contents Principles of Biology 38 Mendelian Inheritance Inheritance follows the rules of probability. Dice and the law of probability. Predicting the likelihood of inheriting a particular trait is similar to predicting the possible outcomes of rolling dice. It is important to understand probability to study genetic inheritance. Adrienne Hart-Davis/Science Source. Topics Covered in this Module Laws of Probability The Punnett Square Major Objectives of this Module Analyze a basic Punnett square problem. Analyze a test cross problem. Determine probabilities in monohybrid and dihybrid crosses. page 198 of 989 4 pages left in this module contents Principles of Biology 38 Mendelian Inheritance Laws of Probability Probability examines how likely it is that an event will occur. Probabilities range from 0, where an event has absolutely no chance of occurring, to 1, where there is no chance that the event will not occur. There are two basic rules to calculating probabilities depending on the type of events. Multiplication rule: Predicts the probability of independent events (a particular event in one experiment does not affect the probability of event in a second experiment). For example, when rolling a dice twice, the outcome in the first roll will not affect the probable outcomes when the dice is rolled a second time. The results of the two dice rolls are independent of each other. Each individual probability is multiplied to obtain an overall probability. Addition rule: Predicts the probability of mutually exclusive events. Mutually exclusive means that if one event occurs (such as a flipped coin comes up as a heads), the other event cannot simultaneously occur (i.e., it cannot be tails). The probability of either of the mutually exclusive events, either A or B, occurring is equal to the probability of event A added to the probability of event B (in the simple coin tossing case, the additions rule tells us that the probability that a flipped coin will be either heads or tails is 100%, because it will turn up heads approximately half of the time and tails the other half). Gregor Mendel studied the patterns of inheritance by breeding pea plants. In one set of experiments, he bred lines of plants that each had a different trait (for example, seed color or shape). He then crossed those lines with different traits to create a monohybrid cross; the offspring (F1) did not have all of the traits that he observed in the parent generation. After crossing individuals in this F1 generation, the traits that had disappeared in the F1 generation reappeared in the offspring of these crosses (F2). However, the trait did not reappear in all of the offspring — on average, it appeared in 1 of 4 plants. He called this trait recessive because it could be masked by the other, dominant trait. What if Mendel had bred only four plants instead of hundreds? Would he have seen this pattern? What does probability have to do with genetics? The products of a genetic cross can be predicted using laws of probability. Mendel's principle of segregation states that two alleles in a parent segregate to form gametes with only one of those alleles through meiosis (Figure 1). The discovery of chromosomes in the latter part of the nineteenth century by Walther Flemming, along with the identification of the cell division process that gives rise to gametes (meiosis), reestablished the importance of Mendel's work. In meiosis, the parent reproductive cell, which contains a pair of each chromosome (called homologous chromosomes), undergoes two steps of division (meiosis I and meiosis II). Each parent cell divides into two cells during meiosis I, and the homologous chromosomes separate, followed by a second division in meiosis II where the two sister chromatids separate, forming four gametes that all contain a single allele (Figure 1). When alleles are described as dominant or recessive, it is important to keep in mind that dominance and recessiveness are not intrinsic allelic properties. Instead, they are effects that may only be measured in relation to the effects of the alternative alleles of the same gene. Figure 1: Mendel's principle of segregation. Homologous chromosomes contain alternative versions of the same allele. In this case, the blue chromosomes contain the allele "G" or "g," and the red chromosomes contain either "W" or "w." Note how the homologous chromosomes and their sister chromatids randomly segregate into the four different gametes based on their positions during meiosis I, leading to the two alternative pathways. The properties of the allele (dominant or recessive) do not influence the location of the alleles in gametes. © 2013 Nature Education All rights reserved. Figure Detail Which allele ends up in which gamete is random. However, one could predict the likelihood of the offspring possessing specific alleles (the genotype), and ultimately the phenotype, using the rules of inheritance that are governed by probability. Probability examines how likely it is that an event will occur. BIOSKILL What Is the Probability of a Coin Toss? Every time one flips a standard (fair) coin, the chance of getting heads is 1 in 2, or 50% (Figure 2). Each toss is one independent probability event — the outcome of one toss does not impact the outcome of any other tosses. You likely understand the reasoning behind the concept of independent probability, but it is not so easy to observe. If one flips a coin twice, there is a good chance that it may be heads both times. If one flips a coin 10 times, there is a chance, but a much smaller chance, that it will be heads all 10 times. During World War II, a South African statistician, John Kerrich, passed some of the time he spent as a prisoner of war flipping a coin 10,000 times. He ended up with heads 50.67% of the time. The more events there are, the more the results reflect the predicted probabilities. Figure 2: Calculating the probability of several independent events. The total probability of multiple independent events is calculated by first determining the probability of one of the independent events. In this case, the probability of a single coin toss resulting in heads is one-half, or 50%. Therefore, the total probability of four coin tosses resulting in heads is ½ × ½ × ½ × ½, or one-half to the fourth power. © 2014 Nature Education All rights reserved. Figure Detail Consider a person flipping a coin four times. What is the chance of getting heads on all four tries? Each of the four tries has a probability of 1/2 that it will be heads. Since none of the flips will impact the outcome of any of the other flips, you multiply the probability of heads for each flip together to determine the probability of four heads in a row. ½ x ½ x ½ x ½ = (½)4 = 1/16 = 0.0625 = 6.25% There is a bit more than a 6% chance of getting heads on all four coin flips. This example demonstrates the multiplication rule for independent probability. BIOSKILL How does the multiplication rule apply to genetics? A recessive trait is expressed only when the allele coding for the recessive trait is inherited from both parents (that is, the organism is homozygous for the recessive trait). If the recessive allele is only inherited from one parent (i.e., the organism is heterozygous), the dominant trait is expressed. In Mendel's pea plants, white flowers were recessive, and purple flowers were dominant. Mendel reasoned that if each parent carried the alleles coding for one dominant and one recessive trait, the offspring had a 50% chance of inheriting the allele coding for the recessive trait from one parent and a 50% chance of inheriting the allele coding for the recessive trait from the other parent. Test Yourself Imagine that two heterozygous plants, each with one dominant allele and one recessive allele, are mated with each other. Since a plant has a 50% chance of inheriting a particular allele from each parent, what is the probability that the offspring of the two plants will inherit two copies of the allele coding for the recessive trait? Submit There is a 25% chance that such a plant will have white flowers and, conversely, a 75% chance that it will have purple flowers. The reason that white flowers, the recessive trait, are less likely is that the recessive trait can only occur under one scenario: namely, the offspring's inheritance of both recessive alleles; conversely, there is a greater chance that the offspring will have purple flowers because purple flowers can result in two different ways: plants that are homozygous for the dominant trait and those that are heterozygous both express the purple flower trait. As a result, when counting plants with different flower colors, we expect that purple flowers should occur in a 3:1 ratio with white flowers. Mendel wanted to test this hypothetical ratio, but he knew that patterns in probability are most reliable when the numbers are large. As a result, he measured hundreds of offspring from multiple monohybrid crosses over many generations of pea plants. IN THIS MODULE Laws of Probability The Punnett Square Summary Test Your Knowledge PRIMARY LITERATURE Using sterile mates and engineered toxins to beat bugs Suppressing resistance to Bt cotton with sterile insect releases. View | Download Classic paper: How scientists cloned the first mammal (1997) Viable offspring derived from fetal and adult mammalian cells. View | Download Classic paper: Fruit fly research reveals how complex organisms form (1980) Mutations affecting segment number and polarity in Drosophila. View | Download SCIENCE ON THE WEB Genetics