12.1: Mendel’s Experiments and the Laws of Probability

12.1.1: Introduction to Mendelian Inheritance

Learning Objective

Describe the traits of pea plants that were studied by Mendel

Key Points

• Mendel studied seven characteristics of the garden pea plants: flower color, seed texture, seed color, stem length, pod color, pod texture, and flower position to develop his Laws of Inheritance.
• Genetics is the study of genes passed from parents to offspring.
• Genes are the basic fundamental units of heredity.

Key Term

genetics

The branch of biology that deals with the transmission and variation of inherited characteristics, in particular chromosomes and DNA.

Gregor Mendel and the Study of Genetics

Genetics is the study of heredity, or the passing of traits from parents to offspring. Gregor Johann Mendel set the framework for genetics long before chromosomes or genes had been identified, at a time when meiosis was not well understood. For his work, Mendel is often referred to as the "father of modern genetics. " Mendel selected a simple biological system, garden peas, and conducted methodical, quantitative analyses using large sample sizes.

Gregor Mendel

Gregor Johann Mendel was a German-speaking Moravian scientist and Augustinian friar who gained posthumous fame as the founder of the modern science of genetics.

Mendel entered the Augustinian St. Thomas's Abbey and began his training as a priest. He began studying heredity using mice, but his bishop did not like one of his friars studying animal sex, so he switched to plants. Mendel grew and studied around 29,000 garden pea plants in a monastery's garden, where he analyzed seven characteristics of the garden pea plants: flower color (purple or white), seed texture (wrinkled or round), seed color (yellow or green), stem length (long or short), pod color (yellow or green), pod texture (inflated or constricted), and flower position (axial or terminal). Based on the appearance, or phenotypes, of the seven traits, Mendel developed genotypes for those traits .

Appearance and genetic makeup of garden pea plant flowers

Based on Mendel's experiments, the genotype of the pea flowers could be determined from the phenotypes of the flowers.

Because of Mendel's work, the fundamental principles of heredity were revealed, which are often referred to as Mendel's Laws of Inheritance. We now know that genes, carried on chromosomes, are the basic functional units of heredity with the capability to be replicated, expressed, or mutated. Today, the postulates put forth by Mendel form the basis of classical, or Mendelian, genetics. Not all genes are transmitted from parents to offspring according to Mendelian genetics, but Mendel's experiments serve as an excellent starting point for thinking about inheritance.

Mendel made all of his observations and findings crossing individual plants. We can now view a human karyotype of all of the chromosomes in an individual to visualize chromosomal abnormalities in offspring, even before birth. Shortly after Mendel proposed that traits were determined by what are now known as genes, other researchers observed that different traits were often inherited together, and thereby deduced that the genes were physically linked by being located on the same chromosome. Mendel's work was the beginning of many of the advances in molecular biology over the years. 

12.1.2: Mendel’s Model System

Learning Objective

Describe the scientific reasons for the success of Mendel’s experimental work

Key Points

• Mendel used true-breeding plants in his experiments. These plants, when self-fertilized, always produce offspring with the same phenotype.
• Pea plants are easily manipulated, grow in one season, and can be grown in large quantities; these qualities allowed Mendel to conduct methodical, quantitative analyses using large sample sizes.
• Based on his experiments with the garden peas, Mendel found that one phenotype was always dominant over another recessive phenotype for the same trait.

Key Terms

phenotype

the observable characteristics of an organism, often resulting from its genetic information or a combination of genetic information and environmental factors

genotype

the specific genetic information of a cell or organism, usually a description of the allele or alleles relating to a specific gene.

true-breeding plant

a plant that always produces offspring of the same phenotype when self-fertilized; one that is homozygous for the trait being followed.

Mendel's Model System

Mendel's seminal work was accomplished using the garden pea, Pisum sativum, to study inheritance . Pea plant reproduction is easily manipulated; large quantities of garden peas could be cultivated simultaneously, allowing Mendel to conclude that his results did not occur simply by chance. The garden pea also grows to maturity within one season; several generations could be evaluated over a relatively short time. 

Pea plants have both male and female parts and can easily be grown in large numbers. For this reason, garden pea plants can either self-pollinate or cross-pollinate with other pea plants. In the absence of outside manipulation, this species naturally self-fertilizes: ova (the eggs) within individual flowers are fertilized by pollen (containing the sperm cell) from the same flower. The sperm and the eggs that produce the next generation of plants both come from the same parent. What's more, the flower petals remain sealed tightly until after pollination, preventing pollination from other plants. The result is highly inbred, or "true-breeding," pea plants. These are plants that always produce offspring that look like the parent. Today, we know that these "true-breeding" plants are homozygous for most traits.

A gardener or researcher, such as Mendel, can cross-pollinate these same plants by manually applying sperm from one plant to the pistil (containing the ova) of another plant. Now the sperm and eggs come from different parent plants. When Mendel cross-pollinated a true-breeding plant that only produced yellow peas with a true-breeding plant that only produced green peas, he found that the first generation of offspring is always all yellow peas. The green pea trait did not show up. However, if this first generation of yellow pea plants were allowed to self-pollinate, the following or second generation had a ratio of 3:1 yellow to green peas. 

In this and all the other pea plant traits Mendel followed, one form of the trait was "dominant" over another so it masked the presence of the other "recessive" form in the first generation after cross-breeding two homozygous plants.. Even if the phenotype (visible form) is hidden, the genotype (allele controlling that form of the trait) can be passed on to next generation and produce the recessive form in the second generation. By experimenting with true-breeding pea plants, Mendel avoided the appearance of unexpected (recombinant) traits in offspring that might occur if the plants were not true breeding.

Mendel's Experiments With Peas

Experimenting with thousands of garden peas, Mendel uncovered the fundamentals of genetics.

12.1.3: Mendelian Crosses

Learning Objective

Identify Mendelian crosses

Key Points

• Mendel carefully controlled his experiments by removing the anthers from the pea plants before they matured.
• First generation pea plants were called parental generation, P₀, while the following generations were called filial, F_n, where n is the number of generations from P₀.
• The ratio of characteristics in the P₀−F₁−F₂ generations became the basis for Mendel's postulates.

Key Terms

filial

of a generation or generations descending from a specific previous one

parental

of the generation of organisms that produce a hybrid

Mendelian Crosses

Mendel performed crosses, which involved mating two true-breeding individuals that have different traits . In the pea, which is a naturally self-pollinating plant, this is done by manually transferring pollen from the anther of a mature pea plant of one variety to the stigma of a separate mature pea plant of the second variety. In plants, pollen carries the male gametes (sperm) to the stigma, a sticky organ that traps pollen and allows the sperm to move down the pistil to the female gametes (ova) below. To prevent the pea plant that was receiving pollen from self-fertilizing and confounding his results, Mendel painstakingly removed all of the anthers from the plant's flowers before they had a chance to mature.

Mendelian Crosses

In one of his experiments on inheritance patterns, Mendel crossed plants that were true-breeding for violet flower color with plants true-breeding for white flower color (the P generation). The resulting hybrids in the F1 generation all had violet flowers. In the F2 generation, approximately three-quarters of the plants had violet flowers, while one-quarter had white flowers.

Plants used in first-generation crosses were called P₀, or parental generation one, plants. Mendel collected the seeds belonging to the P₀ plants that resulted from each cross and grew them the following season. These offspring were called the F₁, or the first filial (filial = offspring, daughter or son), generation. Once Mendel examined the characteristics in the F₁ generation of plants, he allowed them to self-fertilize naturally. He then collected and grew the seeds from the F₁ plants to produce the F₂, or second filial, generation. Mendel's experiments extended beyond the F₂ generation to the F₃ and F₄ generations, and so on, but it was the ratio of characteristics in the P₀−F₁−F₂ generations that were the most intriguing and became the basis for Mendel's postulates.

12.1.4: Garden Pea Characteristics Revealed the Basics of Heredity

Learning Objective

Evaluate the results of F1 and F2 generations from Mendelian crosses of peas

Key Points

• Dominant traits are inherited unchanged from one generation to the next.
• Recessive traits disappear in the first filial generation, but reappear in the second filial generation at a ratio of 3:1, dominant:recessive.
• In the F1 generation, Mendel found that one of the two options for each trait had disappeared (all offspring were identical phenotypes), while in the F₂ generation, the trait reappeared in 1/4 of the offspring (a 3:1 ratio).

Key Terms

hybrid

offspring resulting from cross-breeding different entities, e.g. two different species or two purebred parent strains

recessive

able to be covered up by a dominant trait

dominant

a relationship between alleles of a gene, in which one allele masks the expression (phenotype) of another allele at the same locus

Garden Pea Characteristics Revealed the Basics of Heredity

To fully examine each of the seven traits in garden peas, Mendel generated large numbers of F₁ and F₂ plants, reporting results from 19,959 F₂ plants alone. His findings were consistent.

What results did Mendel find in his crosses for flower color? First, Mendel confirmed that he had plants that bred true for white or violet flower color. Regardless of how many generations Mendel examined, all self-crossed offspring of parents with white flowers had white flowers, and all self-crossed offspring of parents with violet flowers had violet flowers. In addition, Mendel confirmed that, other than flower color, the pea plants were physically identical.

Once these validations were complete, Mendel applied the pollen from a plant with violet flowers to the stigma of a plant with white flowers. After gathering and sowing the seeds that resulted from this cross, Mendel found that 100 percent of the F₁ hybrid generation had violet flowers. Conventional wisdom at that time would have predicted the hybrid flowers to be pale violet or for hybrid plants to have equal numbers of white and violet flowers. In other words, the contrasting parental traits were expected to blend in the offspring. Instead, Mendel's results demonstrated that the white flower trait in the F₁ generation had completely disappeared.

Importantly, Mendel did not stop his experimentation there. He allowed the F₁ plants to self-fertilize and found that, of F₂-generation plants, 705 had violet flowers and 224 had white flowers. This was a ratio of 3.15 violet flowers per one white flower, or approximately 3:1. When Mendel transferred pollen from a plant with violet flowers to the stigma of a plant with white flowers and vice versa, he obtained about the same ratio regardless of which parent, male or female, contributed which trait. This is called a reciprocal cross: a paired cross in which the respective traits of the male and female in one cross become the respective traits of the female and male in the other cross. For the other six characteristics Mendel examined, the F₁ and F₂ generations behaved in the same way as they had for flower color. One of the two traits would disappear completely from the F₁ generation only to reappear in the F₂ generation at a ratio of approximately 3:1 .

Upon compiling his results for many thousands of plants, Mendel concluded that the characteristics could be divided into expressed and latent traits. He called these, respectively, dominant and recessive traits. Dominant traits are those that are inherited unchanged in a hybridization. Recessive traits become latent, or disappear, in the offspring of a hybridization. The recessive trait does, however, reappear in the progeny of the hybrid offspring. An example of a dominant trait is the violet-flower trait. For this same characteristic (flower color), white-colored flowers are a recessive trait. The fact that the recessive trait reappeared in the F₂ generation meant that the traits remained separate (not blended) in the plants of the F₁ generation. Mendel also proposed that plants possessed two copies of the trait for the flower-color characteristic and that each parent transmitted one of its two copies to its offspring, where they came together. Moreover, the physical observation of a dominant trait could mean that the genetic composition of the organism included two dominant versions of the characteristic or that it included one dominant and one recessive version. Conversely, the observation of a recessive trait meant that the organism lacked any dominant versions of this characteristic .

Results of Mendel's Garden Pea Hybridizations

Mendel conducted thousands of experiments and found the same ratios of offspring every time, regardless of which trait he examined.

12.1.5: Rules of Probability for Mendelian Inheritance

Learning Objective

Calculate the probability of traits of pea plants using Mendelian crosses

Key Points

• The Product Rule is used to determine the outcome of an event with two independent events; the probability of the event is the product of the probabilities of each individual event.
• The Sum Rule is used to determine the outcome of an event with two mutually exclusive events from multiple pathways; the probability of the event is the sum of the probabilities of each individual event.
• The Product Rule of probability is used to determine the probability of having both dominant traits in the F₂ progeny; it is the product of the probabilities of having the dominant trait for each characteristic.
• The Sum Rule of probability is used to determine the probability of having one dominant trait in the F2 generation of a dihybrid cross; it is the sum of the probabilities of each individual with that trait.

Key Terms

probability

a number, between 0 and 1, expressing the precise likelihood of an event happening

product rule

the probability of two independent events occurring together can be calculated by multiplying the individual probabilities of each event occurring alone

sum rule

the probability of the occurrence of one event or the other event, of two mutually exclusive events, is the sum of their individual probabilities

Probability Basics

Probabilities are mathematical measures of likelihood. The empirical probability of an event is calculated by dividing the number of times the event occurs by the total number of opportunities for the event to occur. Empirical probabilities come from observations such as those of Mendel. An example of a genetic event is a round seed produced by a pea plant. Mendel demonstrated that the probability of the event "round seed" was guaranteed to occur in the F₁ offspring of true-breeding parents, one of which has round seeds and one of which has wrinkled seeds. When the F₁ plants were subsequently self-crossed, the probability of any given F₂ offspring having round seeds was now three out of four. In other words, in a large population of F₂ offspring chosen at random, 75 percent were expected to have round seeds, whereas 25 percent were expected to have wrinkled seeds. Using large numbers of crosses, Mendel was able to calculate probabilities and use these to predict the outcomes of other crosses.

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 (P_H) or tails (P_T). 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: D₁P_H, D₁P_T, D₂P_H, D₂P_T, D₃P_H, D₃P_T, D₄P_H, D₄P_T, D₅P_H, D₅P_T, D₆P_H, D₆P_T.

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 (P_H) = (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.

Role of probability in segregation of alleles and fertilization

In a genetic cross, the probability of the dominant trait being expressed is dependent upon its frequency. In this case, both parents possessed a dominant and a recessive gene for the trait of flower color. The dominant trait is expressed in 3/4 of the offspring and the recessive trait is expressed in 1/4.

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 (P_H) and the quarter is tails (Q_T), or the quarter is heads (Q_H) and the penny is tails (P_T). Either case fulfills the outcome. We calculate the probability of obtaining one head and one tail as [(P_H) × (Q_T)] + [(Q_H) × (P_T)] = [(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 P_H and Q_T and also the probability of P_T and Q_H, 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.

Attributions

• Introduction to Mendelian Inheritance
  • "Boundless." http://www.boundless.com/. Boundless Learning CC BY-SA 3.0.
  • "Gregor Mendel." https://en.wikipedia.org/wiki/Gregor_Mendel. Wikipedia CC BY-SA 3.0.
  • "genetics." http://en.wiktionary.org/wiki/genetics. Wiktionary CC BY-SA 3.0.
  • "Mapping Genomes." http://cnx.org/contents/GFy_h8cu@9.87:Qq6Y1A16@5/Mapping-Genomes. OpenStax CNX CC BY-SA 3.0.
  • "OpenStax College, Biology. October 16, 2013." http://cnx.org/content/m44474/latest/?collection=col11448/latest. OpenStax CNX CC BY 3.0.
  • "Gregor_Mendel_2.jpg." https://upload.wikimedia.org/wikipedia/commons/b/ba/Gregor_Mendel_2.jpg. Wikipedia Public domain.
  • "Punnett square mendel flowers." http://commons.wikimedia.org/wiki/File:Punnett_square_mendel_flowers.svg. Wikimedia CC BY-SA 3.0.
  • "Gregor_Mendel.png." https://commons.wikimedia.org/wiki/File:Gregor_Mendel.png. Wikipedia Public domain.
• Mendel’s Model System
  • "OpenStax College, Biology. October 16, 2013." http://openstaxcollege.org/textbooks/biology. OpenStax CNX CC BY-SA 3.0.
  • "true-breeding." http://en.wikipedia.org/wiki/true-breeding. Wikipedia CC BY-SA 3.0.
  • "phenotype." http://en.wiktionary.org/wiki/phenotype. Wiktionary CC BY-SA 3.0.
  • "genotype." http://en.wiktionary.org/wiki/genotype. Wiktionary CC BY-SA 3.0.
  • "Human Physiology/Genetics and inheritance." http://en.wikibooks.org/wiki/Human_Physiology/Genetics_and_inheritance. Wikibooks CC BY-SA 3.0.
  • "OpenStax College, Biology. October 16, 2013." http://cnx.org/content/m44476/latest/?collection=col11448/latest. OpenStax CNX CC BY 3.0.
  • "OpenStax College, Introduction. November 1, 2013." http://cnx.org/content/m44474/latest/Figure_12_00_01new.jpg. OpenStax CNX CC BY 3.0.
• Mendelian Crosses
  • "Boundless." http://www.boundless.com/. Boundless Learning CC BY-SA 3.0.
  • "filial." http://en.wiktionary.org/wiki/filial. Wiktionary CC BY-SA 3.0.
  • "parental." http://en.wiktionary.org/wiki/parental. Wiktionary CC BY-SA 3.0.
  • "OpenStax College, Biology. October 16, 2013." http://cnx.org/content/m44476/latest/?collection=col11448/latest. OpenStax CNX CC BY 3.0.
  • "OpenStax College, Mendel’s Experiments and the Laws of Probability. October 16, 2013." http://cnx.org/content/m44476/latest/Figure_12_01_02.jpg. OpenStax CNX CC BY 3.0.
• Garden Pea Characteristics Revealed the Basics of Heredity
  • "Boundless." http://www.boundless.com/. Boundless Learning CC BY-SA 3.0.
  • "dominant." http://en.wikipedia.org/wiki/dominant. Wikipedia CC BY-SA 3.0.
  • "recessive." http://en.wiktionary.org/wiki/recessive. Wiktionary CC BY-SA 3.0.
  • "hybrid." http://en.wiktionary.org/wiki/hybrid. Wiktionary CC BY-SA 3.0.
  • "OpenStax College, Biology. October 16, 2013." http://cnx.org/content/m44476/latest/?collection=col11448/latest. OpenStax CNX CC BY 3.0.
  • "OpenStax College, Mendel’s Experiments and the Laws of Probability. November 1, 2013." http://cnx.org/content/m44476/latest/#tab-ch12-01-01. OpenStax CNX CC BY 3.0.
• Rules of Probability for Mendelian Inheritance
  • "Boundless." http://www.boundless.com/. Boundless Learning CC BY-SA 3.0.
  • "probability." http://en.wiktionary.org/wiki/probability. Wiktionary CC BY-SA 3.0.
  • "OpenStax College, Biology. October 16, 2013." http://cnx.org/content/m44476/latest/?collection=col11448/latest. OpenStax CNX CC BY 3.0.
  • "Punnett square mendel flowers." http://commons.wikimedia.org/wiki/File:Punnett_square_mendel_flowers.svg. Wikimedia CC BY-SA 3.0.