Massinissa Beldjenna 1, Laura (L. R.) Baars 1, Coen (J. G. C.) van Hasselt 1, Tingjie Guo 1
1 Division of Systems Pharmacology and Pharmacy, Leiden Academic Centre for Drug Research, Leiden University (Leiden, Netherlands)
Objectives:
Drug resistance may emerge through pressure exerted by antimicrobial treatments [1], which sometimes lead to acquisition of resistance and selection of resistant subpopulations [2, 3]. Such subpopulations may then proliferate, leading to infection relapse and treatment failure. Proper characterization of resistance dynamics is essential for designing optimal treatment strategies that minimize the risk of selecting multi-drug-resistant bacteria [2].
Here, we compared different resistance modeling techniques using the example of phage-bacteria dynamics [4], as in bacteriophage therapy, bacteria develop genetic resistance rapidly [5]. Resistance development is commonly modeled using a simple ODE model incorporating a rate of mutation (rate-based model without threshold) [6, 7]. This model is however susceptible to the atto-fox problem, namely allowing replication and proliferation of infinitesimal subpopulations below a single cell, and is consequently unable to describe positive eradication of the bacteria. To address this issue, a variation of this model has been proposed by defining a threshold below which bacteria were considered entirely eradicated (rate-based model with threshold) [8]. In the current study, we further evaluate three alternative models that aimed to better represent the stochastic nature of resistance: an ODE model with a time-to-event component of hazard of mutation (TTE-based model), a mixture model (mixture model), and a model with a binomial term of mutation upon replication (hybrid model). These five models were then compared through simulation study under different conditions, assessing the likeliness of regrowth in the bacterial profile due to proliferation of resistant bacteria subpopulations.
Methods:
Bacteria-phage dynamics were implemented based on a modified Cambell viral dynamics model [4, 9]. Bacterial and viral parameters were sampled from typically observed ranges [10, 11]. Resistance development was assumed to be directly proportional to replication [6], and the probability of emergence of a resistant bacterium in a single replication event was noted µ. A range of µ values from 10^(-12) to 10^(-6) was tested. We covered a grid from 10^4 to 10^9 CFU/mL for the bacterial inoculums and 10^2 to 10^10 PFU/mL for the phage inoculums.
For the rate-based model with threshold, the eradication threshold was implemented as post-processing. The TTE-based model was implemented by tracking accumulated hazard and triggering replication of the resistant compartment once the probability of resistance development exceeds a given threshold (set at 50%). For the mixture model, probability to be in either the no-regrowth or regrowth group was derived from the probability of resistance development obtained in the TTE-based model. As for the hybrid model, a binomial term was included in the differential equations to model probability of resistant offspring at each step.
Results:
At high µ values, all models produced nearly indistinguishable profiles with rapid emergence of large resistant subpopulations. At low µ values, the rate-based model without threshold did predict a regrowth due to the atto-fox problem, while all other models predicted no regrowth.
For µ between 10^(-8) and 10^(-6), the five models showed substantially different behaviours. The hybrid model, because of its inherent stochasticity, predicted variable response with complete eradication or regrowth at different timings, even under identical conditions. This stochasticity was consistent with experimental observations from literature [12]. Such variability was not captured with the rate-based and TTE-based models, both being deterministic. Nevertheless, the TTE-based model predicted a probability of regrowth in line with the hybrid model simulations. It also described accurately the most likely profile — namely either the no-regrowth profile if said probability is less than 50%, or a typical regrowth profile. The mixture model managed to describe variability in existence of regrowth [12] but still failed to capture difference in times of regrowth.
Conclusion:
At high probability of resistant mutations, the traditional rate-based model suffices. As for lower probabilities, only the hybrid model fully captures stochasticity, but its complexity precludes parameter estimation. The mixture model offers partial stochastic description, though its inoculum-dependent parameterization limits it to descriptive use. A hybrid of the TTE-based model and the mixture model represents a promising direction for predictive resistance modeling.
References:
[1] Antimicrobial Resistance Collaborators. Global burden of bacterial antimicrobial resistance in 2019: a systematic analysis. Lancet. 2022 Feb 12;399(10325):629-655. doi: 10.1016/S0140-6736(21)02724-0.
[2] Oz T, Guvenek A, Yildiz S, Karaboga E, Tamer YT, Mumcuyan N, Ozan VB, Senturk GH, Cokol M, Yeh P, Toprak E. Strength of selection pressure is an important parameter contributing to the complexity of antibiotic resistance evolution. Mol Biol Evol. 2014 Sep;31(9):2387-401. doi: 10.1093/molbev/msu191.
[3] Fish DN, Piscitelli SC, Danziger LH. Development of resistance during antimicrobial therapy: a review of antibiotic classes and patient characteristics in 173 studies. Pharmacotherapy. 1995 May-Jun;15(3):279-91.
[4] Campbell, A. (1961). Conditions for the Existence of Bacteriophage. Evolution, 15(2), 153–165. doi: 10.2307/2406076.
[5] Oechslin F. Resistance Development to Bacteriophages Occurring during Bacteriophage Therapy. Viruses. 2018 Jun 30;10(7):351. doi: 10.3390/v10070351.
[6] Lenski, R. E., & Levin, B. R. (1985). Constraints on the Coevolution of Bacteria and Virulent Phage: A Model, Some Experiments, and Predictions for Natural Communities. The American Naturalist, 125(4), 585–602.
[7] Leung CYJ, Weitz JS. Modeling the synergistic elimination of bacteria by phage and the innate immune system. J Theor Biol. 2017 Sep 21;429:241-252. doi: 10.1016/j.jtbi.2017.06.037.
[8] Leung CYJ, Weitz JS. Modeling the synergistic elimination of bacteria by phage and the innate immune system. J Theor Biol. 2017 Sep 21;429:241-252. doi: 10.1016/j.jtbi.2017.06.037.
[9] Dominguez-Mirazo M, Harris JD, Demory D, Weitz JS. Accounting for cellular-level variation in lysis: implications for virus-host dynamics. mBio. 2024 Aug 14;15(8):e0137624. doi: 10.1128/mbio.01376-24.
[10] Shao, Y., & Wang, I.-N. (2008). Bacteriophage Adsorption Rate and Optimal Lysis Time. Genetics, 180(1), 471–482. doi: 10.1534/genetics.108.090100.
[11] Wang, I.-N. (2006). Lysis Timing and Bacteriophage Fitness. Genetics, 172(1), 17–26. doi: 10.1534/genetics.105.045922.
[12] Abedon ST. Optical Density-Based Methods in Phage Biology: Titering, Lysis Timing, Host Range, and Phage-Resistance Evolution. Viruses. 2025; 17(12):1573. doi: 10.3390/v17121573.
Reference: PAGE 34 (2026) Abstr 12196 [www.page-meeting.org/?abstract=12196]
Poster: Methodology - New Modelling Approaches