Two different animal species have a spectrum of inter-breeding possibilities:
1) Completely incompatible.
In many cases, it's logistically impossible for the two species to breed; try and imagine a giraffe and a dolphin, for example. Even if one could get the gametes (reproductive cells, e.g. egg and sperm in humans) together, no fertilization would take place.
Eggs have barriers around them containing specific proteins, and in order for a sperm to be able to fertilize the egg, the sperm must have matching enzymes that are capable of penetrating this barrier. You could think of it like a "lock and key." Only sperm of the same species have the correct "key" to enable their DNA to be transferred into the egg and allow fertilization to occur.
2) Fertilization occurs, but the zygote (fertilized egg) isn't viable.
This category covers a quite wide spectrum, but embraces all of the options between fertilization occurring but no subsequent cell division happening (ie remaining a single cell zygote), through to fertilization happening and a "nearly normal" but sterile adult resulting, as is the case with a liger. The factor that determines where on this spectrum the hybrid will fall is the degree of genetic similarity of the parents, particularly with regard to how the essential functional genes are allocated to chromosomes.
If two species have similar functions allocated similarly, to the same complement of chromosomes, you are more likely to end up with a viable adult. The greater the difference in number of chromosomes, and the distribution of essential gene functions on those chromosomes, the less likely that the zygote will be viable.
3) Fertile offspring.
If two species can create a hybrid that is fertile, then they're actually, by definition, not separate species.
Why is fertility so difficult for hybrids?
Fertility is the part of the process that requires the greatest degree of genetic similarity because of the differences between mitosis (asexual cell reproduction) and meiosis (sexual cell reproduction). Mitosis is the process by which a zygote turns into an adult animal by copying its own cells over and over and increasing in size; meiosis is the process by which that adult creates eggs or sperm to contribute to the next generation.
Chromosomes come in pairs; we have two copies of each chromosome, one from each parent. Humans, for example, have 46 chromosomes, consisting of 23 pairs. Both types of cell division, mitosis and meiosis, require chromosomes to line up with the analogue chromosome of their "pair" prior to cell division, then the pairs are "split down the middle," to ensure that each new cell ends up with complementary halves and a full chromosomal complement.
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