We have developed an experimental model system and a complementary mathematical model to explore the impacts of spatial and temporal factors on competition between parasites on a single host. The empirical system consists of two bacteriophages,
X174 and α
3, competing on their common bacterial host, Escherichia coli
C. The mathematical model is an interacting particle system that captures stochastic spatial dynamics down to the level of individual cells. Life-history parameters would predict that
X174 should rapidly exclude α
3 under our experimental conditions, and yet α
3 persists in some experiments, especially those with long incubation times at each passage. We hypothesize that coexistence is the result of evolution of the host species. During each passage, host cells arise that are resistant to phage; in particular,
3-sensitive cells provide a population of hosts that can be infected by the “weaker” phage and that may thereby provide a refuge. Consistent with this, coexistence occurred under conditions that promote the evolution and spread of resistant host cells (long incubation times) but not under conditions in which the spread of resistant host cells are limited (short incubation times). In further support of our hypothesis, addition of a small fraction of
3-sensitive host cells at each passage allows coexistence of the competing phage in the short-incubation treatments. There may, of course, be other factors contributing to the coexistence of
X174 and α
3 in long-incubation experiments. For example, if some α
3-infected cells were to significantly delay lysis, their progeny might appear and be transferred only if incubation periods were sufficiently long.
In simulations, resistant cells were able to form “percolating clusters” in long-incubation, structured-transfer treatments, thus providing a spatial refuge for α
3 in otherwise
X174-dominated territory. This effect is even more striking if one runs the simulations for various values of incubation time. There seems to be a critical value for incubation time; for incubation times below this value, α
3 becomes extinct; for incubation times above this value, α
3 persists (see ). Simulations that allowed for the appearance of mutant cells that are α
X174-sensitive in addition to the usual
3-sensitive cells did not have significantly different outcomes. This was not surprising, because the new mutants would benefit only
X174, which already dominated the spatial landscape. Changing the model to include only α
X174-sensitive mutant cells or to include no mutant cells at all always led to the extinction of α
3, even in long-incubation simulations.
Figure 6 Effect of incubation time on persistence of α3 in structured-transfer simulations. Short incubation times led to the extinction of α3; sufficiently long incubation times led to the coexistence of α3 and X174.
Limitations and Strengths of the Mathematical Model
Obviously, as with any model, there are numerous departures from reality. Nevertheless, the mathematical model employed here is quite complex, with much of this complexity a consequence of accounting for rudimentary spatial structure. The goal is to see whether the model can capture the behavior of competing phages in the experimental system, to help explain the mechanisms responsible for observed behavior, and to suggest directions for future experimental exploration.
In an effort to keep the model relatively simple and the simulation times reasonable, certain properties of the phage/bacteria experimental system were not addressed. For example, plate cultures allow for three-dimensional growth of cells and phage, whereas our mathematical model is based on a flat, two-dimensional lattice. This means, for example, that cell growth cannot proceed past the point at which the lattice fills up, and this happens much more quickly than the time it takes to reach the stationary phase on agar. We have accounted for some of this effect by assuming that cells remain susceptible even after growth has stopped. A full three-dimensional model would require a drastic increase in simulation time.
Despite the fact that our 600 × 600 lattice is considerably larger than most interacting particle system simulations in the literature to date, the spatial scale for simulations is still smaller than it is for experiments. Assuming cell sizes of 1–2 μm and intercellular distances of 1–10 μm, our 600 × 600 lattice translates to a square region of just a few millimeters per side. The area of the petri plates, on the other hand, is 9 cm per side. Another discrepancy between the simulations and the experiments is in the transferred samples. In the experiments, the picker prongs were about 1 mm across and about 3 mm apart. In the model, we chose to keep the sample area proportional to the size of the plate rather than proportional to the cell size. (In the experiments, the prong-to-plate ratio = 1 mm/90 mm ≈ 0.01; in the mathematical model, the sample-to-grid area = 5 sites/600 sites ≈ 0.01). Thus, in the simulations, the transferred samples used a much finer scale for prong size and between-prong distance than in the experimental system.
All of this reinforces the point that one should not think of the mathematical model as an exact replica of the experimental system. The model seeks to incorporate essential features of the real system in an effort to understand the mechanisms that determine important outcomes. The difference of scale, together with the simplifying assumptions of the model and the fact that certain experimental parameters are difficult to measure, means that the findings of the mathematical model should be considered to be more qualitative than quantitative. The goal is not to perfectly capture the dynamics of competition between
X174 and α
3 on a given bacterial host and in a specific experimental system but rather to discover mechanisms that are robust enough to play significant roles in competitions between a variety of viruses or even more general classes of competitors.
Despite the aforementioned limitations, our model captures stochastic and individual details at a microscopic level that differential equation models can only treat in bulk averages. For example, the simulations allowed us to test hypotheses about life-history parameters in determining phage dominance. In addition, we were able to test the hypothesis that coexistence was due to the spread of mutant host cells resistant to the dominant phage, and the simulations allowed us to see how such a mechanism could work in great detail. Unlike the experiments, where observing what is present and inferring mechanisms is fraught with difficulty, the simulations can be observed with absolute clarity, broad ranges of parameters that would be impossible to explore in most experimental situations can be tested, and mechanisms can be controlled and manipulated.
Exploring Experimental Features Not Captured by the Simulations
Although the simulations captured many features of the experimental data, doing so required ongoing input from the experimental system. For example, early simulations suggested that
X174 would outcompete α
3 under our experimental conditions. The observation in the experimental system that the length of incubation affected the outcome of competition led to the hypothesis that host evolution could explain the difference between long and short incubation times. Demonstration that most
X174-resistant phage were α
3 sensitive supported this hypothesis. This is a case where the experiments informed the simulations; including the evolution of host resistance in the simulations led to a more accurate tracking of experimental results. But, although this hypothesis could be tested experimentally by the addition of resistant hosts to the structured competitions, it could be more completely explored in the mathematical model. Simulations of the effects of increasing incubation time on α
3 persistence suggested a transition between α
3 loss and persistence (); as incubation time increased, the extinction time of α
3 increased and eventually extinction was averted. This result could now be explored experimentally.
Another feature of the experiments that was not captured by the simulations was the large oscillations in the frequencies of α3. These oscillations were more pronounced in long-term than in short-term incubations and were especially pronounced in the mixed transfers. We consider three factors that might explain these oscillations: methods of genotyping, host evolution, and phage evolution. Of course, these explanations are not mutually exclusive. Other possibilities include density-dependent phage or host behaviors that were not modeled in this study.
Genotyping Effects on Phage Oscillations
In the experiments, genotyping was performed by replica plating phage on assay hosts with the multiprong picker and scoring how many prongs showed growth on each such host. The assay assessed the presence or absence of phage at each site but did not measure the number of phage. Also, observations on the assay host required time for host and phage growth, and during that time, phage were able to spread to areas inoculated by adjacent prongs on the picker. Unless the strain covered the entire plate, this resulted in an overestimate of the area occupied by each type. In simulations, we can view the spatial distribution of phages with complete clarity immediately after each passage, and thus there is no corresponding overestimate of phage distribution. In the simulations, we can also sample the phage as with a picker and can even vary the area covered by a prong and the distance between prongs. Thus, we have three measures of phage density: the experimental method of “genotyping” and two simulation methods, “exact counting,” which is the actual frequency of phage on the plate after each passage, and “picker counting,” which mimics genotyping. In picker counting, each prong is scored as a 1 or a 0 for the presence or absence of each phage strain, but the transfer to the next passage maintains the actual count and configuration of phage.
Comparing these types of sampling in simulations showed that picker counting had the effect of accentuating the natural oscillations of phage frequencies due to a sampling effect (cf. with ). Enlarging picker prongs and spacing them farther apart in the simulations (which more closely resembled the experimental conditions) resulted in larger oscillations that were more consistent with what was observed in the experiments (). In –, simulation frequencies are determined by picker counting of the sample transferred to the next passage. Although this introduces sampling bias, it more accurately reflects the experimental design.
Another effect of picker counting was a kind of “genotyping bloat” in which the spatial coexistence region appears enlarged because the large phage sample on each prong can lead to a classification of coexistence even if one phage type greatly outnumbers the other; smaller picker prongs lead to smaller coexistence zones (). This effect is even more pronounced with genotyping in the experimental system because phage grow out from each prong. We were able to capture this effect in simulations and thus increase the width of the coexistence regions by allowing for extra spread during picker counting (data not shown). The effects of prong size and distribution on oscillations observed in global densities and on genotyping bloat are examples of experimental artifacts that can be readily explored with appropriate simulation models. Nevertheless, α3 oscillations were more pronounced in the experimental data than in the simulations and were more pronounced with mixed than with structured transfer, suggesting that some factor not included in the simulations contributes to these oscillations.
Host Evolution Effects on Phage Oscillations
The role of resistant cells is also consistent with the observed oscillations in α
3 frequency in the experiments. When α
3 frequencies are low, the maintenance of spatial structure at transfer means that there will be large contiguous regions of the plate containing no α
3 at the start of the next incubation period.
3-sensitive cells appearing in these regions are safe from α
3 infection (and thus early death) during the early stages of the incubation and can reproduce, forming large population clusters. If, later in the incubation period, α
3 comes into contact with a cluster of
X174-resistant cells—for example, if such a cell cluster grows into an area already occupied by α
3 will be able to spread throughout this large population of α
3-sensitive host cells. Of course, these regions likely also contain
X174 from transfers and replication in sensitive E. coli
C, resulting in an area of coexistence. In subsequent transfers, the presence of α
3 in these areas will suppress the expansion of the
3-sensitive clusters, and
X174 will outcompete α
3 in sensitive E. coli
C, resulting in oscillations of α
3 frequency. This process will be most pronounced in long-incubation, structured-transfer experiments, where resistant cells have more opportunity to replicate. Thus, the combination of spatial structure and long incubation time provides a spatial refuge for the weak competitor, allowing coexistence and contributing to oscillations in α
3 frequency. In fact, α
3 oscillations were more pronounced in long-incubation experiments.
Phage Evolution Effects on Phage Oscillations
In this experimental system, it is not only the hosts that are subject to rapid evolution; the phage also evolve in response to the novel host environment, including competition within and between strains. In fact, one expects phage evolution to be more pronounced than host evolution because the resistant hosts are killed at the end of each passage and replaced with naive hosts. The phage, on the other hand, can continue to evolve throughout the experiment. In fact, α3 evolution was evident on the genotyping plates by changes in plaque morphology in some areas on all structured replicates. (The evolution of α3 was confirmed by sequencing but was not pursued for this study.)
Phage evolution would be expected to contribute the most to oscillations in the mixed-transfer experiments because any competitive phage variant arising on the plate would be spread more rapidly in the mixed than in the structured transfers. In fact, α
3 oscillations are very pronounced in the mixed transfers. The short-incubation, mixed-transfer replicates show especially noteworthy oscillations () in that they are synchronous and quite pronounced. The most likely explanation for this pattern is a short arms race between
X174 and α
X174 dominated, almost pushing α
3 to extinction. We speculate that evolution of α
3 led to its resurgence and subsequent evolution of
X174 in response led to defeat and extinction of α
3. Phage evolution has not yet been incorporated into our mathematical model.
Our combined experimental/theoretical approach allowed us to explore a possible mechanism for the unexpected coexistence of competing viral pathogens where one was predicted to be a superior competitor under our experimental conditions. The mechanism for coexistence in this system appears to depend on the evolution of resistance to the dominant competitor by the host species. Additionally, the host that is resistant to the dominant phage must experience sufficient population growth to support a viable subordinate phage population. These results were supported both experimentally and in the simulations, and they suggest that host evolution may play an important role in determining whether competitive exclusion occurs in such spatially structured systems. Future work with this model will focus on the role of phage evolution in the dynamics of competing parasites.