Genetic Crosses

Genetic Crosses - Makin' Babies!

Now that we have established some terminology for basic genetics and have discovered how gametes are made, we can investigate how gametes are fused to form zygotes. When two individuals mate and we are looking at their genetics, we call it a genetic cross. Genetic crosses are ways to predict and represent what offspring between two parents may 'look like' (meaning what their genotype will be).

A. Punnett Squares - What They Really Represent

I don't think I need to tell you how to fill in a Punnett square - almost every high schooler loves them because they're easy to fill in. You simply bring the alleles from the corresponding column and row to the center of the square.

What is more important, however, is what it represents. A Punnett square is a simple statistical tool that allows us to investigate probability. Each allele has a 50% chance of being passed on to offspring. So if a heterozygote (Rr) is making gametes, there is an equal chance (50%) that each gamete will get an R allele or an r allele.

That accounts for one parent's gametes. So those possibilities are placed on the top row. But in sexual reproduction, we obviously have two parents. So we have to do the same for parent #2. In this example, parent #2 is also a heterozygote (Rr). So, there is a 50% chance for parent #2 to pass a R and a 50% chance to pass a r.

So what about what's in the middle? Well, the middle represents possible offspring. More than that, however, each square represents an equal probability of each offspring. So, in this Punnett square for petal color, there is a 25% chance that an offspring's genotype will be BB because there are 4 squares and only one of them is BB.

But let's investigate why 25% is the correct chance for BB using math. In order for an offspring to have a genotype of BB, they had to get a B from one parent AND a B from the other parent. If both of these conditions must be met, then you have to multiply the probabilities. So the chances that the male parent passes a B is 50%, or 1/2. The chances that the female parent passes a B is also 50%, or 1/2. So, if you multiply the probabilities, you get (1/2)(1/2) = (1/4) = 25%

The Product Rule

Mendel demonstrated that the pea-plant characteristics he studied were transmitted as discrete units from parent to offspring. Mendel also determined that different characteristics were transmitted independently of one another and could be considered in separate probability analyses. For instance, performing a cross between a plant with green, wrinkled seeds and a plant with yellow, round seeds produced offspring that had a 3:1 ratio of green:yellow seeds and a 3:1 ratio of round:wrinkled seeds. The characteristics of color and texture did not influence each other.

The product rule of probability can be applied to this phenomenon of the independent transmission of characteristics. It states that the probability of two independent events occurring together can be calculated by multiplying the individual probabilities of each event occurring alone. Imagine that you are rolling a six-sided die (D) and flipping a penny (P) at the same time. The die may roll any number from 1–6 (D#), whereas the penny may turn up heads (PH) or tails (PT). The outcome of rolling the die has no effect on the outcome of flipping the penny and vice versa. There are 12 possible outcomes, and each is expected to occur with equal probability: D1PH, D1PT, D2PH, D2PT, D3PH, D3PT, D4PH, D4PT, D5PH, D5PT, D6PH, D6PT.

Of the 12 possible outcomes, the die has a 2/12 (or 1/6) probability of rolling a two, and the penny has a 6/12 (or 1/2) probability of coming up heads. The probability that you will obtain the combined outcome 2 and heads is: (D2) x (PH) = (1/6) x (1/2) or 1/12. The word “and” is a signal to apply the product rule. Consider how the product rule is applied to a dihybrid: the probability of having both dominant traits in the F2 progeny is the product of the probabilities of having the dominant trait for each characteristic.

The Sum Rule

The sum rule is applied when considering two mutually-exclusive outcomes that can result from more than one pathway. It states that the probability of the occurrence of one event or the other, of two mutually-exclusive events, is the sum of their individual probabilities. The word “or” indicates that you should apply the sum rule. Let’s imagine you are flipping a penny (P) and a quarter (Q). What is the probability of one coin coming up heads and one coming up tails? This can be achieved by two cases: the penny is heads (PH) and the quarter is tails (QT), or the quarter is heads (QH) and the penny is tails (PT). Either case fulfills the outcome. We calculate the probability of obtaining one head and one tail as [(PH) × (QT)] + [(QH) × (PT)] = [(1/2) × (1/2)] + [(1/2) × (1/2)] = 1/2. You should also notice that we used the product rule to calculate the probability of PH and QT and also the probability of PT and QH, before we summed them. The sum rule can be applied to show the probability of having just one dominant trait in the F2 generation of a dihybrid cross.

To use probability laws in practice, it is necessary to work with large sample sizes because small sample sizes are prone to deviations caused by chance. The large quantities of pea plants that Mendel examined allowed him to calculate the probabilities of the traits appearing in his F2 generation. This discovery meant that when parental traits were known, the offspring’s traits could be predicted accurately even before fertilization.

B. Monohybrid Crosses

All of the crosses we have looked at so far are focusing on one trait, so are considered monohybrid crosses. 'Mono-' means one, so we are tracking one trait (one gene). This is the simplest situation, and that is why we started with it. Unfortunately, however, things get a bit more complex...

C. Dihybrid Crosses

A dihybrid cross is exactly what you'd expect - a cross tracking two traits (two genes). However, this becomes much more complicated when it is time to actually create a Punnett square.

To investigate dihybrid crosses, let's take a look at those annoying pea plants again. This time, we will be looking at two traits, of course: seed color and seed shape. Seed color can be either green (G) or yellow (g). Seed shape can be either round (R) or wrinkled (r).

The most complicated dihybrid cross would be between two individuals that are heterozygous for BOTH genes. So, their genotype would be GgRr. Thus, their phenotype would be green, round seeds because green is dominant to yellow and round is dominant to wrinkled.

But, remember, each gene is passed independently (thanks, Mendel!). So, if a green allele (G) is passed, it could be passed alongside either a round (R) allele or a wrinkled (r) allele. So, the possible gamete genotypes for a parent that is GgRr would be: GR, Gr, gR, or gr.

Since there are 4 possible gametes from the parent, we have to make 4 boxes across. But the other parent also has a genotype of GgRr, so has the same possible gametes. So a Punnett square for a dihybrid cross can have 16 squares within it!

Because of the two genes, there are 4 possible phenotypes:

green & round seeds
green & wrinkled seeds
yellow & round seeds
yellow & wrinkled seeds

So, the phenotypic ratio for a dihybrid cross is 9:3:3:1. Be sure that you understand how this ratio is confirmed using this Punnett square. Remember that each box has an equal chance (1/16) of being 'chosen'. Really no genotype is chosen, the boxes just represent the likelihood a given offspring will have that genotype.

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