RAPID COMMUNICATIONS PHYSICAL REVIEW E 89, 030702(R) (2014) Phenotypically heterogeneous populations in spatially heterogeneous environments Pintu Patra and Stefan Klumpp Max Planck Institute of Colloids and Interfaces, Science Park Golm, 14424 Potsdam, Germany (Received 25 July 2013; revised manuscript received 10 December 2013; published 12 March 2014) The spatial expansion of a population in a nonuniform environment may benefit from phenotypic heterogeneity with interconverting subpopulations using different survival strategies. We analyze the crossing of an antibiotic- containing environment by a bacterial population consisting of rapidly growing normal cells and slow-growing, but antibiotic-tolerant persister cells. The dynamics of crossing is characterized by mean first arrival times and is found to be surprisingly complex. It displays three distinct regimes with different scaling behavior that can be understood based on an analytical approximation. Our results suggest that a phenotypically heterogeneous population has a fitness advantage in nonuniform environments and can spread more rapidly than a homogeneous population. DOI: 10.1103/PhysRevE.89.030702 PACS number(s): 87.23.Cc, 87.10.Mn, 87.19.xb, 87.23.Kg Introduction. The development of populations of cells or organisms depends not only on these organisms themselves, but also on their interactions with competing populations and with their environment. Specifically, the spatial structure of the environment can play an important role, for example, by separating populations or providing barriers to the spreading of a species [1–3]. Recently, experimental techniques such as microfluidic habitats [4,5] and range expansion of microbial populations on plates [6–8] and complex interactions between multiple species [9] have been used to study the influence of spatial structures in a quantitative fashion using microbes as model organisms. For example, it has been shown that spatial heterogeneity (different drug concentrations in different organs or concentration gradients for locally administered drugs [10–12]) can both speed up and slow down the emergence of antibiotic resistance in bacteria [13–16]. Another aspect of microbial survival under stressful con- ditions is phenotypic heterogeneity [17,18], i.e., different behaviors (e.g., normal growth, sporulation, competence, persistence) exhibited by genetically identical cells under iden- tical conditions, a feature usually attributed to multistability in the underlying genetic circuitry [19]. A prime example of phenotypic heterogeneity is bacterial persistence, phenotypic tolerance to antibiotics [18,20–23]. A subpopulation of cells is tolerant against antibiotics or other stresses and allows prolonged survival of the population under such conditions. The cells switch stochastically between the normal and the persistent phenotype, thus after the stress is removed, the population can grow back from the surviving persisters (in contrast to resistant mutants, the regrown population remains susceptible to the antibiotic). Persistence has thus been characterized as a bet-hedging strategy, optimal for survival in fluctuating environments [24–26]. In this Rapid Communication, we address a related prob- lem, namely, the role of persisters in the spatial expansion of a bacterial population. The basic idea is that just as persisters allow a population to live through times of stress, they also allow the population to cross regions in space in which the conditions are stressful. Specifically, we consider the case of a population of bacteria expanding from a growth-sustaining environment into another one that is separated from the first by an environment with a high antibiotic concentration (Fig. 1). From a physics perspective this question can be cast as a barrier crossing problem, similar to the type that occur in chemical reaction kinetics, nucleation, and other areas [27,28]. Specifically, there are two different pathways to escape from the initial state with different effective barriers. Thus, we calculate the average time it takes for the cells to reach the third environment and determine the conditions under which the presence of persister cells is beneficial by speeding up the arrival. This dynamics is surprisingly complex with several distinct regimes, which we identify by a combination of stochastic simulations and an analytical theory. We conclude the paper with some remarks concerning possible experiments and an analogy to the crossing of valleys in fitness landscapes. Model. We consider a population with two phenotypes (normal and persisters) in an environment consisting of three connected patches (Fig. 1). These patches may correspond to different organs in a patient, where antibiotic accumulates to different concentrations [29], different chambers in a microfluidic device, or even different patients. The numbers of normal cells and persisters in patch i = 1,2,3 are denoted n i and p i . The first and third patch sustain growth, described as logistic growth with rates μ n and μ p , respectively, and carrying capacity K . The middle patch contains antibiotics, therefore cells migrating into this patch are killed, with death rates, δ n and δ p , and δ p <δ n . Cells migrate between the patches with rate γ , which we take to be independent of phenotype. Switching between the phenotypes is described by rates a and b (normal to persister and vice versa, respectively). These rates are taken to be independent of the environment. Simulation results. Using stochastic simulations, we deter- mined the time after which the first cell crosses the antibiotic- containing environment and arrives in the third patch. In a competitive situation [7], a population that arrives faster will obviously have a fitness advantage compared to other populations. One may expect that the slowly dying persister cells can cross a region of high antibiotic concentration more easily and hence might speed up the population expansion in such a heterogeneous environment. Figure 2 shows results of simulations with realistic growth, death, and switching rates that start with a fully populated first patch. Indeed, a population with persisters (solid red) arrives faster in the third patch than a population without persisters (dashed blue). One also sees that on the time scale on which cells cross the antibiotic barrier, 1539-3755/2014/89(3)/030702(4) 030702-1 ©2014 American Physical Society