What is independent assortment, crossing over, and random fertilization? And how does each increase genetic diversity?
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These are the three primary ways by which sexually reproducing organisms generate diversity in their offspring.
Independent assortment is the concept that alleles are separate from each other when characteristics are being passed down. For example, hair color is independent of height; those two characteristics are assorted independently. It doesn't work in practice for every possible gene because many alleles are linked together on the same chromosome. If they stay on the same chromosome when the genes are passed down, those two traits are not independently assorted and are linked. A good example of this is looking at the female and male phenotypes. Clearly gonad formation is a linked trait with breast formation, body morphology, hormone composition, and a whole set of other traits.
Crossing over describes the process by which chromatids may exchange traits before making gametes. Homologous chromosomes generally have one representative from the father and one representative from the mother. Crossing over allows you to pass a blend of the father's and mother's DNA, rather than simply a replicate of your father's or mother's chromosomes. This is important because it is one of the main processes allowing independent assortment! If on the same chromosome you found hair color and eye color (and some portions of both are on chromosome 19), your only hope of passing down your mother's brown hair with your dad's blue eyes would be crossing over, for example (random colors chosen).
Finally, random fertilization describes the concept that the probability of any set of genes has an equal likelihood of combining to form the offspring. This allows all dominant and recessive traits to compete on equal footing before birth. For example, a human egg is no less likely to be fertilized by a sperm containing the "taco tongue" trait than a sperm containing the "no taco tongue" trait. That being said, there has been a hypothesis put forth that there may be exceptions. One prominent one was that sperm containing the light Y-chromosome to make a boy will have an easier time making it to the egg than sperm containing the bulky X-chromosome.
All of these contribute to genetic diversity in a specific way. The easiest way to consider how they increase diversity is to realize that during fertilization, there is no selection of traits (that comes after fertilization). Each mechanism maximizes the possible combinations of genes to allow for more and more fit organisms to be produced.
This video may be helpful in understanding your question:
I've also attached a diagram which might help you visualize the differences between fertilization types.
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