The law of independent assortment states that the alleles of one gene can segregate independently of the alleles of another gene. Contrast that with his law of segregation: in that case, we stated we were looking at one gene, and stating that only one of its two alleles went into each gamete. In this case, we are comparing the fates of several genes, and noting that the fate of a particular allele of one gene has no bearing on the fate of a particular allele of another gene.
Let's take two hypothetical genes, one on chromosome 7 and the other on chromosome 9, and use it to compare the two laws:
- Mendel's law of segregation states that there are two alleles for the gene on chromosome 7 and two alleles for the gene on chromosome 9. (As it turns out, this is because there are two chromosome 7s and two chromosome 9s - one from your father and one from your mother - and an allele sits on each one.) Only one of each of these pairs will make it into each gamete.
- Mendel's law of independent assortment states that the allele that makes it into a particular gamete from chromosome 9 plays no part in determining which allele from chromosome 7 makes it into the same gamete. In other words, if a particular gamete gets the allele from your 'paternal' chromosome 7, it still has 50:50 odds of getting either the paternal or maternal copy of the allele on chromosome 9.
And how does meiosis explain this? Well, you may recall the it is divided into meiosis I and meiosis II. In meiosis I, one cell is split into two cells, and each cell gets half the original number of chromosomes - that is, the cells are haploid. The body accomplishes this by giving each daughter cell one chromosome from each homologous pair. In other words, there are two chromosome 1s and two chromosome 2s, etc., and each daughter cell produced by meiosis I gets one chromosome 1 and one chromosome 2. However, which one of the pairs gets into a particular daughter cell is random.
To illustrate this, I've drawn a (horrible) diagram below. The big cell has two pairs of homologous chromosomes shown - we can think of them as our two chromosome 1s (the two big 'X's) and two chromosome 2s (the two little 'X's). The chromosomes that you got from your father are shown in red, and those that you got from your mother are in green. As I said, you can see that you got one chromosome out of each homologous pair from each parent.
To the right are the four possible daughter cells. Note that in meiosis I, only two daughter cells are produced (another two are produced by meiosis II), and so this diagram isn't strictly true. Rather, I wanted to show the four possible outcomes of meiosis I, even though only two out of these four can be produced at one time. As you can see, each chromosome has an equal chance of getting into a particular daughter cell, and its odds are independent of the other chromosomes. Maternal chromosomes don't tend to stick with other maternal chromosomes, and so on.
And this, of course, explains what Mendel discovered - his law of independent assortment. The way in which one chromosome (actually, tetrad, for what it's worth) segregates is totally independent of the way in which the other chromosomes segregate. Hence, they 'assort independently'. And, since the various genetic alleles are situated on these chromosomes, they will too.
Hope that made sense...