We sought out to determine the optimal time to induce the E. coli in order to produce the largest quantity of antimicrobial peptides. We hypothesized that the optimal induction time would be around mid-log, when the E. coli are growing fastest and there are enough bacteria to produce a significant amount of peptide before the population levels drop. To test this theory, we created a theoretical model using MATLAB, using E. coli logistical growth combined with exponential decay (due to the antimicrobial peptide) at different induction times, as represented in the graphic below.

We simulated a 24 hour period and determined the optimal time to induce the cells is around mid-log (~8.5 hours). Inducing at this time maximizes production of the peptide. The graph below shows E. coli growth with induction at different times. They follow a logistic growth model until the inducer is added and then there is an exponential decay.

Overlayed with this graph is a plot of total production of of the peptide vs. time of induction, (with induction at every 3 minutes over the 24 hour period). The highest production of peptide over the lifespan of these bacteria is represented by the peak of this plot, which corresponds to induction at mid-log, as we previously hypothesized.
The MATLAB code for our model can be found Here.