Advanced organisms such as animals receive two sets of genes with one set from each parent. While the overall genetic code is the same, the parents often have different versions of the same gene. As a result, the inherited genetic code may contain copies of the two versions; one may be dominant while the other may be recessive.

When a single gene produces a particular trait in an organism, Mendelian inheritance rules apply. They were first proposed by Austrian monk Gregor Mendel in the 19th century and detail how single genes are inherited with a few simple rules. Mendel worked with pea plants and defined dominant and recessive alleles.

Most organism characteristics are not produced by a single gene, though. Instead, many genes influence a characteristic, and some genes affect several organism traits. Because Mendel's simple rules don't apply in such cases, non-Mendelian inheritance deals with these complex processes. Where Mendel assumed that one of the two versions of a gene was dominant, non-Mendelian inheritance accepts that in some cases dominance is incomplete.

Gregor Mendel's work with pea plants focused on observable traits such as flower color and pod shape. Mendel tried to determine what genes produced purple and white flowers and other pea plant traits. He chose traits that were mostly caused by a single gene; as a result, he was able to explain inheritance in simple terms.

His main conclusions were as follows:

• Each organism has two versions of a gene.
• The parents each contribute one version.
• If the two versions are the same, the organism will exhibit the corresponding trait.
• If the two versions are different, the organism will exhibit the dominant trait.

In Medelian inheritance, the two gene versions inherited from parents are called alleles. Alleles can be dominant or recessive. An individual who has one or two dominant alleles will have the trait coded by the dominant gene.

For individuals with two recessive alleles, the recessive trait will appear. According to Mendel, the presence or absence of single genes and their alleles explained which traits were exhibited in the pea plants.

Before Mendel, most scientists thought that traits were passed on as a mixture of the parents' traits. The problem was that often children did not have such a mixture, as when a blue-eyed parent and a brown-eyed parent produced a blue-eyed child.

Mendel proposed that traits were the result of the presence or absence of a dominant allele. His theory is still applicable for traits produced by a single gene.

For example, Mendel proved that pea plants with a short and a long parent did not produce medium-length plants but only short or long plants. Plants that had one parent with smooth and one parent with wrinkly pods did not produce slightly wrinkled pods but either wrinkly or smooth pods.

There was no mixture of traits.

Most traits are produced by multiple genes, though. For example, there are many plants with a range of lengths, not only short and long plants. When a short and a long plant produce an intermediate-length plant, it has to be because of the influence of multiple genes or a lack of complete dominance by the dominant gene.

This kind of inheritance is called non-Mendelian inheritance.

The overall collection of an organism's genes is the genotype while the collection of observable traits produced by the genotype is called the phenotype. Phenotypes are based on the genotype but can be influenced by environmental factors and the behavior of the organism.

For example, a plant may have the genotype for growing tall and bushy, but if it grows in poor soil, it may still be small and sparse.

Organisms that have two dominant or two recessive alleles are homozygous for that gene while those that have a dominant and a recessive allele are heterozygous. This becomes important in non-Mendelian inheritance because homozygous organisms have clear gene expression of the two dominant or recessive alleles and exhibit the corresponding phenotype.

In heterozygous organisms with a dominant and a recessive allele, the dominant/recessive relationship may not be complete, and both alleles may be expressed to a varying degree.

Factors apart from the genotype that influence the phenotype include the following:

• Available resources such as nutrients, space and shelter.
• Toxins such as industrial waste and sewage.
• Radiation, both natural and manmade.
• Temperature extremes.
• Presence of predators.

The interplay of dominant and recessive alleles combined with the environmental factors produces the phenotype from the originating genotype.

The complex nature of non-Mendelian inheritance is based on the fact that many traits are the result of influences from many different genes, environmental factors and organism behavior. In addition to these influences, the alleles of a gene can produce different phenotypes due to the following four mechanisms:

• Codominance: Two alleles of the same gene are expressed and exhibit their trait. For example, a kitten descended from a black cat and a white cat might have alleles for black and white fur and have black and white spots.
  
• Incomplete dominance: A dominant and a recessive allele produce an intermediate trait because the dominance of the dominant allele is incomplete and the recessive allele influences the trait. For example, a plant with a dominant red flower allele and a recessive white flower allele might produce pink flowers.
• Variable expressivity: The alleles for a trait are not always fully expressed. For example, Marfan syndrome is a disorder of the connective tissue throughout the body, but symptoms vary widely because other genes and environmental factors affect gene expression.
• Incomplete penetrance: The individual with a dominant allele does not always exhibit the corresponding trait. The allele is fully expressed but does not cause the phenotype to appear. For example, a gene may make an individual susceptible to cancer, but the cancer only appears when other factors are present.

When incomplete dominance is present for a particular trait, heterozygous offspring may have a mixture of the traits of their parents and display an intermediate phenotype. In humans, skin color is an an example of incomplete dominance because the genes responsible for melanin production and light or dark skin can't establish dominance.

As a result, the offspring often has a skin color that is between the skin tones of the parents.

The mechanism of incomplete dominance has slightly different effects when it appears in single genes versus in a multiple-gene, or polygenic, genotype.

Possible differences in phenotypes resulting from genes with incomplete dominance include the following variations:

• Single heterozygous genes: Neither of the alleles in the dominant/recessive gene pair is completely dominant. A combination of the traits represented by the two alleles results. For example, homozygous snapdragons have red or white flowers, but the heterozygous offspring may have pink flowers.
• Multiple genes: A trait is produced through the effects of many genes. Some alleles have incomplete dominance and contribute a mixture of features to the trait. For example, in human eye color, genes responsible for dark color are not completely dominant and make a dark-colored contribution.
• Other influences: Alleles with incomplete dominance may be affected by other genes or other factors completely separate from the encoded trait. For example, human height is determined by many genetic factors including incomplete dominance, but nutrition also affects growth and individual height.

As a result of these variations, incomplete dominance can result in a large variety of phenotypes and can help explain the continuous variation of many traits.

Mendel did not observe incomplete dominance in his experiments with pea plants, but non-Mendelian inheritance mechanisms, including incomplete dominance, are more common than Mendelian inheritance.

Single traits that are influenced by multiple genes are passed on to offspring through polygenic inheritance. Color in animals is often polygenic, and each gene contributes a little bit to create the overall final phenotype. Within genes, there is an additional difference between alleles, each allele pair bringing a potential four different contributions as well as variations due to degrees of dominance and gene expression.

With so many factors, it is difficult to develop an accurate picture of how a trait is formed and which genes and alleles contribute. Allele pairs are always at the same location or locus on the chromosome, but the genes themselves are harder to find.

A contributing gene could be a linked gene nearby on the chromosome, or it could be at the other end. Some contributing genes might be on other chromosomes, and they may only be expressed under certain circumstances.

Polygenic influences on a trait can include the following:

• Dominant allele.
  
• Two recessive alleles.
• Dominant and recessive allele with incomplete dominance.
• Two codominant alleles.
• Gene not fully expressed because of the influence of other genes.
• Gene fully expressed but with partial penetrance due to environmental factors.

All of these possibilities apply to each of the genes of a trait that has multiple genetic influences. The resulting phenotype can be described in detail, but the exact underlying genetic influences are often less clear.

While Mendel's rules for the inheritance of alleles are generally true and even work at the allele level for traits with multiple genes, the rules for the inheritance of complete polygenic traits are much more complicated. Polygenic traits are influenced by many factors that affect gene expression and penetrance.

Typical examples in humans include the following:

• Skin color: Many genes influence the production of melanin, the pigment responsible for dark skin in humans. Environmental factors such as exposure to sunlight affect skin color as well.
• Eye color: Two main genes are responsible for the darkness and the hue of eye color, but individual eye color varies by darkness, color and range due to the influence of other genes.
• Hair color: Melanin genes also affect the color of hair, but so does exposure to sunlight and age.
• Height: The height of an individual is determined by genes governing growth of bones, size of organs and body shape. Nutrition also influences growth, and other factors such as pharmaceuticals can affect height.

The variation in polygenic traits helps explain the vast differences in phenotypes found in advanced organisms including humans. Instead of a single gene giving rise to a specific trait, the complex mechanisms of polygenic inheritance including incomplete dominance are at the root of a diverse range of characteristics.