Understanding the Dynamics of Antibiotic Resistance in Microbial Communities using Tensor Methods

Spatial heterogeneity plays an important role in the evolution of drug resistance, but relatively little is known about resistance in complex spatial profiles of selection pressure. We have developed a toy model of stochastic microbial dynamics to investigate how different spatial profiles of selection pressure impact the time to fixation of a resistant allele using mean first passage time calculations. While our previous results established that spatial profiles can dramatically speed or slow the emergence of resistance, they provide little information about the trajectory taken by the system to reach fixation. We now expand our analysis to consider the third-order tensor composed of the time to fixation from all possible intermediate states of the system. We develop several methods to deconstruct this tensor into quantities that allow us to gain insight into the evolutionary dynamics of the system as it reaches fixation. We use a 3-D convolution to relate fixation times of neighboring states and a modified CP decomposition to reduce the fixation time tensor into single-microhabitat fixation profiles lacking spatial structure. We demonstrate that these tools allow us to intuitively understand the emergence of fixation in spatially-structured systems.