If mutations are truly random as proposed isn't it possible that there are hundreds of mutations that could arise rather than a mutation of immunity to penecilin?
These other mutations do, indeed, occur. But the population sizes of bacteria used in these experiments are
huge, millions upon millions of bacteria are involved. The reason that the penicillin mutation is the one we detect is that its the one we're looking for.
But my further question is why isn't it possible for the DNA to contain the immunity?
There are people who have immunities that are not active in their offspring but is active in later descendants. Why could that not be possible in bacteria?
Bacteria, unlike people, only have a single copy of their DNA. They are haploid rather than diploid, in the biological parlance.
In a diploid organism there are two copies of each gene. Often one version of the gene will be
dominant, which means it will exert its effect even if there is only one copy of it, and one version of the gene will be
recessive which means that it will only exert its effect if
both copies are this variant. So, what happens when you have a hidden immunity in humans (or other animals) is that both parents carry the recessive gene, but also have a dominant gene so the recessive gene has no effect, and then both pass this recessive copy to their offspring. The offspring, now having two recessive copies, has the immunity.
(An example of this is the CCR5 Δ32 gene which gives near total immunity to HIV-1 infection)
Now, because bacteria aren't diploid, but haploid they have only
one copy of each gene and thus express all of the genes they carry* and cannot have hidden immunity in the same way.
* - strictly, it's more complicated than that because gene expression is under environmental control but that's not relevant in this case.
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