced two
chromatids are generated and then they stick together at the
centromere. Remember that? Now, in mitosis, as we talked about
before, this is followed by one round of chromatid separation.
That's all it is. The chromosomes with their two
chromatids line up along the metaphase plate,
and they get pulled apart during anaphase and telophase.
And importantly the homologs don't pay any attention to one another.
The maternal copy of chromosome one and the paternal copy of chromosome
two ignore each other in mitosis. They could be anywhere along that
metaphase plate. It doesn't matter.
They ignore one another. OK? That's relevant because it's
different in meiosis.
In meiosis, also following one round of chromosome duplication and the
generation of two chromatids per chromosome, this involves one round
of chromosome separation. And when I say chromosome here,
I'm talking about the chromosome which is at this point composed of
two chromatids. In this case the homologs do pay
attention to one another and they separate from one another.
Followed by one round of chromatid separation.
Which is very similar to what happens over here.
OK? So the way you go from having 46 chromosomes to 23 chromosomes is
you go through one round of duplication but two rounds of
separation. The way you maintain 46 chromosomes in mitosis is to go
through one round of duplication and only one round of separation.
Make sense? OK. So let's look at that in detail.
And I won't draw these pictures on the board for you because,
actually, I want to review it in writing one more time.
I'm not going to draw the details of meiosis for you on the board
because it just takes too long, and the book does a perfectly good
job diagramming it for you. And I'll show you those diagrams in
a second. The terms that we use in meiosis are the same terms that we
use in mitosis. But there are two rounds,
as I mentioned, and they're broken down by meiosis one and meiosis two.
In meiosis one you have a prophase, and it's called prophase one.
You have a prometaphase, prometaphase one.
A metaphase where the chromosomes align on the metaphase plate.
And anaphase where the chromosomes get pulled apart by the mitotic
spindle, actually, in this case the meiotic spindle.
And finally a telophase where they get pulled all the way to the poles.
And then there's a second round, meiosis two, where the same terms
are applied but they're denoted with twos.
So there's a second phase of prophase, of prometaphase,
metaphase, anaphase and telophase. And this is where the chromatids
align on the chromosome plate and get separated.
OK? Now, there's a very key event, and I'm going to draw it for you
because it's so important, that takes place during prophase of
meiosis one. And this is the distinguishing feature between
meiosis and mitosis. As I said in mitosis,
the homologs, the two copies of chromosome one,
the maternal one and the paternal one for example ignore one another.
That's not true in mitosis, in meiosis. In meiosis they bind to
one another. So imagine again our, I think I called this the paternal
copy of chromosome whatever, one previously. It had four genes
that we showed previously. It's represented as two chromatids,
which are associated at the centromere. And then you have
another homolog, the maternal homolog.
It likewise has the same four genes. It's been duplicated so it's
represented as two chromatids held together at the centromere.
In prophase of meiosis one, these homologs pair.
And it's the pairing of the homologs
that allows the two to separate faithfully during meiosis one.
They also interact in an interesting way that we'll come to
in a moment. They don't just pair. They actually undergo interactions
between each other which allows them to exchange sequences which is
critical for generating diversity amongst our chromosomes.
So now I'm going to take you through these steps as shown in your book.
This is figure 9.14 in your book. We're going to go through meiosis
one first. So in prophase of meiosis one, we've undergone
chromosome duplication already into the chromatids.
Now the chromosomes are going to condense, and you can see that here.
And importantly during prophase the two chromosomes,
represented by the four chromatids align.
In this picture you're seeing two different chromosomes,
a larger one and a smaller one. And you can see that the maternal
and paternal copies have paired with one another. OK?
So we get pairs of homologs. Still later in prophase they
interact in this other fashion that I just alluded to where they
actually exchange information. They exchange genetic information.
And we'll go through this in a little bit more detail in a bit.
At metaphase of meiosis one, metaphase one,
the homologs line up on the metaphase plate with one homolog to
the left, the other homolog to the right, one homolog to the left,
the other homolog to the right. This is one of two possible
configurations that could take place here. In fact,
in another cell undergoing metaphase one, the red one might be on the
left, the blue one on the right, but here the blue one might be one
the left, the red one might be on the right.
And that's important. You'll see why later.
So the homologs line up along the metaphase plate.
And now during anaphase one the homologs separate from one another.
The two chromatids of each homolog separate from one another,
and that's completed in telophase. Now, without an intervening round
of DNA duplication, this does not involve another round
of S phase, we go straight into meiosis two where again the DNA
condenses, a mitotic spindle is built.
And now, just like in mitosis, the two chromatids line up along the
metaphase plate with one chromatid to the top, the other chromatid to
the bottom, one chromatid to the top, the other to the bottom.
And then they get separated during anaphase and telophase
of meiosis two. The end product of that are germ
cells which are now haploid. They have only one copy of each of
the chromosomes, one of the big ones and one of the
small ones. So that's how you go from a cell that has two copies of
each to a cell that has one copy of each. One round of DNA duplication,
two rounds of chromosome or chromatid separation.
Now, importantly you'll see if you can squint a little bit,
that the chromosomes that come out of this process of meiosis don't
always look, in fact, never look like the chromosomes that
come in. They undergo this process of exchange of genetic information.
This process of exchange of genetic
information is called meiotic recombination.
And I'll show it to you in a figure
in a second. But it generates first recombinant chromatids,
chromatids that look different from the chromatids that were generated
initially. That's why they're called recombinant.
And these, when resolved during meiosis two, generate recombinant
chromosomes.
So, again, if we imagine our two homologs of a given chromosome with
the four genes that we talked about previously lined up along the length
of those chromosomes, during this process of meiotic
recombination --
-- new versions of the chromosomes are produced. We can generate a
chromosome which has a little bit of Dad's DNA and a little
bit of Mom's DNA. A chromosome that was generated
specifically in you. It's your unique contribution to
the offspring. It's not solely what you got from
Mom or Dad. It's your unique version of that chromosome that's a
hybrid between what you got from Mom and what you go from Dad.
And this process of meiotic recombination allows for the
increased diversity in the population. You don't just pass on
what you inherited. You actually mix it up a little bit
and pass on new combinations of the alleles that you got from your
parents. So this is shown in greater detail here.
And, again, gone through in much detail in your book.
This is figure 9. 6. Here we are in prophase of
meiosis one. The homologs have paired and they've undergone a
genetic interaction. The DNA of the paternal and the
maternal chromatids have literally exchanged information.
That's what this crossover is called. It's referred to as a
chiasma. And you can literally see it in the electron microscope.
When this gets resolved, like a cut and paste reaction.
When this gets resolved, you generate new chromatids.
This chromatid has mostly red sequence but a little bit of blue
sequence. And this chromatid has mostly blue sequence and a little
bit of red sequence. If you think about that with
respect to genes in our analogy over here, the red chromosome had alleles
that we'll refer to as the white versions of gene one,
two, three and four. The blue homolog had the black
versions of genes one, two, three and four. The products
of this reaction include chromatids that look just like the parental
ones, a red chromosome with white one, two, three and four,
a black chromosome with, a blue chromosome with black one,
two, three and four, but also new chromatids that didn't exist
previously, a red one with white one, two, three but a black four,
and a blue one with black one two and three and a white four.
New versions of the chromosomes. And this is important. It's
important for you to understand how genetics works,
but it's also important for the species to generate increased
diversity which when acted upon by natural selection forces might
select out more fit organisms. So that is mitosis, meiosis and
meiotic recombination. And we'll be drawing on your
knowledge of meiosis and mitotic recombination to understand the
principles of genetics. And that will be the next topic
that we turn to. Before I do so I want to just
mention that this process of meiosis doesn't always work perfectly.
No biological system works perfectly. There's always some
error rate. And that's true in meiosis in germ cell development,
although it's supposed to be the case that during metaphase one the
homologs, which are paired together, separate to the two daughters.
That doesn't always happen. Occasionally errors take place such
that one daughter gets nothing and the other daughter gets both copies.
This is a problem because now when these guys go through meiosis two
they have two copies of a given chromosome instead of just one.
And if those germ cells, in this case they're eggs,
and that's usually the problem in humans where it occurs,
if those eggs get fused by a sperm that carries its own copy of that
chromosome the zygote, instead of having just two copies,
has three. And this is a situation called trisomy,
an extra copy of a given chromosome. The opposite can happen as well.
This guy, this egg, if it were to be fertilized would
have only a single copy of that given chromosome.
That would be monosomy, single instead of two. This does
happen during human development. In the vast majority of cases when
it happens it's incompatible with early development and the fetus
aborts. There are a few examples where the fetus can actually make it
quite late in gestation and even be born but then doesn't
thrive thereafter. But there's one example where the
fetus comes to term, is born and the individual can
survive, and that's chromosome 21 trisomy or Down syndrome.
This happens when during meiosis one nondisjunction takes place such
that two copies of chromosome 21 wind up in an egg.
Sperm comes along and delivers another copy of chromosome 21.
Now that zygote is trisomic for chromosome 21.
And you can see the karyotype here.
Everything else is diploid but this has three copies of chromosome 21.
And this leads to a characteristic defect in development which results
in an individual with very characteristic features.
And this extra copy of 21 is responsible for this too much gene
product from the genes on chromosome 21. And I don't remember how many
there are. There are 600 or so genes on chromosome 21.
Having too much of some or some of them, some combination of them,
too much product from those leads to defects in development,
which results in this very clear phenotype.
So meiosis usually insures that you get just one copy of each
chromosomes, but it doesn't always work. And this can be one
consequence. OK. Let's take a breath and change
gears. So we're going to move now from mitosis and meiosis to genetics
and genetic principles, and specifically Mendelian genetics.
So what we want to now begin to understand is besides the mechanics,
what are the consequences? What happens when you inherit alleles of
given genes? What is the effect of the inheritance of those alleles and
what are the principles that guide that? Why is it that when a child
is born of two blue-eyed people that child has blue eyes?
When a child is born of a brown-eyed person and blue-eyes
person that child is likely to have brown eyes, but not always.
Maybe they'd have blue eyes. What governs the presentation of a
given trait based on the genes that that individual inherits?
Or more precisely the alleles of the genes that individual inherits.
You might actually be able to see that the murderer can be seen in the
reflection of this person's eye right here. You've got to look
carefully. We can learn a lot about this by looking at inheritance
patterns in humans. And that's increasingly powerful
given our knowledge of the human genome today. There's a lot more we
can do in human genetics today than we could have done ten years ago.
But the field has actually been brought along to this point using
genetics in model organisms. And this is a fly, a fruit fly,
drosophila melanogaster, a major genetic organism in biology.
And you can see again that you can isolate flies with different colored
eyes, red eyes, white eyes, dark eyes.
And you can understand the principles of genetics by crossing,
mating flies together and looking at what happens in the eye color of the
offspring of flies that start off with a given eye color.
Now, the principles that guide us in our thinking about genetics derive
from this individual here whose name is Gregor Mendel.
And the principles that he laid down are referred to as Mendelian
inheritance. And he, through work that I'll briefly
summarize, developed certain laws of inheritance which turned
out to be true. And we use them even now as we think
about how genes and alleles get passed on through the generations.
Mendel lived in the 1800s. He did his work around 1850s,
1860s. His organism was the pea plant and the peas that give rise to
the pea plant. And he spent a lot of time
observing peas and pea plants, breading or crossing pea plants
together, looking at the products of those crosses and coming
up with these theories. He published a paper on that work in,
I think, 1865, and it was roundly ignored.
Nobody paid any attention. I should have mentioned that he was an
Austrian monk. He lived up in the hills in Austria
and did his work largely in seclusion. But he did,
in fact, publish a paper in 1865 reporting his principles of
inheritance. And he was largely ignored until about 50 years later
when other workers started to do similar things and basically
rediscovered these principles that he had laid down so many years
before. And we now give him credit for coming up with these principles
in the first place. So Mendel focused on traits,
traits that he could observe in the pea plant or in the peas themselves.
And this is an example of the traits that he might look at.
You may know that peas can come in different shapes and textures.
There are peas that are smooth. There are peas that are wrinkled.
And Mendel wondered what controlled whether a pea was smooth
or wrinkled. If you crossed a pea plant that
would have produced smooth peas with a pea plant that would have produced
wrinkled peas, do you get smooth or wrinkled peas?
The principles that guided the thinking previously were that this
was some sort of a mixture. If you had one type and another
type, the offspring would have some intermediate type.
There was sort of a mixing of genetic information,
not really referred to as genetic information at the time,
but heritable information. And Mendel wondered whether that
was true. And then he actually determined that
for most things that was not true. One type determined the appearance
of the trait in the offspring. In order for you to understand what
Mendel did, you have to understand a little bit about pea plants and peas.
One thing you need to know is that peas are the embryo.
They are the product of fertilization.
And when you plant a pea it will develop into a pea plant.
And you can score traits either in the pea itself,
in the embryo, smooth, wrinkled, dark green, light green,
or in the pea plant that results from that pea.
Does it have red flowers or white flowers or pink flowers?
And importantly you also can fertilize one pea plant from another.
You can take the germ cells, the male germ cells or male gametes
from one pea plant, and purposely fertilize a female
gamete of another pea plant. You can carry out these very precise
crosses. Or you can do it within the same pea plant.
You can take the male germ cells or gametes and fertilize the female
gametes from that same plant, so-called self-pollination. OK?
And through this methodology, Mendel was able to very precisely
control what the pea plants looked like and how he could
manipulate them.
Through this process of self-pollination,
Mendel was able to generate what he called “pure breading strains”.
That is they always produced the same trait. If you cross them
together, if you fertilize, self-fertilize them, you always got
the same result, always smooth peas or always
wrinkled peas or always red flowers or always white flowers.
They were pure. There wasn't any heterogeneity.
And we're going to refer to these in our discussion as
parental strains.
And kind of an example of a question that Mendel would want to know is if
you took two parental strains and he crossed pollinated from one that
always produced smooth peas, and here we're talking about this X
represents crossing fertilization, with a plant that always produced
wrinkled peas, what did the offspring look like?
What did the pea look like in the product of that cross?
These parental strains are referred to as P. The product of a cross
between two parental strains is called the F1 or first
filial generation.
And the question Mendel wanted to know was what is the P type in the
F1? The answer turns out, for this particular example,
to be smooth peas. And the question is why? Based on this kind of
observation, Mendel generated a hypothesis.
He suggested that traits in the peas,
as well as in the subsequent plants, arise from the inheritance of two
units. These units we would think of now as
two alleles of a given gene. He wasn't thinking of genes or
alleles. He was just thinking of what number of things was
contributing to this particular trait. And that these two units
were delivered to the embryo from each parent.
And importantly the parent then has
two units so that each parent delivers one of its two alleles to
the offspring via the production of germ cells.
So everybody has two. They pass on one of those two to
their germ cells. And then the embryo is the product
of two germ cells coming together generating, once again,
something with two units. OK?
So let's think about this with respect to pea plants,
and specifically the shape of the peas. So the trait that we're
interested in is smooth or wrinkled peas.
The gene, in our language,
is the S gene. It is going to determine whether or not the pea is
smooth or wrinkled. And this gene comes in two
varieties, two alleles. One is called big S and the other
is called little S. In the cross that was done,
the plant that always generated smooth peas, which was a pure
breeding strain, had the same allele in both copies.
It was diploid for the big S allele. Both of its chromosomes that carry
the S gene had the big S version of that gene.
And the wrinkled plant, the plant that produced wrinkled
peas carried two copies of the other allele, little S.
These plants produced smooth peas. These plants produced wrinkled peas.
When these plants produced germ cells, what allele of this gene gets
put into those germ cells? Big S. It's the only one there.
So when these plants undergo meiosis they're going to produce
germ cells that carry in them the big S gene. When these plants
undergo meiosis what allele of the S gene did they put in their
germ cells? Little S. These haploid germ cells have the
little S gene. Now, when I cross,
I take one of these germ cells, mix it with one of those germ cells
in the cross, what gene, what genes, what alleles does that
offspring inherit? It gets a big S from here and it
gets a little S from here. OK? And what do I observe when I
make an organism, a pea, that looks like that?
What does the trait look like? It's smooth. OK.
Now, I've drawn details without given you some of the relevant
nomenclature. The parental strain over here had big S,
big S. We call that the genotype. A genotype refers to which alleles
you have; your genotype. The genotype of the other parental
strain is S, S. And the genotype of the offspring is
big S, little S. That's the genotype.
If you have two of the same alleles it is called being homozygous.
This also has two of the same alleles so it is also homozygous for
the other allele but still homozygous.
If you have one allele of one type and the other allele you are called
heterozygous. OK. So that's some relevant terminology.
Now, in contrast to the genotype, we also refer to the phenotype. And
the phenotype refers to the manifestation of the trait.
What you look like. Regardless of what the gene
combination or allele combination you have, what do you actually look
like? So what is the phenotype of these peas? Smooth.
And what is the phenotype of these peas? They're wrinkled.
And what is the phenotype of these peas? They're smooth.
Based on the patterns that he observed when he crossed two pure
breading strains and generated offspring that resembled one of the
two pure breading strains, Mendel suggested that in this
particular cross the big S unit, we would call it allele, is dominant
over little S. If you have a big S allele,
regardless of what you have as the other allele, you're going to have
the big S trait which is smooth. Big S is dominant over little S.
And a related term, little S is recessive to big S.
The only way you observe the phenotype associated with the little
S allele is if you're homozygous for the little S. If you're
heterozygous, as in the case in the middle, you manifest the phenotype
associated with the other allele. Dominance and recessive.
OK. Now, based on these ideas of inheritance of a single unit or
allele from the parents who carry two and the concept of dominance and
recessiveness, you can come up with methods for
determining the frequency of observing genotypes and phenotypes
in crosses such as this. So I want to consider now not a
parental cross giving rise to an F1, but rather a cross between F1. We
call this a hybrid cross. And because we're focusing on one
gene we actually call it a monohybrid cross.
The genotypes of the plants that
we're going to start with in the F1 generation are as we described
previously. They're heterozygous big S, little S.
And that's true for one plant and the other.
They're both F1s. They're both heterozygous for big S,
little S. If we now cross these together, we can think about what
germ cells these plants produce. So what are the four products of
meiosis look like from this? What do you get in the meiotic
products starting with these genotypes?
Two of them get big S, two of them get little S.
And likewise over here. Two of the germ cells get big S,
two of the germ cells get little S. Therefore, half of the germ cells
are big S, half are little S. If I think about the frequency of
big S germ cells it's one-half. If I think of the frequency of
little S germ cells it's one-half. And likewise over here, half of the
germ cells are big S, half of the germ cells are little S.
Based on this idea I can generate a grid --
-- which will allow me to calculate
the frequency of offspring that have four different combinations.
If this germ cell combines with this germ cell I'll be big S,
big S. If this germ cell combines with this germ cell the genotype
will be big S, little S. If this germ cell
combines with this germ cell the genotype will be little
S, little S. And if this germ cell combines with
this germ cell the genotype will be big S, little S.
This box, which is a useful device for calculating the frequency of the
genotypes that you observe in different crosses,
is referred to as a Punnett Square.
So let's think about what the numbers would be in such a
monohybrid cross. There are three possible genotypes.
If you look at the Punnett Square there are three possible genotypes.
You can be homozygous big S, big S. You can be homozygous little S,
little S. Or you can be heterozygous big S, little S.
Those are the three possible genotypes from a cross such as this.
The ratio of those genotypes, if you look in the Punnett Square is
there is one of these for every one of these and two of these.
The ratio of the three genotypes is 1:2:1 as determined by this
kind of calculation. The reason that you get that ratio
is that a quarter of these, sorry. Half of these germ cells are
big S, half of these germ cells are big S, so a quarter plus,
times a quarter, sorry, sorry, sorry, I said that wrong. Half of
these germ cells are big S. Half of those germ cells are big S.
So a half times a half is a quarter. Half of these germ cells are little
S. Half of these germ cells are little S. The probability of
getting one of each is a half times a half or a quarter.
For big S, little S you can do it two different ways,
so it's a quarter plus a quarter or a half.
The ratio of these fractions is
1:2:1 which determines the products that you see. Now,
before I let you go, I want you think about the
phenotypes. The phenotypes that you observe in these peas is smooth,
in these peas is smooth and in these peas wrinkled.
So the ratio of phenotypes is 2:1. You'll need to think about why
those, sorry, sorry, two plus one is three,
3:1. You'll need to think about why those principles give rise to these
calculations and these frequencies.