AP Biology Practice 2 - Using Mathematics Appropriately

Summary

Mr. Andersen's video emphasizes the critical role of mathematics in AP Biology, underpinning all scientific disciplines. He discusses the exponential growth in genomics and the impact of chaos theory, computing power, and in silico experiments on mathematical biology. Key equations from the AP curriculum are highlighted, including Hardy-Weinberg Equilibrium, solute potential, and Chi-Squared analysis, alongside practical examples of applying mathematical routines to understand biological phenomena. The video encourages students to develop a mathematical 'sense' for deeper insights into biology.

Takeaways

• 📚 Mathematics is integral to AP Biology as it helps in understanding and applying biological concepts.
• 🌱 The foundation of all sciences, including biology, is mathematics, which is evident in the progression from biology to biochemistry, chemistry, physics, and finally mathematics.
• 📈 The field of mathematical biology is growing rapidly, driven by advancements in genomics, chaos theory, computing power, and in silico laboratory experiments.
• 🧬 Genomic sequencing is becoming cheaper and more prevalent, leading to an exponential growth in the amount of genetic data available for analysis.
• 🌿 Chaos theory and mathematical patterns, such as the Rule of Thirty, can predict complex biological phenomena like the growth patterns of ferns and cone snails.
• 💻 Moore's Law illustrates the exponential increase in computing power, which enables more complex mathematical modeling and simulations in biology.
• 🧪 In silico experiments allow for the simulation of biological processes in a computer, offering advantages like handling large data sets and avoiding ethical concerns associated with live subjects.
• 🔍 Four key equations in the new AP Biology curriculum are crucial for understanding different biological concepts: Hardy-Weinberg Equilibrium, solute potential, Chi-Squared analysis, and population growth models.
• 📉 Students should be able to justify the selection of mathematical routines, calculate mean rates of population growth, and estimate quantities describing natural phenomena using mathematical formulas.
• 📝 The ability to apply mathematical formulas and algorithms is essential for success in AP Biology, including understanding allele frequencies, calculating slopes for growth rates, and predicting changes in osmotic conditions.
• 🔢 Charles Darwin emphasized the importance of mathematics in scientific discovery, suggesting that it provides a new sense for understanding the natural world.

Q & A

Why is mathematics important in AP Biology according to Mr. Andersen?

Mathematics is crucial in AP Biology because it forms the foundation of all sciences, including biology. It is essential for understanding the core concepts and for the emerging field of mathematical biology, which is driven by genomics, chaos theory, increasing computing power, and in silico laboratory experiments.

What is the significance of the Hardy-Weinberg Equilibrium in the context of the video?

The Hardy-Weinberg Equilibrium is an important formula in the field of evolution. It allows students to determine the allele frequencies in a population based on certain measurements, which is a key concept in understanding genetic changes over time.

How does the decrease in the cost of sequencing DNA relate to the growth of mathematical biology?

The decrease in the cost of sequencing DNA has led to an exponential growth in the amount of genetic data available. This data explosion necessitates the use of mathematical models and computational tools to analyze and interpret the genetic information, thus contributing to the growth of mathematical biology.

What is Moore's Law and how does it relate to the advancement of mathematical biology?

Moore's Law is the observation that the number of transistors on a microprocessor doubles approximately every two years, leading to faster and cheaper computers over time. This increase in computing power enables more complex mathematical modeling and simulations in biological research.

What is the concept of 'in silico' laboratory experiments mentioned in the script?

In silico laboratory experiments refer to the simulation of life processes on a computer. This method allows researchers to perform complex experiments and analyze large amounts of data without the ethical concerns or limitations associated with in vivo or in vitro experiments.

What are the four important equations Mr. Andersen mentions in the new AP Biology curriculum?

The four important equations are related to the four big ideas in the AP Biology curriculum: Hardy-Weinberg Equilibrium for evolution, solute potential for free energy, Chi-Squared analysis for information, and exponential and logistic growth models for predicting population changes.

How can the Hardy-Weinberg Equilibrium be used to determine allele frequencies in a population?

The Hardy-Weinberg Equilibrium can be used to calculate the frequency of alleles (q) in a population by using the formula q^2 + 2pq + p^2 = 1, where q and p represent the frequencies of the two alleles. By measuring the phenotypic frequencies, one can solve for the allele frequencies.

What is the Rule of Thirty mentioned in the video and how does it predict life patterns?

The Rule of Thirty is a simple mathematical pattern created by Stephen Wolfram that can predict complex biological patterns, such as the growth patterns of ferns or the shell patterns on a cone snail. It demonstrates the power of mathematics in predicting natural phenomena.

How does Mr. Andersen suggest students prepare for the AP Biology test regarding mathematical skills?

Mr. Andersen suggests that students should familiarize themselves with the formula sheet provided for the AP Biology test and practice using a calculator to apply mathematical routines, equations, and algorithms to biological scenarios.

What are the three mathematical understandings Mr. Andersen mentions that students should have?

The three mathematical understandings are: 1) Justifying the selection of a mathematical routine, 2) Applying mathematical algorithms or equations to solve biological problems, and 3) Estimating quantities that describe natural phenomena using mathematical models.

How does Mr. Andersen use the example of the peppered moth to illustrate the application of mathematics in biology?

Mr. Andersen uses the example of the peppered moth's coloration change during the Industrial Revolution to demonstrate how to calculate allele frequencies using the Hardy-Weinberg Equilibrium. By analyzing the population percentages of dark and light moths over time, students can estimate the frequency of the light allele (q).

What is the concept of osmosis as applied in the dialysis tube example in the script?

Osmosis is the movement of water across a semipermeable membrane from a region of lower solute concentration to a region of higher solute concentration. In the dialysis tube example, the change in mass of the tube is predicted based on the difference in solute concentrations inside and outside the tube, illustrating the principles of osmosis.

How does Charles Darwin's perspective on mathematics relate to the study of biology?

Charles Darwin believed that every new body of discovery is mathematical in form, suggesting that mathematics provides a fundamental framework for understanding natural phenomena. He also suggested that mathematics gives one a new sense, implying that it enhances one's ability to perceive and analyze biological processes.

Outlines

00:00

🔢 Importance of Mathematics in AP Biology

Mr. Andersen introduces the significance of mathematical application in AP Biology, emphasizing that biology is fundamentally rooted in mathematics. He discusses the emergence of mathematical biology, driven by genomics, chaos theory, increasing computing power, and in silico experiments. He highlights the exponential growth in gene sequencing and the decreasing costs, illustrating the role of mathematics in predicting life patterns and the importance of understanding key formulas in the new AP curriculum.

05:02

📊 Applying Mathematics to Understand Biological Phenomena

This paragraph delves into the practical application of mathematics in biology, focusing on justifying the selection of mathematical routines, calculating mean rates of population growth, and estimating natural phenomena. Mr. Andersen provides examples, including calculating allele frequencies in peppered moths using Hardy-Weinberg Equilibrium, determining growth rates from bacterial density graphs, and predicting changes in mass within dialysis tubing due to osmosis. He encourages familiarity with the AP formula sheet and the use of a calculator, concluding with Darwin's insights on the indispensability of mathematics in biological discovery.

Keywords

💡AP Biology

AP Biology refers to the Advanced Placement Biology course and exam, which is a rigorous college-level curriculum designed to prepare high school students for university-level biology studies. In the script, Mr. Andersen emphasizes the importance of understanding not only the biological content but also the mathematical applications within the subject, as it is crucial for excelling in the AP Biology test and for grasping the scientific concepts deeply.

💡Mathematical Biology

Mathematical Biology is an interdisciplinary field that applies mathematical tools and methods to understand biological phenomena. The script discusses how this field is growing due to advancements in genomics and computational power, and it plays a significant role in modern biology, as it allows for the modeling and prediction of biological processes.

💡Genomics

Genomics is the study of all the genes in an organism, their functions, and their interactions. The script mentions the exponential growth in the sequencing of DNA and how it has become cheaper, which has contributed to the expansion of mathematical biology by providing vast amounts of genetic data that can be analyzed and modeled mathematically.

💡Chaos Theory

Chaos Theory is a branch of mathematics that deals with the behavior of dynamical systems that are highly sensitive to initial conditions, an aspect of which is the unpredictability of certain systems. In the script, it is mentioned in the context of using mathematics and computers to predict patterns in life, such as the growth patterns of ferns and the patterns on a cone snail's shell.

💡Moore's Law

Moore's Law is the observation that the number of transistors on a microprocessor doubles approximately every two years, leading to an increase in computational power and a decrease in cost. The script uses Moore's Law to illustrate the rapid growth in computing capabilities, which has facilitated the development of mathematical biology by enabling more complex simulations and analyses.

💡In Silico

In Silico refers to the use of computer simulations to conduct experiments and analyze data. In the script, Mr. Andersen explains that in silico laboratory experiments allow for the simulation of biological processes without the need for physical experiments, offering advantages such as the ability to process large amounts of data and avoiding ethical concerns associated with live subjects.

💡Hardy-Weinberg Equilibrium

Hardy-Weinberg Equilibrium is a principle in population genetics that states that in a large, randomly mating population with no evolutionary influences, the frequency of alleles and genotypes will remain constant from generation to generation. The script uses this concept as an example of a mathematical formula that students should be familiar with to analyze allele frequencies in a population.

💡Water Potential

Water Potential is a measure of the potential energy of water in a system, which is influenced by factors such as solute concentration, pressure, and temperature. The script discusses how understanding water potential is important in the context of osmosis and can be calculated using a mathematical formula involving the negative ionization constant, molar concentration, pressure constant, and temperature.

💡Chi-Squared Analysis

Chi-Squared Analysis is a statistical method used to determine whether there is a significant difference between the expected frequencies and the observed frequencies in a study. In the script, it is mentioned as a tool for comparing expected and observed results in genetic crosses, which is an important mathematical application in the study of genetics.

💡Population Growth Models

Population Growth Models are mathematical models that describe how populations change in size over time. The script discusses two types of models: exponential growth, which assumes a constant growth rate, and logistic growth, which accounts for limited resources and eventually reaches a carrying capacity. These models are essential for understanding the dynamics of population changes in biology.

💡Slope

Slope is a concept in mathematics that describes the steepness and direction of a line. In the context of the script, calculating the slope is used to determine the rate of population growth in a graph representing bacterial growth over time, illustrating the application of mathematical concepts to interpret biological data.

💡Osmosis

Osmosis is the movement of solvent molecules, such as water, across a semipermeable membrane from an area of lower solute concentration to an area of higher solute concentration. The script discusses how understanding osmosis is crucial for predicting changes in mass within a dialysis tube and for understanding cellular processes, such as the movement of water into and out of cells.

Highlights

The importance of using mathematics in AP Biology is emphasized, as it is essential for understanding the core of all sciences.

Biology is built upon biochemistry, chemistry, physics, and ultimately mathematics, highlighting the foundational role of mathematics in the study of life.

The emergence of mathematical biology, driven by advancements in genomics, chaos theory, computing power, and in silico laboratory experiments.

The exponential growth in gene sequencing and the decreasing cost, making personal genome sequencing more accessible.

Chaos theory's application in using mathematics and computers to predict life patterns, such as ferns and cone snail patterns.

Moore's Law and its impact on the exponential increase in computing power, making simulations and data analysis more efficient.

The concept of in silico experiments, simulating life in a computer, offering advantages like handling large data sets and avoiding ethical concerns.

The necessity for students to be familiar with four key equations in the new AP Biology curriculum.

Hardy-Weinberg Equilibrium as a crucial formula for understanding allele frequencies in populations.

The significance of solute potential and water potential in understanding free energy and osmosis.

The use of Chi-Squared analysis in comparing expected and observed results in genetic data.

Population growth prediction formulas, including exponential and logistic growth, and their relation to carrying capacity.

The importance of understanding how to justify the selection of a mathematical routine, as demonstrated through sample problems.

Calculating allele frequencies using Hardy-Weinberg Equilibrium to determine the frequency of light alleles in peppered moths over time.

Using slope to calculate the mean rate of population growth in bacterial growth experiments.

Estimating quantities that describe natural phenomena, such as predicting the final mass changes in dialysis tubes with different sucrose concentrations.

The practical application of mathematics in estimating the concentration of potato cores in osmosis experiments.

Darwin's perspective on the importance of mathematics in scientific discovery and its role in providing a 'new sense' for understanding the natural world.

Transcripts

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.