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 Genetic analysis predates Gregor Mendel, but Mendel's laws form the theoretical basis of our understanding of the genetics of inheritance.

Mendel made two innovations to the science of genetics:

1.  

2. developed pure lines
3. counted his results and kept statistical notes

Pure Line - a population that breeds true for a particular trait [this was an important innovation because any non-pure (segregating) generation would and did confuse the results of genetic experiments]

The principles of heredity were written by the Augustinian monk Gregor Mendel in 1865. Mendel discovered that by crossing white flower and purple flower plants, the result was not a blend. Rather than being a mix of the two, the offspring was purple flowered. He then conceived the idea of heredity units, which he called "factors", one which is a recessive characteristic and the other dominant. Mendel said that factors, later called genes, normally occur in pairs in ordinary body cells, yet segregate during the formation of sex cells. Each member of the pair becomes part of the separate sex cell. The dominant gene, such as the purple flower in Mendel's plants, will hide the recessive gene, the white flower. After Mendel self-fertilized the F1 generation and obtained the 3:1 ratio, he correctly theorized that genes can be paired in three different ways for each trait; AA, aa, and Aa. The capital A represents the dominant factor and lowercase a represents the recessive. (The last combination listed above, Aa, will occur roughly twice as often as each of the other two, as it can be made in two different ways, Aa or aA.)

Mendel stated that each individual has two factors for each trait, one from each parent. The two factors may or may not contain the same information. If the two factors are identical, the individual is called homozygous for the trait. If the two factors have different information, the individual is called heterozygous. The alternative forms of a factor are called alleles. The genotype of an individual is made up of the many alleles it possesses. An individual's physical appearance, or phenotype, is determined by its alleles as well as by its environment. An individual possesses two alleles for each trait; one allele is given by the female parent and the other by the male parent. They are passed on when an individual matures and produces gametes: egg and sperm. When gametes form, the paired alleles separate randomly so that each gamete receives a copy of one of the two alleles. The presence of an allele doesn't promise that the trait will be expressed in the individual that possesses it. In heterozygous individuals the only allele that is expressed is the dominant. The recessive allele is present but its expression is hidden.

Results from Mendel's Experiments

Mendel's Conclusions

1. The hereditary determinants are of a particulate nature. These determinants are called genes.

    

2. Each parent has a gene pair in each cell for each trait studied. The F₁ from a cross of two pure lines contains one allele for the dominant phenotype and one for the recessive phenotype. These two alleles comprise the gene pair.

    

3. One member of the gene pair segregates into a gamete, thus each gamete only carries one member of the gene pair.

    

4. Gametes unite at random and irrespective of the other gene pairs involved.

Mendel picked common garden pea plants for the focus of his research because they can be grown easily in large numbers and their reproduction can be manipulated.  Pea plants have both male and female reproductive organs.  As a result, they can either self-pollinate themselves or cross-pollinate with another plant.  In his experiments, Mendel was able to selectively cross-pollinate purebred  plants with particular traits and observe the outcome over many generations.  This was the basis for his conclusions about the nature of genetic inheritance.

Mendel's Laws:

Law of Segregation (The "First Law")

The Law of Segregation states that when any individual produces gametes, the copies of a gene separate, so that each gamete receives only one copy. A gamete will receive one allele or the other. The direct proof of this was later found when the process of meiosis came to be known. In meiosis the paternal and maternal chromosomes get separated and the alleles with the characters are segregated into two different gametes.

Mendel established two laws to summerize his findings. Mendels second law, the Law of Independent Assortment, states that alleles of different genes assort independently of one another during gamete formation.

Here is a picture of how Mendelian Genetics looks like using a flower as in example and the color of the flowers are the products :

This picture shows Mendal's theory about incomplete dominance.  This happens when all of the alleles are dominant, and they blend when they are combined.  Incomplete dominance is a form of intermediate inheritance in which one allele for a specific trait is not completely dominant over the other allele. This results in a combined phenotype.  An example is in cross-pollination experiments between red and white snapdragon plants, the resulting offspring are pink. The dominant allele that produces the red color is not completely expressed over the recessive allele that produces the white color.

Here's an example of what happens when two pink flowers (a color in result from incomplete dominance) produce more flowers:

Here is a concise, clear and simple explanation of heredity and punnett squares:

Following is a video of Gregor Mendel and genetics.

This website gives information and details on the pea plant experiment that Mendel was famous for. It is the foundation for determining heredity today, with its use of observing different traits and using parent plants to determine potential outcomes for offpring

http://www.juliantrubin.com/bigten/mendelexperiments.html

 The dominance relationship between alleles for each trait was already known to Mendel when he made this cross. The purpose of the dihybrid cross was to determine if any relationship existed between different allelic pairs.

Mendelian Trait: A Mendelian Trait is a specific characteristic of an organism that is determined by one section of a chromosome. Mutations in these sections can cause disease. A mutated mendelian trait causes a change in the inheritance line. The places where these mutations would occur are known as loci (plural for locus), and a disease can be caused by a change in one locus, like sick cell anemia, or in several loci, like arthritis. 

Mendel also performed crosses in which he followed the segregation of two genes. These experiments formed the basis of his discovery of his second law, the law of independent assortment.

Mendel's Second Law - the law of independent assortment; during gamete formation the segregation of the alleles of one allelic pair is independent of the segregation of the alleles of another allelic pair

As with the monohybrid crosses, Mendel confirmed the results of his second law by performing a backcross - F₁ dihybrid x recessive parent.

In genetics there are two types of alleles on any given locus of any choromosome. One type is the dominant allele. This allele will always be the determinant of an organisms phenotype, or its outward appearance. Dominant traits in humans include pigment of the eye (not blue), a straight thumb, and a widows peak. Even if the organism is heterozygous for the trait, the dominant gene will always be expressed.

The Second type of allele is the recessive allele. This allele will be masked in the presence of a dominant allele and only showns when it is homozygous with another recessive allele. When the phenotype of two parent organisms is of the dominant trait and they produce a daughter organism with a recessive trait, it must be assumed that both parents were heterozygous: In the equation Rr x Rr, there is a 25 percent chance that an rr will be produced. 

Mendel's studies on genetics had already been in universal practice by farmers. His studies were only a proof that by breeding a pair of animals with simliar traits that the offspring would share those traits. Mendel furthered our understanding of genetic inheritance by establishing patterns of recessive and dominant traits. Today, these laws of genetics are used to breed cows that produce more milk, pure bred dogs and all sorts of specialized animals.

Mendelian Genetics Definitions

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• Allele - one alternative form of a given allelic pair; tall and dwarf are the alleles for the height of a pea plant; more than two alleles can exist for any specific gene, but only two of them will be found within any individual

   

• Allelic pair - the combination of two alleles which comprise the gene pair

   

• Homozygote - an individual which contains only one allele at the allelic pair; for example DD is homozygous dominant and dd is homozygous recessive; pure lines are homozygous for the gene of interest

   

• Heterozygote - an individual which contains one of each member of the gene pair; for example the Dd heterozygote

   

• Genotype - the specific allelic combination for a certain gene or set of genes

• Phenotype - Literally means "The form that is shown", it is the physical appearance of a specific trait that the subject takes on because of its specific allelic combination

• Backcross - the cross of an F₁ hybrid to one of the homozygous parents; for pea plant height the cross would be Dd x DD or Dd x dd; most often, though a backcross is a cross to a fully recessive parent
• Testcross - the cross of any individual to a homozygous recessive parent; used to determine if the individual is homozygous dominant or heterozygous. So far, all the discussion has concentrated on monohybrid crosses.
• Monohybrid cross - a cross between parents that differ at a single gene pair (usually AA x aa)
• Monohybrid - the offspring of two parents that are homozygous for alternate alleles of a gene pair
• Remember --- a monohybrid cross is not the cross of two monohybrids. Monohybrids are good for describing the relationship between alleles. When an allele is homozygous it will show its phenotype. It is the phenotype of the heterozygote which permits us to determine the relationship of the alleles.
• Dominance - the ability of one allele to express its phenotype at the expense of an alternate allele; the major form of interaction between alleles; generally the dominant allele will make a gene product that the recessive can not; therefore the dominant allele will express itself whenever it is present
• Dominant Allele - The allele that expresses itself at the expense of an alternate allele, comes from the cross of two pure lines
• Recessive Allele - An allele whose expression is supressed in the presence of a dominant allele

This video explains Mendal's experiments and how they explain inherited traits.  This video really helped me because someone is explaining Mendal's theories and why he performed his experiments. 

This video shows how to make and use a punnett square.  It explains the ways traits are inherited through families.  This video really helped me understand dominant and recessive traits.

This is a brilliant link that goes into great detail in describing Mendel's experiments, beginning with a excellent description of punnet squares and Mendel's seed crosses: http://www.sumanasinc.com/webcontent/animations/content/mendel/mendel.html 

http://www.ndsu.edu/pubweb/~mcclean/plsc431/mendel/mendel1.htm

^This website is not an animation, but it does give a lot of good definitions and examples of Mendal's ideas and the Mendalian genetics.