Hi. It's Mr. Andersen and this
is AP Biology Science Practice 2. It's on
using mathematics appropriately. Remember
in AP Biology you not only have to know the
content that is biology. You have to know
how to apply that. And so if you want to do
well on the AP Bio test in the spring you
really have to know how to use mathematics
and apply it. But there is a bigger reason
why. Number is that everything is built on
mathematics. In other words if we look at
biology, that's just going to be the upper
level of how life works. But if you dig into
biology you quickly find biochemistry. Underneath
that is chemistry. Underneath that is physics.
And finally we have mathematics. And so all
sciences at their core root have mathematics.
And in biology it's very crucial because we
are having the explosion of this new field
in biology called mathematical biology. It's
driven by a number of things. One of those
is genomics. We're sequencing so much DNA
right now and it's getting cheaper and cheaper.
So this right here shows the exponential growth
in genes being submitted to the gene bank.
This is the cost it takes to sequence one
million nucleotides, which is approaching
zero. You can see here, zero dollars. And
then this is the cost to sequence a human
genome, which started in the millions but
is getting cheaper and cheaper everyday. So
in the future you'll probably be able to have
your own genome sequenced. Even in a biology
class. We also had the creation of what's
called chaos theory. And so we're using mathematics
and computers to predict life. Like this is
a computer simulation of what a fern may be.
This is a simple mathematical pattern called
the Rule of Thirty that was created by Stephen
Wolfram. But it can predict even the patterns
on this cone snail here. And so mathematics
can predict life. We're also seeing an explosion
in computing power. This is Moore's Law, the
idea that every two years the number of transistors
on a processor, a microprocessor, is going
to double. That means that our computers get
faster and faster and cheaper and cheaper
over time. And then finally we have the creation
of laboratory experiments in silico. What
does that mean? Well if I said I'm doing some
lab work in vitro, that means I'm growing
things in a test tube or in a petri dish.
If I do it in vivo that means in a living
organism. In this case, in a lab rat. But
in silico means that I'm simulating life in
a computer. In this case we're simulating
dendrite growth. And so what are some advantages
of that? You can crunch a huge amount of numbers.
And also you don't have those ethical concerns
that you would have dealing with a mouse.
And so or a rat. And so math is very important.
And in the new AP curriculum there are four
pretty big and pretty important equations
that you should be familiar with in each of
the four different big ideas. And so I went
through and chose one formula for each of
these. In evolution I went with Hardy-Weinberg
Equilibrium. You should be able to apply that.
Figure out what the alleles are based in a
population just based on making some measurements.
I'll put some links here to videos I've made
so if you don't understand Hardy-Weinberg
you could go watch that. In free energy we
should really have an understanding of solute
potential, which leads to water potential.
In this case it's caused by the negative ionization
constant times molar concentration times pressure
constant times temperature. At information,
when we're looking at genetic data, in this
case we're looking at the results of a cross,
we could have our predicted or our expected
results. And then we could compare that to
our observed results using the Chi-Squared
analysis. Again, I'll put a link here if you're
confused by that. And even predicting population
growth. Here we have exponential and then
logistic growth. But there are formulas that
predict how that change is going to occur
and how it's eventually going to approach
K or carrying capacity. And so there's a formula
sheet, remember that goes along with the AP
test. But you want to be familiar with it
and don't just hope that it's going to get
you through the test. You also want to have
a calculator that you can use. And so I came
up with three questions that are tied to the
three understandings that you should have
in the area of mathematics. And so the first
thing that they want you to be able to do
is justify the selection of a mathematical
routine. And so I've given you a sample problem
here. You can pause the video right now, read
through it, and then try to answer the question.
Or you could just have me do it. And so what
we're looking at is coloration in the peppered
moth. Remember that's famous in evolution
because it showed evolution. It showed the
moths turning from a light color to a dark
color with the Industrial Revolution and a
change in coloration on the trees. And so
coloration in peppered moths is cause by a
single allele. The allele for dark bodied
moths is big D. And it's dominant. So what
they're asking in this question, what's the
approximate frequency of the light allele
in 1840 and 1900. So they're looking for D.
In this case that would be q. Okay, how do
you solve that? Well if you look right here,
here's where the evolution took place. When
we get a change in the population. It's important
that you always read the axis on a graph.
They're giving us the percent population of
the dark bodied moth. And so remember the
dark, since it's dominant, doesn't tell us
much. We're going to want to get to the percent
of light colored moths. And so they're looking
right here in 1840. And then in 1900. Well
the math is a little bit easier up here. I
could do it in my head. So in 1900 we had
51 percent of the moths being dark, that means
49 percent of the moths were going to be light.
So that's going to be my q squared value if
I take the square root of that I'm going to
get a q of 0.7. And so the right answer in
1900 would be around 0.7. You'll be able to
use your calculator on the test and then you're
going to grid-in those answers. But I couldn't
have gotten there if I didn't know, okay,
we're looking at allele frequency, let's us
Hardy-Weinberg Equilibrium. Okay in this question
they're asking us to calculate the mean rate
of population growth between 80 and 176 minutes.
And so this is bacterial growth. You can see
that the density went from zero almost towards
one. And so if you're ever asked to do a growth
rate, that's going to simply be the slope
of that line. And slope is very important.
Especially if we're looking at our data. And
so from 80 to 176 minutes, how do I calculate
slope? Well, that's going to be the rise over
the run. So I would draw in a little triangle,
calculate the rise. In this case it's going
to be about 0.72 change in population density.
My time or my run is going to be around 96
minutes. And so if I divide my rise by the
run I'm going to get 0.0075 density increase
per minute. And so what am I doing here? I'm
applying a mathematical routine. In this case
it's slope. But you're also going to have
to be able to apply any kind of an equation
or any kind of a mathematical algorithm that
you're going to find on that sheet. In this
case it's slope, but it might be standard
error. It might be mean or range. And so you
really want to be comfortable using your calculator
and using the formal sheet. If we look at
the last one, they want you to be able to
estimate quantities that describe natural
phenomena. And so this is a question I came
up with. Predict the following masses of the
final masses or the following dialysis tube.
So we've got different concentrations, sucrose
concentrations, inside this dialysis tubing.
They've given you the mass of each of those.
And then we're going to put it in the concentration
of 0.4 molar sucrose. And they want you to
predict the mass at the end. Well the easiest
one would be to start with this one. 0.4 and
0.4. There's going to be movement of water
back and forth through osmosis, but our net
change, this should be around 30.84. If you
look at the next one, we're putting more concentrated
in less concentrated, in other words we're
putting hypertonic inside hypotonic. And so
what's going to happen there? Well we're going
to get a movement of water into the dialysis
tubing. So I would predict that this one is
going to go up, a number, let's say 30. This
one is going to go down. And this one is going
to go down even more. Because if we've got
distilled water inside here, we put it inside
0.4, then the water's going to be flowing
out. Why is that important? Well that's how
cells work remember. And why isn't the sucrose
moving? It's too big for the dialysis tubing.
Let's try another estimation. This is the
actual osmosis lab that a lot of people do.
So what we did was took potato cores. Put
them in different concentrations of sucrose
solution. Then we measured their percent mass
change. And so sometimes those potatoes got
bigger. Sometimes they got smaller. But they're
asking us to estimate the concentration of
the potatoes. Well, if you put a potato that
has the same concentration of the surroundings,
there's going to be isotonic. There's not
going to be any change. So I could look at
a zero percent change in mass and that's going
to be around, I don't know, 0.28. Something
like that, molar sucrose. That's going to
be my guess for the concentration of potatoes.
At least in our lab. And so again, applying
mathematics is incredibly important. A good
place to start might be with that formula
sheet for AP biology and then just working
through each of them. Why is it important?
Well even Darwin, a long time ago, before
we had computers and before we had genomics,
he said two things in relation to math. Number
one, "Every new body of discovery is mathematical
in form. In other words it's at that root,
because there's no other guidance that we
have." And he also said, "Mathematics seems
to endow one with something like a new sense."
And so if you want to have this extra sense
that Darwin's talking about you better learn
your mathematics and I hope that was helpful.
