Figure 2: The results of sequential therapy on AIEC bacterial colonies are illustrated. The averages of the original and replicate experimental test results for both sets a and b were taken to visualize the results in the figure above. The X axis presents all of the types of antibiotics that the AIEC colonies were exposed to and the Y axis represented the average number of surviving AIEC colonies after treatment. For sets a and b, all 7 plates listed were exposed to the same antibiotics but in different order. Set A represented the weak-strong antibiotic therapy and set B represented the strong-weak antibiotic therapy. The plates with LB+ Kan + Spe tended to yield the most AIEC colonies of any sequential antibiotic treatments no matter what order antibiotics were given in. However, the AIEC colonies treated with LB+P/S+AMP tended yield the lowest number of colonies no matter what order the treatments were given. The standard error of the mean (SEM) calculated for set a is 4.196 and for set b is 4.87.
Regarding measuring the effect of antibiotic treatments on AIEC under sequential therapy and the evolution of MDR in the strains, a population dynamics model regarding the evolution of MDR in the AIEC bacterial population was made.
The model was made with the assumption that bacteria acquire resistance mutations against the first and the second antibiotics with rates α1 and α2, and that an antibiotic-resistant allele confers a cost of resistance in the absence of that antibiotic (10). Other assumptions that are made is that mutations that confer resistance against the first antibiotic are deleterious with selection coefficient c 1 in the absence of the following antibiotic and that the fitness cost of mutation will confer resistance to the second antibiotic, which corresponds to variable c 2. Other variables used in the population dynamics model include ω which is number of MDR cells in AIEC, r1 which is the intrinsic growth rate, B which is the antibiotic value for each antibiotic and is a value dependent on whether a strong or weak antibiotic is used, ω refers to the colony population number, and f which is the specific value associated with each following antibiotic.
In the first phase of the experiment, it was assumed that the AIEC bacterial population grew and by the end of this phase, it reached the mutation-selection balance (10), α1/c 1, and that there were no MDR mutants.
During this specific portion of the phase the population then grew for time t in the presence of the first antibiotic, and then grew for time t again after the second antibiotic was applied on a new plate with the bacterial samples.
An assumption regarding my experiment is that MDR tended to evolve best if on average there was at least one cell harboring resistance to both antibiotics at the time the second antibiotic was applied. Therefore if ω15(t ) is the expected number of MDR cells at time during the second growth phase, the equation for MDR evolution is ω15 (T)≥1.
Relating the aforementioned equation to variables α1, α2, c 1, and c 2 in terms of MDR evolution, an assumption that must be made is that the population stays below its carrying capacity and grows exponentially for time T under the presence of the first antibiotic. Therefore the term ω1(t ) which refers to the expected number of bacteria resistant to the first antibiotic only at time t , can be described and defined by these equations:˙
                                                                                ω1=r1ω1−α2ω1,
                                                                         ω12=(r1−c2)ω12+α2ω1
Which then changes to this equation assuming that conditions stay the same at the beginning of the second growth phase:
                                                                              ω1(0)=ω0α1/c1,
                                                                                      ω15(0)=0
Relating the previously mentioned equations to the intrinsic growth rate (6) of single-drug-resistant bacteria in the presence of the first antibiotic then expands the equation for:
                                                    
                                                            ω1(t)=(ω0α1/c1) * e^(r1−α2)t,
                        ω15(t)= ((ω0α1/c1)μ2 /(α2−c2)) * (e^(r1−c2)t − e^(r1−α2)t).
which then becomes: ((ω0α1α2* e^(r1−α2)t / (c1)) * (1− e^−(c2−α2)t / (c2−α2))) / ≥ 1 which
can reevaluated as:
                                               c1 ≤ Bf(c2) where B = ((ω0α1α2) * e^(r1−α2)t* (t))
                                                           and f(c)= (1− (e^−(c−α2)t)) /((c−α2)t)
In the experiment, it is possible to relate the one strong and one weak antibiotic used per LB agar plate as, X and Y , in which resistance incurs the costs CX and CY , respectively. Without loss of generality then CX > CY . Relating the aforementioned equation into a term that relates to set A where the weak antibiotic is applied before the strong antibiotic would yield the equation C Y≤ Bf(C X) when simplified. As for the case relating to set B were the strong antibiotic is applied before the weak antibiotic, the equation that would be yielded is the equationC X ≤ Bf(C Y) when simplified. For the mathematical model below, only the equations C y ≤ Bf(C X) and C x ≤ Bf(C Y) are relevant since the aforementioned equations are just the unsimplified versions of C y ≤ Bf(C X) and C x ≤ Bf(C Y).