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15.3: Mendel’s Experiments and the Laws of Probability - Biology


Learning Objectives

By the end of this section you will be able to:,

  • Describe the scientific reasons for the success of Mendel’s experimental work
  • Describe the expected outcomes of monohybrid crosses involving dominant and recessive alleles
  • Apply the sum and product rules to calculate probabilities

Johann Gregor Mendel (1822–1884) (Figure 1) was a lifelong learner, teacher, scientist, and man of faith. As a young adult, he joined the Augustinian Abbey of St. Thomas in Brno in what is now the Czech Republic. Supported by the monastery, he taught physics, botany, and natural science courses at the secondary and university levels. In 1856, he began a decade-long research pursuit involving inheritance patterns in honeybees and plants, ultimately settling on pea plants as his primary model system (a system with convenient characteristics used to study a specific biological phenomenon to be applied to other systems). In 1865, Mendel presented the results of his experiments with nearly 30,000 pea plants to the local Natural History Society. He demonstrated that traits are transmitted faithfully from parents to offspring independently of other traits and in dominant and recessive patterns. In 1866, he published his work, Experiments in Plant Hybridization,[1] in the proceedings of the Natural History Society of Brünn.

Mendel’s work went virtually unnoticed by the scientific community that believed, incorrectly, that the process of inheritance involved a blending of parental traits that produced an intermediate physical appearance in offspring; this hypothetical process appeared to be correct because of what we know now as continuous variation. Continuous variation results from the action of many genes to determine a characteristic like human height. Offspring appear to be a “blend” of their parents’ traits when we look at characteristics that exhibit continuous variation. The blending theory of inheritance asserted that the original parental traits were lost or absorbed by the blending in the offspring, but we now know that this is not the case. Mendel was the first researcher to see it. Instead of continuous characteristics, Mendel worked with traits that were inherited in distinct classes (specifically, violet versus white flowers); this is referred to as discontinuous variation. Mendel’s choice of these kinds of traits allowed him to see experimentally that the traits were not blended in the offspring, nor were they absorbed, but rather that they kept their distinctness and could be passed on. In 1868, Mendel became abbot of the monastery and exchanged his scientific pursuits for his pastoral duties. He was not recognized for his extraordinary scientific contributions during his lifetime. In fact, it was not until 1900 that his work was rediscovered, reproduced, and revitalized by scientists on the brink of discovering the chromosomal basis of heredity.

Mendel’s Model System

Mendel’s seminal work was accomplished using the garden pea, Pisum sativum, to study inheritance. This species naturally self-fertilizes, such that pollen encounters ova within individual flowers. 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. By experimenting with true-breeding pea plants, Mendel avoided the appearance of unexpected traits in offspring that might occur if the plants were not true breeding. The garden pea also grows to maturity within one season, meaning that several generations could be evaluated over a relatively short time. Finally, large quantities of garden peas could be cultivated simultaneously, allowing Mendel to conclude that his results did not come about simply by chance.

Mendelian Crosses

Mendel performed hybridizations, which involve mating two true-breeding individuals that have different traits. In the pea, which is naturally self-pollinating, 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.

Plants used in first-generation crosses were called P0, or parental generation one, plants (Figure). 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.

Garden Pea Characteristics Revealed the Basics of Heredity

In his 1865 publication, Mendel reported the results of his crosses involving seven different characteristics, each with two contrasting traits. A trait is defined as a variation in the physical appearance of a heritable characteristic. The characteristics included plant height, seed texture, seed color, flower color, pea pod size, pea pod color, and flower position. For the characteristic of flower color, for example, the two contrasting traits were white versus violet. To fully examine each characteristic, 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 (Table 1).

Table 1. The Results of Mendel’s Garden Pea Hybridizations
CharacteristicContrasting P0 TraitsF1 Offspring TraitsF2 Offspring TraitsF2 Trait Ratios
Flower colorViolet vs. white100 percent violet
  • 705 violet
  • 224 white
3.15:1
Flower positionAxial vs. terminal100 percent axial
  • 651 axial
  • 207 terminal
3.14:1
Plant heightTall vs. dwarf100 percent tall
  • 787 tall
  • 277 dwarf
2.84:1
Seed textureRound vs. wrinkled100 percent round
  • 5,474 round
  • 1,850 wrinkled
2.96:1
Seed colorYellow vs. green100 percent yellow
  • 6,022 yellow
  • 2,001 green
3.01:1
Pea pod textureInflated vs. constricted100 percent inflated
  • 882 inflated
  • 299 constricted
2.95:1
Pea pod colorGreen vs. yellow100 percent green
  • 428 green
  • 152 yellow
2.82:1

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.

So why did Mendel repeatedly obtain 3:1 ratios in his crosses? To understand how Mendel deduced the basic mechanisms of inheritance that lead to such ratios, we must first review the laws of probability.

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. It is also possible to calculate theoretical probabilities by dividing the number of times that an event is expected to occur by the number of times that it could occur. Empirical probabilities come from observations, like those of Mendel. Theoretical probabilities come from knowing how the events are produced and assuming that the probabilities of individual outcomes are equal. A probability of one for some event indicates that it is guaranteed to occur, whereas a probability of zero indicates that it is guaranteed not to occur. An example of a genetic event is a round seed produced by a pea plant. In his experiment, Mendel demonstrated that the probability of the event “round seed” occurring was one 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 and Sum Rule

Mendel demonstrated that the pea-plant characteristics he studied were transmitted as discrete units from parent to offspring. As will be discussed, Mendel also determined that different characteristics, like seed color and seed texture, 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 still produced offspring that had a 3:1 ratio of green:yellow seeds (ignoring seed texture) and a 3:1 ratio of round:wrinkled seeds (ignoring seed color). 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. The product rule states that the probability of two independent events occurring together can be calculated by multiplying the individual probabilities of each event occurring alone. To demonstrate the product rule, 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 of this action (Table 2), and each event is expected to occur with equal probability.

Table 2. Twelve Equally Likely Outcomes of Rolling a Die and Flipping a Penny
Rolling DieFlipping Penny
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. By the product rule, the probability that you will obtain the combined outcome 2 and heads is: (D2) × (PH) = (1/6) × (1/2) or 1/12 (Table). Notice the word “and” in the description of the probability. The “and” is a signal to apply the product rule. For example, consider how the product rule is applied to the dihybrid cross: 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, as shown here:

On the other hand, the sum rule of probability is applied when considering two mutually exclusive outcomes that can come about by more than one pathway. The sum rule states that the probability of the occurrence of one event or the other event, of two mutually exclusive events, is the sum of their individual probabilities. Notice the word “or” in the description of the probability. The “or” indicates that you should apply the sum rule. In this case, 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 coin coming up tails? This outcome can be achieved by two cases: the penny may be heads (PH) and the quarter may be tails (QT), or the quarter may be heads (QH) and the penny may be tails (PT). Either case fulfills the outcome. By the sum rule, 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 (Table). 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. Again, the sum rule can be applied to show the probability of having just one dominant trait in the F2 generation of a dihybrid cross:

Table 3. The Product Rule and Sum Rule
Product RuleSum Rule
For independent events A and B, the probability (P) of them both occurring (A and B) is (PA × PB)For mutually exclusive events A and B, the probability (P) that at least one occurs (A or B) is (PA + PB)

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 calculate the probabilities of the traits appearing in his F2 generation. As you will learn, this discovery meant that when parental traits were known, the offspring’s traits could be predicted accurately even before fertilization.

Section Summary

Working with garden pea plants, Mendel found that crosses between parents that differed by one trait produced F1 offspring that all expressed the traits of one parent. Observable traits are referred to as dominant, and non-expressed traits are described as recessive. When the offspring in Mendel’s experiment were self-crossed, the F2 offspring exhibited the dominant trait or the recessive trait in a 3:1 ratio, confirming that the recessive trait had been transmitted faithfully from the original P0 parent. Reciprocal crosses generated identical F1 and F2 offspring ratios. By examining sample sizes, Mendel showed that his crosses behaved reproducibly according to the laws of probability, and that the traits were inherited as independent events.

Two rules in probability can be used to find the expected proportions of offspring of different traits from different crosses. To find the probability of two or more independent events occurring together, apply the product rule and multiply the probabilities of the individual events. The use of the word “and” suggests the appropriate application of the product rule. To find the probability of two or more events occurring in combination, apply the sum rule and add their individual probabilities together. The use of the word “or” suggests the appropriate application of the sum rule.

A Open Assessments element has been excluded from this version of the text. You can view it online here: pb.libretexts.org/fob2/?p=216

Additional Self Check Questions

  1. Describe one of the reasons why the garden pea was an excellent choice of model system for studying inheritance.
  2. How would you perform a reciprocal cross for the characteristic of stem height in the garden pea?

Answers

  1. The garden pea is sessile and has flowers that close tightly during self-pollination. These features help to prevent accidental or unintentional fertilizations that could have diminished the accuracy of Mendel’s data.
  2. Two sets of P0 parents would be used. In the first cross, pollen would be transferred from a true-breeding tall plant to the stigma of a true-breeding dwarf plant. In the second cross, pollen would be transferred from a true-breeding dwarf plant to the stigma of a true-breeding tall plant. For each cross, F1 and F2 offspring would be analyzed to determine if offspring traits were affected according to which parent donated each trait.

Try It

blending theory of inheritance: hypothetical inheritance pattern in which parental traits are blended together in the offspring to produce an intermediate physical appearance

continuous variation: inheritance pattern in which a character shows a range of trait values with small gradations rather than large gaps between them

discontinuous variation: inheritance pattern in which traits are distinct and are transmitted independently of one another

dominant: trait which confers the same physical appearance whether an individual has two copies of the trait or one copy of the dominant trait and one copy of the recessive trait

F1: first filial generation in a cross; the offspring of the parental generation

F2: second filial generation produced when F1 individuals are self-crossed or fertilized with each other

hybridization: process of mating two individuals that differ with the goal of achieving a certain characteristic in their offspring

model system: species or biological system used to study a specific biological phenomenon to be applied to other different species

P0: parental generation in a cross

product rule: probability of two independent events occurring simultaneously can be calculated by multiplying the individual probabilities of each event occurring alone

recessive: trait that appears “latent” or non-expressed when the individual also carries a dominant trait for that same characteristic; when present as two identical copies, the recessive trait is expressed

reciprocal cross: 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

sum rule: probability of the occurrence of at least one of two mutually exclusive events is the sum of their individual probabilities

trait: variation in the physical appearance of a heritable characteristic



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. It is also possible to calculate theoretical probabilities by dividing the number of times that an event is expected to occur by the number of times that it could occur. Empirical probabilities come from observations, like those of Mendel. Theoretical probabilities, on the other hand, come from knowing how the events are produced and assuming that the probabilities of individual outcomes are equal. A probability of one for some event indicates that it is guaranteed to occur, whereas a probability of zero indicates that it is guaranteed not to occur. An example of a genetic event is a round seed produced by a pea plant.

In one experiment, Mendel demonstrated that the probability of the event “round seed” occurring was one 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.


Understanding of Inheritance in the Mid-1800s

From the standpoint of basic qualifications, Mendel was perfectly positioned to make a major breakthrough in the then-all-but-nonexistent field of genetics, and he was blessed with both the environment and the patience to get done what he needed to do. Mendel would end up growing and studying nearly 29,000 pea plants between 1856 and 1863.

When Mendel first began his work with pea plants, the scientific concept of heredity was rooted in the concept of blended inheritance, which held that parental traits were somehow mixed into offspring in the manner of different-colored paints, producing a result that was not quite the mother and not quite the father every time, but that clearly resembled both.

Mendel was intuitively aware from his informal observation of plants that if there was any merit to this idea, it certainly didn't apply to the botanical world.

Mendel was not interested in the appearance of his pea plants per se. He examined them in order to understand which characteristics could be passed on to future generations and exactly how this occurred at a functional level, even if he didn't have the literal tools to see what was occurring at the molecular level.


The Second Law of Thermodynamics

A living cell’s primary tasks of obtaining, transforming, and using energy to do work may seem simple. However, the second law of thermodynamics explains why these tasks are harder than they appear. None of the energy transfers that we have discussed, along with all energy transfers and transformations in the universe, is completely efficient. In every energy transfer, some amount of energy is lost in a form that is unusable. In most cases, this form is heat energy. Thermodynamically, scientists define heat energy as energy that transfers from one system to another that is not doing work. For example, when an airplane flies through the air, it loses some of its energy as heat energy due to friction with the surrounding air. This friction actually heats the air by temporarily increasing air molecule speed. Likewise, some energy is lost as heat energy during cellular metabolic reactions. This is good for warm-blooded creatures like us, because heat energy helps to maintain our body temperature. Strictly speaking, no energy transfer is completely efficient, because some energy is lost in an unusable form.

An important concept in physical systems is that of order and disorder (or randomness). The more energy that a system loses to its surroundings, the less ordered and more random the system. Scientists refer to the measure of randomness or disorder within a system as entropy . High entropy means high disorder and low energy ((Figure)). To better understand entropy, think of a student’s bedroom. If no energy or work were put into it, the room would quickly become messy. It would exist in a very disordered state, one of high entropy. Energy must be put into the system, in the form of the student doing work and putting everything away, in order to bring the room back to a state of cleanliness and order. This state is one of low entropy. Similarly, a car or house must be constantly maintained with work in order to keep it in an ordered state. Left alone, a house’s or car’s entropy gradually increases through rust and degradation. Molecules and chemical reactions have varying amounts of entropy as well. For example, as chemical reactions reach a state of equilibrium, entropy increases, and as molecules at a high concentration in one place diffuse and spread out, entropy also increases.


59 Mendel’s Experiments and the Laws of Probability

By the end of this section, you will be able to do the following:

  • Describe the scientific reasons for the success of Mendel’s experimental work
  • Describe the expected outcomes of monohybrid crosses involving dominant and recessive alleles
  • Apply the sum and product rules to calculate probabilities

Johann Gregor Mendel (1822–1884) ((Figure)) was a lifelong learner, teacher, scientist, and man of faith. As a young adult, he joined the Augustinian Abbey of St. Thomas in Brno in what is now the Czech Republic. Supported by the monastery, he taught physics, botany, and natural science courses at the secondary and university levels. In 1856, he began a decade-long research pursuit involving inheritance patterns in honeybees and plants, ultimately settling on pea plants as his primary model system (a system with convenient characteristics used to study a specific biological phenomenon to be applied to other systems). In 1865, Mendel presented the results of his experiments with nearly 30,000 pea plants to the local Natural History Society. He demonstrated that traits are transmitted from parents to offspring independently of other traits and in dominant and recessive patterns. In 1866, he published his work, Experiments in Plant Hybridization, 1 in the proceedings of the Natural History Society of Brünn.

Mendel’s work went virtually unnoticed by the scientific community, which believed, incorrectly, that the process of inheritance involved a blending of parental traits that produced an intermediate physical appearance in offspring. The blending theory of inheritance asserted that the original parental traits were lost or absorbed by the blending in the offspring, but we now know that this is not the case. This hypothetical process appeared to be correct because of what we know now as continuous variation. Continuous variation results from the action of many genes to determine a characteristic like human height. Offspring appear to be a “blend” of their parents’ traits.

Instead of continuous characteristics, Mendel worked with traits that were inherited in distinct classes (specifically, violet versus white flowers) this is referred to as discontinuous variation . Mendel’s choice of these kinds of traits allowed him to see experimentally that the traits were not blended in the offspring, nor were they absorbed, but rather that they kept their distinctness and could be passed on. In 1868, Mendel became abbot of the monastery and exchanged his scientific pursuits for his pastoral duties. He was not recognized for his extraordinary scientific contributions during his lifetime. In fact, it was not until 1900 that his work was rediscovered, reproduced, and revitalized by scientists on the brink of discovering the chromosomal basis of heredity.

Mendel’s Model System

Mendel’s seminal work was accomplished using the garden pea, Pisum sativum, to study inheritance. This species naturally self-fertilizes, such that pollen encounters ova within individual flowers. 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. By experimenting with true-breeding pea plants, Mendel avoided the appearance of unexpected traits in offspring that might occur if the plants were not true breeding. The garden pea also grows to maturity within one season, meaning that several generations could be evaluated over a relatively short time. Finally, large quantities of garden peas could be cultivated simultaneously, allowing Mendel to conclude that his results did not come about simply by chance.

Mendelian Crosses

Mendel performed hybridizations , which involve mating two true-breeding individuals that have different traits. In the pea, which is naturally self-pollinating, 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.

Plants used in first-generation crosses were called P0 , or parental generation one ((Figure)). After each cross, Mendel collected the seeds belonging to the P0 plants 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.

Garden Pea Characteristics Revealed the Basics of Heredity

In his 1865 publication, Mendel reported the results of his crosses involving seven different characteristics, each with two contrasting traits. A trait is defined as a variation in the physical appearance of a heritable characteristic. The characteristics included plant height, seed texture, seed color, flower color, pea pod size, pea pod color, and flower position. For the characteristic of flower color, for example, the two contrasting traits were white versus violet. To fully examine each characteristic, 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 (the blending theory) 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 ((Figure)).

  • 705 violet
  • 224 white
  • 651 axial
  • 207 terminal
  • 787 tall
  • 277 dwarf
  • 5,474 round
  • 1,850 wrinkled
  • 6,022 yellow
  • 2,001 green
  • 882 inflated
  • 299 constricted
  • 428 green
  • 152 yellow

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.

So why did Mendel repeatedly obtain 3:1 ratios in his crosses? To understand how Mendel deduced the basic mechanisms of inheritance that lead to such ratios, we must first review the laws of probability.

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. It is also possible to calculate theoretical probabilities by dividing the number of times that an event is expected to occur by the number of times that it could occur. Empirical probabilities come from observations, like those of Mendel. Theoretical probabilities, on the other hand, come from knowing how the events are produced and assuming that the probabilities of individual outcomes are equal. A probability of one for some event indicates that it is guaranteed to occur, whereas a probability of zero indicates that it is guaranteed not to occur. An example of a genetic event is a round seed produced by a pea plant.

In one experiment, Mendel demonstrated that the probability of the event “round seed” occurring was one 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 and Sum Rule

Mendel demonstrated that pea plants transmit characteristics as discrete units from parent to offspring. As will be discussed, Mendel also determined that different characteristics, like seed color and seed texture, 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 still produced offspring that had a 3:1 ratio of green:yellow seeds (ignoring seed texture) and a 3:1 ratio of round:wrinkled seeds (ignoring seed color). 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. The product rule states that the probability of two independent events occurring together can be calculated by multiplying the individual probabilities of each event occurring alone. To demonstrate the product rule, 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 of this action ((Figure)), and each event is expected to occur with equal probability.

Twelve Equally Likely Outcomes of Rolling a Die and Flipping a Penny
Rolling Die Flipping Penny
D1 PH
D1 PT
D2 PH
D2 PT
D3 PH
D3 PT
D4 PH
D4 PT
D5 PH
D5 PT
D6 PH
D6 PT

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. By the product rule, the probability that you will obtain the combined outcome 2 and heads is: (D2) x (PH) = (1/6) x (1/2) or 1/12 ((Figure)). Notice the word “and” in the description of the probability. The “and” is a signal to apply the product rule. For example, consider how the product rule is applied to the dihybrid cross: 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, as shown here:

On the other hand, the sum rule of probability is applied when considering two mutually exclusive outcomes that can come about by more than one pathway. The sum rule states that the probability of the occurrence of one event or the other event, of two mutually exclusive events, is the sum of their individual probabilities. Notice the word “or” in the description of the probability. The “or” indicates that you should apply the sum rule. In this case, 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 coin coming up tails? This outcome can be achieved by two cases: the penny may be heads (PH) and the quarter may be tails (QT), or the quarter may be heads (QH) and the penny may be tails (PT). Either case fulfills the outcome. By the sum rule, 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 ((Figure)). 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. Again, the sum rule can be applied to show the probability of having just one dominant trait in the F2 generation of a dihybrid cross:

The Product Rule and Sum Rule
Product Rule Sum Rule
For independent events A and B, the probability (P) of them both occurring (A and B) is (PA × PB) For mutually exclusive events A and B, the probability (P) that at least one occurs (A or B) is (PA + PB)

To use probability laws in practice, we must 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 calculate the probabilities of the traits appearing in his F2 generation. As you will learn, this discovery meant that when parental traits were known, the offspring’s traits could be predicted accurately even before fertilization.

Section Summary

Working with garden pea plants, Mendel found that crosses between parents that differed by one trait produced F1 offspring that all expressed the traits of one parent. Observable traits are referred to as dominant, and non-expressed traits are described as recessive. When the offspring in Mendel’s experiment were self-crossed, the F2 offspring exhibited the dominant trait or the recessive trait in a 3:1 ratio, confirming that the recessive trait had been transmitted faithfully from the original P0 parent. Reciprocal crosses generated identical F1 and F2 offspring ratios. By examining sample sizes, Mendel showed that his crosses behaved reproducibly according to the laws of probability, and that the traits were inherited as independent events.

Two rules in probability can be used to find the expected proportions of offspring of different traits from different crosses. To find the probability of two or more independent events occurring together, apply the product rule and multiply the probabilities of the individual events. The use of the word “and” suggests the appropriate application of the product rule. To find the probability of two or more events occurring in combination, apply the sum rule and add their individual probabilities together. The use of the word “or” suggests the appropriate application of the sum rule.

Review Questions

Mendel performed hybridizations by transferring pollen from the _______ of the male plant to the female ova.


Mendelian Crosses

Mendel performed hybridizations, which involve mating two true-breeding individuals that have different traits. In the pea, which is naturally self-pollinating, 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.

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, and one quarter had white flowers.


Working with garden pea plants, Mendel found that crosses between parents that differed by one trait produced F offspring that all expressed the traits of one parent.

Gregor mendel, focuses on inheritance, which is a type of biological inheritance, found inheritance of trait in pea plants, that each pair of alleles segregates individually during the production of gametes egg and sperm. This law is called "independent Assortment". Mendel's analytical studies put emphasis on three laws of inheritance: the law of segregation that is first law the law of independance is second law, the law of dominance is the third law of inheritance. Mendel's first law of segregation provided locus segregation into separate gametes While the second law of independent assortment, demonstrates that the alleles of one gene assemble into gametes freely and independently of the alleles of another gene. Mendel's third law of dominance, claims that the one of the cause or factor for having a pair of inherited traits will be dominant while the other recessive. Mendel's in his experiment reveals seven different traits: pea (shape and color), pod (shape and color), flower (color and position) and plant size. Mendel's studied alternative kinds of factors, which are now recognised as genes, and the alternative "forms" are now called alleles.

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You might think that Mendel's discoveries would have made a big impact on science as soon as he made them. But you would be wrong. Why? Because Mendel's work was largely ignored. Mendel was far ahead of his time and working from a remote monastery. He had no reputation among the scientific community and no previously published work.

Mendel&rsquos work, titled Experiments in Plant Hybridization, was published in 1866, and sent to prominent libraries in several countries, as well as 133 natural science associations. Mendel himself even sent carefully marked experiment kits to Karl von Nageli, the leading botanist of the day. The result - it was almost completely ignored. Von Nageli instead sent hawkweed seeds to Mendel, which he thought was a better plant for studying heredity. Unfortunately hawkweed reproduces asexually, resulting in genetically identical clones of the parent.

Charles Darwin published his landmark book on evolution in 1869, not long after Mendel had discovered his laws. Unfortunately, Darwin knew nothing of Mendel's discoveries and didn&rsquot understand heredity. This made his arguments about evolution less convincing to many people. This example demonstrates the importance for scientists to communicate the results of their investigations.

Rediscovering Mendel&rsquos Work

Mendel&rsquos work was virtually unknown until 1900. In that year, three different European scientists &mdash named Hugo De Vries, Carl Correns, and Erich Von Tschermak-Seysenegg &mdash independently arrived at Mendel&rsquos laws. All three had done experiments similar to Mendel&rsquos. They came to the same conclusions that he had drawn almost half a century earlier. Only then was Mendel&rsquos actual work rediscovered.

As scientists learned more about heredity - the passing of traits from parents to offspring - over the next few decades, they were able to describe Mendel&rsquos ideas about inheritance in terms of genes. In this way, the field of genetics was born.

Genetics of Inheritance

Today, we know that characteristics of organisms are controlled by genes on chromosomes (see the Figure below). The position of a gene on a chromosome is called its locus. In sexually reproducing organisms, each individual has two copies of the same gene, as there are two versions of the same chromosome (homologous chromosomes). One copy comes from each parent. The gene for a characteristic may have different versions, but the different versions are always at the same locus. The different versions are called alleles. For example, in pea plants, there is a purple-flower allele (B) and a white-flower allele (b). Different alleles account for much of the variation in the characteristics of organisms.

Chromosome, Gene, Locus, and Allele. This diagram shows how the concepts of chromosome, gene, locus, and allele are related. What is the different between a gene and a locus? Between a gene and an allele?

During meiosis, homologous chromosomes separate and go to different gametes. Thus, the two alleles for each gene also go to different gametes. At the same time, different chromosomes assort independently. As a result, alleles for different genes assort independently as well. In these ways, alleles are shuffled and recombined in each parent&rsquos gametes.

Genotype and Phenotype

When gametes unite during fertilization, the resulting zygote inherits two alleles for each gene. One allele comes from each parent. The alleles an individual inherits make up the individual&rsquos genotype. The two alleles may be the same or different. As shown in the Table below, an organism with two alleles of the same type (BB or bb) is called a homozygote. An organism with two different alleles (Bb) is called a heterozygote. This results in three possible genotypes.

Alleles Genotypes Phenotypes
BB (homozygote) purple flowers
B (purple) Bb (heterozygote) purple flowers
b (white) bb (homozygote) white flowers

The expression of an organism&rsquos genotype produces its phenotype. The phenotype refers to the organism&rsquos characteristics, such as purple or white flowers. As you can see from the Table above, different genotypes may produce the same phenotype. For example, BB and Bb genotypes both produce plants with purple flowers. Why does this happen? In a Bb heterozygote, only the B allele is expressed, so the b allele doesn&rsquot influence the phenotype. In general, when only one of two alleles is expressed in the phenotype, the expressed allele is called the dominant allele. The allele that isn&rsquot expressed is called the recessive allele.

How Mendel Worked Backward to Get Ahead

Mendel used hundreds or even thousands of pea plants in each experiment he did. Therefore, his results were very close to those you would expect based on the rules of probability (see "Probability" concept). For example, in one of his first experiments with flower color, there were 929 plants in the F2 generation. Of these, 705 (76 percent) had purple flowers and 224 (24 percent) had white flowers. Thus, Mendel&rsquos results were very close to the 75 percent purple and 25 percent white you would expect by the laws of probability for this type of cross.

Of course, Mendel had only phenotypes to work with. He knew nothing about genes and genotypes. Instead, he had to work backward from phenotypes and their percents in offspring to understand inheritance. From the results of his first set of experiments, Mendel realized that there must be two factors controlling each of the characteristics he studied, with one of the factors being dominant to the other. He also realized that the two factors separate and go to different gametes and later recombine in the offspring. This is an example of Mendel&rsquos good luck. All of the characteristics he studied happened to be inherited in this way.

Mendel also was lucky when he did his second set of experiments. He happened to pick characteristics that are inherited independently of one another. We now know that these characteristics are controlled by genes on nonhomologous chromosomes. What if Mendel had studied characteristics controlled by genes on homologous chromosomes? Would they be inherited together? If so, how do you think this would have affected Mendel&rsquos conclusions? Would he have been able to develop his second law of inheritance?


Mendel’s Model System

Mendel’s seminal work was accomplished using the garden pea, Pisum sativum, to study inheritance. This species naturally self-fertilizes, such that pollen encounters ova within individual flowers. 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. By experimenting with true-breeding pea plants, Mendel avoided the appearance of unexpected traits in offspring that might occur if the plants were not true breeding. The garden pea also grows to maturity within one season, meaning that several generations could be evaluated over a relatively short time. Finally, large quantities of garden peas could be cultivated simultaneously, allowing Mendel to conclude that his results did not come about simply by chance.


Glossary

Blending theory of inheritance

hypothetical inheritance pattern in which parental traits are blended together in the offspring to produce an intermediate physical appearance

Continuous variation

inheritance pattern in which a character shows a range of trait values with small gradations rather than large gaps between them

Discontinuous variation

inheritance pattern in which traits are distinct and are transmitted independently of one another

Dominant

trait which confers the same physical appearance whether an individual has two copies of the trait or one copy of the dominant trait and one copy of the recessive trait

first filial generation in a cross the offspring of the parental generation

second filial generation produced when F1 individuals are self-crossed or fertilized with each other

Hybridization

process of mating two individuals that differ with the goal of achieving a certain characteristic in their offspring

Model system

species or biological system used to study a specific biological phenomenon to be applied to other different species

parental generation in a cross

Product rule

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

Recessive

trait that appears “latent” or non-expressed when the individual also carries a dominant trait for that same characteristic when present as two identical copies, the recessive trait is expressed

Reciprocal cross

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

Sum rule

probability of the occurrence of at least one of two mutually exclusive events is the sum of their individual probabilities


Watch the video: Mendelian Inheritance (January 2022).