The Adaptation Speed of Bacteria under the Gradient of Antibiotics Concentration

The evolution of bacteria is strongly influenced by environmental heterogeneity, particularly antibiotic concentration gradients that shape resistance through mutation and selection. Experiments [1] have shown that bacteria acquire resistance stepwise under gradients, involving multiple mutations rather than a single event. Since antibiotic levels vary among tissues, adaptation under heterogeneous environments is biologically relevant. However, how the steepness of the gradient affects the evolutionary rate remains poorly understood. Previous theoretical studies [2], such as the stochastic staircase model, treated the gradient as a fixed parameter, without considering its scaling properties.

To address this, the model is extended by incorporating the concept of First Passage Time to introduce a scaling framework that links gradient steepness to transition dynamics. This formulation enables quantitative comparison between systems with different environmental heterogeneities. Numerical analysis using stochastic simulations reveals that adaptation time exhibits a nontrivial dependence on the gradient parameter, showing the existence of an optimal gradient that minimizes the time required for resistance acquisition.

[1] Baym M. et al. Science 353(6304), 1147–1151 (2016).

[2] Hermsen R., Deris J. B. & Hwa T. Proc. Natl. Acad. Sci. U.S.A. 109(27), 10775–10780 (2012).