Little did Pascal and de Méré know that the same rules of probability that govern throwing a die would also apply to the traits of offspring when two parents get together. It wasn’t until about 200 years later that Gregor Mendel would start his work with pea plants and became founder of the field of genetics. Genetics represents a beautiful illustration of the unity of mathematics and biology.

In this exercise, poker chips are used to represent genes. Students draw tree diagrams to illustrate the chance for inheritance of multiple traits. This activity is borrowed with permission from Biology in a Box

Grade Level: 9^th-12^th grade
Subject Matter: Genetics, Probability, Mathematics

Download a pdf of this lesson plan>>

Vocabulary

Trait: a genetically determined characteristic
Chromosome: a single piece of DNA containing information for many genes as well as proteins, an organized structure located in the cell nucleus
Allele: one of two or more alternate forms of a gene, located on the same place on the chromosome
Inheritance: the passing on of gene alleles from parents to offspring
Genotype: the specific allele make-up of an organism or cell
Probability: the likelihood an event will occur
Independent Events: two events (occurances) that may happen that do not depend on the other to occur
Random: equal chance that any possible event may occur

One can use probability to analyze the way an offspring might inherit a single trait, where there are only two steps or tasks.  The first task is to randomly pick the allele that came from the female parent, and the second task was to randomly pick the allele that came from the male parent.  It is easy to list and count all of the possible outcomes.  In considering how an offspring might inherit multiple traits, in which many more tasks (with many more outcomes) are involved, using tree diagrams and the counting principle is another helpful way to determine the probability of an event, and an alternative to drawing a Punnett square.

The counting principle: If there are x ways to perform one task, and y ways to perform a second task, then there are xy ways to perform both tasks.

Suppose for example that we flip two coins.  Then there are 2 possible results from the flip of the first coin (heads or tails), and 2 possible results from the flip of the second coin, so there are 2×2 = 4 possible results when flipping both coins.

1. Suppose that you have three shirts (green, orange, and yellow) and two pairs of pants (blue and white).  How many different outfits can you assemble from these clothing items?

Tree diagrams

• A tree diagram is a visual aid that helps us find and count all possible outcomes of an experiment.  For example, we can construct a tree diagram for the experiment from question 1.

• A tree diagram can also help us to find the probability that an event occurs.  For example, suppose that when you get up in the morning, you grab a shirt from your closet without turning on the light. What is the probability that your shirt is yellow?

• Answer: Since you don’t turn on the light, you are equally likely to grab any of the shirts in your closet. Therefore, the probability that your shirt is yellow is the quotient of the number of outcomes with a yellow shirt divided by the total number of outcomes.  Looking at the tree diagram we see that there are two outcomes with a yellow shirt and six outcomes total, so the probability that your shirt is yellow is 2/6 = 1/3.

2.  What is the probability that your shirt is yellow or orange?

What if you got another green shirt as a birthday gift?  This would change the probabilities of each outcome, but we can still use a tree diagram to help us do so, using the rules of probability.  Now you would have four shirts instead of three, so the probabilities of getting each possible outfit would change, since the probabilities of grabbing a shirt of a particular color are no longer all equal.

• What are the probabilities of getting each color shirt now that you have received this gift?

• Answer: The probability that you would draw a green shirt at random now would be equal to 2/4 = 1/2, since you now have four shirts, and two of them are green.  The probability of drawing an orange shirt would now be equal to 1/4, and the probability of drawing a yellow shirt would also be 1/4.

Though you could have reflected this by drawing another tree diagram, and adding an additional branch to represent the new green shirt, another way of calculating the probabilities of the overall outcome (the final outfit) without having to draw extra branches would be to write the probabilities of the result of each task on our tree diagram.  We can also then use these to calculate the probabilities of each of the outcomes.  Look at the new tree diagram below for an example.

Recall the counting principle: that if events A and B are independent, P(A and B)= P(A)P(B)  Since the shirt that you grab from your closet at random has no effect on the pants that you grab at random (these events are independent!), you can calculate the probabilities of each outcome on your tree diagram by simply multiplying the probabilities of the branches along the branches leading to that particular outcome, as shown below:

Now, let’s apply this information to the topic at hand, the inheritance of multiple traits.  Think of a cross between two parents, both with the genotype CC’TT’. C and C’ are two different alleles for color, and T and T’ are different alleles for size.

3.  How many different tasks are involved in determining the genotype of an offspring in this cross?  List each of these tasks.

4.  In how many ways can each of the tasks from the previous question be performed?

5.  Construct a tree diagram to show all of the possible outcomes of this cross.

6.  How many outcomes are possible for this experiment?

7. How many possible genotypes could the offspring produced by this cross have?  (Remember that the genotype is the set of alleles that an individual has with no regard to which parent donated which allele.)

Activity Materials

• One “Female Parent” Box and One “Male Parent” Box per group
• Poker chips – 2 colors, 2 thicknesses, 3 of each (total of 12) per box
• Blindfold

What to do

• Now we will simulate the inheritance of multiple traits by an offspring produced by a cross between parents of randomly determined genotypes.
• Find the boxes labeled Female (representing the female parent’s gene pool) and Male (representing the male parent’s gene pool).
• Make sure each box has equal numbers of thick red, thin red, thick white, and thin white chips (3 of each)..
• Put on your blindfold and find your parent’s genotypes by picking two chips each from the respective Female and Male boxes. Record the genotypes of the parents on a sheet of paper.
• Repeat this experiment two more times, simulating new parent genotypes for each cross as described above.
• Answer 3-7  above for each of your crosses.

8. Let A be the event that the offspring has at least one C allele, and B be the event that the offspring has at least one T allele.  Use each of your tree diagrams to find P(A), P(B), P(A and B), and P(A)P(B) from each of your crosses.  Do your answers support Mendel’s Law of Independent Assortment?  That is, do they support the hypothesis that alleles for the color and thickness genes are inherited independently?  Explain why or why not.

9. (Critical thinking!)  Why was it important that you made sure that both the F and M boxes contained equal numbers of thick red, thin red, thick white, and thin white chips?

Answers
1.   3×2=6

2. 

3.  There are four tasks involved in the determination of an offspring’s genotype:

1. The female parent must donate an allele for color.
2. The female parent must donate an allele for size.
3. The male parent must donate an allele for color.
4. The male parent must donate an allele for size.

You could have also said that there were two tasks involved: the production of a gamete by the female parent, and the production of a gamete by the male parent.  However, just keep in mind that production of each gamete includes the donation of an allele for both color and size, which would mean that each of the “gamete production” tasks could have more possible ways that they could be performed, each involving a combination of size and color alleles!

4.  This question could be answered this way …

1. Female parent’s donation of color allele: 2 ways (C or C’)
2. Male parent’s donation of color allele: 2 ways (C or C’)
3. Female parent’s donation of size allele: 2 ways (T or T’)
4. Male parent’s donation of size allele: 2 ways (T or T’)

Or, if you considered this to be two tasks, the production of a gamete by each parent, the total number of ways each task can be performed are as follows:

1. Female parent’s gamete produced: 4 ways (CT, CT’, C’T, C’T’)
2. Male parent’s gamete produced: 4 ways (CT, CT’, C’T, C’T’)

5. In the tree diagram on the next page, the first branch point represents the possible alleles for color donated by one parent.  The next set of branches represents the possible alleles for color donated by the other parent, given the allele for color donated by the first parent.  The next set of branches represents possible alleles for thickness donated by one parent, given the alleles donated by both parents for color, and the final set of branches represents the possible alleles for thickness donated by the other parent, given the alleles donated by both parents for color, and the allele for thickness donated by the first parent.  Finally, the outcomes after the arrows represent the resulting genotypes of the offspring in each scenario.

6.  There are a total of 16 possible outcomes for this experiment.

7. As you can see in the tree diagram above, even though there are sixteen possible outcomes, some of those outcomes are the same, but just represent different ways that those outcomes could have occurred.  There are only nine possible offspring genotypes (CCTT, CCTT’, CCT’T’, CC’TT, CC’TT’, CC’T’T’, C’C’TT, C’C’TT’, C’C’T’T’) from the cross of two parents both with the genotype CC’TT’, but some of these outcomes are more likely than others!

8. You should have found in all cases that P(A)P(B) = P(A and B). Recall that this is equivalent to the statement that the events A and B are independent.

9. The reason that it was important for there to be equal numbers of each of these types of chips is because if there had been, for example, one less thick white chip, this would have represented a situation where the two traits (color and size) did NOT assort independently, which was not the point of this exercise!  In other words, if you were one thick white chip short, this would have represented that the red (C) and thick (T) alleles are more likely to go together during gamete production (which does happen sometimes with certain genes!).

Extended Links:
Khan Academy’s Intro to Heredity
http://www.youtube.com/watch?v=eEUvRrhmcxM&feature=relmfu

The Tech Museum’s Understanding Genetics Page: includes a calculator for determining the probability of your offspring’s eye color
http://www.thetech.org/genetics/

___________________

In Biology by Numbers, learn about the ways math can solve biological problems. Produced by the National Institute for Mathematical and Biological Synthesis (NIMBioS). NIMBioS brings together researchers from around the world to collaborate across disciplinary boundaries to investigate solutions to basic and applied problems in the life sciences. NIMBioS is sponsored by the National Science Foundation, the U.S. Department of Homeland Security, and the U.S. Department of Agriculture with additional support from The University of Tennessee, Knoxville.

An adult blackmargined aphid under a dissecting scope. Photo by Jessica Honaker.

Last week, Kristie wrote a post on solifuges, some of the coolest arachnids around! This week, I'll introduce you to the awesome insect that I studied for my Masters research – the blackmargined aphid.

But what is an aphid? Aphids are very small insects, and they have rounded, soft bodies. They live on plant stems and undersides of leaves, and have a long mouthpart called a proboscis that acts like a needle to jam into the plant tissues to suck out the juices. Another totally cool feature of aphids is a pair of cornicles on their abdomen. They remind me of little biological exhaust pipes – but they secrete a kind of wax that coats the bodies of the aphids to keep them from drying out.

There is an easy way to distinguish blackmargined aphids from other species. The outer margins of their wings are lined with black (hence their name)!

You can see the black margins on the aphid's wings! Photo by Bart Drees, TAMU.

One of the coolest things about aphids is their watery waste. It’s called honeydew. It’s mostly made up of sugars from the plant that the aphid didn’t process. And for blackmargined aphid, their honeydew is about 95% sugar. Part of my research dealt with quantifying how much honeydew blackmargined aphids produced, and figuring out how much energy was being removed from pecan trees.

So honeydew is both good and bad. It’s bad because if there is a lot of honeydew produced by a heavy infestation of aphids, it coats the pecan leaves. Because of the high sugar content, it’s the perfect place for sooty mold to grow. Growth of sooty mold means not enough sunlight gets to the leaves and photosynthesis is reduced. But it can be a good thing (in moderation!), too. The honeydew attracts lots of beneficial insects to the trees. These “natural enemies” feed on pest species – including blackmargined aphids – that have the potential to harm the tree.

Blackmargined aphids only feed on pecan trees. Both species are native to North America, and have co-evolved for thousands of years! Today, pecan is being cultivated in other parts of the world as well, and the blackmargined aphid has moved along with it.

Distribution of pecan and blackmargined aphid in the United States. Photo by USDA NRCS.

Fun Facts:

• Most of the year, blackmargined aphid females reproduce by parthenogenesis, or cloning. But once the weather starts to get cold, male aphids develop and mate with females. Eggs are laid, and overwinter to hatch the following spring.

• Each adult blackmargined aphid can produced up to 32 generations of offspring.

• Despite the high sugar content, honeydew doesn’t actually taste sweet.

• Blackmargined aphids hold their wings flat against their backs, while other yellow pecan aphids hold their wings pitched over their bodies.

When I first started my research, I didn't really know much about aphids. But once I started studying them and learning more about their behavior and biology, the more fascinating they became. Now, they're one of my favorite insects!

If you have pecans around where you live, check some of the leaves around May - June to see if you can find blackmargined aphids. You can also see if you have other species on different plants. Just look for the squishy bodies and cornicles!

--Jess

Here's what the SciStarter blog says about the project:

When someone collapses and stops breathing, an automated external defibrillator or AED can save their life. [Home AEDs are available for purchase.] In Philadelphia, PA, a city with about 1.5 million people, AEDs are all around us. Near our homes, workplaces, and even grocery stores! Currently, there is no comprehensive map, and, as a result, AEDs are often not used when they are most needed. With the crowdsourced information collected from this contest, the organizers will build a map of AED locations in Philadelphia that can inform 911 services and the public.

The MyHeartMap contest will officially go live January 31, 2012 at 9am! Until then, you can download the app from the iPhone store and Android marketplace and start submitting entries. Clues will be posted at the project website myheartmap.org and philly.org. The contest closes on March 13, 2012, at 6pm ET!

There are three ways to play:

1. Find and photograph the most AEDs in Philadelphia County before March 13, 2012 and win the $10,000 grand prize. The team or individual that finds the most “confirmed,” “eligible” AEDs by the contest end date will receive the grand prize of $10,000.

2. Be the first to submit a photograph of a “Golden”AED and win $50. The organizers have identified between 20 and 200 AEDs in Philadelphia County as “Golden” AEDs. These are unmarked, and you won’t know it’s a winner when you photograph it. Clues will be posted at the MyHeartMap project website.

3. Want to help but not compete for a prize? Submit addresses of locations without AEDs or that you wish had an AED – this is just for fun, and it will help with the map.

Darlene Cavelier, SciStarter founder, had the opportunity to chat about this project with Raina Merchant, Assistant Professor in the Department of Emergency Medicine at the University of Pennsylvania Perelman School of Medicine. Here's the interview (also posted on the SciStarter blog.)

SciStarter: Why did you start this project, Raina?

Raina: I wanted everyone who had a cardiac arrest to survive and have a second chance at life. For this to happen, lots of different components of the chain of survival have to be in place. AEDs are an important link in the chain, and this seemed like a good place to start and then build on. I was disturbed to learn that although AEDs were in public places all over the world, no one knew where they were, and, in the event of an emergency, I couldn’t use my phone or emergency services to locate them. I wanted to approach the AED problem using a novel approach that engaged the public through technology, phones, and social media.

I learned about the DARPA Network Challenge (DNC) to locate red balloons from a colleague and thought that it seemed like a great approach to apply to a public health problem. The DNC showed that with the proper incentives, today’s “networked society” is able to virtually mobilize to help solve a challenge and importantly to innovate. When this challenge relates to the American public health system and the well-being of our citizens, there is the opportunity for the response to be equally strong if not better. Ultimately, studying how social networking can augment traditional research methods offers significant promise in approaching public health challenges that are essentially stuck and need a paradigm shift to advance

SciStarter: What do you hope to accomplish?

Raina: Our goals are to: Build the first U.S. city crowdsourced map of AEDs that can be made available to the 911 center and the public via a mobile phone. Gain a better understanding of the distribution of AEDs in Philadelphia so that we can determine optimal AED placement. Develop crowdsourcing and social media metrics related to data collection, validation, and surveillance. Use the information learned from this project to expand the MyHeartMap challenge to other US cities and then the rest of the U.S.

SciStarter: What persuaded you to make it participatory (involve the public)?

Raina: The traditional approach for locating AEDs would involve hiring a large team of research assistants to search for AEDs. This approach would be costly and time-consuming and wouldn’t help with improving the public’s awareness of AEDs in their environment. There are lots of examples of how citizen science projects can engage the public to help with data collection – the result is often a more empowered public and new ways of approaching health challenges.

SciStarter: Any concerns about the quality of data?

Raina: We recognize that collecting data about AEDs is difficult and that some data entries with low quality data will be intentional and others unintentional.

SciStarter: And surprising developments?

Raina: Several teams have contacted us and indicated that they are going to participate long-distance and collect data without being in Philadelphia. They are going to rely completely on data entered through social networks. We’re excited to see if this strategy works as this could provide important insights about how organizations can evaluate data they haven’t visualized.

SciStarter: Take a wild guess at how many defibs will be accounted for in total and how many will be spotted by the winner?

Raina: We think there are about 5000 AEDs in Philadelphia. We hope that winner uses a creative strategy to find most of them. I’ll guess 4997!

SciStarter: What’s next?

Raina: We hope to develop subsequent challenges in Philadelphia and then expand to collect AED data across the country. We’re also looking to use other social media tools like Facebook, Twitter, Foursquare, Gigwalk, Tumblr, Google Insights etc…..to engage the public to help us study and solve important public health challenges.

So, if you are in Philadelphia, look into the MyHeartMap Challenge. It will help make a map. It could even save a life!

For any problem there's always a solution. How does one go about solving the problem of a deaf man who has had his fingers amputated after a car accident? (If a deaf man loses his fingers, he has a hard time communicating using sign language.) Dan Didrick solved this problem by inventing the X-Finger.

The X-Finger is an important innovation in the field of prosthetics. Of course, it didn’t happen overnight, 10 years of work went in to this product before it was officially launched.

So what's so special about this invention? According to Didrick Medical, it is the first artificial finger that is designed for partial finger amputees. The best benefit of the X-Finger is that it is powered by the body so there is no need for any power supplies. Not only is it easy to use, but it is also lightweight -- and it actually resembles realistic fingers due to the thermoplastic cosmetic skin that is placed over the actual mechanical finger.

You can see the X-finger in action in this video:

Video posted by BusinessWire

A story on the invention posted at ASME.org explains why Didrick thinks there's a big market for his device:

What’s little realized, he said, is how many children lose fingers. The largest group of people who lose fingers outside the workplace are children under five, who undergo finger amputation due to accidents like slamming them in a car door.

He also has learned that one out of 200 people will lose one or more fingers within their lifetime. That statistic takes into account people living all over the world.

Sometimes, even though we try our best, we just don’t have control over some situations. Amputations usually result from these kinds of situations. Inventions like the X-finger keep changing the world and give people hope.

Read more about the X-Finger at:
Everyday Prosthetic Fingers
Didrick Medical