Sustainability of Virulence in a Phage-Bacterial Ecosystem

Abstract

Virulent phages and their bacterial hosts represent an unusual sort of predator-prey system where each time a prey is eaten, hundreds of new predators are born. It is puzzling how, despite the apparent effectiveness of the phage predators, they manage to avoid driving their bacterial prey to extinction. Here we consider a phage-bacterial ecosystem on a two-dimensional (2-d) surface and show that homogeneous space in itself enhances coexistence. We analyze different behavioral mechanisms that can facilitate coexistence in a spatial environment. For example, we find that when the latent times of the phage are allowed to evolve, selection favors “mediocre killers,” since voracious phage rapidly deplete local resources and go extinct. Our model system thus emphasizes the differences between short-term proliferation and long-term ecosystem sustainability.

The replication strategies of phages fall into two major categories: virulent and temperate. A temperate phage has the ability to integrate its DNA into the host chromosome, where it is then replicated along with the bacterial DNA during cell division. This strategy allows the phage to slow down or completely stop exploitation of the bacteria, thus reducing the risk of driving its host to extinction. A virulent phage lacks this ability, and it is not fully understood how they manage to coexist with their bacterial prey (4, 19). Consider, for example, the highly effective T4 phage. For the sake of argument, let us assume a burst size of 100 offspring upon lysis. On average, not more than a single phage out of each burst of 100 should survive to infect another bacterium, or else the phage would rapidly outgrow the bacteria and drive them to extinction. The half-life (t_1/2) of a free T4 phage particle has been measured to be approximately 10 days in LB at 37°C (6). Therefore, on average, at least t_1/2 × log₂(100) ≈ 2 months should pass between infections to prevent runaway phage growth—a time span that seems highly unreasonable for many of the environments where phage and bacteria interact, such as soil or biofilm. Even a more considered calculation, inserting the above half-life measurement into more realistic Lotka-Volterra-like predator-prey models (9) does not change the conclusion that T4 and other virulent phages appear to be far too effective predators for coexistence to be feasible. It is, however, an undisputed fact that virulent phages and bacteria have coexisted for eons and do so still, everywhere around us and inside us. One possible explanation for this puzzle is that bacteria constantly evolve resistance to existing phages and that the phages evolve to attack resistant bacteria in a continuous arms race. This “Red Queen” argument (23) has, however, been criticized on the grounds that the rates of evolution of phages and bacteria are not symmetric (17, 12). Recent measurements support this: in soil, phages appear to be “ahead of the bacteria in the coevolutionary arms race” (24). We therefore wish to explore mechanisms other than bacterial resistance that may promote coexistence between virulent phages and bacteria.

Historically, phage-bacterial ecosystem models have ignored the issue of space, utilizing zero-dimensional approaches, such as ordinary differential equations (e.g., see references 1, 5, 13, 14, 15, and 21). However, many real phage-bacterial ecosystems are found in environments with a complex spatial structure, such as soil, biofilms, or wounds in animal and plant tissue. Schrag and Mittler (20) showed that coexistence between virulent phage and bacteria is feasible in a chemostat but not in serial cultures, due to the heterogeneous nature of the environment in the chemostat. Further, experiments done by Brockhurst et al. (3) indicate that reduced phage dispersal can prolong coexistence for virulent phage and bacteria in spatial environments by creating ephemeral refuges for the bacteria. Kerr et al. (10) introduced a simple cellular automaton to model fragmented populations of phage and bacteria in which coexistence was more easily achieved when migration was spatially restricted. Thus, the main extension to the simple predator-prey framework that we examine will be to add a spatial dimension.

We construct and compare two phage-bacterial ecosystem models: one model where the phage and bacteria exist in a two-dimensional space, such as the surface of an agar gel (referred to as the “spatial model”), and the other model where the phage and bacteria are repeatedly mixed, mimicking serial cultures or a well-mixed broth (referred to as the “well-mixed model”). We show that space does indeed enhance coexistence. We then move on to explore other mechanisms that phage could incorporate into their behavior to further enhance coexistence. These can broadly be classified as “hardwired” (where every phage follows the same deterministic strategy) versus “adaptive” (where each phage potentially behaves differently, thus allowing the population to explore different options).

We have chosen to look at three specific mechanisms as examples of these categories: (i) phage effectiveness would be reduced if they were unable to register whether they were infecting live, infected, or dead bacteria (a hardwired behavior); (ii) phage could prolong their latent time, concurrently increasing burst size, depending on the number of multiple infections (also a hardwired behavior, but a more “active” sort, where each phage senses and responds to information from the environment; T4 is known to use such a lysis inhibition strategy), and (iii) phage offspring could have altered latent times due to mutations in the holin genes (an adaptive behavior). We will compare each of these mechanisms in the spatial and well-mixed models to investigate whether the heterogeneity possible in a spatial environment affects the outcome.

METHODS

Rules of the spatial model.

In the spatial model, virulent phage and bacteria interact on an L × L grid of locations, or “sites.” Each site in the grid can either be empty or occupied by a single bacterium (each grid site thus has a carrying capacity of one bacterium). The bacterium may be healthy or infected. In addition, there can be any number of free phage particles at that site. Time proceeds in discrete steps. Precise timers control bacterial cell division and the lysis of an infected bacterium, which releases a burst of free phage. Other processes are random, e.g., death and diffusion of phage, and are modeled as Poisson processes.

In each time step, the following can happen.

(i) Bacterial replication.

A bacterium with at least one empty adjacent site will attempt to divide in every time step after the current time has become greater than the value of its replication timer. The probability of replication is set to be proportional to the number of empty neighbor sites. Once a bacterium divides, one daughter cell remains in the original site, and the other is placed randomly in one of the adjacent empty sites. The replication timers of both cells are reset to the current time plus replication time (T), a parameter which thus sets the growth rate of the bacteria.

(ii) Bacterial infection.

A healthy bacterium that shares its site with some free phage may be infected with a probability p_α, that depends on the number of phage at the site, the infection rate per phage per bacterium (α), and the decay rate of the phage (δ). The number of free phage at that site is then reduced by one, and the lysis timer of the newly infected bacterium is set to τ (the latent time of the infecting phage) and starts counting down from that value.

(iii) Bacterial lysis.

An infected bacterium will die when its lysis timer has counted down to zero. The number of phage at that site increases, upon lysis, by the burst size (β).

(iv) Phage decay.

Free phage die with a probability p_δ per phage, which depends on the phage decay rate (δ).

(v) Phage diffusion.

Each free phage may jump to a neighboring site with a probability p_λ, which sets the phage diffusion constant.

The burst size increases with latent time: β = γ(τ − ɛ). This formula models the constant rate of replication (γ) of phage, after a minimum preparatory time (ɛ), usually referred to as the eclipse time. The values of the parameters and the size of the basic time step depend on the choice of phage and bacterial species. With Escherichia coli, a reasonable choice is a time step of 1 min, a replication time (Τ) of 30 min (i.e., 30 time steps), and an area of 1 μm² per grid site.

W chose p_λ so as to keep the phage diffusion constant (D) fixed at D ≈ 1/4 (site area)/(time step), meaning it would take on average 104 time steps for a phage to move across the grid size of L = 100 that we use (too large a diffusion constant would make the system well mixed, negating the purpose of our study). With T4 in mind, we will fix ɛ at 10 time steps and γ at 7 (time step)⁻¹, resulting in (nonzero) burst sizes ranging from 7 (at τ = 11 time steps) to 280 phage (at τ = 50 time steps). However, for the basic spatial model, τ is fixed at 30 time steps. For the phage infection rate, α, we will explore a range of values, between 0.0001 and 5 (site area)/(time step). For T4, for example, an infection rate of α = 5 μm² min⁻¹ means that a phage closer than 1 μm to an E. coli bacterium would infect, on average, within 12 s, which is fast but not unrealistic. With this range of α values, we need δ to be up to 5 (time step)⁻¹ to see coexistence, as explained in the Results section below. p_α and p_δ are calculated from the values of δ and α by assuming the processes to be Poissonian random processes (see the supplemental material for further details of model rules for bacterial replication, infection, and lysis and for derivation of the probabilities p_α, p_δ, and p_λ).

Rules of the well-mixed model.

The well-mixed model is very similar to the spatial model, except that (i) upon bacterial cell division, newborn bacteria are placed at random empty grid sites, and (ii) newborn phages, released when an infected bacterium is lysed, are randomly placed all over the grid. This results in continuous mixing of the phage and bacteria populations while at the same time ensuring that the two models are as similar as possible to allow for straightforward comparison.

RESULTS

Quantifying coexistence.

The color map in Fig. 1 shows the average steady-state bacterial density per grid site, B, for simulations of the spatial model with various combinations of δ (degradation rate of phage) and α (infection rate per phage per bacterium). In the deep-red region to the right in Fig. 1, the phage are so inefficient that they die out and the bacteria subsequently grow to carrying capacity. In the deep-blue region on the left, the phage are so efficient that they drive the bacteria to extinction and then die out themselves. In the middle region, where 0 < B < 1, the bacteria and phage coexist stably. The size of this region in the δ-α parameter plane quantifies how easily coexistence is achieved in the models we examine, since δ and α are the parameters that determine the overall predatory effectiveness of the phage. It is interesting to note that coexistence requires much higher values of δ (1 to 4 min⁻¹ for, say T4) than has been measured in laboratory conditions. This suggests that the effective death rate for phage may be much higher in real ecosystems than in the laboratory.

FIG. 1.

(Top) Coexistence region in the δ-α plane. δ is the degradation rate of the phage, and α is the infection rate for a phage that occupies the same lattice site as a bacterium (see Methods). The color map shows the average bacterial density in steady state. There is coexistence only in the zone in the middle, where phage are neither too effective nor too ineffective. The white dashed line shows a theoretical estimate for the threshold at which exactly one phage per burst survives long enough to find and infect a bacterium (see the supplemental material). The jaggedness of the boundaries, in this and subsequent plots, arises because only a single simulation was done for each δ-α pair. Doing more simulations does not significantly alter the position and shape of the coexistence region. (Bottom) Snapshots of the ecosystem at the points marked A to D in the top panel. Healthy bacteria, infected bacteria, and dead bacteria are shown. Phage are not shown.

The typical dynamics of the spatial model involve one or more bacterial colonies that grow at a rate determined by their replication time. These colonies are invaded by phage that move in traveling infection fronts that sweep through the colonies. The speed of the infection front depends on the effectiveness of the phage, i.e., α and δ. If the phage die too quickly or infect very inefficiently, they die out. Conversely, if the phage live a long time or infect quickly, then the infection front may propagate even faster than the bacterial growth front. Within the coexistence region, there is considerable variation in the dynamics of the ecosystem, as shown in the four snapshots in Fig. 1. At point A, right at the edge of the coexistence region, the phage infection front in fact travels faster than the bacterial growth front. Nevertheless, there is coexistence because the infection fronts leave behind healthy bacteria often enough to keep the bacterial population from going extinct. However, at point A, there is considerable variation in bacterial density with time, as the bacteria typically form a small number of big colonies which are then decimated by the fast-moving infection fronts. Increasing δ or decreasing α from point A moves the system deeper into the coexistence region to points B and C, respectively, where there is a higher average bacterial density. Point B, in stark contrast to point A, is characterized by many small intermixed domains of bacteria and phage, and their populations are quite stable with relatively small fluctuations. At point C, bacteria survive a passing infection front more often than at point A (because of the lower infection rate α), and therefore, the bacterial domains are smaller and more dispersed than at point A. Qualitatively similar patterns and dynamics are observed as one moves along isocolor lines (i.e., lines of constant bacterial density) to lower α and δ values. Thus, the dynamics at point D are very similar to the dynamics at point C. At very small δ values (δ ≤ 10⁻⁴), however, the system starts behaving like a well-mixed ecosystem because the phage are able to diffuse across the entire grid before either dying or infecting.

Space helps coexistence.

Figure 2 compares the coexistence regions for the spatial and well-mixed models, keeping all parameters other than δ and α fixed at their default values. The coexistence region is approximately 20% smaller for the well-mixed model than for the spatial model. The right boundary of the coexistence region coincides for both models and is situated where the time between infections is so long that on average only one phage per burst survives. The left boundary, however, is situated further to the left for the spatial model than for the well-mixed model, meaning that when there is space, the bacteria can coexist with far more effective phage than in the well-mixed model. In the well-mixed model, the left boundary corresponds, in fact, to the onset of high-amplitude oscillations in the populations. These oscillations cause the bacterial numbers to periodically fall to extremely low levels. When this happens, the few bacteria left have a finite probability of all of the bacteria becoming infected before they divide, so that, sooner or later, the bacteria go extinct. For the same parameter values, the spatial model shows damped or low-amplitude oscillations and therefore coexistence.

FIG. 2.

Space helps coexistence. (Top) Coexistence regions for the spatial and well-mixed models plotted on top of each other. In the white region, there is coexistence only in the spatial model. In the gray region, there is coexistence in both models. The area of the gray region is around 20% smaller than the area of the white region. 2D space, two-dimensional space. (Bottom) The green bars and curves show the total number of free phage in the spatial and well-mixed models as a function of time for the parameters corresponding to the point marked E in the top panel. The population quickly settles at a stable level, with some fluctuations, for the spatial model. In contrast, the well-mixed model exhibits oscillations with increasing amplitude that eventually drive the bacterial population, and subsequently the phage, to extinction.

Behavioral mechanism that enhances coexistence. (i) Hardwired phage behavior.

Figure 3 shows the coexistence regions when two hardwired mechanisms are implemented in the spatial model. Both impede phage infection and dispersal, but in different ways. First, the top panel in Fig. 3 shows what happens if phage simply cannot distinguish between healthy and infected/dead bacteria—they infect whatever they come into contact with, and when that is a dead or previously infected bacterium, the phage dies. (We extended the spatial model to keep dead bacteria around for a certain characteristic time, before the site holding them becomes empty [see the supplemental material].) Traditionally, phage-bacterial models ignore the interaction of phage with dead and infected bacteria (16, 13, 21, 5). It has, however, been proposed that the build up of bacterial debris could hinder phage diffusion, protect live bacteria, and enhance coexistence (1, 17). This is indeed the effect we see in the top panel in Fig. 3 at the left boundary of the coexistence region. In contrast, the right boundary is unaffected because here the phage population is relatively low, on the verge of extinction, while the bacterial population is very close to the carrying capacity so infection of previously infected or dead bacteria is rare.

FIG. 3.

Effects of two hardwired phage strategies on the coexistence region. (Top) Phage infect live, dead, and infected bacteria alike. (Bottom) Multiple infections of the same bacterial cell result in delayed lysis. In both plots, the white region corresponds to parameters where there is coexistence in the spatial model both with and without the phage strategies, while the gray region shows where there is coexistence only when the corresponding phage strategy is implemented.

The bottom panel in Figure 3 shows the effect of a more “active” strategy, where the phage can detect multiple infections and delay lysis. T4 is known to use such lysis inhibition (2, 7). Through a mechanism involving the anti-holin rI, T4 delays lysis by 5 to 10 min whenever the cell becomes superinfected with other T4 phage (Ryland Young, personal communication). (We implement this in the spatial model by allowing phage to infect already infected cells. Whenever this happens, lysis is postponed by 8 time steps. However, we set an upper limit of 200 time steps beyond which lysis cannot be postponed. This gives a maximum burst size of 1,330 phage, which approximately corresponds to the phage production allowed by the resources available in a single bacterium [7].) This mechanism also boosts coexistence, as shown in the bottom panel in Fig. 3. Again, the right boundary is unaffected because superinfections are rare here.

The behavior of infecting dead and infected bacteria effectively increases δ for the phage, whereas delaying lysis upon superinfection effectively decreases α (by reducing burst size per phage). Either way, the result is to shift the boundary of the coexistence region to the left compared to the basic spatial model.

(ii) Adaptive phage behavior.

Another strategy we explored was to allow the latent times of the phage to mutate. The phage proteins that cause lysis, holins, control the time of lysis with very high precision (± 1 min), and point mutations within the holin gene can significantly alter the lysis time without changing the precision (26). We allowed a small fraction of phage progeny to mutate to have a different latent time (and therefore also a different burst size) from the parent phage. (The latent times of 0.5% of the phage from each burst are chosen randomly and uniformly from the range 0 to 50 time steps. The other 99.5% inherit the same latent time as the parent phage. Additionally, 0.5% of the bursts are comprised entirely of latent time mutants [see the supplemental material for the biological reasoning behind these rules].) The top panel in Fig. 4 shows that implementing this adaptive mechanism enhances coexistence in the spatial model.

FIG. 4.

Effect of an adaptive phage strategy where latent times are allowed to mutate. (Top) The white region corresponds to parameters where there is coexistence in the spatial model both with and without latent time mutability, while the gray region shows where there is coexistence only when latent time mutability is implemented. (Bottom) The gray region corresponds to parameters where there is coexistence in the well-mixed model both with and without latent time mutability, while the white region shows where there is coexistence only when latent time mutability is not implemented.

It is not intuitively obvious why this strategy helps. Consider that if the bacterial density is kept fixed, the phage will evolve to all have the same “optimal” latent time that maximizes the rate with which they spread in that density (see the supplemental material, where we show how the optimal latent time depends on the bacterial density; also see reference 9). At the maximum bacterial density of one per site, which is what an infection front typically encounters in the spatial model, the infection front of the optimal phage actually moves faster than the growth front of the bacterial colony.

One then wonders why the host population is not wiped out by the appearance of these optimal “efficient killers,” resulting in an overall reduction of coexistence compared to the standard spatial model.

The reason this does not happen is that when an “optimal” phage mutant arises in a colony, it quickly wipes it out and subsequently goes extinct if it cannot quickly find another colony nearby to infect. Thus, when the bacterial population is split into many small colonies, there is effective selection against very efficient phage. In turn, the very existence of phage with different latent times makes the system self-organize to have a larger number of small bacterial colonies compared to the basic spatial model, as shown in Fig. 5d.

FIG. 5.

Close-up views of ecosystem snapshots of the spatial model for parameters corresponding to point A in Fig. 1. (a) Basic spatial model (phage latent time is fixed at τ = 30). (b) Phage infect live, dead, and infected bacteria alike (phage latent time is fixed at τ = 30). (c) Basic spatial model with delayed lysis upon superinfection (phage latent time is τ = 30 but increases with 8 time steps upon each super infection). (d) Basic spatial model with latent time mutability (phage latent time τ can mutate to any value in the range 11 to 50 time steps). Light red/orange cells are bacteria infected by phage with shorter latent times, while dark red/brown cells are bacteria infected by phage with long latent times. Free phage are not shown.

In contrast, in the well-mixed model, overefficient phage that arise have access to the entire bacterial population, so there is no negative selection to restrain them. This, along with the increased oscillations we observe when implementing the adaptive strategy in the well-mixed model, makes coexistence harder to achieve than in the absence of latent time mutability (Fig. 4, bottom panel).

DISCUSSION

Spatial heterogeneity boosts coexistence.

The comparison between the spatial and well-mixed models shows that space boosts coexistence—even uniform two-dimensional space, without any built-in heterogeneities, such as permanent bacterial refuges. Spatial heterogeneity arises spontaneously as a result of the dynamic interaction between the bacterial growth front and the propagating phage infection front and is crucial for enhancing coexistence. In the well-mixed model, which lacks this heterogeneity, the infection and burst events are more prone to happen in sync for the whole system, often resulting in large-amplitude oscillations that destroy coexistence. In the spatial model, each small bacterial colony might experience oscillations or big population fluctuations, but on a larger spatial scale, these average out because the life cycles of the phage attacking separate colonies quickly become desynchronized and uncorrelated.

Looking at Fig. 1, moving from point A deeper into the coexistence region, to point B (by increasing δ) or point C (by decreasing α), results in more heterogeneity in a snapshot of the system. When phage infect dead or previously infected bacteria, their δ is effectively increased, and when phage delay lysis upon superinfection, their α is effectively decreased. Thus, one would expect both behavioral mechanisms to increase heterogeneity compared to the basic spatial model. This is exactly what we see in Fig. 5, which shows snapshots of the ecosystem for the basic spatial model and the different strategies, for the same parameter values.

That shielding by dead bacteria enhances coexistence has been observed before in models that lack space (1, 17). However, in these models, to see a significant effect, the dead bacteria must remain in the system for quite long times. In our spatial model, the enhancement of coexistence is much more dramatic. Even when the degradation rate of the dead bacteria is such that we cannot see any enhancement of coexistence in the well-mixed model (see the supplemental material), we still see a distinct enhancement in the spatial model. This is because the free phage and dead bacteria are typically colocalized here—both are “created” by the same events.

The mechanism of lysis inhibition also works in slightly different ways in the spatial and well-mixed models. It has been previously argued that this mechanism could enhance coexistence in the following way: the original infecting phage interpret superinfection as a sign that phage outnumber host cells in the external environment (18), whereupon delaying lysis gives the few bacteria left alive out there an additional chance to reproduce, thereby reducing the risk of driving them to extinction (22). This reasoning breaks down in the well-mixed case because lysis inhibition also creates ticking “time bombs”; multiply superinfected bacteria that release a huge number of phage when they eventually burst, which counteract the effect of allowing bacteria more time to replicate. In the spatial model, however, these time bombs are typically left behind by the moving infection front, so when they do lyse and release a huge number of phage, these phage are generally relatively far from susceptible bacteria. (We observe some enhancement of coexistence in the well-mixed model also when lysis inhibition is implemented [see the supplemental material], which occurs because the strategy of delaying lysis desynchronizes burst events and therefore dampens oscillations.)

Survival of the mediocre killers.

One of the most interesting aspects of the adaptive strategy in a spatial setting is that it exhibits selection against the most efficient killers since these deplete resources locally and subsequently die out. This part of our study thus emphasizes that one must be careful in assessing what is “optimal” behavior for a phage. Calculations that try to determine optimal latent times, for instance, often take the short-term view of maximizing the phage population growth rate (25, 27). Recognizing the risks of making assumptions of this kind has led others to suggest extending the notion of fitness to include “environmental inheritance” (8). Our study supports this point of view: for long-term survival in a spatial environment, virulent phage must ensure that their offspring inherit an environment with sufficient resources. Space promotes survival of mediocre killers.