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Punnett Squares as Genetic Tools. A Punnett square is a diagram representing all possible outcomes of a genetic cross. The genotypes of all possible gametes from the male and female parents are arranged along two adjacent sides of a square, and each box in the matrix is then used to represent the genotype resulting from the union of the two gametes at the heads of the intersecting rows. By the law of independent assortment, all possible combinations are equally likely, so the frequency of a given genotype among the boxes represents the frequency of that genotype among the progeny of the genetic cross represented by the Punnett square. Figure $20-34$ shows the Punnett squares for two crosses of pea plants. The genetic characters involved are seed color (a) One-factor cross (b) Two-factor cross FIGURE $20-34$ Punnett Squares. See Problem $20-7$ (where $Y$ is the allele for yellow seeds and $y$ for green seeds) and seed shape (where $R$ is the allele for round seeds and $r$ for wrinkled seeds). The Punnett square in Figure $20-34$ a represents a one-factor cross between parent plants that are both heterozygous for seed color $(Y y \times Y y) .$ The Punnett square in Figure $20-34 b$ is a two-factor cross between plants heterozygous for both seed color $(Y y)$ and seed shape $(R r)$ (a) Using the Punnett square of Figure $20-34 a$, explain the 3: 1 phenotypic ratio Mendel observed for the offspring of such a cross. (b) Explain why the Punnett square of Figure $20-34$ b is a $4 \times 4$ matrix with 16 genotypes. In general, what is the mathematical relationship between the number of heterozygous allelic pairs being considered and the number of different kinds of gametes? (c) How does the Punnett square of Figure $20-34$ b reflect Mendel's law of independent assortment? (d) Complete the Punnett square of Figure $20-34$ b by writing in each of the possible progeny genotypes. How many different genotypes will be found in the progeny? In what ratios? (e) For the case of Figure $20-34$ b, how many different phenotypes will be found in the progeny? In what ratios?

Name: Problem 7, Chapter 20: Becker's World of the Cell
Uploaded: 2020-11-10T21:12:22-08:00
Duration: 12 min 25 s
Channel: Bryan Lynn

Punnett Squares as Genetic Tools. A Punnett square is a diagram representing all possible outcomes of a genetic cross. The genotypes of all possible gametes from the male and female parents are arranged along two adjacent sides of a square, and each box in the matrix is then used to represent the genotype resulting from the union of the two gametes at the heads of the intersecting rows. By the law of independent assortment, all possible combinations are equally likely, so the frequency of a given genotype among the boxes represents the frequency of that genotype among the progeny of the genetic cross represented by the Punnett square.
Figure $20-34$ shows the Punnett squares for two crosses of pea plants. The genetic characters involved are seed color
(a) One-factor cross
(b) Two-factor cross
FIGURE $20-34$ Punnett Squares. See Problem $20-7$
(where $Y$ is the allele for yellow seeds and $y$ for green seeds) and seed shape (where $R$ is the allele for round seeds and $r$ for wrinkled seeds). The Punnett square in Figure $20-34$ a represents a one-factor cross between parent plants that are both heterozygous for seed color $(Y y \times Y y) .$ The Punnett square in Figure $20-34 b$ is a two-factor cross between plants heterozygous for both seed color $(Y y)$ and seed shape $(R r)$
(a) Using the Punnett square of Figure $20-34 a$, explain the 3: 1 phenotypic ratio Mendel observed for the offspring of such a cross.
(b) Explain why the Punnett square of Figure $20-34$ b is a $4 \times 4$ matrix with 16 genotypes. In general, what is the mathematical relationship between the number of heterozygous allelic pairs being considered and the number of different kinds of gametes?
(c) How does the Punnett square of Figure $20-34$ b reflect Mendel's law of independent assortment?
(d) Complete the Punnett square of Figure $20-34$ b by writing in each of the possible progeny genotypes. How many different genotypes will be found in the progeny? In what ratios?
(e) For the case of Figure $20-34$ b, how many different phenotypes will be found in the progeny? In what ratios?

Becker's World of the Cell

Jeff Hardin, Gregory… 8th Edition

Chapter 20, Problem 7

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Solved by Expert Bryan Lynn

11/10/2020

Step 1

The phenotypes corresponding to these genotypes are yellow seeds for YY, Yy, and yY, and green seeds for yy. Since there are three yellow seed genotypes and one green seed genotype, the phenotypic ratio observed for the offspring of this cross is 3:1 Show more…

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Punnett Squares as Genetic Tools. A Punnett square is a diagram representing all possible outcomes of a genetic cross. The genotypes of all possible gametes from the male and female parents are arranged along two adjacent sides of a square, and each box in the matrix is then used to represent the genotype resulting from the union of the two gametes at the heads of the intersecting rows. By the law of independent assortment, all possible combinations are equally likely, so the frequency of a given genotype among the boxes represents the frequency of that genotype among the progeny of the genetic cross represented by the Punnett square. Figure $20-34$ shows the Punnett squares for two crosses of pea plants. The genetic characters involved are seed color (a) One-factor cross (b) Two-factor cross FIGURE $20-34$ Punnett Squares. See Problem $20-7$ (where $Y$ is the allele for yellow seeds and $y$ for green seeds) and seed shape (where $R$ is the allele for round seeds and $r$ for wrinkled seeds). The Punnett square in Figure $20-34$ a represents a one-factor cross between parent plants that are both heterozygous for seed color $(Y y \times Y y) .$ The Punnett square in Figure $20-34 b$ is a two-factor cross between plants heterozygous for both seed color $(Y y)$ and seed shape $(R r)$ (a) Using the Punnett square of Figure $20-34 a$, explain the 3: 1 phenotypic ratio Mendel observed for the offspring of such a cross. (b) Explain why the Punnett square of Figure $20-34$ b is a $4 \times 4$ matrix with 16 genotypes. In general, what is the mathematical relationship between the number of heterozygous allelic pairs being considered and the number of different kinds of gametes? (c) How does the Punnett square of Figure $20-34$ b reflect Mendel's law of independent assortment? (d) Complete the Punnett square of Figure $20-34$ b by writing in each of the possible progeny genotypes. How many different genotypes will be found in the progeny? In what ratios? (e) For the case of Figure $20-34$ b, how many different phenotypes will be found in the progeny? In what ratios?

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Key Concepts

Combinatorial Mathematics in Genetic Crosses

In genetics, the number of different gametes produced by an organism with multiple heterozygous pairs of alleles is determined by the formula 2?, where n is the number of heterozygous loci. This relationship explains why crosses involving multiple gene pairs result in exponentially larger numbers of potential genotype combinations, as seen in expanded Punnett squares.

Genotypic and Phenotypic Ratios

Genotypic ratios represent the relative frequencies of the different genetic makeups among the offspring, while phenotypic ratios show the distribution of observable traits based on dominant and recessive relationships. These ratios provide insight into the probability and outcomes of particular genetic crosses.

Mendel's Law of Independent Assortment

Mendel's law of independent assortment states that alleles for different genes segregate independently during gamete formation. This principle underlies the rationale for constructing multi-dimensional Punnett squares, as it ensures that each combination of alleles is equally probable when traits are unlinked.

One-Factor Cross

A one-factor cross deals with the inheritance of a single trait controlled by one gene pair with dominant and recessive alleles. This type of cross is often used to illustrate basic Mendelian inheritance patterns, such as dominant-recessive relationships and the resulting phenotypic ratios.

Punnett Square

A Punnett square is a diagrammatic tool used to organize and visualize the combinations of alleles from the gametes of two parents. It systematically shows how alleles can combine and helps predict the possible genotypes and phenotypes of offspring resulting from a cross.

Two-Factor Cross

A two-factor cross examines the simultaneous inheritance of two different traits, each governed by its own gene pair. In such crosses, the Punnett square expands to account for multiple allele combinations, demonstrating how traits can segregate into more complex genotypic and phenotypic ratios.

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Transcript

00:01 Okay, so here we're still working on mendel's experiments and heredity.

00:04 We're going to use the probability method to find our genotypes.

00:09 And in this case, we are doing a cross from a heterozygous and a homozygous recessive.

00:23 So if we actually go ahead and do this cross, we have four possibilities for genes from the heterozygous.

00:36 But just the one from the homozygous.

00:41 So we're having a smaller pool of possibilities here.

01:00 Okay, and so our total number is going to be out of four, whereas if we were doing the two, across between two of the homozygous, i guess, we would, of course, have the 16.

01:13 And so when we want to look at the probability method, we're looking for a recessive in a.

01:24 Okay, so how many possibilities of combinations are there for b? well, there's just two.

01:29 There's the heterozygous or the recessive.

01:36 Okay, so there's two of these, and we would multiply that by the one possibility for a, which is what we're looking for.

01:45 Okay, so that's going to give us two out of four, because the total number is four here, and that would be one -half.

01:52 Our homozygous, i guess, recessive in a.

01:54 So now we want to look for dominant in both genes.

02:08 Okay, so in this instance, how many ways can the ab gene be dominant? and that only occurs in one way.

02:24 Multiply that by the number of ways the b gene can be dominant.

02:27 And again, they can only be dominant in this heterosigous form.

02:31 So that means one -fourth are going to be dominant in both.

02:37 If we want recessive in both, we're basically going to do the same thing, right? there's one way that a can be recessive in this combination.

02:45 There's one way that b can be recessive in this combination.

02:49 So we get one fourth again.

02:53 And then, just to be a little bit more complicated, we're looking for recessive in either a or b.

03:00 Okay, so for this one, if we start with recessive in a, there'd be one way it can be recessive in a, and we multiply that by both ways that be.

03:17 Can be, which is going to be either headers, i guess, or recessive.

03:22 We're going to add that to the two ways that, basically all the ways that a can be multiplied by the recessive b allele.

03:46 Okay, then we have to be careful here because we're double counting when both are recessive.

03:57 So we also need to subtract one to make sure we don't double count.

04:03 So we're subtracting one time.

04:05 Basically, we're subtracting the number of times that we get aab, which in this crosses 1.

04:11 Okay, so that's 2 plus 2 minus 1 is going to be 3 out of 4.

04:20 Should have recessive features.

04:24 And either a or b.

04:25 If we go back to our little grid, we can see that that's true.

04:31 Okay, and so then moving on, b in some ways it's trickier, and in some ways it's easier.

04:37 We're doing the standard cross, where it's heterozygous in every gene.

04:49 Okay, and in this case, we have three genes, so it's going to be a total of 64 that we're working with.

04:59 And the first thing we want to know is recessive in all traits.

05:02 So we want to find the probability of a, a, b, b, c.

05:09 Okay, so there's one way we can get a, a, multiply that by the.

05:16 So one way we can get recessive b times the one way we can get recessive c, and that will be one out of 64.

05:26 So next we want recessive in c and dominant in the other two.

05:35 Okay, so we're looking for a, anything, dominant in b, and then recessive in c.

05:44 So there's three ways we can have dominance in a.

05:48 Homozygous or kind of both flavors of heterazegis...

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