What this empirical example demonstrates is that the sequence of mutations must occur in an order of ever increasing fitness in order for the evolutionary process to have a reasonable chance of occurring. In addition, this example demonstrates that there is more than a single sequential order, which can occur. In other words, not every member of the population must have the same sequence of mutations in order to evolve resistance to the antibiotic selection pressure. The population of bacteria has subdivided into subpopulations, each taking their own trajectory to achieve resistance to this particular selection pressure.
If we label one subpopulation ‘1’, that subpopulation must get mutation A1 followed by mutation B1, in turn followed by mutation C1, then D1 and finally E1 in order to evolve resistance to the antibiotic selection pressure. If we label another subpopulation ‘2’, that subpopulation must get a different set of mutations, which we can label A2 followed by mutation B2, in turn followed by mutation C2, then D2 and finally E2 in order to evolve resistance to the antibiotic selection pressure. Each of the subpopulations that Weinreich and his co-authors describe has their own set of mutations, which lead to the evolution of a high-resistance-lactamase allele. Each of the subpopulations are evolving independently of the other subpopulations. Once a particular subpopulation starts on an evolutionary trajectory, the replication of members from that subpopulation to not contribute to trials for the next beneficial mutation in a different subpopulation on a different evolutionary trajectory.
In order to evolve a high-resistance-lactamase allele, a bacterial subpopulation must accumulate five mutations in a particular sequence. We will now address how a subpopulation can accumulate these five mutations. We will drop the numeric and call these five mutations A, B, C, D and E, and the five mutations must occur in that order. The first mathematical question is to identify the trial and possible outcomes for that trial in this stochastic process. We describe this mathematics in the following section.