Bacterial defiance as a form of prodrug failure

Brandon Alexander Holt; Gabriel A. Kwong

To quantitatively understand how the rate of prodrug activation competes with the number 15 of living bacteria, we built a mathematical model using a system of nonlinear ordinary differential equations (ODE). In our system, the three dynamic populations are the Bacteria, B, the Locked drug, L, and the Unlocked drug, U, for which we formulated governing ODEs by considering the system parameters that affect populational change over time. For this prodrug system, increased E. coli growth results in increased proteolytic activation of AMP, which in turn lyses bacteria to 20 create a negative feedback loop. Therefore, we modeled the bacteria population B as increasing (i.e., green dashed arrows) the rate of enzymatic drug conversion (i.e., L to U) and the unlocked drug population U as increasing the rate of bacterial death (Fig 2A) . To account for the fact that bacterial growth rate, r, slows down as environmental resources become limiting (i.e., carrying capacity, Bmax), we used a logistic growth model 35 , which produces an S-shaped curve and has 25 . CC-BY-NC-ND 4.0 International license is made available under a The copyright holder for this preprint (which was not peer-reviewed) is the author/funder. It . https://doi.org/10.1101/556951 doi: bioRxiv preprint been used extensively in biology to study population expansion 36 and tumor growth 37 Here we assumed our system constituted a well-mixed solution of freely diffusing substrates (i.e., locked drug) in large excess, which were valid assumptions for this study since AMP prodrug was present 10 at concentrations ~10 2 micromolar in an aqueous environment. Because the total amount of drug is conserved, we defined the MM activation rate of unlocked drug, U, as opposite of the degradation rate of locked drug L (Equation 1.3) . We further included a term to account for the loss of active AMPs according to a proportionality constant, a * , that represents the number AMP required to kill one E. coli. This term was necessary as AMPs lyse E. coli by intercalating with 15 bacterial membranes 39 and therefore are removed from the system and unable to target additional bacteria.