With our modern understanding of genes, chromosomes, and cellular reproduction, we can explain the biological basis of Mendel’s observations and make pretty accurate predictions about the offspring that any given cross (short for crossbreeding) will produce.

Each of the traits that Mendel observed in his pea plants came in one of two varieties; modern science calls any gene that gives rise to more than one version of the same trait an allele. So, for example, the tall gene and the short gene are different alleles (variations) of the height gene.

Every somatic cell contains two complete sets of chromosomes, one from each parent. Now you can understand why homologous chromosomes are similar, but not identical: although they contain the same genes, they may not contain the same alleles for these genes.

Going back to Mendel’s plants, we can now say that all of his true-breeding plants contained two of the same alleles for each of the observed genes. Tall plants in this P generation had two alleles for tallness (TT), and short P generation plants had two alleles for shortness (tt). Anytime an organism’s two alleles for a specific trait are identical, that the individual is said to be homozygous (“homo” means same) for that trait.

On the other hand, crossing the tall and short plants to produce F₁ hybrids created a generation of plants with one tall allele and one short allele (Tt). An organism with two opposing alleles for a single gene is said to be heterozygous for that trait.

Although the P generation of pure-breeding tall plants looked the same as their hybrid F₁ offspring, the P and F₁ generations did not have identical genetic makeups. The genetic makeup of a certain trait (e.g., TT, Tt, or tt) is called its genotype, while the physical expression of these traits (e.g., short or tall) is called a phenotype.

For any given trait, an organism’s genotype will indicate alleles from both parents, while the phenotype only indicates the allelic form that is physically expressed in that individual. This distinction between genetic makeup and physical appearance explains the apparent “disappearance” of the recessive alleles in the F₁ generation. Mendel’s results for the F₂ generation can also be reinterpreted in light of these new distinctions. Mendel’s results showed that 75 percent of the F₂ offspring exhibited the dominant phenotype, a ratio of 3:1 dominant to recessive. But from a genetic perspective, the breakdown would actually be around 25 percent homozygous dominant (TT), 50 percent heterozygous with a dominant phenotype (Tt), and 25 percent homozygous recessive (tt)—a ratio of 1:2:1.

The Punnett square is a convenient graphical method for representing the genotypes of the parental gametes and all the possible offspring they produce. The Punnett square below shows the mating of two F₁ hybrids (Aa genotype). We call this mating a monohybrid cross, because it involves only one gene. According to the law of segregation, two possible gametes are formed: A and a. The paternal gametes are listed as columns across the top of the square, and maternal gametes are listed as rows down the left side of the square. Combining the gametes in the intersecting boxes provides the genotypes of all possible offspring.

In this case, 25 percent of the F₂ offspring will be AA, 50 percent will be Aa, and 25 percent will be aa. Both AA and Aa will have the dominant phenotype, giving the 3:1 ratio (75 percent to 25 percent) of dominant to recessive phenotypes that Mendel observed.

For the SAT II Biology, if you are given the genotypes of two parents, you should be able to predict the genotypes and phenotypes of their offspring by using a Punnett square.

After finishing his monohybrid crosses, Mendel moved on to dihybrid crosses, in which he bred pure, parental varieties that had two traits distinguishing them from each other. He wanted to determine whether the inheritance of one trait was connected in any way to the inheritance of the other.

The color and shape of the pea seeds provided two convenient traits to study. The seeds were either yellow or green, with yellow dominant; in shape, they were either round or wrinkled, with round dominant. Mendel crossed double dominant (phenotype yellow and round, genotype RRYY) plants with double recessive (phenotype green and wrinkled, genotype rryy) plants. As expected, the F₁ generation consisted of hybrid offspring all with the double dominant (round yellow) phenotype and a heterozygous genotype (RrYy). The key test came in the proportions of different phenotypes in the F₂ generation. If the inheritance of one trait did not influence the inheritance of the other, then each parent should make equal numbers of the four possible gametes, and sixteen different genotypes would be equally represented in the offspring. As seen in the Punnett square below, there should be four different phenotypes (yellow and round, green and round, yellow and wrinkled, green and wrinkled) occurring in the proportions 9:3:3:1.

Mendel’s phenotype counts of F₂ seeds did indeed show the 9:3:3:1 proportions anticipated in the Punnett square for the dihybrid cross. From these results, he concluded that the inheritance of one trait was unrelated to the inheritance of a second trait. The units from any one hereditary pair segregate into the gametes independently of the segregation of the units from any other pair. This principle is known as the law of independent assortment.

Drawing Punnett squares is a helpful way to visualize simple genetics problems, but with problems involving several different genes, it is often easier to use the rules of probability. (A Punnett square for a three-gene hybrid cross would have 64 squares!) There are two rules of probability that you will need to solve genetics problems. First, the probability of an outcome that depends on the occurrence of two or more independent events is obtained by multiplying together the probability of each necessary independent event. This is the and rule of probability:

In contrast, if an outcome depends on the occurrence of any one of several mutually exclusive alternatives, then the probability of the outcome is obtained by adding together the probabilities of the alternatives. This is the or rule of probability:

As an example, we can calculate the probability of getting an 11 when rolling two dice, die A and die B. In order to roll an 11, we need a 5 and a 6. The probability of rolling a 5 on die A and a 6 on die B is But we can also roll an 11 with a 6 on die A and a 5 on die B. This is a mutually exclusive alternative to the first roll we considered; its probability is also ¹/ ₃₆. Since either A5, B6 or A6, B5 gives us a total of 11, the final probability of rolling an 11 using two dice is ¹ /₃₆ + ¹/₃₆ = ²/₃₆ = ¹/₁₈.

Moving from gambling to genetics, we can calculate the probability that a cross between genotypes AABBCc and aaBbCc will produce an offspring with genotype AaBbcc. Taking one gene at a time, the probability of the Aa combination is a perfect 1, since an AA and aa cross can produce only Aa offspring.

The probability of the Bb combination is ¹/₂, because the BB and Bb cross will produce Bb offspring 50 percent of the time.

The probability of the cc combination is ¹/₄, because the Cc and Cc cross gives cc offspring 25 percent of the time.

Since Aa and Bb and cc must occur to produce our desired outcome, the probability is

A test cross is the means by which a scientist can determine whether an individual with a dominant phenotype has a homozygous (AA) or heterozygous (Aa) dominant genotype. The test cross involves mating the individual with the dominant phenotype to an individual with a recessive (aa) phenotype and observing the offspring produced. If the individual being tested is homozygous dominant, then all offspring will have a dominant phenotype, since all the offspring will have at least one A allele and the A is dominant.

If the tested individual is heterozygous dominant, then half of the offspring will show the dominant phenotype, while the other half show the recessive phenotype.

Mendel’s law of dominance is generally true, but there are many exceptions to the law. In some instances, instead of a heterozygote expressing only one of two alleles, both alleles could be partially expressed. For example, the flower color of the four o’clock plant is determined by a single gene with two alleles: plants homozygous for the R₁ allele have red flowers, while plants homozygous for the R₂ allele have white flowers. If interbred, the heterozygous R₁R₂ plants have pink flowers. Incomplete dominance is the term used to describe the situation in which the heterozygote phenotype is intermediate between the two homozygous phenotypes.

If the heterozygote form simultaneously expresses both alleles fully, then the relationship between the two alleles is called codominance. An example of codominance appears in human blood type. Blood type is determined by two alleles, A and B, that code for the presence of antigen A and antigen B on the surface of red blood cells. Allele A and B are codominant. If only the allele A is present, then only antigen A exists on the blood cell. If only allele B is present, then only antigen B exists on the blood cell. If both alleles A and B are present, neither dominates the other and both antigens appear on the red blood cell. A third allele, i, is recessive: if only it appears, then the blood is of type O. The following is a summary of the genotypes that result in the four different blood types:

Fortunately for Mendel, the genes encoding his selected traits did not reside close together on the same chromosome. If they had, his dihybrid cross results would have been much more confusing, and he might not have discovered the law of independent assortment. The law of independent assortment holds true as long as two different genes are on separate chromosomes. When the genes are on separate chromosomes, the two alleles of one gene (A and a) will segregate into gametes independently of the two alleles of the other gene (B and b). Equal numbers of four different gametes will result: AB, aB, Ab, ab. But if the two genes are on the same chromosome, then they will be linked and will segregate together during meoisis, producing only two kinds of gametes.

For instance, if the genes for seed shape and seed color were on the same chromosome and a homozygous double dominant (yellow and round, RRYY) plant was crossed with a homozygous double recessive (green and wrinkled, rryy), the F₁ hybrid offspring, as usual, would be double heterozygous dominant (yellow and round, RrYy). However, since in this example the R and Y are linked together on the chromosome inherited from the dominant parent, with r and y linked together on the other chromosome, only two different gametes can be formed: RY and ry. Therefore, instead of 16 different genotypes in the F₂ offspring, only three are possible: RRYY, RrYy, rryy. And instead of four different phenotypes, only the original two will exist. Notice that the inheritance pattern now resembles that seen in a monohybrid cross, with a 3:1 phenotypic ratio, rather than the 9:3:3:1 ratio expected from the dihybrid cross. If physically linked on a single chromosome, the round and yellow alleles would segregate together, and the wrinkled and green alleles would segregate together: no round green seeds or wrinkled yellow seeds would ever appear.

The above explanation, however, neglects the influence of the crossing over of genetic material that occurs during meiosis. The farther away two genes are from one another, the more likely an exchange point for crossing over will form between them. At these exchange points, the alleles of one gene switch to the opposite homologous chromosome, while the other gene alleles remain with their original chromosomes. When alleles switch places like this, the resulting gametes are called recombinant. In the example above, the original parental gametes would be RY and ry, while the recombinant gametes would be Ry and rY. Thus four different kinds of gametes will be formed, instead of only two formed when the genes were linked.

If two genes are extremely close together, crossing over will almost never occur between them, and recombinant gametes will almost never form. If they are very far apart on the chromosome, crossing over will almost certainly occur between them, and recombinant gametes will form just as often as if the genes were on different chromosomes (50 percent of the time). If the genes are at an intermediate distance from each other, crossing over may sometimes occur between them and sometimes not. Therefore, the percentage of recombinant gametes (reflected in the percentage of recombinant offspring) correlates with the distance between two genes on a chromosome. By comparing the recombination rates of multiple different pairs of genes on the same chromosome, the relative position of each gene along the chromosome can be determined. This method of ordering genes on a chromosome is called a linkage map.