Mendel, an Austrian monk, worked on basic concepts in genetics which were not recognized until after his death. Rediscovery of Mendelian genetics paved the path for modern genetics. His concepts heavily relied on test crosses and the rules of probability. The addition and multiplication rules are applied to mutually elusive and independent cases, respectively. However, modern genetics is an exception from Mendel’s genetics. Discovery of multigene inheritance, multiple alleles, dominance variations and epistasis has changed the face of classical genetics to a great deal.
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## Foundation of Classical Genetics

Reproduction is a feature of living beings which allows them to transfer biological traits to future generations through the phenomenon of hereditary. Hereditary information is transferred from parents to offspring’s through genes.

Foundation of classical genetics was laid by an Austrian monk, Gregor Mendel. His work on plant hybridization was published in 1866 which became the basis of genetics although his work did not get recognition and was not welcomed at that stage.

The era of classical genetics began with rediscovery of Mendelism in 1900 till the discovery of DNA in 1953 by Watson and Crick. These discoveries led to the understanding of genes and their functions at a molecular level. Mendel put forth his experimental conclusions which were later established as ‘Mendel’s laws of inheritance’.

### Mendel’s first law

Law of segregation: pair of genes, i.e., alleles separate from each other during gamete formation and each gamete carries only one allele.

### Mendel’s second law

Law of independent assortment: Genes for different traits assort independently with respect to each other during gamete formation.

## Application of Basic Probability for Solving Genetics’ Problems

Probability is the likely frequency of an event to occur over the series of possibilities or it is quantifying likelihood of an event to occur. If an event doesn’t occur, probability is 0, while if it will occur probability is equal to 1, i.e., there is a 100 % chance of an event to occur. For an uncertain event which may or may not occur, the value of probability is between 0 and 1.

According to the hypothesis put forth by Mendel, passing of an allele to the gamete is random, i.e., chance of both alleles entering any gamete is equal. Moreover, post-fertilization combination of gametes is also made at random.

The concept of probability plays a major role in Mendelian genetics due to randomness of the hereditary events. Combinations are mostly of the following two types:

1. Mutually exclusive
2. Independent

Mutually exclusive combinations: In such combination occurrence of one event nullifies the probability of occurrence of the other events.

Independent combinations: Possibility of an event to occur or not to occur does not impact the possibility of occurrence of any other event that means these events are independent of each other.

Two different rules of probability are applied to predict the outcomes of both mutually exclusive and independent events.

Probability of occurrence of either one or the other ‘mutually exclusive’ events is equal to the sum of probability of occurrence of each individual event. Rule of addition is applied to only mutually exclusive events. Addition rule is applied to ‘either/or’ cases only.

For example: The probability of either allele P (probability y) or allele p (probability z) is y+z, i.e., ½ + ½ = 1.

One thing to consider is that the addition rule cannot be applied to allele P and allele Q of two different genes, as in this case both alleles would not be mutually exclusive.

### Multiplication Rule

Probability of occurrence of an ‘independent’ event is equal to the product of probability of occurrence of each individual event. Segregation of genes produce equal numbers of alleles which will assort independently. Outcomes of such crosses are predicted through the multiplication rule of probability. The multiplication rule is applied to ‘both … and’ cases.

For example: If genes Pp (probability y) and Qq (probability z) are assorting independently. The probability of gamete PQ will be calculated as: y*z i.e. ½ * ½ = ¼ = 0.25

## Application of Probability to Test Crosses

Phenotype can be explained while observing an individual as opposed to genotype. For a tall plant genotype can be either homozygous (TT) or heterozygous (Tt).

• Homozygous is the ‘pure breed’ as only one type of gametes are produced.
• Heterozygous are known as hybrids and produces two different types of gametes, i.e., half of gametes will carry gene T and the other half will carry gene t.

In order to identify the genotype of dominant phenotypes of plants, Mendel performed tests and used test crosses to confirm validity of his conclusions. He devised a system where an individual of an unknown genotype by a dominant phenotype is crossed with a homozygous recessive individual to reveal the genotype of the unknown.

### Punnett square

Punnett square was devised by Reginald C. Punnett who discovered the phenomenon of gene-linkage to learn about genetics and problem solving. By making a Punnett square, all possible random fertilization events and probable genotypes and phenotypes can be viewed. However, it gets hard to apply it to six gene problems and so on as it is just the visual representation of possible combinations.

Test cross is used to determine the homozygosity or heterozygosity of an individual, i.e., the unknown genotype. By the application of addition and multiplication rules of probability to test cross experiments; dihybrid, trihybrid and other cumbersome test crosses can be solved.

## Probability for Predicting Monohybrids and Dihybrid Test Cross

### Monohybrid cross

Probability of appearance of a homozygous or a heterozygous trait in next generation can be predicted by the application of the addition rule to a monohybrid test cross. In the following example, dominant combination is (BB) for brown eye color and (bb) for the recessive trait  blue eye color. A heterozygous individual will show brown eye color (Bb). Its frequency can be calculated as follows:

Image: “A Punnett Square showing a BB x Bb cross for eye colour. Here, a homozygous dominant brown-eyed parent and a w:heterozygous brown-eyed parent produce 50% homozygous dominant brown-eyed offspring and 50% heterozygous brown-eyed offspring. The example discussed in the text is concerned with the cross test between a Bb and a Bb ” by Purpy Pupple. License: CC BY-SA 3.0

Results: probability of a homozygote and a heterozygote in F2 generation if both parents are heterozygous.

• 25% of the offspring will have a homozygous dominant genotype (BB).
• 25% of the offspring will have a homozygous recessive genotype (bb).
• 50% of the offspring will have a heterozygous genotype (Bb).

### Dihybrid and trihybrid crosses

Dihybrid cross or trihybrid cross where more than one trait are under consideration; probabilities are based on possibilities of monohybrid crosses. Dihybrid cross can be easily understood by making two separate monohybrid crosses.

The two traits are considered to be inherited independently the probability of a seed to be green or yellow is independent of the seed being round or wrinkled. However, yellow and round characteristics of the seed are dominant. Independent assortment of these traits allowed application of product rule of probability to the case of dihybrid cross.

Image: “RrYy x RrYy Dihybrid Cross Tree Method” by Tim DeJulio. License: Public Domain

## Prediction of an Unknown Genotype Through Test Cross

Unknown genotype of an individual can be predicted through the method devised by Mendel, i.e., creating a cross between homozygous recessive individual and the individual with unknown genotype but dominant phenotype. This method will uncover the unknown genotype. This signifies the importance of monohybrid test cross as it helps with detection of recessive alleles which impacts negatively on the population.

In the following example, brown is the dominant eye color while blue is the recessive. An individual with an unknown genotype but brown colored eyes is crossed with homozygous recessive individual.

Results can be predicted as follows:

• If 50 % individuals have brown colored eyes and 50 % has blue colored eyes, it indicates that the unknown genotype is a heterozygote (Bb).
• If all individuals in the next generation have brown colored eyes, it indicates that the unknown genotype is a homozygote (BB).

## Limitations to Mendel’s Laws

### Unexpected phenotypes

Limitation to Mendel’s laws is posed by the fact that nearly all genes for the traits that he studied were fortunately localized on different chromosomes or at a significant distance on the same chromosome, hence, outcomes were predictable. In a case where more pairs of genes are tracked, different and unexpected phenotypes appear due to the phenomenon of crossing over.

### Continuous variations

In natural populations many traits are seen to appear as continuously varying due to polygenic inheritance. Polygenic inheritance arises when multiple genes are responsible for the appearance of a single characteristic or trait. Environmental factors together with the number of genes responsible for a single trait result in huge variation in phenotype of a trait in a natural population.

An example of such traits is eye color, skin color and height etc. The amount of pigment is higher in brown eyes while lesser is found in blue, green or grey eyes. Such continuously varying traits are represented typically by a bell graph.

### Pleiotropy

At times a single gene impacts more than one phenotype or trait of a person which is known as pleiotropy. These traits can be unrelated and these alleles are transmitted in the same way as other non-pleiotropic alleles.

This phenomenon is also responsible for multiple disorders in humans like phenyl keto urea and Marfan disease. In Marfan disease height, eye lens, fingers and heart seem to function abnormally. At times it can be responsible for embryonic lethality as shown in the following example.

Image: “Punnett square illustrating the lethal yellow coat allele in mice.” by Jcfidy. License: CC BY-SA 4.0

### Dominance

The phenomenon of dominance is applied to diploid individuals where at least two alleles for the same gene are present. Mendel believed that one allele is fully dominant over the other gene. However, this is not the case, dominance is not always complete; at times both alleles express themselves to give rise to a different phenotype. Internal and environmental factors influence the expression of genes.

### Co-dominance

A gene can have more than one copy of alleles, i.e., more than two allelic variants for the same gene, the phenotype in this case is also different than the one predicted via Mendalism. In such a case where multiple alleles are present and different alleles are fully expressed, it is known as co-dominance.

For example, in case of blood group types, alleles iA and iB for antigen A and B are dominant over recessive allele i0; they are fully expressed in heterozygous individuals with blood group AB.

The following is an example of co-dominance from Roan cattle for the coat color.

Image: “This diagram shows co-dominance. In this example a white bull (WW) mates with a red cow (RR), and their offspring exhibit co-dominance expressing both white and red hairs.” by Hhughes15. License: CC BY-SA 4.0

### Incomplete dominance

When an intermediate phenotype is observed in a heterozygous individual it is known as incomplete dominance. It happens when the dominant allele is unable to completely mask the effect of recessive allele. However, Mendel did not support appearance of such blended phenotype.

Example: When two snap dragon plants are crossed where one plant had red flowers and other had white flowers; offspring generation showed appearance of pink flower, i.e., an intermediate phenotype. This is an example of incomplete dominance.

### Overdominance

When a heterozygote offspring of the two homozygote parents shows phenotype beyond the range of the parents, it is known as over dominance.

For example: two homozygous plants with fruit length 10 cm (SS) and 20 cm (ss) are crossed. The next generation is a heterozygote (Ss) and 30 cm fruit length is seen then it is a case of over dominance.

### Impact of environment over phenotypes

At times changes in expression of genes is observed as a result of changing environmental factors like temperature etc.

Himalayan rabbits

Himalayan rabbits have black hair on tail, ears, nose and legs. However, hair on trunk are white in color. The reason for this phenomenon is the temperature sensitive expression of genes for tyrosinase enzyme. Himalayan rabbits are homozygous for a mutant form of tyrosinase enzyme; it is responsible for production of melanin pigment which give darker color to the hair.

This enzyme only works when environmental temperature is below 33° C (91.4.F). The trunk region stores a fair amount of heat, hence, hair on the trunk appears light. Siamese cats appear lighter in summers and darker during winters for the same reason; they have the same temperature sensitive version of tyrosinase enzyme.

### Epistasis alters phenotypic ratios

Epistasis is defined as interaction of genes at different loci. Expression of one gene is dependent on the presence of a specific genotype on other loci. It is also responsible for complete suppression of a mutant phenotype in some cases.

For example, the homozygous parent generation of white flowers, where two alleles are responsible for flower color are cross-fertilized to give rise to purple colored flower in F1 generation. Further crossing results in appearance of both white and purple flowers in F2 generation. Hence, epistasis can change the expected phenotypic ratios. Appearance of purple color in F1 generation is due to the interdependence of enzymes as a result of epistatic phenomenon.

## Modern Genetics is Beyond Mendelism

Mendelian genetics laid the foundation for modern genetics and it holds good for many traits. It is good for basic principles of genetics but modern genetics is a whole new story and an exception from Mendelian genetics due to the following aspects:

• Mendel’s one-gene-one character hypothesis is not universal due to multigene inheritance where many genes control a single character.
• Independent assortment of genes is not applicable to closely linked genes which are inherited together due to localization at a close proximity.
• Due to epistasis Mendel’s assumption that one gene cannot influence the other, cannot be applied universally.
• Mendel put forth the concept of two alleles for one a single gene is also no more valid due to the emergence of the concept of multiple alleles.
• Mendel suggested that characters can either be dominant or recessive, but incomplete dominance, co-dominance and overdominance don’t comply with his concept.