Role of bacterial persistence in spatial population expansion

Bacterial persistence, tolerance to antibiotics via stochastic phenotype switching, provides a survival strategy and a fitness advantage in temporally fluctuating environments. Here we study its possible benefit in spatially varying environments using a Fisher wave approach. We study the spatial expansion of a population with stochastic switching between two phenotypes in spatially homogeneous conditions and in the presence of an antibiotic barrier. Our analytical results show that the expansion speed in growth-supporting conditions depends on the fraction of persister cells at the leading edge of the population wave. The leading edge contains a small fraction of persister cells, keeping the effect on the expansion speed minimal. The fraction of persisters increases gradually in the interior of the wave. This persister pool benefits the population when it is stalled by an antibiotic environment. In that case, the presence of persister enables the population to spread deeper into the antibiotic region and to cross an antibiotic region more rapidly. Further we observe that optimal switching rates maximize the expansion speed of the population in spatially varying environments with alternating regions of growth permitting conditions and antibiotics. Overall, our results show that stochastic switching can promote population expansion in the presence of antibiotic barriers or other stressful environments.

Figure 1

(a) The model equations display traveling waves moving with a constant expansion speed. The total population density (black line) wave has contributions from two subpopulation waves corresponding to normal (red broken line) and persister cells (blue dashed line). We have used the following parameters for numerical calculations: ${\mu }_n=1h^{-1}$, ${\mu }_n=0.1h^{-1}$, $D_n=D_p=100\mu {\text{m}}^2{h}^{-1}$, $a=b=0.02h^{-1}$. (b) The subpopulation ratio ($p/n$) at the leading edge and back of the wave follows the theoretically predicted ratios for the exponential growth (dark gray dashed line) and stationary phase (gray dashed line), respectively. (c) Expansion speed (open squares) and persister fraction at the leading edge (closed squares) as functions of the phenotype switching rates (for $a=b$). The solid lines show the corresponding analytical results. (d) The decrease in the expansion speed with the increase in persister fraction at the leading edge (open squares) follows the constitutive relation given by Eq. (4).

Figure 2

(a) Expansion speed transitions from a slow (persister-dominated) to a fast (normal cell-dominated) mode as seen in our numerical simulations (cross points) that start with a high persister fraction. The front position as a function of time exhibits the two linear regimes with slopes that match the analytic expressions for the fast (solid line) and slow (broken line) expansion speed. (b) During the speed transition, the fractions of the two phenotypes change; the population shifts from a persister-dominated (broken line) to a normal-cell dominated (solid line) wave.

Figure 3

(a) Steady-state profile of the total population inside an antibiotic region (gray area) for different phenotype switching rates, $a=b={10}^{-1}{h}^{-1}$ (solid red line), ${10}^{-2}{h}^{-1}$ (black broken line), and ${10}^{-3}{h}^{-1}$ (dashed blue line). The inset shows that the population decay is the slowest for an intermediate phenotype switching rate. We have used ${\mu }_n^s=-1h^{-1},{\mu }_n^s=-0.1h^{-1}$ and other parameters are the same as in Fig. 1. (b) The penetration depth of the population wave into the antibiotic region as a function of the switching rates $a$ and $b$. It is computed as the distance of the position where $n\left(x_e\right)+p\left(x_e\right)=N_{\mathrm{tip}}$ ($N_{\mathrm{tip}}=0.005$ in the simulations). (c) For equal rates $a=b$, the plot shows the existence of a phenotype switching rate, for which the extent of the wave into the antibiotic region is maximal.

Figure 4

Expansion in symmetric periodic environments. (a) Expansion speed of the population for different switching rates ($a,b$) for a short environmental width ($X_G=X_S=X=50\mu \text{m}$). The expansion speed decreases with the increase in $b$ for fixed $a$ values. The maximum value of the expansion rate lies in the region with the lowest switching rates. (b) Same as in (a) for a long environmental width ($X_G=X_S=X=100\mu \text{m}$). (c) The switching rates ($a_{\mathrm{opt}},b_{\mathrm{opt}}$) corresponding to the maximal expansion speed (obtained from maps like in b) varies inversely with the environmental widths ($X$). (d) The ratio between the maximum expansion speed in the presence of persistence and the speed without persistence for different environmental widths. The benefit (where the ratio is $>1$) from persistence exists only after a threshold value for the environment's width. Note that the plotted values are based on the maximal expansion speed observed in the simulated parameter range. Values $<1$ indicate that the limiting case without persisters (not included in the simulated parameter range) gives the maximal expansion speed.

Figure 5

Expansion in asymmetric periodic environments. (a) Optimal switching rates ($a_{\mathrm{opt}}$ shown by squares, $b_{\mathrm{opt}}$ shown by circles) as a function of the antibiotic region width for a fixed growth region width ($X_G=400\mu \text{m}$ and $X_G=200\mu \text{m}$ in solid and broken lines, respectively) (b) Ratio of expansion speed with optimal switching rates and without any phenotype switching for different antibiotic region widths. The inset shows the absolute value of the maximum speed for $X_A=200\mu \text{m}$ and $X_A=400\mu \text{m}$.

Figure 6

Phenotypic flux distribution along the wave in the moving frame of reference ($z$). In the populated regions (where $n+p=1$), the phenotypic flux $J$ (solid black line) increases exponentially from the back of the wave to the front, in excellent agreement with the analytical expression for $J$ (dashed black line).

Figure 7

(a) Persister fraction at the boundary of growth and antibiotic regions as a function of the phenotype switching rate ($a=b$, open circles). For large switching rate, the persister fraction approaches $f_{p,\max }=0.5$ due to equal switching rates. (b) Two characteristic decay constants of the population density in the antibiotic region, ${\kappa }_{+}$ (solid line) and ${\kappa }_{-}$ (dashed line). Both are decreasing functions of the switching rate. (c) The impact of the switching rate on the persister fraction at the boundary and on ${\kappa }_{-}$ is sufficient to explain the maximum in the extent of the wave as a function of the phenotype switching rate (open circles are from the numerical calculation). The solid line uses Eq. (A9) together with the numerical values of $f_p^0$.

Figure 8

(a) Expanding population stalled by an antibiotic region. Initial conditions: $n+p=1$ with $n/p=b/a$ for $x<5\text{mm,}$ otherwise 0. (b) Established population decaying due to the addition of the antibiotic region. Initial conditions: $n+p=1$ with $n/p=b/a$ for $x<75\text{mm,}$ otherwise 0. (c) Population density profile for the case (a) and case (b) overlap each other, indicating that steady state is independent of initial condition. The switching rates are $a={10}^{-3}h$ and $a={10}^{-4}h$.

Figure 9

Antibiotic barrier crossing time of the population as a function of the phenotype switching rate $a$ for different barrier widths. The crossing time is normalised with respect to the barrier width. The inset schematic depicts the simulation setup. A long growth region is chosen to allow the population to reach a steady state (for each $a$ value) before facing the antibiotic barrier.