Metapopulation model

For the sake of concreteness, we consider a two-dimensional (2D) microbial metapopulation that can be envisioned as a grid of linear size L, containing demes (or sites) labelled by a vector where (with L = 20 in our examples), and periodic boundary conditions [18,20,75–77,97]. The demes of this spatially explicit model are connected to their four nearest neighbours via cell migration, at per capita rate proportional to the migration parameter m (later simply referred to as “migration rate”), and are subject to a constant input rate of an antimicrobial drug [29,30,35]. Each deme has the same carrying capacity, denoted by K, and at time t consists of a well-mixed subpopulation of cells of type S that are drug-sensitive and microbes of the AMR-resistant strain R, with deme size , while the total number of R/S cells at time t across the metapopulation is and the overall time-fluctuating number of microbes is , see Fig 1.

Antimicrobial resistance can often be seen as a form of cooperative behaviour [21,43,56], for example in the case of -lactam antibiotics, microbes with resistance gene-bearing plasmids produce a -lactamase resistance enzyme hydrolysing the antimicrobial drug in their surroundings [21,26,28,43,56,98,99]. In this context, when there are enough resistant microbes, the local concentration of resistance enzymes can reduce the drug concentration below the minimum inhibitory concentration (MIC), so that antimicrobial resistance acts as a public good, protecting both resistant and sensitive cells. Resistant microbes can either secrete the resistance enzymes (extracellular enzymes, typically produced by Gram-positive bacteria) or retain them within the cell (intracellular enzymes, Gram-negative bacteria) [100,101]. Our theoretical model can account for both scenarios as it relies on the catalytic inactivation of the antimicrobial drug, either inside or outside R cells [56,102,103]. In any case, the public good (shared drug protection) can be interpreted as encoding the local decrease in the concentration of the active drug [56] (Introduction; see Fig 1).

Here, we consider a metapopulation model where S and R cells compete in each deme for finite resources in a time-fluctuating environment and in the presence of an antimicrobial drug. We assume that R cells share the benefit of drug protection with S cells within a deme when the local number of resistant cells reaches or exceeds a certain fixed cooperation threshold that is constant across demes, i.e., R cells act as cooperators in deme when [26,43,56] (i.e., drug inactivation occurs at a faster timescale than microbial replication, each deme has a constant drug influx/outflux, and the deme volume is fixed; see Extra and intracellular drug inactivation in the subsection “Robustness of results across parameters and scenarios” of Discussion, and [28] for the case of a cooperation threshold set by the fraction of R cells in a single deme due to a time-varying volume). We therefore assume that microbes of the cooperative resistant strain R have the same constant growth fitness in all demes, where the parameter s (with 0 < s < 1) represents a resistance-production metabolic cost. Moreover, cells of type S that are sensitive to the drug have a baseline fitness f_S = 1 when , and when , where s < a < 1, which denotes the growth fitness reduction caused by the drug [26,28] (see Population size and microbial parameter values in the Discussion subsection “Robustness of results across parameters and scenarios” for further considerations on biostatic and biocidal drug action). Hence, with , the fitness of the R strain exceeds that of S when , whereas the R type has a lower fitness than S, with , when . Both strains, R and S, are subject to the same single-deme carrying capacity (K), which is homogeneous across all demes of the metapopulation, at all times.

Here, we particularly focus on the metapopulation’s eco-evolutionary dynamics under slow migration, a biologically relevant dispersal regime known to increase population fragmentation and hence influence its evolution and diversity [18,104–107] (see Population size and microbial parameter values in Discussion for further details on the slow migration regime). Environmental variability is here encoded in the time-varying carrying capacity of each single deme, K(t), which changes simultaneously across all demes, and that we assume to switch endlessly between values representing mild and harsh conditions [51–54], see Fig 1A and below. (A detailed discussion of the time-fluctuating K(t) is provided in Environmental assumptions in the Discussion subsection “Robustness of results across parameters and scenarios”).

Intra- and inter-deme processes: Bacterial division, death, and cell migration

In close relation to the Moran process [108–111], a reference model in biology for evolutionary processes in finite populations [112], the intra-deme dynamics within a lattice site is represented by a birth-death process defined by the reactions and of birth (cell division) and death of R/S cells, occurring at local transition rates [26–28,51–53,79]

(1)

where is the average fitness in deme at time t. The continuous time variable t is measured in units of microbial replication cycles, i.e., microbial generations (see “Background” below and Results, as well as S1 Appendix Sec. 3).

The inter-deme dynamics on the 2D grid stems from the migration of one cell of R/S type from the site to one of its four nearest-neighbour demes denoted by . Cells’ dispersal in microbial populations is generally density-dependent, with movement often directed towards areas that are rich in resources [113], but simpler assumptions are commonly used [18,20,79,114]. Here, we have considered two forms of migration: 1) We have first assumed a local density-dependent per-capita migration rate , with increasing migration rate as the deme’s population size approaches the carrying capacity (less available resources in ). 2) We have also studied the simpler case of a constant per capita migration rate m, corresponding to the same dispersal in all spatial directions (symmetric migration) of all R and S cells in a deme . The inter-deme dynamics is therefore implemented by picking randomly a cell (R or S) from deme and moving it into a nearest-neighbour according to the reactions occurring at the migration transition rates

(2a)

(2b)

where and are the two forms of local migration rates (respectively, with density-dependent and density-independent per capita rate). We have found that the specific form of migration does not qualitatively affect our main findings, see Discussion. For notational simplicity, we may refer to m as the “migration rate”, being it clear from the context which of the two forms of migration, or , is used.

Environmental variability

Microbial populations generally live in time-varying environments, and are often subject to conditions changing suddenly and drastically, e.g., experiencing cycles of harsh and mild environmental states [41,50,55,80–92], see Fig 1A. Here, environmental variability is encoded in the time-variation of the binary carrying capacity [26–28,51–55,78,79]

(3)

which can take the values (see Environmental assumptions in the Discussion subsection “Robustness of results across parameters and scenarios” for further discussion and interpretation of K(t)). The carrying capacity is thus driven by the coloured dichotomous Markov noise (DMN), also called telegraph process, that switches between according to at rate when [115–117]. It is convenient to write in terms of the mean switching rate and switching bias , where , and indicates that, on average, more time is spent in the environmental state than in ( corresponds to symmetric switching) [26–28,53,54]. In all our simulations, the DMN is at stationarity, and is therefore initialised from its long-time distribution, see S1 Appendix Sec. 3. All the DMN instantaneous correlations are thus time-independent while its auto-covariance reads [53,115–117], where denotes the ensemble average and is the finite correlation time (when ). Following Eq. (3), the carrying capacity switches back and forth at rates between a value K = K₊ () corresponding to a mild environment, e.g., where there is abundance of nutrients and/or lack of toxins, and () under harsh environmental conditions (e.g., lack of nutrients, abundance of toxins) according to , and thus describes (“feast and famine”) cycles of mild and harsh conditions. As the DMN, the time-fluctuating K(t) is always at stationarity: its expected value is , and its auto-covariance is [53,115–117]. Accordingly, in our simulations the initial value of the carrying capacity is drawn from its stationary distribution, i.e., with a probability , see S1 Appendix Sec. 3. The randomly time-switching K(t) drives the deme size of all demes simultaneously, and is hence responsible for the coupling of demographic fluctuations with environmental variability [27,51–55,78,79]. This effect is particularly important when the dynamics is characterised by population bottlenecks [26,28,79]; see below and S1 Appendix Sec. 1.2.3.

The stochastic metapopulation model is therefore a continuous-time multivariate Markov process – defined by the transition rates given by Eqs. (1), (2a) and (2b) – that satisfies the master equation given in S1 Appendix Sec. 1.1. The individual-based dynamics encoded in Eqs. (1), (2a) and (2b) has been simulated using the Monte Carlo method described in S1 Appendix Sec. 3. It is worth noting that N, N_R/S, and all depend on the deme , time t, and environmental state . However, for notational simplicity, we often drop the explicit dependence on some or all of the variables , and .

Background: Eco-evolutionary dynamics in an isolated deme

Since the metapopulation consists of a grid of connected demes, all with the same carrying capacity K(t), it is useful to review and summarise the properties of the eco-evolutionary dynamics in a single isolated deme (when m = 0), studied in Ref. [26] (see also Refs. [28,51,52]). Further technical details can be found in S1 Appendix Sec. 1.2.

Mean-field approximation of the eco-evolutionary dynamics of an isolated deme.

In an isolated deme, there is only cell division and death according to the intra-deme processes with rates given by Eq. (1). Upon ignoring all fluctuations, the mean-field dynamics in an isolated deme subject to a carrying capacity of constant and very large value K = K₀ is characterised by the rate equations for the deme size N and the local fraction of resistant cells [51,52] (see details in S1 Appendix Sec. 1.2.1):

(4)

where the dot indicates the time derivative. The logistic rate equation for N predicts the relaxation of the deme size towards the carrying capacity on a timescale (one microbial generation). The fraction of R cells is coupled to N: x decreases when , and increases otherwise (shared protection). Since 0 < s < a < 1, in this mean-field picture, x approaches on a timescale ( and ) [26,28]. In our examples, we have yielding a clear timescale separation between the dynamics of the deme size and its make-up: N and x are respectively the fast and slow variables; see S1 Appendix Sec. 1.2.1.

Eco-evolutionary dynamics in an isolated deme subject to a fluctuating environment.

When the deme size is sufficiently large to neglect demographic fluctuations and randomness only stems from environmental variability via Eq. (3), the deme size dynamics is well approximated by the piecewise deterministic Markov process (N-PDMP) [26–28,51–55,78,79,118] defined by

(5)

In the realm of the N-PDMP approximation, the deme size thus satisfies a deterministic logistic equation in each environmental state , subject to the time-switching carrying capacity, see Eq. (3) (and details in S1 Appendix Sec. 1.2.3). The properties of the N-PDMP, defined by Eq. (5) and discussed in S1 Appendix Sec. 1.2.3, shed light on how the deme size changes with the rate of environmental changes. In particular, N tracks K(t) when the environmental switching is slower than the logistic dynamics ( and ). In this biologically relevant intermediate switching regime [94,119], the deme experiences a bottleneck whenever the carrying capacity switches from K₊ to and its size is drastically reduced, see Fig 1A, with subpopulation prone to fluctuation-driven phenomena [10,26,28,50–52,86,87,120].

Critical migration rate and bottleneck strength to eradicate antimicrobial resistance

To study under which circumstances environmental variability coupled to demographic fluctuations leads to the eradication of resistance in the two-dimensional metapopulation, we focus on the regime of intermediate environmental switching, with and , and assume . In this biologically relevant regime [94,119] (see Population size and microbial parameter values in the Discussion subsection “Robustness of results across parameters and scenarios”), each deme experiences a sequence of bottlenecks of strength , occurring at a rate , accompanied by “transient dips” in the number of cells, and the demes generally consist of a majority of S cells when it is in the mild state (K = K₊) [26,28] (“Background” in Model & Methods; see also S1 Appendix Sec. 1.2.3 and Fig H in S1 Appendix Sec. 5.4). In addition, we consider a slow/moderate migration rate, with (see proper definitions just after Eq. (6)). The demes of the metapopulations are thus neither entirely isolated (m = 0), nor fully connected (). We know that, in this intermediate environmental switching regime, fluctuation-driven eradication of R is likely to occur in isolated demes [26], i.e., in metapopulations with (see S1 Appendix Secs. 1.2.3 and 5.1), while the probability of long-lived coexistence of R and S is expected to increase with m (the latter as in Figs B and C in S1 Appendix Sec. 2, for constant and very slow/fast switching environments, respectively). In this context, when , fluctuations can clear resistance in some demes, but these R-free demes can be recolonised following migration events from neighbouring sites still containing R cells; see Figs 1B, 2 and 3, and Figs D and E in S1 Appendix Sec. 5.1. The effect of fluctuations caused by environmental bottlenecks is thus countered by migration, and it is not obvious whether eradication of resistance can arise in the spatial metapopulation.

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Fig 2. The eradication of R cells depends on the bottleneck strength and migration rate.

The shared parameters in all panels are , , L = 20, a = 0.25, s = 0.1, N_th = 40, and (Model & Methods) with migration according to Eq. (2a). Other parameters are as listed in Table 1. Panel A: Heatmap of the probability P(N_R(t)= 0) of total extinction of R (resistant) cells as a function of bottleneck strength, , and migration rate m at time t = 500. Each value pair represents an ensemble average of independent simulations, where we show the fraction of realisations resulting in complete extinction of R (resistant) microbes after 500 microbial generations (standard error of the mean in P(N_R(t = 500)=0) below 4%; see S1 Appendix Sec. 3.3.2). The colour bar ranges from light to dark red, where darkest red indicates complete R extinction in all 200 simulations at time t = 500, P(N_R(t = 500)=0)=1. The green line is the theoretical prediction of Eq. (6) and the white dashed vertical line indicates an axis break separating m = 0 and (Model & Methods). The black and white annotated letters point to the specific values used in the outer panels. Panels B-H: Typical example trajectories of the fraction of demes without R cells, up to t = 500 microbial generations (gen.), defined by Eq. (7) and corresponding to the fixation of S in the metapopulation. (The fraction of demes without S cells, , is vanishingly small and unnoticeable.) Here, is shown as a function of time (microbial generations) for the value pairs indicated in Panel A (see S1 Appendix Sec. 3).

https://doi.org/10.1371/journal.pcbi.1013997.g002

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Fig 3. A closer look to individual demes: Migration and intermediate environmental switches shape local eradication of R cells.

Example eco-evolutionary dynamics of the metapopulation in a single simulation realisation. Parameters are K₊=2000, , , and m = 0.001, with density-dependent migration according to Eq. (2a); other parameters are as in Table 1. Panels A-F: Snapshots of the metapopulation at six microbial generation times . Red pixels indicate R-free demes (containing only S cells) and pink pixels are demes where R and S cells coexist. The two demes, and , whose time composition is tracked in Panels G and H are indicated by a black border. Panel A shows the metapopulation a few generations after an environmental bottleneck. From panels A to B no bottleneck occurs, and many S-only demes are recolonised by R cells (many red pixels become pink). Between B and C, the metapopulation experiences a bottleneck causing a burst of local R extinctions (with burst of randomly located red pixels, see also the spike of in Panel I). Panel D: Pink clusters spread across the grid due to the migration of R cells causing many recolonisation events ( in Panel I decreases for ). Panels E-F: After a sequence of bottlenecks starting at , the number of S-only demes increases overwhelmingly across the grid ( in Panel I), and resistance persists only in a few demes where R and S coexist. See S3 Movie and S1 Appendix Sec. 4 for a video of the full spatial metapopulation dynamics for this example realisation and its detailed description. Panels G-H: Temporal evolution of the fraction of resistant cells and , up to t = 500 microbial generations (gen.), in the example demes and indicated as highlighted pixels in Panels A-F. Green bands indicate periods in the harsh environment (where ); harsh periods shorter than 1 microbial generation are not shown (S1 Appendix Sec. 3.3.1). Each transition from white background to a green band indicates an environmental bottleneck. The deme of panel G first exhibits R/S coexistence, followed by fluctuation-driven R eradication at due to environmental bottlenecks. In Panel H, similar dynamical development is followed by the restoration of resistance through recolonisation of the deme by R cells, as indicated by the blue spikes at long times (, Discussion). Panel I: Temporal evolution of the fraction of demes without R cells (red pixels within Panels A-F, see Eq. (7)). From left to right, the dashed vertical lines indicate the corresponding snapshot times in Panels A-F. Green background areas as in Panels G-H.

https://doi.org/10.1371/journal.pcbi.1013997.g003

We ask for what migration rates can environmental and demographic fluctuations clear resistance. The first central question that we address is therefore whether there is a critical migration rate m_c above which the fluctuation-driven eradication of resistance is unlikely, and below which it is either possible or likely. Since the amplitude of the fluctuations generated by the bottlenecks increases with their strength, we expect m_c to be an increasing function of .

We have computed the probability P(N_R(t)=0) that there are no resistant cells across the entire metapopulation after a time t (by sampling realizations, see S1 Appendix Sec. 3). In Fig 2A we report P(N_R(t)=0) as a function of bottleneck strength and migration rate m when , finding when m is below a certain value. Similar results are found in Fig 4B, 4C, and 4D for other environmental parameters at different times t. This indicates the existence of a trade-off between the rate of migration and bottleneck strength, see Figs 2A and 4A, 4B, 4C, and 4D: For a given bottleneck strength , when the migration rate is below the critical value m_c, shown as red/dark phases here, the fluctuations caused by bottlenecks can clear resistance across the whole metapopulation in a finite time (that scales with 1/s, see below). A few bottlenecks thus suffice to eradicate R, as shown in, e.g., Fig 2C, and Fig F (panels C, D, and F) in S1 Appendix, where each red spike corresponds to a bottleneck (see “Breaking it down” subsection below). An approximate expression for m_c is obtained by matching the total number of R migration events across the metapopulation, during the time between two successive bottlenecks, with the number of new R-free demes due to a bottleneck, yielding (see the derivation at the end of this subsection)

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Fig 4. Near-optimal conditions for resistance clearance: Slow migration can speed up and enhance the eradication of R cells.

Temporal evolution of the heatmap showing the probability P(N_R(t)= 0) of R extinction as a function of bottleneck strength, , and migration rate m (implemented according to Eq. (2a)) at t = 200 (Panel A), t = 300 (Panel B), t = 400 (Panel C), and t = 500 (Panel D) with environmental switching rate and bias ; other parameters are as in Table 1. As in Fig 2A, each value pair is an ensemble average over 200 independent metapopulation simulations and the P(N_R(t)= 0) colour bar ranges from light to dark red indicating the fraction of simulations that have eradicated R cells at each snapshot in time (standard error of the mean in P(N_R(t)=0) is below 4%; see S1 Appendix Sec. 3.3.2). The green and dashed white lines represent the theoretical prediction of Eq. (6) and an eye-guiding axis break, respectively (as in Fig 2A). The golden lines in Panels D-E show , with in the (upper) region between the golden and green lines, according to Eq. (9). The grey horizontal lines in Panels A-E indicate the example bottleneck strength used in Panel E. Panel E: Probability of R extinction P(N_R(t)= 0) as a function of migration rate m at bottleneck strength for t = 100, 200, 300, 400, 500 microbial generations (bottom to top). Solid lines (full symbols at m = 0) show results averaged over 200 realisations; shaded areas (error bars at m = 0) indicate binomial confidence interval computed via the Wald interval (see S1 Appendix Sec. 3.3.2). Panel F: 90th and 95th percentile ( and respectively) of R eradication times as function of the migration rate with a bottleneck strength (see S1 Appendix Sec. 3.3.3). Panel F shows a single minimum at corresponding to .

https://doi.org/10.1371/journal.pcbi.1013997.g004

(6)

whose graph is shown in the green curve of Fig 2A where it approximately captures the border between the red/white phases and how m_c increases with (see also Fig 4A, 4B, 4C, and 4D where m_c ranges from to ). Here, the regime of slow migration is defined by (with moderate migration when ).

Further light into the phenomenon of fluctuation-driven R eradication is shed by computing the fraction of demes that consist only of S/R microbes at time t:

(7)

where is the indicator function defined as if and otherwise. Since each deme is never empty, also corresponds to the fraction of demes without any R/S cells, i.e., thus gives the fraction of R-free demes across the metapopulation at time t. In Figs 2 and 3, and those of S1 Appendix, correspond to the fraction of R/S-free demes in a single realization of the metapopulation. In the results of Fig 2B, 2C, 2D, 2E, 2F, 2G, and 2H (and Fig 3I), increases sharply coincidentally with each bottleneck and then transiently decreases due to the recolonisation of R-free demes via migration, whereas at all times. When bottlenecks are strong and ( and m in the red/dark phase of Fig 2A), recolonisation cannot counter bottlenecks and eventually with the eradication of resistance from all demes; see Figs 2C, 2D, 2E and 3I, as well as Figs E (panels E-H), F (panels C, D, and F), and G (panels D and F) in S1 Appendix, while (R-only demes are very unlikely, but see the blue lines in Figs E (panel D), F (panel E), and G (panels C and E) in S1 Appendix when ). When m > m_c (without scaling as the system size, see below), remains finite, while generally regardless of m. This indicates the persistence of R in the metapopulation, which then consists of demes where R and S coexist, and other demes that are resistance free (S-free demes are very unlikely, , when m > m_c; see Fig E (panels I-L) in S1 Appendix).

The fluctuation-driven eradication of R across the two-dimensional metapopulation hence requires intermediate environmental switching, strong enough bottlenecks, and slow migration, which can be summarised by the necessary conditions

(8)

where the first condition indicates that the demes are not fully connected. In the limit of fast migration, here defined as (see below), the metapopulation can be regarded as L² fully connected demes (island model [121,122]), all subject to the same fluctuating carrying capacity given by Eq. (3). The fraction of R cells just after an environmental bottleneck still fluctuates about in each deme (same x as in the mild environment, see Eq (4)). All L² demes experience the same carrying capacity during the bottleneck. The approximate total number of R cells across the metapopulation right after a bottleneck is thus . When , resistance can typically be eradicated only if all R cells can be eliminated simultaneously during a single bottleneck. This is because ensures quick and efficient deme mixing between each bottleneck: If R is not eradicated from each deme after a single strong bottleneck, fast migration restores resistance in all demes before the next bottleneck (as opposed to the case of slow migration, where R-free demes can gradually accumulate after several consecutive bottlenecks). The eradication of R can occur under fast migration when following a single bottleneck, i.e., for . Hence, when and , fluctuation-driven eradication of resistance occurs for very strong bottlenecks, , regardless of the actual value of the migration rate.

Breaking it down: bottlenecks and fluctuations vs. spatial mixing

To further understand the joint influence of bottlenecks and migration on R eradication, we analyse typical single realisations of the metapopulation spatio-temporal dynamics when it is subject to intermediate switching rate and slow migration (m < m_c), and experiences bottlenecks of moderate strength; see Fig 3 where the parameters , m = 0.001, and (see also S3 Movie and S1 Appendix Sec. 4) satisfy the R fluctuation-driven eradication conditions of (8).

This resistance clearance mechanism, driven by bottlenecks and fluctuations, occurs randomly across the grid (scattered red sites in Fig 3A). The microbial composition of each deme fluctuates due to the homogeneous environmental variability (K switches simultaneously in time across all demes of the grid) and random birth-death events. In the regime defined by the conditions (8), strong bottlenecks cause demographic fluctuations that, after enough time, lead to R eradication in some demes (e.g., after t = 70 in Fig 3A and 3G). However, resistant cells can randomly migrate from neighbouring demes, recolonising R-free demes and favouring the spread of microbial coexistence across the metapopulation (pink clusters in Fig 3A and 3B). The fraction of R cells in a deme recolonised by resistance is characterised by spikes after a period of extinction, corresponding to R recolonisation events (e.g., at in Fig 3H, see also pixel in Fig 3A, 3B, 3C, 3D, and 3E). In summary, the occurrence of bottlenecks increases the fraction of R-free demes across the grid (one spike in Fig 3I at each bottleneck), whereas R recolonisation gradually reduces , leading to a sequence of spikes and decreases of (Fig 3I). Spikes are higher the stronger the bottlenecks (larger ), while the decrease of steepens for faster migration (higher values of m). The typical stages of dynamics are thus: (i) an environmental bottleneck eradicates R in some demes causing to spike, (ii) some of these demes are then recolonised by R cells through migration, and decreases. This is then followed by another bottleneck, that restarts the cycle of spikes and decreases of . The succession of steps (i) and (ii) as the environment switches back and forth, eventually leads to either R eradication (when m < m_c) or to the persistence of resistance (long-term balance of spikes and decreases of ). In the example of Fig 3 for a single realization of the metapopulation, the number of R-free demes steadily increases with the number of bottlenecks and, migration not being fast enough to restore resistance across the grid, R cells are eventually eradicated from the entire metapopulation (increasingly more red R-free demes in Fig 3E and 3F than in Fig 3A, 3B, 3C, and 3D; when in Fig 3I; see also S3 Movie and S1 Appendix Sec. 4). This is consistent with the metapopulation realisation of Fig 3 satisfying the conditions (8). See S1 Appendix Sec. 5.4 and Fig H in S1 Appendix for the dynamics of the absolute number of S and R cells in this example realisation of Fig 3.

When sensitive and resistant cells locally coexist on the grid, the demes containing R and S microbes (pink pixels in Fig 3A, 3B, 3C, 3D, 3E, and 3F) consist approximately of and cells of type R and S, respectively (see Fig H in S1 Appendix Sec. 5.4). In the examples considered here, prior to a bottleneck, coexisting demes in the mild environment (where ) are thus made up of an overwhelming majority of S cells, see Fig H in S1 Appendix and S2-S4 Movies, which is consistent with an R “containment strategy” [124].

Slow migration can speed up and enhance R eradication: Near-optimal conditions for resistance clearance

We have disentangled the trade-off between the population bottleneck strength and migration rate m. To further clarify the interplay between fluctuations and migration in the metapopulation, we investigate how the probability P(N_R(t)=0) that there are no resistant cells across the entire grid after a time t depends on m and over time (Fig 4), and how it changes for different values of the switching rate (Fig J in S1 Appendix Sec. 5.6). Under the conditions of (8), the probability P(N_R(t)=0) of overall resistance eradication increases in time (red/dark phases in Fig 4A, 4B, 4C, and 4D): the R eradication mechanism driven by strong bottlenecks overcomes microbial mixing, and the red/dark phase expands in time until reaching its border where (see Eqs. (6) and (8)).

Remarkably, in Fig 4 we find that after some time (), R cells are most likely to be eradicated from the metapopulation under slow but non-zero migration (in Fig 4B, 4C, and 4D red regions are darker for than ; see S1-S2 Movies and S1 Appendix Secs. 4, and Figs D and E in S1 Appendix Sec. 5.1). In Fig 4E, the probability of R eradication P(N_R(t)=0) for increases steadily with m before reaching a plateau near 1 for ( in Fig 4E), and then sharply decreases as m exceeds m_c. Since , most migration events in the intermediate regime defined by the conditions of (8) occur when demes are in the mild environmental state, where the number of microbes in each deme is typically large: Thus, and most individuals are of type S, with (Model & Methods and Fig 1A, and Fig H in S1 Appendix Sec. 5.4). We hence estimate that the rate of migration per deme in the switching regime (see conditions (8)) is roughly mK₊, and consists mostly of sensitive individuals moving into a neighbouring deme. In this context, the impact of migration is particularly significant for the eradication of R cells when the rate of cell migration per deme (mostly of S during the mild environmental state), approximately mK₊, is comparable to the rate at which bottlenecks arise (Model & Methods; see also S1 Appendix Sec. 1.2.1). In fact, when , the R-dominated demes (that have by chance been taken over by R) can be efficiently recolonised by S cells, and can then be eventually cleared from resistance by the fluctuation-driven mechanism caused by strong bottlenecks, as illustrated in S1 Appendix Sec. 5.1 and Figs D (panel C) and E (panels E-H) in S1 Appendix. Matching the rates at which bottlenecks and the S-recolonisation of R-dominated demes occur, yields the condition for which migration can efficiently help promote the fluctuation-driven clearance of resistance; see yellow lines in Fig 4C and 4D. R-dominated demes are not effectively recolonised when the migration rate is lower than , and therefore the probability of R eradication when is the same as for m = 0 (see Population size and microbial parameter values in the Discussion subsection “Robustness of results across parameters and scenarios”; see also S1 Appendix Sec. 5.1).

The probability P(N_R(t)=0) of R eradication for is an increasing function of t at fixed migration rate (Fig 4E). In fact, as this environmental regime is characterised by a sequence of strong bottlenecks, each of which can be seen as an attempt to eradicate R (“Background” in Model & Methods), the clearance of resistance for any is certain in the long run, i.e., . However, maximising the probability clearance of resistance in the shortest possible time is of great biological and clinical significance, e.g., to devise efficient antibacterial treatments [32,41,125]. This means that it is important to determine when the eradication of resistance is both likely and rapid. The second central question that we ask is therefore: What are the conditions ensuring a quasi-certain clearance of resistance in the shortest possible time t^*?

To address this important problem, we have determined the migration rate , satisfying the conditions (8), for which the time for the eradication of R, here denoted by , is minimal. As detailed below, in Fig 4F we determine and , corresponding to the near-optimal conditions for the clearance of resistance, for the example of Fig 4 when the bottlenecks strength is (largest value considered in Fig 4). To this end we have computed as a function of m in the range (all other parameters being kept fixed). is thus the shortest time after which there is at least a 90% chance that resistance has been cleared from the metapopulation. Similarly, we have also determined giving the minimal time for which the R clearance probability exceeds 0.95. Therefore, and give respectively the 90% and 95% percentile of R eradication times (see S1 Appendix Sec. 3.2). The results of Fig 4F show that has a single minimum value at essentially the same migration parameter for both 90% and 95% percentiles. Since this is generally the case for strong enough bottlenecks, to shorten the notation and unless specified otherwise, we henceforth refer to simply as . In the example of Fig 4F, we find , and increases sharply when m > m_c while (not shown in Fig 4F, we have verified that ). Hence, the fluctuation-driven eradication of R is most efficient for , when the probability of resistance clearance after microbial generations is close to 1 (see vs. m in Fig 4F). Since the eradication of R is here driven by the strong bottlenecks at an average frequency (see “Background” in Model & Methods), scales as 1/s, i.e., . We have verified that these findings are robust since similar results are obtained for other percentiles and values of .

Together with the necessary requirements of (8), we thus obtain the following near-optimal conditions for the quasi-certain fluctuation-driven eradication of resistance from the metapopulation (for K₊/K_- fixed):

(9)

with after . Under these conditions, the probability of eradicating resistant cells from the metapopulation after is near optimal: For a migration rate (with the other parameters fixed and satisfying the conditions (9)), the probability of R eradication reaches a set value close to one (Fig 4F), with the fraction of R-free demes across the metapopulation thus approaching 1 in a time (see Fig E (panel H) in S1 Appendix). Remarkably, this means that, under the near-optimal conditions of (9), slow migration enhances the eradication of resistance compared to the non-spatial case (m = 0, Model & Methods). (Moreover, when , the near-optimal conditions of (9) extend to all m, see Derivation of the critical migration rate m_c above). The condition is shown as the region within the golden and green lines in Fig 4D and 4E, and corresponds to the near-optimal values found in Fig 4F. Interestingly, this range of migration rates are of the same order as those studied in Refs. [29,35], and are consistent with typical microfluidic experiments [107,126] (see Translation to the laboratory in the Discussion subsection “Future directions”). It is worth noting that the conditions of (8) and (9) are essentially independent of the spatial dimension of the metapopulation and hence the fluctuation-driven eradication of resistance is a phenomenon expected to hold on lattices of any dimension; see Impact of the spatial dimension in the Discussion subsection “Robustness of results across parameters and scenarios”, S1 Appendix Sec. 5.5, and Fig I in S1 Appendix.

We have also studied the probability of R eradication P(N_R(t)=0) as a function of and m for a range of slow, intermediate, and fast switching rates and different values of switching bias , confirming that R eradication occurs chiefly for (Fig J in S1 Appendix Sec. 5.6).

These results demonstrate that not only the fluctuation-driven eradication of the resistant strain R arise in the two-dimensional metapopulation under the conditions (8), but that slow migration () actually speeds up the clearance of resistance and it can even enhance the probability of R elimination (Discussion and S2 Movie and S1 Appendix Sec. 4), with the best conditions for the fluctuation-driven eradication of R given by the relations (9), and corresponding to the clearance of resistance from the grid being almost certain in a near-optimal time .

An intuitive explanation for why slow migration can promote the fluctuation-driven eradication of resistance is illustrated by Figs D and E in S1 Appendix Sec. 5.1: in the absence of migration, when R cells randomly take over a deme during periods in the harsh environment, with the low carrying capacity (blue in Fig E (panels A-D) in S1 Appendix; see also S1 Movie and S1 Appendix Sec. 4), resistance cannot be eradicated from that deme in isolation. However, slow migration allows for sensitive cells to recolonise that deme, from which it is then possible to clear resistance by means of the above fluctuation-driven eradication mechanism (Figs D (panel C) and E (panels E-H) in S1 Appendix).

Robustness of results across parameters and scenarios

Population size and microbial parameter values.

We have carried out extensive stochastic simulations of the ensuing metapopulation dynamics, and repeatedly tracked the simultaneous temporal evolution of up to a total of microbes distributed across L² = 400 spatial demes through hundreds of realisations and thousands of different combinations of environmental parameters and migration rates (Table 1 and S1 Appendix Sec. 3). This is a rather large metapopulation model, even though many experiments are carried out with even bigger microbial populations [41,128,129]. In addition, some in-vivo host-associated metapopulations are naturally fragmented into a limited number of small demes, e.g., and in mouse lymph nodes [22,79,130,131] and and in mouse intestine crypts [132], resulting in system sizes comparable to those considered here. Our results have been obtained by neglecting the occurrence of mutations, which is acceptable since R fluctuation-driven eradication typically occurs on a faster timescale (see Introduction and conditions (9)) compared to mutations [22,133,134]. We also note that the resistance mutation rate has been shown to decrease with the population density [133,134], suggesting that our results can hold in large bacterial populations. It is worth stressing that, according to the conditions of (8), fluctuation-driven eradication of R is expected whenever (and ), regardless of the population size and spatial dimension of the metapopulation (see Impact of the spatial dimension below, S1 Appendix Sec. 5.5, and Fig I in S1 Appendix). Remarkably, this condition is satisfied by values characterising realistic microbial communities, e.g., [26] (S1 Appendix Sec. 1.2.3). Furthermore, we note that the indicative values used in our examples for the extra metabolic cost of resistance (s = 0.1) and the biostatic impact of the antimicrobial drug (a = 0.25), are biologically plausible parameter values, with similar figures used in existing studies [26,135,136]. The values of s and a would typically decrease on a long evolutionary timescale due to compensatory mutations (typically after more than microbial generations in a low mutation regime [22]); however, they can be considered to remain constant on the shorter timescale considered here (a few hundred microbial generations). Additionally, results are robust against changes in values of s and a as long as [26] (“Background” in Model & Methods). Here we find mathematically convenient to represent the action of the drug as limiting the growth of S cells (bacteriostatic scenario) despite -lactams being typically bactericidal drugs (increasing cell death). This choice is acceptable in the regime of low drug concentration considered here, where bactericidal and bacteriostatic antibiotics have a similar action [137,138]. We note that higher drug concentration would increase the cooperation threshold N_th (more resistant cells needed to protect S individuals), thus increasing the chances that resistant cells spread and fix in many demes (see S1 Appendix Sec. 2).

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Table 1. Summary of simulation parameters for Figs 2-4 and S1-S5 Movies.

Parameters kept fixed are listed by a single value, other parameters are listed as ranges. The average number of sensitive cells S per deme at t = 0 () equals the metapopulation’s carrying capacity at t = 0 minus the constant threshold value for cooperation, , which depends on whether the system begins in a harsh or mild environment, (S1 Appendix Sec. 3.1). See Fig J (and Fig C) in S1 Appendix for the extended range in and , and S1 Appendix Sec. 5.5 for the discussion of results obtained on a periodic one-dimensional lattice (cycle) of length L = 100.

https://doi.org/10.1371/journal.pcbi.1013997.t001

We have explored a substantially broad range of values of migration rate m, spanning four orders of magnitude (), in addition to the benchmark case of no migration (m = 0) corresponding to a metapopulation of isolated demes. (Here, is effectively equivalent to m = 0; see Figs 2A, 4A, 4B, 4C, 4D, and 4E). Slow to moderate migration, , corresponds, on average, to the local migration of 0.001% to 10% of cells during each microbial generation, ranging from effective deme isolation to a significant mixing via dispersal. This wide range of migration rates is consistent with diverse experimental settings (see Translation to the laboratory in “Future directions”, below), from standard laboratory chemostats to microfluidic devices, e.g., [107,126], as well as with theoretical studies, e.g., [14,29,35]. Moreover, in line with general principles [39,96], we have shown that for a migration rate beyond the critical value m_c (fast migration) dispersal strengthens strain coexistence (Figs 2 and 4, and Figs B (panels B and C), C, D (panel D), and E (panels I-L) in S1 Appendix). In our examples, the near-optimal migration rate for the fluctuation-driven eradication of R is in the range (slow migration; see below), corresponding on average to one migrant per cells every generation. Overall, this study spans a broad range of migration rates, from isolated sites to fully connected demes, and identifies the parameter regimes under which fluctuation-driven eradication of resistance occurs (see “Critical migration rate” in Results).

Concordance of results with existing literature

On the one hand, most related studies that investigate the impact of fluctuating environments have focused on two-strain competition dynamics in well-mixed communities, either in the absence of public goods or when cooperative behaviour benefits both strains [51–55,144], including the case of time-varying nutrient and toxin concentrations [27]. The eco-evolutionary dynamics of cooperative AMR in well-mixed populations, under binary time-varying environmental conditions, has been studied in Refs. [26,28], that are directly relevant for this work and revealed when fluctuations can lead to the eradication of resistant cells.

On the other hand, the impact of spatial structure—such as star graphs and island models—has been typically studied in static environments under slow migration [18], with non-cooperative AMR considered in Ref. [22] and Ref. [15] experimentally investigating the spread of an antibiotic-resistant mutant through a star graph. Other studies have examined the dynamics of bacterial colonies of resistant and sensitive cells undergoing range expansion in constant environments [142,145,146]. Remarkably, recent metapopulation studies have investigated growth-and-dilution cycles coupled to fast cell migration [20], and competing wild-type and mutant cells subject to feast-and-famine cycles on metapopulation lattices [79]. However, metapopulation studies that include some form of environmental stochasticity have generally not addressed cooperative resistance. Instead, they typically focus on single-strain populations, island models, global migration, or spatially heterogeneous metapopulations [58–63].

Moreover, a substantial body of literature has focused on rescue dynamics, chiefly on demographic rescue in structured metapopulations [57], which occurs when a declining population is rescued from local extinction by an influx of individuals migrating from neighbouring demes – for instance, by restoring resistant cells in R-free demes after strong bottlenecks (Fig 3). Interestingly, Ref. [14] reported that intermediate migration rates (similar to our slow/moderate regime) maximise species persistence time in paired batch cultures undergoing growth–migration–dilution cycles. This results from recolonisation events following local extinctions and is aligned with findings from other computational studies of non-cooperative rescue dynamics [65–68] and with experimental observations [69–73]. Here, we have found that slow-but-nonzero migration rates – of similar magnitude to those in Ref. [14] – enhance the fluctuation-driven eradication of resistance (Fig 4 and Figs D and E in S1 Appendix Sec. 5.1), rather than rescuing resistance. These results are however compatible with those of Ref. [14]. As discussed in “Slow migration” (Results), here slow migration promotes S-cell recolonisation of R-only demes – analogous to the rescue dynamics of Ref. [14] – but subsequently renders these demes prone to fluctuation-driven eradication of resistance when the environment switches from mild to harsh conditions (Figs D and E in S1 Appendix Sec. 5.1). Note that the fluctuation-driven eradication mechanism does not arise in Ref. [14], because the latter involves two mutually cooperative strains resistant to two drugs, and considers growth–dilution cycles rather than the binary environmental switching studied here.

The fluctuation-driven eradication mechanism was unveiled in Ref. [26]. While it relies on biologically realistic assumptions (see above), some of these do not correspond to commonly used laboratory setups, which explains why this phenomenon has not yet been tested experimentally. Specifically, most experiments with environmental bottlenecks involve serial dilution protocols, where each cycle consists of a period of exponential growth followed by an instantaneous dilution step [15,147] (see Translation to the laboratory in “Future directions”, below). Such systems generally do not exhibit fluctuation-driven eradication of R, since this phenomenon typically requires the population to spend finite periods in the harsh environment (see “Background” in Model & Methods and [26]). This also applies to studies of microbial cooperation [56], including those investigating the effects of “dilution shocks” [13] or “disturbance events” [87], as well as cooperative antimicrobial resistance [14]. Moreover, as we discuss below in Translation to the laboratory, there is currently no clear correspondence between chemostat-inspired models and those based on serial dilution experiments. Although recent chemostat experiments have analysed the effects of environmental fluctuations in well-mixed populations switching between different nutrient sources [25], these did not consider cooperative resistance and therefore did not display fluctuation-driven eradication. Similar environmental fluctuations to those considered here have also been implemented in metapopulation microfluidic experiments, e.g., in Ref. [93] for phenotypic switching in a single strain.

Remarkably, the authors of Ref. [20] investigated a spatially structured (non-cooperative) two-strain model consisting of subpopulations connected by migration and subject to bottlenecks arising from growth–dilution cycles. They found that slow migration can amplify selection for the fittest strain, whereas fast migration tends to suppress selection. However, the lack of cooperation and the serial dilution dynamics of the system considered in Ref. [20] (inspired by batch culture setups) cannot give rise to fluctuation-driven eradication. It is worth noting that the authors of Ref. [22] studied resistance rescue in a metapopulation of sensitive cells and non-cooperative drug-resistant mutants, showing that spatial structure can facilitate the survival of resistance. Since the model of Ref. [22] is non-cooperative and subject to a single environmental change (when a biostatic drug is added), it does not exhibit fluctuation-driven resistance eradication.

Cooperative resistance in spatially structured metapopulations has also been studied experimentally, for instance in the range expansion experiments of Refs. [142,145,146], which were performed under constant environmental conditions. As discussed above, these are conditions under which fluctuation-driven resistance eradication cannot occur (S1 Appendix Sec. 2 and Fig B in S1 Appendix). Moreover, as detailed above, Ref. [14] investigated cooperative AMR in a spatial setup where no fluctuation-driven eradication is expected. We also note that Ref. [21] focuses on cooperative resistance rescue under environmental fragmentation into random, independent subpopulations (without migration). While this model cannot exhibit fluctuation-driven eradication (as it involves a single strain), the authors of Ref. [21] show that habitat fragmentation enhances resistance rescue.

To the best of our knowledge, no previous study has examined the optimal conditions for eradicating cooperative drug-resistant cells in a stochastic metapopulation composed of sensitive and resistant strains, where demes are connected by local migration and subject to global feast–famine cycles. Here, we present the first metapopulation study showing that slow-but-nonzero migration helps eradicate cooperative AMR in time-fluctuating environments. This is in stark contrast – yet fully consistent (see above) – with previous results showing that slow migration, in environmental conditions different from those considered here, helps maintain cooperative AMR [14] and non-cooperative strain competition [65–73].

Future directions

Code availability

The C++ code used to generate the data and the Python and Matlab codes to process and visualize the data within this work can be found at the Open Science Framework repository (Lluís Hernández-Navarro, Kenneth Distefano, Uwe C. Täuber, and Mauro Mobilia. 2024. Supplementary data, code, and movies for “Slow spatial migration can help eradicate cooperative antimicrobial resistance in time-varying environments.” OSF. https://doi.org/10.17605/OSF.IO/EPB28).