Superinfection exclusion: A viral strategy with short-term benefits and long-term drawbacks.

Viral superinfection occurs when multiple viral particles subsequently infect the same host. In nature, several viral species are found to have evolved diverse mechanisms to prevent superinfection (superinfection exclusion) but how this strategic choice impacts the fate of mutations in the viral population remains unclear. Using stochastic simulations, we find that genetic drift is suppressed when superinfection occurs, thus facilitating the fixation of beneficial mutations and the removal of deleterious ones. Interestingly, we also find that the competitive (dis)advantage associated with variations in life history parameters is not necessarily captured by the viral growth rate for either infection strategy. Putting these together, we then show that a mutant with superinfection exclusion will easily overtake a superinfecting population even if the latter has a much higher growth rate. Our findings suggest that while superinfection exclusion can negatively impact the long-term adaptation of a viral population, in the short-term it is ultimately a winning strategy.

Introduction

Bacteriophages (phages) are viruses that infect and replicate within bacteria. Much like many other viruses, reproduction in lytic phage is typically characterised by the following key steps: adsorption to a host cell, entry of the viral genetic material, hijacking of the host machinery, intracellular production of new phage, and finally, the release of progeny upon cell lysis. Phages represent one of the most ubiquitous and diverse organisms on the planet, and competition for viable host can lead to different strains or even species of phage superinfecting or co-infecting the same bacterial cell, ultimately resulting in the production of more than one type of phage (Fig 1a) [1-3]. In the following, we define infection terminology in line with Turner & Duffy [4], such that co-infection occurs when two or more phage have successfully infected a single bacteria, and superinfection occurs when there is a delay between infection by the first and second phage. Therefore, all cells which have been successfully superinfected can be said to be co-infected [4]. To account for different usages throughout the literature and across fields, we also refer to multiple infections, to indicate any case where multiple viruses exist within a single host simultaneously.

Fig 1

Modelling setup.

(a): In superinfection-excluding scenarios, all of the progeny released as the cell lyses are copies of the initial infecting phage, whereas when superinfecting is permitted, the progeny are split between both types of phage. (b): During superinfection, pseudo-populations p and p are used to represent the growth of phage inside the host cells. These populations increase by 1 whenever a phage infects the host, and each population increases by some fraction of its rate β/τ determined by the relative size of the populations in the previous step. (c): An example realisation of the simulation. The resident phage population initially grows until it reaches a steady state, at which point a mutant phage is introduced to the population, and the simulation is run until extinction or fixation of the mutant.

Interestingly, several phages have evolved mechanisms that prevent superinfection (superinfection exclusion). This can be achieved at the early stage of infection, by preventing further adsorption of phage, or at a later stage, by preventing the successful injection of subsequent phage DNA [5, 6]. For instance, bacteriophage T5 encodes a lipoprotein (Llp) that is synthesised by the host at the start of infection and prevents further adsorption events by blocking the outer membrane receptor site (FhuA protein) [7, 8]. Bacteriophage T4 encodes two proteins, Imm and Sp, that prevent superinfection by other T-even phages by inhibiting the degradation of bacterial peptidoglycan, whose presence hinders the DNA transfer across the membrane [9, 10].

Given that populations which allow and prevent superinfection both exist in the wild, it is natural to wonder what impact either strategy has on the evolution of viral populations. This question has been studied in various systems from the perspective of intracellular interactions and competition [11-19]. Multiple infections allow for the exchange of genetic material between viruses through recombination, which can increase diversity and improve the efficiency of selection, but may also decrease fitness by promoting the presence of deleterious mutants at low frequencies [20-22]. Additionally, in RNA viruses with segmented genomes, multiple infections can lead to hybrid offspring containing re-assorted mixtures of the parental segments (reassortment). This mechanism can in principle improve selection efficiency, as re-assorted segments may generate highly deleterious variants that will be easily out-competed by the rest of the population [23]. Multiple infections can also lead to viral complementation, where defective viruses can benefit from superior products generated by ordinary viruses inside the host [23-27]. This process increases the diversity of the population, but also allows cheating individuals to persist in the viral population for long times [23, 24].

The likelihood of multiple infections occurring increases with the number of free phage available per viable host—multiplicity of infection (MOI)—and several experimental systems have been used to study the impact of MOI on viral dynamics [25, 26, 28–32]. For instance, high MOI in RNA phage ϕ6 has been shown to result in a behaviour conforming to the Prisoner’s Dilemma strategy in game theory, and a reduction in viral diversity [28–31, 33]. Theoretically, the same question has been investigated in different scenarios [34], in particular in the context of human immunodeficiency virus (HIV) infections [20, 21, 35–40]. These studies have focused on determining whether multiple infections preferentially occur simultaneously or sequentially, in an effort to explain experimental data, and on the role of recombination in the acquisition of drug resistance, showing that its impact depends on the effective population size. The role of MOI has also been studied in terms of diversity and evolution of the viral population [20, 21, 37, 41–46], with theoretical predictions suggesting that multiple infection favours increased virulence, and that within-host interactions can lead to a more diverse population.

Despite the active work in the area, several fundamental questions on the role of superinfection exclusion on viral dynamics remain unanswered. First, while decreasing MOI in viral populations that allow superinfection decreases the likelihood of superinfection, it does not introduce a superinfection exclusion mechanism that prevents superinfection altogether, making it difficult to draw conclusions about the (dis)advantages of this viral strategy. Second, little is known about how the occurrence of superinfection alone, before even accounting for the additional effects of any intracellular interactions, impacts the evolution of viral populations, particularly when it comes to fundamental evolutionary outcomes such as mutant fixation probabilities. A quantitative understanding of this baseline behaviour is necessary to evaluate the impact of the many additional intracellular interactions that can occur (recombination, defective viruses, etc.). The limited work in this area has shown that in the absence of intracellular interactions, high MOI in superinfecting viral populations can promote the presence of disadvantageous mutants in the “short term,” and obstruct it in the “long term” [47, 48], but how the evolutionary outcomes in each case depend on the parameters describing the viral life-cycle (adsorption rate, lysis time and burst size) and the (dis)advantages of either strategy remain unclear.

Here, we explore how allowing or preventing superinfection impacts the evolutionary fate of neutral and non-neutral variants in a simulated well-mixed phage population with constant, but limited, availability of host. We choose to focus on superinfection exclusion mechanisms that allow secondary adsorption events, but prevent DNA insertion, so that in isolation the phage growth dynamics is the same in the two cases and a direct comparison between the (dis)advantages of the two strategies is more straightforward. We first quantify the effective population size of superinfecting (S) and superinfection-excluding (SX) populations to estimate how these strategies affect genetic drift. We then turn our attention to the effect of non-neutral mutations on (i) the phage growth rate in isolation and (ii) their ability to out-compete the wild-type. Having characterised both the neutral dynamics and the fitness of different variants, we put both aspects together to explore the balance between drift and selection in superinfecting and superinfection-excluding populations, showing that selection is consistently more efficient in superinfecting populations. Finally, we study the evolutionary fate of a mutation which changes whether an individual is capable of preventing superinfection or not. Overall, this work establishes a baseline expectation for how the simple occurrence of superinfection impacts fundamental evolutionary outcomes and provides insights into the selective pressure experienced by viral populations with limited, but constant host density.

Results

Computational modelling framework

We study the evolutionary fate of phage mutants using a stochastic agent-based model. We simulate a well-mixed population of phages V interacting with a population of host bacteria that is kept at a constant density, similarly to a turbidostat [49, 50]. Each phage has a defining set of life history parameters, namely an adsorption rate α, a lysis time τ and a burst size β, and each bacteria can either be in an uninfected B or an infected I state.

In each simulation time-step, adsorption, phage replication within the host and lysis occur. The number of infecting phage V in each step is drawn from a Poisson distribution whose mean corresponds to the expected value αV(B + I) in a well-mixed population. The infecting phage are removed from the pool of free phage, and V bacteria, whether infected or uninfected, are chosen uniformly and with replacement to be the infection target. In both superinfecting and superinfection-excluding scenarios, the final lysis time τ of the host is set by the first phage to infect it and it is treated as deterministic to limit the number of model parameters. This choice was made for the sake of simplicity, given the complex and varied nature of superinfection mechanisms [1-3]. A preliminary analysis of the effect of stochasticity in lysis time is presented in S1 Appendix. In the case where multiple phage infect the same host in a single time-step, the ‘first’ phage is chosen uniformly among those infecting the host. Phage replication within the host post-adsorption depends on whether superinfection is allowed or prevented:

Absence of superinfection

τ steps after the first adsorption event, the bacteria will lyse, releasing new phage into the pool of free phage. The number of phage released Y is drawn from a Poisson distribution with mean β.

Presence of superinfection

Pseudo-populations tracking the growth of phage inside the host are used (see Fig 1b). Because here we focus on the case of two superinfecting phage populations, this results in two pseudo-populations p and p. During the intermediate steps between the first adsorption event and lysis, in the case where there is only one type of phage inside the host, that population will grow at a constant rate β/τ, where β and τ are both specific to the type of phage (i.e. p grows at rate β/τ and p grows at rate β/τ). This is to reflect previous reports of a positive linear relationship between lysis time and burst size [51]. In the event where both types of phage are present within the host, to reflect the intracellular competition for the host’s resources, each population increases by only a fraction of its potential β/τ determined by the size of each population at that time, i.e. p increases by an amount β/τ × p/(p + p) and p increases by an amount β/τ × p/(p + p). At the point of lysis, the total number of phage released Y is drawn from a Poisson distribution with mean p + p − V, where V represents the number of viruses that infected the host prior to lysis. This is to ensure that, in the event where a cell is only infected by 1 type of phage, its mean burst size remains β, regardless of how many phages had infected the cell until that point. The number of phage released of one type Y is then drawn from a binomial distribution with Y attempts and probability p/(p + p) of success, with any remaining phage being the other type (Y = Y − Y).

Following lysis, the lysed bacteria are immediately replaced with a new, uninfected host, resulting in a bacterial population of constant size. We also introduce a decay, or removal, of free phage at rate δ, which accounts for natural phage decay and the outflow of the turbidostat system.

Simulations were initialised with B0 uninfected bacteria and 2B0 “resident” phage, and then run until the phage, uninfected bacteria and infected bacteria populations each reached steady state values (V, B and I respectively), as determined by their running average (Fig 1c). This steady state arises due to a balance between phage production and loss and it is independent of the initial number of phages (S1 Fig).

Superinfection leads to a larger effective population size

First, we find that genetic diversity consistently declines faster in populations that prevent superinfection, indicating a smaller effective population size N when compared to superinfecting populations (see Methods). This can be intuitively understood by considering that in the superinfecting scenario, each phage has more opportunity to successfully infect a host cell, since secondary infections can result in the production of some offspring when the cell lyses. Therefore, more phage are able to contribute to the next generation, thereby slowing down diversity loss.

In addition, Fig 2 shows that in both superinfecting and superinfection-excluding populations higher adsorption rate and burst size, and shorter lysis time result in larger effective populations. This observation is, however, partially attributable to the change in total phage population N = (V + βI), where V indicates the steady state free phage population, I indicates the steady state number of infected bacteria, and so βI represents the number of phage that inevitably will join the free phage population.

Fig 2

Effective population size.

The effective population size in both superinfecting (S) and superinfection-excluding (SX) populations as a function of adsorption rate α, burst size β and lysis time τ. Effective population size are also shown scaled by the size of the total phage population N = (V + βI). Parameters used were α = 3 × 10−6, β = 100 and τ = 15 unless otherwise stated. Throughout, δ = 0.1 and B0 = 1000. Error bars are plotted but are too small to see. The data is obtained from an average of at least 1000 independent simulations.

Indeed, adsorption rate and lysis time impact both the effective and actual population sizes in the same way (i.e. N/N ≈ const.). By contrast, larger burst sizes increase the effective population size less than the actual population size (Fig 2), resulting in a decrease of N/N. This can be interpreted by noticing that while increasing burst size results in more phage, the number of phage that can actually contribute to the next generation (i.e. the effective population size) is limited by the number of bacteria that are available. Therefore, as burst size is increased, a larger fraction of phage become wasted.

Neutral mutants are consistently more likely to fix in superinfecting populations

To continue our characterisation of the neutral dynamics in both superinfecting and superinfection-excluding populations, we turn to the fixation probabilities of neutral mutants, and determine how they depend on the phage infection parameters.

Because the total phage population size depends on the life history parameters, the initial mutant frequency corresponding to one mutant phage inoculated in the population also varies with life history parameters. To account for this effect, we re-scale the fixation probability by the initial frequency of the mutant , which is the same in superinfecting and superinfection-excluding populations. Fig 3 shows that as the parameters are varied, indicating that the total number of phages for a given set of parameters is the main controller of neutral dynamics. Indeed, we find that the impact of the life history parameters on the probability of fixation is what one would intuitively expect (S2 Appendix): larger adsorption rate and burst size, and shorter lysis time, increase the steady-state size of the phage population, and reduce P. By describing the average behaviour of our simulations with a system of ordinary differential equations (ODEs), we confirm that the ODE solution for the total phage population at steady-state N is the same as in the stochastic model (S2 Appendix).

Fig 3

Fixation of neutral mutants.

Probability of mutant fixation P in the superinfecting (S) and non superinfection excluding (SX) scenarios, scaled by the initial frequency of the mutant , as a function of adsorption rate α, burst size β and lysis time τ. Dashed lines indicate the average of the data for both the superinfecting (blue) and superinfection-excluding (red) scenarios. These lines indicate that neutral mutants in superinfecting populations experience a small advantage over mutants in an equivalent superinfection-excluding population. Unscaled P data can be seen in S2 Appendix. Unless otherwise stated, the parameters used were α = 3 × 10−6, β = 100, τ = 15, δ = 0.1 and B0 = 1000. The error in our estimate of the fixation probability ΔP is given by , where n and n represent the total number of simulations and the number of simulations where the mutant fixes respectively. The data is obtained from a minimum of 14 million independent simulations.

Fig 3 also shows that, on average, neutral mutants in the superinfecting scenario are more likely to fix than mutants in an equivalent superinfection-excluding population (blue and red dashed lines in Fig 3 respectively). This result agrees with that found by Wodarz et al. [48], who showed that in a superinfecting viral population, higher multiplicities of infection slightly favoured rare neutral and disadvantageous mutants in the short term. The intuition behind this observation can be explained in the following way: at the moment that the mutant is introduced, all infected cells are infected by the resident phage. In the superinfecting scenario, the mutant population can therefore grow by infecting an uninfected cell, or by infecting an already infected cell, as this secondary infection will lead to some fraction of the burst size being allocated to the mutant type. While resident phage can replicate by infecting either types of host, the resident population cannot further grow by infecting previously infected cells. This is because all infected cells are already exclusively infected with resident phage, and superinfection of resident infected cells by more resident phage does not result in any more resident phage being produced. As a result, superinfection increases the mutant’s chance of survival in the early stages in comparison to the superinfection-excluding counterpart, similarly to conditions of high vs. low MOI [48].

Higher growth rate does not translate into competitive advantage

To investigate the evolutionary fate of non-neutral mutations, we first characterise how phage growth rate and competitive fitness is affected by changes to the phage life history parameters, i.e., adsorption rate α, burst size β and the lysis time τ, relative to the values used in our neutral simulations (Fig 3).

S2 Fig shows that increasing burst size or adsorption rate results in a larger selective advantage both in isolation and in direct competition (see Methods). However, while variations in burst size affect similarly the phage growth rate in isolation and its (dis)advantage in a competitive setting (s ≈ s, Fig 4), variations in adsorption rate lead to a stronger competitive (dis)advantage than what would be predicted by the growth rate (|s| < |s|). The intuition behind this result is that increasing adsorption rate becomes particularly advantageous in a competitive environment, as being the first virus to infect a host allows the virus to have largely (superinfection scenario) or completely (superinfection exclusion scenario) exclusive access to the host resources.

Fig 4

Competitive vs isolated selective advantage.

The selective advantage in a competitive setting s as a function of the change in growth rate s, when changing adsorption rate α, burst size β and lysis time τ. Straight line fits are shown as dashed lines, with gradient σ such that s = σs. From the above data we find σ = 1.2324, σ = 1.2764, σ = 1.0432, σ = 0.9134, σ = 0.3057 and σ ≈ 0. Resident parameters used were α = 3 × 10−6, β = 100 and τ = 15. As before δ = 0.1 and B0 = 1000. s determined from 500 simulations, and s determined from 200 simulations. Error bars are given by the standard error on the mean of the simulations. Error bars on x axis have been omitted for clarity, but are shown in S2 Fig.

The impact of altering lysis time τ is surprising. S2 Fig shows that increasing τ results in a reduced growth rate, as intuition suggests. Yet, in the superinfection-excluding scenario no discernible impact on s is observed (Fig 4). This result is supported by our ODE model (S2 Appendix), which shows that once the system is at steady-state, alterations to lysis time offer no advantage to one phage over the other (S3 Fig). We believe that this is a special feature of a turbidostat setting, as lysed hosts are immediately replaced by uninfected cells, providing the same number of viable hosts independently of the time needed by the phage to lyse them. By contrast, in the superinfecting case, we are able to observe a selective (dis)advantage in direct competition, although at a significantly reduced level compared to the change in growth rate. We believe that this effect arises because, while the extracellular competition is limited by the turbidostat setup, in the superinfecting scenario there is the opportunity for some intracellular competition to occur, as mutants will grow at different rates inside the host, resulting in different numbers of phage (both in total and proportionally) being released upon lysis. We leave a full characterisation of the relationship between growth rate in isolation and competitive fitness to future works.

Superinfection results in more efficient selection

Having characterised how changes to the phage infection parameters alter first genetic drift and second fitness, we now put both ingredients together and investigate the dynamics of non-neutral mutants. To this end, we simulate a resident phage population to steady state, introduce a single non-neutral mutant and then run the simulation until extinction or fixation occurs.

In agreement with our observations regarding the difference between growth rate and competitive fitness, we find that the value of s is not sufficient to determine the fixation probability of the corresponding mutant (Fig 5): a mutation associated with a higher adsorption rate α increases the mutant’s chance to fix more than a mutation which alters the burst size β and leads to the same growth rate. We also find that beneficial mutations are consistently more likely to fix (and deleterious mutations more likely to go extinct) in superinfecting populations (red) than superinfection-excluding populations (blue). This suggests that superinfection improves selection efficiency, by more readily fixing beneficial mutations and purging deleterious ones.

Fig 5

Fixation of non-neutral mutants.

Probability of mutant fixation P as a function of selective growth advantage s. Points indicate simulation results, while lines indicate theoretically predicted values in a Moran model with equivalent parameters (Eq 1). Data points for the α and β mutants have been omitted from the right hand panel for clarity. The error in our estimate of the fixation probability ΔP is given by , where n and n represent the total number of simulations and the number of simulations where the mutant fixes respectively. Error bars in the x-axis represent the errors on the growth rate fitness s that each burst size corresponds to. These are calculated by fitting a linear relation to growth rate measurements such that s = m(β − β). The fractional error on the s is then equal to the fractional error on the fitted gradient m. The data is obtained from a minimum of 5 million independent simulations.

To provide a theoretical framework to our findings, we compare the simulation data to the fixation probabilities one would expect in a corresponding Moran model. For small selective advantage s, the probability of fixation is given by where f0 is the initial frequency of the mutant in the population with effective population size N [52, 53]. Our earlier results on neutral dynamics and fitness provide independent measurements of the parameters in Eq 1 for different values of α, β and τ: from our initial condition (i.e., 1/N, where N is the steady-state phage population size when the mutant is introduced); N is measured from the decay of heterozygosity (Fig 2); and s = σs is derived from our measurements of the relationship between competitive and growth rate advantage (Fig 4). These theoretical predictions are plotted without additional fitting parameters as lines in Fig 5.

Fig 5 shows that the theoretical prediction from the appropriately parameterised Moran model matches the simulation data remarkably well, despite the complex internal infection dynamic (see S3 Appendix for quantitative comparison). We note, however, that the simulation data consistently fails to intersect at the same point when s = 0 in the superinfecting scenario. This is because of the effect outlined in Fig 3, where rare mutants initially experience a slight advantage in the superinfecting scenario because they are able to increase in number by infecting both uninfected and infected cells. To test the validity of our findings across parameter space, we also perform all of the above analysis with different resident parameters, obtaining similar results (S4 Appendix).

Superinfection exclusion slows down adaptability in the long run, but is a winning strategy in the short term

Our findings imply that, even in the absence of intra-cellular processes such as recombination, superinfection results in more efficient selection, so that beneficial mutations are relatively more likely to fix, and deleterious ones are more likely to be purged, leading to a fitter overall population in the long run. From the point of view of viral adaptation, allowing superinfection ultimately seems like the better long-term strategy. It is therefore puzzling why several natural phage populations have developed sophisticated mechanisms to prevent superinfection, particularly given that employing these mechanisms is expected to come with a biological cost, such as reduced burst size [54, 55] or increased lysis time [56].

To address this question, we consider the fate of mutations that either (i) remove the mutant’s ability to prevent superinfection in a resident superinfection-excluding population or (ii) provide the mutant the ability to prevent superinfection in a resident superinfecting population. Fig 6 shows that if the mutant is neutral (β = β = 100), then the superinfection-excluding mutant is two orders of magnitude more likely to fix than the expectation based on its initial frequency , and that, by contrast, the superinfecting mutant is at least two orders of magnitude more likely to go extinct. It should be noted that we actually find no instances of mutant fixation in this case, but our detection power is limited by the number of simulation runs. Here, we run at least 20 million simulations, and we can thus infer that P ≪ 10−7. This indicates that mutants which are able to prevent superinfection experience a very strong selective advantage over their superinfecting counterparts, and vice-versa.

Fig 6

Mutations which alter the ability to prevent superinfection.

(a) The probability P of a mutant which prevents superinfection fixing in a population that allows it, as a function of mutant burst size β. (b) The probability P of a mutant which allows superinfection fixing in a population that prevents it, as a function of mutant burst size β. It can be seen that the superinfecting mutant requires a significantly increased burst size to fix, and conversely the superinfection-excluding mutant can fix, even if its burst size is greatly reduced. The error in our estimate of the fixation probability ΔP is given by , where n and n represent the total number of simulations and the number of simulations where the mutant fixes respectively. Error bars in the x-axis represent the errors on the growth rate fitness s that each burst size corresponds to. These are calculated by fitting a linear relation to growth rate measurements such that s = m(β − β). The fractional error on the s is then equal to the fractional error on the fitted gradient m. The fixation data is obtained from a minimum of 20 million independent simulations.

To account for the possibility that superinfection exclusion comes at a cost in phage growth, as preventing superinfection likely requires the production of extra proteins, the resources for which could otherwise have gone to the production of more phage, we consider the case where superinfection exclusion is associated with a reduction in burst size [54]. Remarkably, we find that even when preventing superinfection carries a burden of 7% reduction in burst size (s < −7%), the superinfection-excluding mutant still fixes more often than a neutral superinfecting mutant (Fig 6). Conversely, a minimum of 8% increase in burst size (s > 8%) is necessary to give a superinfecting mutant any chance of fixing in a superinfection-excluding population. This indicates that while allowing superinfection increases selection efficiency at the population level, preventing it is ultimately a winning strategy in the short term, partially explaining why superinfection exclusion is so common in nature [5, 6].

Discussion

In this work, we have considered the impact of either allowing or preventing superinfection on the evolution of viral populations. Using a stochastic agent-based model of viral infection, we have shown that allowing superinfection reduces the strength of genetic drift, leading to an increase in effective population size. Weaker fluctuations result in a higher efficiency of selection in viral populations, with beneficial mutations fixing more frequently, and deleterious ones more readily being purged from the population. Despite the long term, population-wide benefit of allowing superinfection, we find that if a mutant arises which is capable of preventing superinfection, it will fix remarkably easily, even if its growth rate is heavily compromised. Conversely, if the whole population is capable of preventing superinfection, mutants which allow it will have almost no chance of ever succeeding.

The evolutionary impact of superinfection (and more generally multiple infections) has most often focused on the role of intracellular interactions and competition [11–14, 16–19], such as genetic recombination and reassortment [20-23], and viral complementation [23-27]. A prevalent finding (amongst others) is that recombination and reassortment can improve the efficiency of selection in viral populations which do not exclude superinfection. Remarkably, our work demonstrates that the basic occurrence of superinfection alone, absent of any recombination or reassortment, is capable of increasing the selection efficiency. In this context, our results provide a useful baseline for comparison when trying to assess the significance of each of these more complex effects, which may or may not be present in different situations.

An unexpected finding of this work is that in the turbidostat system we consider, while increased adsorption rate and burst size both increase the fitness of the phage population in all respects, in the superinfecting scenario lysis time plays a significantly reduced role in the competitive (dis)advantage experienced once the system has reached a steady-state, and in the superinfection-excluding scenario it plays no role whatsoever. While it has been demonstrated previously that changes to fecundity and generation time can have different impacts on mutation fixation probability, even when they have the same impact on long-term growth rate [57], our result is somewhat in contrast with previous studies showing that well-mixed liquid cultures with an abundance of hosts generally select for higher adsorption rates and lower lysis times [51, 58–60]. The key difference between such liquid cultures and the turbidostat system we model here is that in the former host cells are not maintained at a constant density, but the phage population continues to grow until no bacteria remain. This finding illustrates how the presence or absence of a co-existing steady-state between phage and bacteria completely alters the selective pressure on the phage with important implications for studies into the co-evolution of phage and bacteria populations using continuous culturing set-ups [61-63]. In particular, our results suggest that in an evolutionary experiment in a turbidostat, the virus should evolve towards very large burst size even if this feature comes at the cost of longer lysis times, especially if superinfection exclusion occurs [59]. Reciprocally, detecting a selective pressure on lysis time could be used to identify potential phages that allow superinfection, as, in this case, a shorter lysis time is slightly advantageous all else being equal.

Following this, it is natural to wonder how the (dis)advantages and impact of either strategy depends on the selective pressure experienced in different environments. The relationship between viral fitness and the phage life-history parameters (adsorption rate, lysis time and burst size) has been shown to be very context-dependent in both well-mixed and spatially structured settings. For instance, as noted previously, well-mixed settings generally favour higher adsorption rates [64], but in spatially structured settings phage with lower adsorption rates are more successful [65, 66]. Additionally, it has been shown previously that eco-evolutionary feedbacks at the edge of expanding viral populations can result in travelling waves with vastly different effective population sizes [67]. Given that competition for resources (i.e. viable hosts) in spatially structured environments is local rather than global, phage are more likely to be in competition with other genetically identical phage released by nearby cells. It is therefore possible that superinfection exclusion proves less useful in this context than in well-mixed environments where competition is global and phage are more likely to encounter other genetically different viruses. All of this points at the role of superinfection strategies and other social viral behaviour on the eco-evolutionary dynamics of spatially expanding viral populations as an exciting avenue for future research.

Methods

Measuring effective population size of the phage population

Consistently with previous work [52], we expect that the neutral standing diversity of the phage population, quantified by the heterozygosity H, will decay exponentially at long times due to genetic drift, so that (S4 Fig), with the decay rate in units of generations being expressed in terms of an effective population size 2/N (Moran model [52]).

We track the viral heterozygosity H as a function of time, which in a biallelic viral population is given by where f and (1 − f) represent the frequencies of two neutral viral alleles in the population, and 〈…〉 indicates the average over independent simulations. H(t) can be understood to be the time-dependent probability that two individuals chosen from the population are genetically distinct.

To determine the generation time T, we first calculate the net reproduction rate R0, which represents the number of offspring an individual would be expected to produce if it passed through its lifetime conforming to the age-specific fertility and mortality rates of the population at a given time (i.e. taking into account the fact that some individuals die before reproducing) [68]. R0 can be calculated as where l represents the proportion of individuals (in our case, phage) surviving to age t, and m represents the average number of offspring produced at age t.

There are two mechanisms in our simulations by which phages can ‘die’ when superinfection exclusion applies: either by decaying with rate δ, or by adsorbing to an infected host with rate αI. In a sufficiently small timestep Δt, these rates correspond to a proportion δΔt and αIΔt of the total phage, respectively. Equivalently, these can be considered to be the probability that any single phage will die in the same period. As a result, the probability of a phage surviving to age t is l = (1 − δΔt − αIΔt).

The average number of offspring m produced at age t is 0 if t < τ, because we assume that no phage is released before the lysis time. For t > τ, m is given by the probability of successfully infecting a viable host in a timestep Δt, τ time earlier (αBΔt), multiplied by the yield of new phage (β − 1).

In the limit where Δt → 0, this will result in a net reproductive rate of the form where the integral starts at τ because no offspring are produced prior to that point.

Then the generation time T, defined as the average interval between the birth of an individual and the birth of its offspring, is Here, we will use resident phage parameters α = 3 × 10−6, τ = 15, δ = 0.1 and a total bacterial population of B0 = 1000, which leads to I = 681 and a generation time of T = 24.8. This generation time is also supported by stochastic simulations of the phage adsorption and death processes (S5 Appendix). Throughout this work, we use the same generation time for both superinfecting and superinfection-excluding populations (more details in S5 Appendix).

For comparison, coliphage T7 in liquid culture typically has parameters of τ ≈ 10 − 20 min, α ≈ 10−9 ml/min and B0 ≈ 106 − 108 ml−1, thereby yielding an αB0 ≈ 10−3−10−1 min−1 [59, 69]. These values are comparable to our own if we equate 1 timestep = 1 min, and so τ = 15 min and αB0 = 3 × 10−3 min−1, such that the relative timescales in our simulation remain consistent. The reason behind choosing a larger adsorption rate and smaller bacteria population is purely practical, as the alternative would lead to unreasonably long computational times. Given these values, our choice of decay rate δ is made such that steady-state population sizes are reached.

Measuring mutant fitness and growth rate

We start by defining a selective advantage s in terms of the exponential growth rate r of the mutant phage population relative to that of the resident phage r [70]: The exponential growth rate is determined by simulating the growth of the corresponding phage population in isolation, and performing a linear fit to the log-transformed phage number as a function of time, which is then averaged over 500 independent simulations. It should be noted that as there is only one type of phage in these simulations, the growth rate of both superinfecting and superinfection-excluding populations is the same.

We also characterised the fitness of mutants in a competitive setting, by simulating a resident population until steady state, and then replacing 50% of the population with the mutant. In this direct competition scenario, we determine the selective (dis)advantage s of the mutant phage by tracking the relative growth of mutant and resident populations, so that as V(t = 0) = V(t = 0). s is determined from the average of 200 simulations. Importantly, in contrast to s, this competitive selective advantage (s) can in principle differ between superinfecting (s) and superinfection-excluding (s) phage populations. In the absence of any interactions between the two competing phage populations, s and s are typically expected to be the same.

Measuring mutant probability of fixation

To measure fixation probabilities of individual mutations, we allow our simulations to reach steady state, we then introduce a single mutant phage into the free phage population, and run the simulation until either fixation or extinction occurs. This process is repeated at least 5 million times for each set of parameters. The probability of mutant fixation P is determined from the fraction of simulations where the mutant fixed, n, over the total number of simulations run, n (i.e. P = n/n). Assuming a binomial distribution, the error in our estimate of the number of fixation events Δn is given by . Consequently, our error in the estimate of fixation probability ΔP is given by . It can be easily verified that in the case where n ≪ n, as we have here, the error approaches as would be found in a Poisson distribution.

Steady-states are independent of intial conditions.

The steady-state phage population V reached does not depend on the initial number of phage V0 in the simulations. In all, α = 3 × 10−6, β = 100, τ = 15, δ = 0.1 and B0 = 1000.

(EPS)

Click here for additional data file.

s as a function of phage life-history parameters.

The selective advantage s relative to a resident phage that results from a change to adsorption rate α, burst size β and lysis time τ. This is measured both in terms of the effect on the isolated growth rate of the mutant (s, Eq 8), and in terms of the change in frequency in a population initiated with 50% mutant and 50% resident (s and s, Eq 9). Resident parameters used were α = 3 × 10−6, β = 100 and τ = 15. As before δ = 0.1 and B0 = 1000. s determined from 500 simulations, and s determined from 200 simulations. Error bars are given by the standard error on the mean of the simulations.

(EPS)

Click here for additional data file.

s in the ODE model.

The relative change in frequency of two populations in the ODE model (indicating the average behaviour in the stochastic model). It can be seen that once at steady-state, changing lysis time τ has no effect. Parameters used were α = 3 × 10−6, β = 100 and τ = 15 unless otherwise stated. As throughout, δ = 0.1 and B0 = 1000.

(EPS)

Click here for additional data file.

Example decay in heterozygosity.

Linear fit to log transformed heterozygosity data, with slope Λ ≡ 2/N revealing that allowing superinfection (red) results in a larger effective population size compared to the case where superinfection is prevented (blue). Parameters used were α = 3 × 10−6, β = 100, τ = 15, δ = 0.1 and B0 = 1000. Data obtained is the average of 1000 independent simulations.

(EPS)

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Stochasticity in lysis time.

Here we discuss the decision to not incorporate stochasticity in lysis time in the model presented in the main text.

(PDF)

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ODE description of model.

The average behaviour of the model used in the main text is described by a set of ordinary differential equations (ODEs), showing good agreement with our stochastic simulations.

(PDF)

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Comparison with expectation from Moran model.

A quantitative comparison between the fixation probabilities obtained in our stochastic simulations with those that would be predicted in a similarly parameterised Moran model.

(PDF)

Click here for additional data file.

Repeat measurements with β = 70.

Here we repeat a subset of the measurements carried out in the main text with different resident phage parameters, in this instance β = 70.

(PDF)

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Calculation of generation time.

Here we support the generation time calculated in the main text with results of stochastic simulations. We also include a more detailed discussion about the differences in generation time between superinfecting and superinfection-excluding populations.

(PDF)

Click here for additional data file.

21 Jan 2022

Dear Mr. Hunter,

Thank you very much for submitting your manuscript "Superinfection exclusion: a viral strategy with short-term benefits and long-term drawbacks" for consideration at PLOS Computational Biology.

As with all papers reviewed by the journal, your manuscript was reviewed by members of the editorial board and by several independent reviewers. In light of the reviews (below this email), we would like to invite the resubmission of a significantly-revised version that takes into account the reviewers' comments.

The reviewers are broadly enthusiastic about this work, though Reviewer 2 raises an important issue about the effect of certain assumptions upon the validity of the results. This issue, alongside other issues raised by both reviewers, should be dealt with in a revised version of the manuscript.

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The reviewers are broadly enthusiastic about this work, though Reviewer 2 raises an important issue about the effect of certain assumptions upon the validity of the results. This issue, alongside other issues raised by both reviewers, should be dealt with in a revised version of the manuscript.

Reviewer's Responses to Questions

Comments to the Authors:

Please note here if the review is uploaded as an attachment.

Reviewer #1: Overall, I really enjoyed this paper. The question of whether superinfection exclusion poses advantages or disadvantages to the evolution of phage populations is a longstanding one, but the authors manage to present their approach in a way that appears straightforward while also making a nice contribution to the field. I find that their results justify their conclusions, though I have several minor points that I think could be addressed to make the work even more clear. There were a few areas where I think additional exposition would help the reader understand the authors’ points.

I found the introduction well-written and it poses the questions the authors aim to address very clearly. I think that the breadth of literature covered in the introduction does a good job of summarizing the gaps of knowledge in this sub-field and where progress has been made so far. I would also recommend citing the sociovirology review written by Sam Diaz-Munoz, Rafael Sanjuan, and Stuart West (10.1016/j.chom.2017.09.012), which is in my view one of the more important pieces written on the topic.

Lines 85-86: The authors state that adsorption that happens at times later than the first adsorption event do not contribute DNA to the host cell in the superinfection excluding scenarios. Does this mean that the phages that enter later are removed from the free phage population, but do not get replicated? This would be in contrast, I think, to what is described in the Superinfecting section that begins on line 126 where subsequent adsorption events increase the within-cell population until lysis. Lines 123 to 125 may be a good place to clarify.

Lines 121-123: The lysis time is not drawn from a distribution with parameter tau, unlike burst size which has mean beta, or the number of phage that adsorb in a timestep which depends on alpha. Why lysis time would not be stochastic, but burst size and adsorption are stochastic, is not clear to me so I would appreciate it if the authors motivated this choice.

Lines 194-197: The juxtaposition here is throwing me. Why would a neutral mutant have two cell types available to it while the wild-type can only infect uninfected cells? Is it because we are at steady state when the mutant infects, so it has multiple cell-states available, whereas the wild-type before steady state only expanded through uninfected cells? Please elaborate.

Line 206: You introduce parameter s here, but I don’t think it was defined in the methods. Is it just the general term for s_growth and s_comp? If so, please state.

Line 252: How do sigma and phi relate to each other? You describe them as fitting and scaling parameters for the various s variables, but their roles are unclear.

Figure 6: It would be nice if you listed the number of simulations ran for each of your figures and their subplots. It is in the methods for some of the work but I don’t think it encompasses all of the figures. looks like 1/10^6.5 based on 6b if the leftmost point has been adjusted given your absence of observed fixation. I do appreciate that they state the limitation that they did not observe fixation here, though would the value plotted be an upper bound instead of a lower bound like the authors write? More simulations to observe a rare fixation event would decrease the estimated probability, no?

Figure 6: The error bars need to be clarified. I assume it has something to do with s_growth reduction but it really isn’t clear to me how the bars relate to the points as they seem to be varying different parameters, but wouldn’t an additional dimension need to be displayed to adequately show the exploration of this parameter space?

Lines 328-330: I think this is an excellent point.

Line 404: Is this alpha value consistent with what is explored in the figures? Here you state that the empirical value is on the order of 10^-9, but the parameters explored in the figures are on the order of 10^-6.

I thought that the Methods section was very well written – nice work. Particularly the section on calculating the fixation probability.

I also found several spelling errors, so I think the authors should give a careful read if they resubmit. Examples:

Figure 1 caption: “superinfecting in permitted” should instead be “is permitted”

Line 332: “our results suggests” should be “suggest”

Reviewer #2: The review is uploaded as an attachment.

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Have the authors made all data and (if applicable) computational code underlying the findings in their manuscript fully available?

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Reviewer #1: Yes

Reviewer #2: No: The authors provide a link to their code in the manuscript, but for some reason it cannot be accessed.

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Reviewer #2: No

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Submitted filename: Comments to the authors.pdf

Click here for additional data file.

31 Mar 2022

Submitted filename: Reply_PLOS_mth47.docx

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20 Apr 2022

Dear Mr. Hunter,

We are pleased to inform you that your manuscript 'Superinfection exclusion: a viral strategy with short-term benefits and long-term drawbacks' has been provisionally accepted for publication in PLOS Computational Biology.

Before your manuscript can be formally accepted you will need to complete some formatting changes, which you will receive in a follow up email. A member of our team will be in touch with a set of requests.

Please note that your manuscript will not be scheduled for publication until you have made the required changes, so a swift response is appreciated.  We recommend that further changes be considered in line with the suggestions of reviewer 2.

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Thank you again for supporting Open Access publishing; we are looking forward to publishing your work in PLOS Computational Biology.

Best regards,

Christopher Illingworth

Guest Editor

PLOS Computational Biology

Ville Mustonen

Deputy Editor

PLOS Computational Biology

***********************************************************

Your manuscript has been considered again by the reviewers, who are happy that your work merits publication in PLoS Computational Biology. In the preparation of a final manuscript we recommend consideration be given to the thoughts raised by reviewer 2, but we are happy for changes to be made at the authors' discretion.

Reviewer's Responses to Questions

Comments to the Authors:

Please note here if the review is uploaded as an attachment.

Reviewer #2: Hunter and Fusco have thoroughly revised their manuscript in accordance with the issues and suggestions raised by reviewer 1 and myself. At least in my case, in this new version of the paper, the authors have resolved most of my concerns. The issues regarding the absence of intracellular competence in the previous computational framework have been addressed as I suggested, except that the time to the cell lysis is still defined by the first infecting phage. However, I understand that this is probably the computationally simplest approach and I doubt very much that a different setting would qualitatively change the conclusions of the paper.

The authors have also introduced changes in relation to my criticisms about non-necessity of high MOI regimens for superinfection to occur and have introduced a terminology for the use of coinfection and superinfection terms in accordance with the definitions proposed previously by Turner and Duffy 2008. This improves the readability of the article, because although, under the definitions used, any superinfection event can be considered a coinfection event, the opposite is not true, and the inconvenient use of superinfection is avoided when coinfection in a stricter sense (i.e. coinfection without superinfection) would also be valid or even more appropriate (e.g. see lines 47, 51, and 55). In these cases, the term multiple infection has been introduced, which encompasses cases of strict coinfection as well as superinfection. However, by defining the term "multiple infection" within the host (i.e. “any case where multiple viruses exist within a single host simultaneously”, lines 32-34) it is not necessarily confined to the cellular level, although it is in the case of phages, which is the focus of the article.

Overall, because of these and all the other changes, corrections and additional information included in the new manuscript, I think the article is now acceptable to publish. However, I still want to mention two things that have raised some doubts.

(I) As the authors mention in line 236, I find the null effect of tau on fitness in competition (s_comp) in the superinfection exclusion scenario surprising. Although I am confident that the result is correct, the explanation about the nature of the turbidostat system (lines 241-243) does not entirely satisfy me. If both resident and mutant viruses share adsorption rate and burst size, but the mutant lyses cells faster, even though the number of lysed cells is immediately available as susceptible cells, shouldn't this mutant phage become more common in the population as it carries out more infection cycles in the same time frame? By doing do it would be expected to produce a greater number of viral progeny and susceptible cells than the phage with longer tau.

(II) For the calculation of the error (I assume SEM) in estimated fixation probabilities, it is considered that the probability of fixation fits a binomial distribution (lines 454-456). Under this assumption I don't see why error is defined as sqrt(nfix)/n. I would understand that this would be the case if the variance equaled the mean (which is nfix/n). Maybe I am wrong, but I understand that in each simulation can be considered a Bernoulli trial, so the variance in the Pfix parameter should be Pfix(1-Pfix) and the SEM sqrt(Pfix(1-Pfix)/n). Therefore, shouldn't you arrive at an expression, in the terms used in the article, such that sqrt((n*nfix-nfix^2)/n^3))?

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Have the authors made all data and (if applicable) computational code underlying the findings in their manuscript fully available?

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Reviewer #2: Yes

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Reviewer #2: No

5 May 2022

PCOMPBIOL-D-21-02234R1

Superinfection exclusion: a viral strategy with short-term benefits and long-term drawbacks

Dear Dr Hunter,

I am pleased to inform you that your manuscript has been formally accepted for publication in PLOS Computational Biology. Your manuscript is now with our production department and you will be notified of the publication date in due course.

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