THE TRANSMISSION-VIRULENCE TRADE-OFF AND SUPERINFECTION: COMMENTS TO SMITH

Authors


A recent study by smith (2011) challenges the transmission-virulence trade-off hypothesis of virulence evolution. Although we think that the experimental results are of good quality and informative, we disagree with their interpretation; we do not think they can be used to support or refute this hypothesis. In this commentary, we express two conceptual and one methodological reservations. We also propose a mathematical analysis, which we hope can help in clarifying the problems and in appropriately formalizing empirical tests.

The transmission-virulence trade-off hypothesis is one of the few adaptive explanations of virulence evolution. It states that virulence evolution is driven by an overall positive correlation between parasite transmission and virulence. Virulence may evolve to an intermediate optimal level if transmission is a saturating function of virulence, meaning that beyond the optimal level further increases of virulence will be selected against because they decrease parasite fitness (Anderson and May, 1982; Ewald, 1983; Alizon et al., 2009). If the trade-off shape is different, then virulence will evolve either to become maximal or minimal. Most mathematical models involving the trade-off hypothesis have been developed for horizontally and directly transmitted parasites that decrease host survival. Adding other routes of transmission or adding more detailed processes in the model that interact with host survival (e.g., immunopathology, Day et al., 2007) are predicted to affect the exact value of the optimal level of virulence.

It has become increasingly popular to empirically challenge the trade-off hypothesis, which is obviously desirable. There are two ways in which one may wish to question this hypothesis. The first is to ask whether it has any relevance in explaining the level of virulence of a given parasite. This can be done by evaluating the empirical support for a relationship between transmission and virulence across different genotypes of the same parasite. If, for example, there is no relationship between transmission and virulence in a biological system then the trade-off hypothesis is irrelevant in explaining the levels of virulence of that system. This approach, which we refer to as the “assumptions challenge,” led to further empirical support for the trade-off hypothesis (Fraser et al., 2007; de Roode et al., 2008).

A second, more subtle, kind of challenge is to ask what is the importance of the trade-off hypothesis among a number of other hypotheses in explaining levels of virulence of a given or across several host–parasite systems. The trade-off hypothesis may be relevant but still relatively unimportant. Other processes may under most biologically relevant conditions have a larger influence on virulence levels. Obviously for such tests the experimental procedure will depend on the alternative mechanisms. We refer to this as the “predictions challenge” because it is performed by evolving populations under different conditions that should lead to different evolutionary trajectories of virulence if there is a transmission-virulence trade-off.

SMITH’S STUDY

Smith challenges the trade-off hypothesis by studying the evolution of virulence of bacterial plasmids in the presence or absence of uninfected bacteria. Under both conditions plasmids are transmitted vertically. Moreover, under the first condition plasmids should be mostly transmitted by infecting uninfected bacteria, what smith calls “infectious transmission.” Under the latter condition, plasmids can only be transmitted through superinfection, that is by infecting already infected bacteria and replacing the resident plasmid. The first situation is meant to represent a situation in which plasmid virulence evolves only under the forces involved in the trade-off hypothesis, that is the relationship between virulence and transmission. The second situation, horizontal transmission through superinfection only, would represent an alternative to the trade-off hypothesis. Smith thus develops a predictions challenge as described above: by evaluating the relative evolution of virulence under the two conditions, he aims at evaluating the relative role of the trade-off hypothesis in explaining the evolution of virulence of these plasmids.

His main conclusion is the following: “Highly virulent plasmid genotypes evolved in all populations of the no-immigration treatment and became abundant enough to be sampled (Fig. 4), to reduce population growth (Fig. 2), and to select for resistant hosts (Fig. 6). These results are inconsistent with the trade-off hypothesis, which predicts that virulent plasmid mutants should have been selected against in no-immigration treatment populations because infectious transmission only increases plasmid fitness when there are uninfected hosts.”

The reading of this article motivated this note because even though we find that the experimental results obtained by smith on the evolution of plasmid virulence are of very good quality and informative, we disagree in many respects with their interpretation.

THE CONCEPTUAL PROBLEMS

We have one methodological and two conceptual problems. We first expose the conceptual problems, then develop a mathematical framework that allows to better visualize and understand them. Finally, we discuss the methodological problem.

Our first concern is that smith opposes the trade-off and within-host processes in general, as illustrated by two sentences in page 2: “A major alternative to the trade-off hypothesis is that virulence is determined by competition among pathogens within hosts.” and “Either way, pathogens are only predicted to maximize R0 if within-host competition is an unimportant component of pathogen fitness. Most applications of the trade-off hypothesis thus assume that within-host competition is an unimportant determinant of virulence evolution.” First, complexifying the epidemiology (e.g., adding vertical transmission or frequency-dependent processes) can indeed render R0 inappropriate to estimate parasite fitness at the between-host level. However, this is not a problem for the trade-off hypothesis, it is a problem of whether R0 adequately represents parasite fitness. Second, although within-host processes can be opposed to the trade-off hypothesis, it is not true that they are always opposable. For example, most multiple infection models assume a trade-off and within-host processes are readily included in evolutionary epidemiology (Nowak and May, 1994; van Baalen and Sabelis, 1995; Gandon et al., 2002; Alizon, 2008).

Opposing infectious transmission to superinfection is particularly questionable as the latter can be seen as yet another transmission route: infecting infected hosts instead of uninfected hosts. Thus opposing these two processes does not necessarily imply opposing superinfection to the trade-off hypothesis. To oppose superinfection to the trade-off hypothesis, one needs to show that it has consequences on virulence in a way that is independent from transmission. We do not see anything in smith (2011) aiming at separating such effects. In particular, we believe that the experimental measures of superinfection include infectious transmission effects. In the mathematical analysis we propose below, we try to make the relationship between the two processes clear, and present the issues related to assumptions or predictions challenges.

Our second, minor, concern has to do with how virulence is expressed and measured. In his study, smith states that virulence is expressed as a reduction in the reproductive rate of bacteria (p. 2). However, in the experiments, virulence is measured as the reduction of host fitness, that is a measure that in principle includes effects on both reproduction and survival. The distinction between effects on reproduction and survival is difficult to make in bacteria: they die when they reproduce, and decreases of generation time also decrease life duration and hence the duration of infection. Nevertheless, it is important to keep in mind that the trait on which virulence is expressed may affect the predictions, and we briefly show how in the mathematical model exposed below.

To address these two concerns (both processes acting simultaneously and virulence definition), we develop a mathematical model to study the evolution of virulence of a parasite that can be transmitted horizontally and vertically (see the Appendix for the equations governing epidemiological dynamics). It is inspired from the model developed by smith with three main differences. First, we use an “epidemiology Price equation” approach (Day and Proulx, 2004; Day and Gandon, 2006), which predicts the evolutionary dynamics of a trait in response to a perturbation depending on the correlations between the trait of interest and other traits. Perturbations may correspond to for example the addition of uninfected hosts, and the trait of interest can be parasite virulence, as in smith’s experimental setup. In his model, smith assumes no correlation between any pair of traits, and in particular no correlation between virulence and infectious transmission. Second, we use a logistic growth rate for bacteria instead of modeling glucose concentration directly, which allows us to simplify the system. Finally, the model we develop ignores the transfers, which would reflect conditions met in a chemostat. The second and third points should not affect the results qualitatively.

Results

SELECTIVE PRESSURES ON VIRULENCE EVOLUTION

We assume a classical susceptible-infected (SI) epidemiological model with superinfection in which we allow for vertical transmission (Fig. 1). In modeling superinfection, we assumed that it can be decomposed in three processes (see Appendix for details): superinfection leading to the replacement of plasmid j by plasmid i will occur at a rate βiεi(1 −ρj) defined by the rate at which i is infectiously transmitted (βi), the rate at which it takes hosts over (εi) and the rate at which the resident plasmids resist superinfection (ρj). We assume for simplicity that these three processes are independent. Of course this can be modified to match specific biological situations.

Figure 1.

Epidemiological model. Susceptible hosts (S) reproduce at a rate that decreases with total population size (N). They can be infected by parasites of strain i or j at a rate βi or βj. Infected hosts die at a rate α and suffer a decrease s in their reproduction rate. Finally, infected hosts can infect hosts already infected by another strain at a rate that depends on their infectious transmission rate (βi) and their within-host competitiveness (εi). Note that the resistance of parasite jj) decreases the superinfection rate of other parasite strains.

The Price equation approach (Day and Proulx, 2004) allows us to predict the evolution of the parasite host exploitation strategy (denoted x), which is here taken to be the decrease in host fitness (as measured in the experiments). In the following, we will refer to x as virulence. After some calculations based on Day and Gandon (2006) described in Appendix, we find that the rate of change of the average value of virulence (inline image) in the virus population is governed by the equation

image(1)

where the σ indicate genetic covariances between two plasmid traits, r is the reproduction rate of uninfected bacteria, K is the bacteria maximum population size, U is the density of uninfected bacteria, and IT is the total density of infected bacteria.

The four terms in the right-hand side of equation (1) represent how virulence evolution is affected by the relationship between virulence (x) and, respectively, parasite effects on host fecundity (s), on host survival (α), “infectious transmission” (β), and finally superinfection. Genetic covariances between plasmid traits allow us to predict how virulence should evolve in response to perturbations in the host population. More precisely:

  • • σx, s is the correlation between virulence and the decrease in host fecundity. If the parasite does not affect host fecundity this term is nil. If there is an effect, we expect it to be positive because plasmids do not increase bacteria reproduction rate in smith’s experiment. For a host population close to its carrying capacity, this term becomes negligible (decreasing host reproduction has little effect because hosts have few offspring). On the other hand, for an expanding population, decreasing host reproduction is very costly for the parasite because of its vertical transmission.
  • • σx, α is the correlation between virulence and the parasite-induced decrease in host survival rate. If the parasite does not affect host survival this term is nil. Otherwise, we expect it to be positive.
  • • σx, β is the correlation between virulence and horizontal transmission to susceptible hosts (infectious transmission). This is usually what is referred to as the transmission-virulence trade-off. If there is a trade-off, we expect the correlation to be positive; otherwise zero virulence would maximize transmission. Note that the reverse condition is not necessarily true: there can be a positive correlation with transmission being a saturating function of virulence (which is necessary to have an optimal intermediate level of virulence).
  • • σx, βε and σx, ρ are both related to superinfection (i.e., transmission to infected hosts). The former term reflects the correlation between virulence and the ability to take over already infected hosts. The latter is the correlation between virulence and the ability to resist superinfection by another strain. We expect both of them to be positive; otherwise there would be no cost on the evolution of these parameters. Note that this term is scaled by the total number of infected hosts.

DISENTANGLING TRANSMISSION AND SUPERINFECTION

The main feature of equation (1) we wish to highlight is that the infectious transmission rate (β) occurs in the superinfection term. This illustrates our concern that the transmission-virulence trade-off and superinfection are difficult to oppose. More precisely, the term that raises the most acute problem is σx, βε. The fact that two variables (βi and εi) are aggregated into the covariance with virulence (x) opens the door to potential interactions between traits. In other words, even if there is no correlation between virulence and the infectious transmission rate (σx, β= 0) and there is a correlation between virulence and within-host competitiveness (σx, ε > 0), the infectious transmission rate could still have an effect through its interaction with within-host competitiveness. In fact, the only situation where there is no problem is when all parasites have the same infectious transmission rate (β). As soon as there is variance both in β and in ε, superinfection and the transmission-virulence trade-off become inextricable.

Let us now consider the predictions in terms of virulence evolution under the two conditions tested experimentally by smith, that is either high input of uninfected hosts (U≫ 0) or no input of uninfected hosts (U≈ 0). If U= 0 (N treatment), then the rate of increase in virulence is given by

image(2)

If the covariances are positive, the first two terms of this equation are negative, which implies that if virulence is observed in the N treatment, it is due to superinfection.

If U > 0 (I treatment), two extra terms appear (Uσx, β and Urσx, s/K) and the rate of change in virulence is given by equation (1). We thus expect virulence to be higher in the I treatment than in the N treatment if it increases transmission to uninfected hosts or if it decreases host fecundity. This means that even if there was no effect of virulence on the infectious transmission rate (σx, β= 0), we would still expect the input of uninfected hosts to have a positive effect on virulence, if virulence is at least partially expressed through decreases of fecundity (σx, s > 0).

Results are summarized in Table 1. Smith’s result that virulence is observed in the N treatment means that superinfection must be at play (i.e., σx, βε > 0 and/or σx, ρ > 0). However, it is impossible to rule out the trade-off (i.e., to perform an assumptions challenge by showing that σx, β= 0) because the case in which both processes are at play yields identical results as the case with only superinfection.

Table 1.  Qualitative predictions for all scenarios. The number of + indicates the level of virulence. In some cases, virulence can be observed or not depending on whether superinfection alone is sufficient to overcome the cost of host death. For the I treatment, we assume that U is sufficiently large that terms affecting the rate of change of virulence negatively are likely to be overcome.
  Superinfection onlyTrade-off onlyBoth
 No effect
x, βx, βεx, ρ= 0)
x, βε > 0 and/or σx, ρ > 0, σx, β= 0)x, βεx, ρ= 0, σx, β > 0)x, βε > 0 and/or σx, ρ > 0, σx, β > 0)
N treatment (U= 0)00 or +00 or +
I treatment (U≫ 0)00 or ++  

SURVIVAL VERSUS REPRODUCTION

One of the concerns we have with smith’s study is that virulence is measured through a decrease in growth rate but he refers to it as a decrease in reproduction rate. If virulence is only expressed through a decrease in host reproduction (i.e., σx, α= 0), then the expression for the rate of change in virulence is

image(3)

In this case, the term linking host mortality to virulence disappears. Because this term negatively affects virulence evolution, virulence is more likely to be observed in all the treatments. Note that again, increasing the availability of uninfected hosts is expected to increase virulence both with or without a trade-off.

On the contrary, if virulence is only expressed through host survival (i.e., σx, s= 0) or if the parasite is not transmitted vertically, then the rate of change in virulence is governed by

image(4)

In this case only is the testing of the role of the trade-off possible with the N and I treatments, because increasing the availability of uninfected hosts should increase virulence only if there is a correlation between virulence and infectious transmission.

THE METHODOLOGICAL PROBLEM

Setting the conceptual problems mentioned above aside, we concentrate on a methodological problem that we think is crucial to the goal of smith (2011). The goal is to evaluate the relative importance of infectious transmission and superinfection in determining virulence evolution. Unfortunately, as the author states in the “plasmid sampling” section: “It should be noted that this sampling method is likely biased toward plasmids with high infectious transmission rates. Sampled plasmids may therefore not reflect the full distribution of traits in experimental populations.” We do not know how important that bias may be, but potentially it has important consequences on the measures and interpretation of results. Preferentially sampling plasmids with high infectious transmission rates not only biases the mean but also strongly reduces the variance of transmission rates. This bias may explain the much smaller range of variation observed for infectious transmission rates relative to superinfection rates (see Fig. 5 in smith, 2011), and potentially affects the magnitude of the correlation between virulence and infectious transmission.

Discussion

Contrary to what many believe, the trade-off hypothesis is compatible with other processes. In his study, smith presents several times superinfection and the trade-off as mutually exclusive hypotheses (although in other sections of the article this is not the case), which goes against existing theory on multiple infections and virulence evolution (Nowak and May, 1994; van Baalen and Sabelis, 1995; Gandon et al., 2002). In fact, it has been shown that many processes such as immunopathology (Day et al., 2007) or vaccination (Gandon et al., 2001) alter the optimal level of virulence without invalidating the trade-off hypothesis.

Predicting virulence evolution can be nontrivial for parasites with complicated life cycles. Here, plasmids can be transmitted both vertically and horizontally. In the latter case, they can infect naïve hosts or superinfect, thus making a mathematical framework necessary. Such a framework highlights two problems with smith’s study. First, it is impossible with this experimental setup to conclude that transmission is irrelevant to virulence evolution, although one may conclude that superinfection affects virulence (because plasmids decrease host fitness in all the treatments). The second, minor, problem is that smith actually measures virulence as a decrease in host growth rate (by competing infected and uninfected cells) and not in reproduction as stated in his text. Unfortunately, model predictions may depend on how virulence is expressed. For instance, if one assumes no effect of parasite infection on host survival, then for high inputs of uninfected hosts there are no selective pressures against virulence. On the contrary, infection-induced reductions of host survival will always negatively affect the evolution of virulence.

Smith challenges the trade-off hypothesis with what we called a “predictions challenge”, that is, he tries to evaluate the relative importance of the two processes by evolving the populations under different conditions. We show that in the case of superinfection this is likely to be complicated because, unless the infectious transmission rate is constant among plasmid strains, virulence evolution depends on the interaction between infectious transmission rate and within-host competitiveness such that the superinfection and the trade-off hypothesis become inextricable. From our point of view, an “assumptions challenge,” that is, a challenge based on evaluating the genetic correlations between traits, seems more promising with this experimental set-up because it considers each correlation separately, thus allowing to control for interaction terms. For example, by dividing smith’s superinfection measure of strain i (as shown in his Fig. 5b) by its infectious transmission measure (as shown in his Fig. 5a), one can obtain an estimation of the within-host competitiveness of this strain (εi) scaled by the resistance of the ancestral plasmid used for the superinfection measure. Further analyses to study the correlations between virulence and within-host competitiveness or between virulence and resistance could yield interesting results. However, this will require to address the methodological bias mentioned by smith in the Materials and Methods section of his paper: the sampling design selects strains with high infectious transmission, which likely modifies the correlation between virulence and infectious transmission.

In the end, smith’s experimental results on virulence evolution do show that superinfection is necessary to explain patterns of virulence evolution but they do not invalidate the trade-off hypothesis. These results also provide empirical support for a positive correlation between virulence and superinfection, which is an assumption at the root of many epidemiology models (e.g., Gandon et al., 2001, 2002). We hope that this discussion will help clarifying ways in which the respective roles of within-host competitive ability and the trade-off hypothesis can be adequately evaluated.


Associate Editor: P. Turner

ACKNOWLEDGMENTS

We are indebted to T. Day for pointing out the simplicity of the Price equation approach to us. We also thank S. Lion, j. smith, and P. Vale for helpful comments. We thank the CNRS and the IRD for funding.

Appendix

APPENDIX :  The Model

The “epidemiology Price equation” approach we used was introduced by Day and Gandon (2006) following the framework developed by Day and Proulx (2004). The idea is to model the parasite population as a diverse population to follow the dynamics of average trait values. The great advantage of this approach is that it follows variations in population densities and in trait value at the same time. This allows us to track short-term evolution.

The dynamics of the system with uninfected bacteria (U) and bacteria infected by a plasmid strain i (Ii) is governed by the following set of ODEs:

image((A-1a))
image((A-1b))

where λ is the input rate of uninfected hosts, r is the base-line bacteria reproduction rate, N is the total population size, K is the maximum bacterial population size, βi is the infection rate of strain i, si is the decrease in fecundity due to the infection, αi is the decrease in survival due to the infection, εi is the within-host competitiveness of strain i (i.e., its ability to take over an infected host), and ρi is its ability to resist superinfection by another strain. Note that as Smith we assume perfect vertical transmission.

The equation of interest is equation (A-1b). If we introduce the average value of any parasite trait y as inline image, where IT is the total density of infected hosts, it can be simplified into

image((A-2))

The dynamics of the total density of infected hosts (IT=∑iIi) is given by

image((A-3))

Note that superinfection terms cancel out as expected because superinfection does not imply any increase or any loss of infected hosts.

We want to study the dynamics of a parasite-related trait x over time. By definition,

image((A-4))

where pi=Ii/IT is the fraction of infected hosts infected by strain i.

From equation(A-4),

image((A-5))

If we assume that the trait of strain i does not change over time, inline image. Furthermore,

image((A-6))

By combining equations (A-2) and (A-3,) we find after some calculations that

image((A-7))

Going back to equation (A-5) and assuming that inline image, we have

image((A-8))

By definition, inline image is the covariance between x and y. Equation (A-8) can thus be written more simply as

image((A-9))

where the σ are the covariances.

Ancillary