Selective Pressure vs Spontaneous Mutation

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Spontaneous mutations are "natural" changes to the base pair sequence of the genetic material of an organism and is usually either deleterious or neutral to the organism. Occasionally, natural selection (a selective pressure) will lead to an increase in the gene frequency of a spontaneous mutation which is advantageous to the organism (ie, the first cameleon that somewhat altered the color of its integument to blend in with the surroundings which helped to avoid its prey).

There are problems with your example of the bacterial infection but there are some teaching points. After the original person is treated with the antibiotic because they are clinically well, we can assume that the critical amount of bacteria (since it was an antibiotic as opposed to an antiviral or antifungal agent) was destroyed. Could any of that species be left? Yes, usually once the critical amount has been destroyed, the body's immune system will clean up most of the rest. Could a mutation have occurred? Yes, it is possible (more likely if the antibiotic was taken over a relatively long period of time).

So we will assume for the purpose of your example that a beneficial mutation occured for the bacterium and it is now immune from the old antibitic x.

You said that the original person gets sick from the co-worker with a different disease which we will assume to mean a different bacterium but of a "kind" of bacteria of the same family as the first since they are both sensitive to the same antibiotic.

Now the original working has lots of this new baterium but almost none of the old mutated one. Why does antibiotic x not help? The issue of conjugation is "pretty" from an intellectual standpoint but extremely unlikely in such an example since it is unnecessarily complicated.

It is more likely just to say that over time, antibiotic x destroyed the 2nd bacterial species in the original worker and that the selective pressure favored the mutated first species' growth leading to untreated infection. Again, because these are humans that are infected (as opposed to following bacterial growth/mutations/selective pressures in a petri dish)just from a practical standpoint, the antibiotic use must be prolonged.