Tuning Spatial Profiles of Selection Pressure to Modulate the Evolution of Drug Resistance

Spatial heterogeneity plays an important role in the evolution of drug resistance. While recent studies have indicated that spatial gradients of selection pressure can accelerate resistance evolution, much less is known about evolution in more complex spatial profiles. Here we use a stochastic toy model of drug resistance to investigate how different spatial profiles of selection pressure impact the time to fixation of a resistant allele. Using mean first passage time calculations, we show that spatial heterogeneity accelerates resistance evolution when the rate of spatial migration is sufficiently large relative to mutation but slows fixation for small migration rates. Interestingly, there exists an intermediate regime—characterized by comparable rates of migration and mutation—in which the rate of fixation can be either accelerated or decelerated depending on the spatial profile, even when spatially averaged selection pressure remains constant. Finally, we demonstrate that optimal tuning of the spatial profile can dramatically slow the spread and fixation of resistant subpopulations, even in the absence of a fitness cost for resistance. Our results may lay the groundwork for optimized, spatially resolved drug dosing strategies for mitigating the effects of drug resistance.

Figure 1

(a) Stochastic model for emergence and spread of resistant cells (red) in a spatially extended population of sensitive cells (green). Each spatial habitat ($x_i$) contains $N$ total cells. Cells migrate at a rate $\beta$ between neighboring habitats, and sensitive cells mutate at a rate $\mu$ to resistant cells. The spatial distribution of selection pressure is characterized by a background value ($s_0$) and a peak height ($\delta s$). (b) Example plot of the mean fixation time for different landscapes with $\mu =5\times {10}^{-3}$, $\beta =0.08$, $N=25$, and $⟨s⟩=0.167$. The time to fixation can be either faster (green) or slower (red) than the spatially homogeneous landscape with $\delta s=0$. Inset: Selection landscapes for $\delta s=-0.2$ (left) and $\delta s=0.5$ (right).

Figure 2

Spatial heterogeneity can speed or slow fixation depending on the rates of migration ($\beta$) and mutation ($\mu$). (a) Phase diagram illustrates region of parameter space where the homogeneous landscape leads to a maximum (light blue), minimum (dark blue), or intermediate (medium blue) value of the in fixation time. MFPT calculations were performed for the indicated values of $\beta$ and $\mu$ and for $-0.2\le \delta s\le 0.5$ in steps of 0.1. (b) Sample fixation curves in the regions where heterogeneity slows fixation (left-hand panel, diamonds; $\beta ={10}^{-4}$, $\mu ={10}^{-4}$) or accelerates fixation (right-hand panel, squares; $\beta =5\times {10}^{-2}$, $\mu ={10}^{-4}$). Solid curves indicate analytical approximations. (c) Gray shaded region indicates fixation time ${\tau }_f$ from every initial state [$n^{*}(x_0),n^{*}(x_1),n^{*}(x_2)$], where $n^{*}(x_i)$ is the initial number of mutants at position $x_i$. Red curves show mean fixation time over all initial states with a given total mutant fraction. Vertical arrows represent time to achieve a total mutant fraction of $1/5$ (${\tau }_1$, blue) and time to go from that fraction to fixation (${\tau }_2$, green). Left to right panels: Increasing $\beta$ at a fixed value of $\mu ={10}^{-4}$; plots correspond to symbols on phase diagram in (a). $N=25$ and $⟨s⟩=0.167$ in all panels.

Figure 3

(a) Schematic: A subpopulation of resistant mutants (red) arises at a particular spatial location. How can one choose the spatial distribution of selection pressure (i.e., drug concentration) to maximize the time to fixation? (b) Heterogeneity can significantly speed or slow fixation starting from an initial resistant subpopulation consisting of $N/2$ cells in the center habitat ($\mu ={10}^{-5}$, $\beta =8\times {10}^{-3}$). Points, exact calculation; solid line, analytical approximation. (c) The optimal spatial heterogeneity ($\delta s$) leading to the slowest mean fixation time from an initial state of $(0,N/2,0)$. Depending on the specific parameter regime, the optimal selection pressure profile is the one with the largest possible valley consistent with $⟨s⟩$ (black) or the one having the largest possible peak (white). Red solid line, analytical approximation. (d) Relative magnitude of ${\tau }_f^{\delta s_{\max }}$ (mean fixation time at maximum value of $\delta s$) and ${\tau }_f^{\delta s_{\min }}$ (mean fixation time at minimum value of $\delta s$) as mutation rate decreases at constant migration rate [green arrow, (c)]. Points, exact calculation; solid line, analytical approximation. $N=24$ and $⟨s⟩=0.167$ in all panels. Analytical approximation is given in Eq. (S24) of SM [47].