The authors have declared that no competing interests exist.

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.

Viral social behaviour has recently been receiving increasing attention in the context of ecological and evolutionary dynamics of viral populations. One fascinating and still relatively poorly understood example is superinfection or co-infection, which occur when multiple viruses infect the same host. Among bacteriophages, a wide range of mechanisms have been discovered that enable phage to prevent superinfection (superinfection exclusion) even at the cost of using precious resources for this purpose. What is the evolutionary impact of this strategic choice and why do so many phages exhibit this behaviour? Here, we conduct an extensive simulation study of a phage population to address this question. In particular, we investigate the fate of viral mutations arising in an environment with a constant supply of bacterial hosts designed to mimic a “turbidostat,” as these are increasingly being used in laboratory evolution experiments. Our results show that allowing superinfection in the long-term yields a population which is more capable of adapting to changes in the environment. However, when in direct competition, mutants capable of preventing superinfection experience a very large advantage over their superinfecting counterparts, even if this ability comes at a significant cost to their growth rate. This indicates that while preventing superinfection can negatively impact the long-term prospects of a viral population, in the short-term it is ultimately a winning strategy.

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 (

(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 _{a} and _{b} 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

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 [

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 [

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 [

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

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.

We study the evolutionary fate of phage mutants using a stochastic agent-based model. We simulate a well-mixed population of phages

In each simulation time-step, adsorption, phage replication within the host and lysis occur. The number of infecting phage _{I} in each step is drawn from a Poisson distribution whose mean corresponds to the expected value _{I} 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

Pseudo-populations tracking the growth of phage inside the host are used (see _{a} and _{b}. 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 _{a} grows at rate _{a}/_{a} and _{b} grows at rate _{b}/_{b}). This is to reflect previous reports of a positive linear relationship between lysis time and burst size [_{a} increases by an amount _{a}/_{a} × _{a}/(_{a} + _{b}) and _{b} increases by an amount _{b}/_{b} × _{b}/(_{a} + _{b}). At the point of lysis, the total number of phage released _{a} + _{b} − _{n}, where _{n} 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 _{a} is then drawn from a binomial distribution with _{a}/(_{a} + _{b}) of success, with any remaining phage being the other type (_{b} = _{a}).

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

Simulations were initialised with _{0} uninfected bacteria and 2_{0} “resident” phage, and then run until the phage, uninfected bacteria and infected bacteria populations each reached steady state values (_{ss}, _{ss} and _{ss} respectively), as determined by their running average (

First, we find that genetic diversity consistently declines faster in populations that prevent superinfection, indicating a smaller effective population size _{e} when compared to superinfecting populations (see

In addition, _{T} = (_{ss} + _{ss}), where _{ss} indicates the steady state free phage population, _{ss} indicates the steady state number of infected bacteria, and so _{ss} represents the number of phage that inevitably will join the free phage population.

The effective population size in both superinfecting (S) and superinfection-excluding (SX) populations as a function of adsorption rate _{T} = (_{ss} + _{ss}). Parameters used were ^{−6}, _{0} = 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. _{e}/_{T} ≈ const.). By contrast, larger burst sizes increase the effective population size less than the actual population size (_{e}/_{T}. 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.

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 _{fix}. 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 _{T} is the same as in the stochastic model (

Probability of mutant fixation _{fix} in the superinfecting (S) and non superinfection excluding (SX) scenarios, scaled by the initial frequency of the mutant _{fix} data can be seen in ^{−6}, _{0} = 1000. The error in our estimate of the fixation probability Δ_{fix} is given by _{fix} 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.

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

_{growth} ≈ _{comp}, _{growth}| < |_{comp}|). The intuition behind this result is that increasing adsorption rate becomes particularly advantageous in a competitive environment, as being the

The selective advantage in a competitive setting _{comp} as a function of the change in growth rate _{growth}, when changing adsorption rate _{comp} = _{growth}. From the above data we find _{Sα} = 1.2324, _{SXα} = 1.2764, _{Sβ} = 1.0432, _{SXβ} = 0.9134, _{Sτ} = 0.3057 and _{SXτ} ≈ 0. Resident parameters used were ^{−6}, _{0} = 1000. _{growth} determined from 500 simulations, and _{comp} determined from 200 simulations. Error bars are given by the standard error on the mean of the simulations. Error bars on

The impact of altering lysis time _{comp} is observed (

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 _{growth} is not sufficient to determine the fixation probability of the corresponding mutant (

Probability of mutant fixation _{fix} as a function of selective growth advantage _{growth}. Points indicate simulation results, while lines indicate theoretically predicted values in a Moran model with equivalent parameters (_{fix} is given by _{fix} represent the total number of simulations and the number of simulations where the mutant fixes respectively. Error bars in the _{growth} that each burst size corresponds to. These are calculated by fitting a linear relation to growth rate measurements such that _{growth} = _{mut} − _{res}). The fractional error on the _{growth} is then equal to the fractional error on the fitted gradient

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 _{comp}, the probability of fixation is given by
_{0} is the initial frequency of the mutant in the population with effective population size _{e} [_{T}, where _{T} is the steady-state phage population size when the mutant is introduced); _{e} is measured from the decay of heterozygosity (_{comp} = _{growth} is derived from our measurements of the relationship between competitive and growth rate advantage (

_{growth} = 0 in the superinfecting scenario. This is because of the effect outlined in

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 [

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. _{mut} = _{res} = 100), then the superinfection-excluding mutant is two orders of magnitude more likely to fix than the expectation based on its initial frequency _{fix} ≪ 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.

(a) The probability _{fix} of a mutant which prevents superinfection fixing in a population that allows it, as a function of mutant burst size _{mut}. (b) The probability _{fix} of a mutant which allows superinfection fixing in a population that prevents it, as a function of mutant burst size _{mut}. 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 Δ_{fix} is given by _{fix} represent the total number of simulations and the number of simulations where the mutant fixes respectively. Error bars in the _{growth} that each burst size corresponds to. These are calculated by fitting a linear relation to growth rate measurements such that _{growth} = _{mut} − _{res}). The fractional error on the _{growth} is then equal to the fractional error on the fitted gradient

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 [_{growth} < −7%), the superinfection-excluding mutant still fixes more often than a neutral superinfecting mutant (_{growth} > 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 [

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 [

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 [

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 [

Consistently with previous work [_{e} (Moran model [

We track the viral heterozygosity

To determine the generation time _{0}, 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) [_{0} can be calculated as
_{t} represents the proportion of individuals (in our case, phage) surviving to age _{t} represents the average number of offspring produced at age

There are two mechanisms in our simulations by which phages can ‘die’ when superinfection exclusion applies: either by decaying with rate _{ss}. In a sufficiently small timestep Δ_{ss}Δ_{t} = (1 − _{ss}Δ^{t/Δt}.

The average number of offspring _{t} produced at age _{t} is given by the probability of successfully infecting a viable host in a timestep Δ_{ss}Δ

In the limit where Δ

Then the generation time ^{−6}, _{0} = 1000, which leads to _{ss} = 681 and a generation time of

For comparison, coliphage T7 in liquid culture typically has parameters of ^{−9} ml/min and _{0} ≈ 10^{6} − 10^{8} ml^{−1}, thereby yielding an _{0} ≈ 10^{−3}−10^{−1} min^{−1} [_{0} = 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

We start by defining a selective advantage _{growth} in terms of the exponential growth rate _{mut} of the mutant phage population relative to that of the resident phage _{res} [

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 _{comp} of the mutant phage by tracking the relative growth of mutant and resident populations, so that
_{mut}(_{res}(_{comp} is determined from the average of 200 simulations. Importantly, in contrast to _{growth}, this competitive selective advantage (_{comp}) can in principle differ between superinfecting (_{S}) and superinfection-excluding (_{SX}) phage populations. In the absence of any interactions between the two competing phage populations, _{growth} and _{comp} are typically expected to be the same.

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 _{fix} is determined from the fraction of simulations where the mutant fixed, _{fix}, over the total number of simulations run, _{fix} = _{fix}/_{fix} is given by _{fix} is given by _{fix} ≪

The steady-state phage population _{ss} reached does not depend on the initial number of phage _{0} in the simulations. In all, ^{−6}, _{0} = 1000.

(EPS)

The selective advantage _{growth}, _{SX} and _{S}, ^{−6}, _{0} = 1000. _{growth} determined from 500 simulations, and _{comp} determined from 200 simulations. Error bars are given by the standard error on the mean of the simulations.

(EPS)

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 ^{−6}, _{0} = 1000.

(EPS)

Linear fit to log transformed heterozygosity data, with slope Λ ≡ 2/_{e} revealing that allowing superinfection (red) results in a larger effective population size compared to the case where superinfection is prevented (blue). Parameters used were ^{−6}, _{0} = 1000. Data obtained is the average of 1000 independent simulations.

(EPS)

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

(PDF)

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)

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)

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

(PDF)

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)

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.

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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

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”

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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.

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Reviewer's Responses to Questions

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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PCOMPBIOL-D-21-02234R1

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

Dear Dr Hunter,

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