Open Access

Phenotypic mixing and hiding may contribute to memory in viral quasispecies

BMC Microbiology20033:11

DOI: 10.1186/1471-2180-3-11

Received: 28 April 2003

Accepted: 9 June 2003

Published: 9 June 2003

Abstract

Background

In a number of recent experiments with food-and-mouth disease virus, a deleterious mutant, RED, was found to avoid extinction and remain in the population for long periods of time. Since RED characterizes the past evolutionary history of the population, this observation was called quasispecies memory. While the quasispecies theory predicts the existence of these memory genomes, there is a disagreement between the expected and observed mutant frequency values. Therefore, the origin of quasispecies memory is not fully understood.

Results

We propose and analyze a simple model of complementation between the wild type virus and a mutant that has an impaired ability of cell entry, the likely cause of fitness differences between wild type and RED mutants. The mutant will go extinct unless it is recreated from the wild type through mutations. However, under phenotypic mixing-and-hiding as a mechanism of complementation, the time to extinction in the absence of mutations increases with increasing multiplicity of infection (m.o.i.). If the RED mutant is constantly recreated by mutations, then its frequency at equilibrium under selection-mutation balance also increases with increasing m.o.i. At high m.o.i., a large fraction of mutant genomes are encapsidated with wild-type protein, which enables them to infect cells as efficiently as the wild type virions, and thus increases their fitness to the wild-type level. Moreover, even at low m.o.i. the equilibrium frequency of the mutant is higher than predicted by the standard quasispecies model, because a fraction of mutant virions generated from wild-type parents will also be encapsidated by wild-type protein.

Conclusions

Our model predicts that phenotypic hiding will strongly influence the population dynamics of viruses, particularly at high m.o.i., and will also have important effects on the mutation-selection balance at low m.o.i. The delay in mutant extinction and increase in mutant frequencies at equilibrium may, at least in part, explain memory in quasispecies populations.

Background

RNA viruses are important pathogens in humans, animals, and plants. Their short generation time and high mutation rates allow them to adapt rapidly to changing environmental conditions, and make them ever-changing targets for anti-viral therapies or vaccinations. RNA viruses form highly polymorphic clouds of mutants, so-called quasispecies [13]. At viral loads of 108 or more virions in an infected host, viral quasispecies in vivo often contain all possible single-point mutants from the consensus sequence, as well as a sizeable fraction of two- or three-point mutants [4]. Besides their importance as pathogens, RNA viruses have in recent years also become one of the main tools for experimental verification of theoretical population genetics and evolutionary theory in general (reviewed in [3]). This line of research can draw on a substantial body of literature on the theory of quasispecies dynamics and its relation to standard population genetics [516].

The quasispecies theory has been useful to understand RNA virus evolution, but the assumptions underlying the model do not always reflect the biology of RNA viruses. In some instances the original model has been corrected to account for discrepancies, such as infinite vs. finite population sizes [7, 9, 16]. However, other factors need to be considered as well. For instance, the standard quasispecies model assumes low multiplicity of infection (m.o.i.), that is, each cell is assumed to be infected by at most one virion. At high m.o.i., when several virions coinfect a single cell, the virions can recombine [1719] or reassort (for segmented viruses) [2025]. Second, the model assumes that the genotype of a virion fully determines its fitness. However, this assumption is not always justified, since many viral functions can be provided in trans – an effect referred to as complementation. Thus, the phenotype of a virion may not reflect its genotype [2629]. Phenotypic mixing and hiding is a particular case of complementation involving surface proteins (above citations). It occurs either at high m.o.i., when two or more different mutants or virus strains freely exchange genetic sequences and surface proteins inside a cell, or at low m.o.i., when a mutant offspring is encapsidated by its parent's proteins.

No general theory is available for the evolutionary dynamic of a viral quasispecies under the influence of virus density as it changes with changing m.o.i. However, some particular cases have been studied [3032], mostly related to recombination and reassortment [3336] and to the dynamic of defective interfering particles [3741]. In this paper, we modify the quasispecies model by incorporating phenotypic hiding between wild-type virus and a deleterious mutant. We assume that the mutant differs from the wild type in a surface protein necessary for cell entry, so that the fitness difference between two virions is solely determined by the surface proteins, and not by the particular genomes they carry. We show that phenotypic hiding of the mutant genome behind wild-type capsids increases the mutant frequency above the level predicted by the standard quasispecies model. This increase in mutant frequency is most pronounced at high m.o.i., but exists also at very low m.o.i. Moreover, we show that the decrease of mutant frequency from an initially high level down to the level of mutation-selection balance is slowed down substantially at high m.o.i. because of phenotypic hiding.

We propose that our model may explain some of the quantitative differences observed between theoretical predictions of the quasispecies model and experimental data published in the literature [4245]. Foot-and-mouth disease virus (FMDV) has been one of the most productive models of viral quasispecies dynamics (reviewed in [46]). FMDV belongs to the Picornaviridea family. It has a non-segmented, positive strand genome of approximately 8.5 Kb encapsidated by viral proteins, VP1, VP2, VP3 and VP4. A conserved Arginine-Glycine-Aspartic acid (RGD) triplet in βG-βH loop of VP1 recognizes the αvβ3, αvβ6 or αvβ1 integrin as viral receptor (reviewed in Ref. [47]), the first necessary step for virus entry. Work carried out with different RGD virus mutants and mimic peptides [4852] clearly demonstrates the critical role that the RGD triplet has in virus entry, and its little tolerance to variation. Among the few RGD mutants that have been isolated, on is RED, a low fitness mutant in which the glycine is replace by glutamic acid [53]. Ruíz-Jarabo and co-workers found that after revertant RGD genomes became dominant, and mutant genomes were no longer detected in the consensus sequence, RED was able to remain in the population for very long times and at surprisingly high frequencies [4245]. There is a qualitative agreement between the predictions of the quasispecies model and this result, but there is also a quantitative discrepancy. RED memory genomes can be observed at much higher frequency than expected. However, phenotypic hiding promotes a decelerated decay of mutant genomes in the population, and this decelerated decay could be interpreted as a memory effect, in agreement with the results by Ruíz-Jarabo et al. The research design of the original work [4245] does not allow us to conclusively accept or reject our model as the correct explanation for their observations, but the values of mutant frequencies predicted by our model are consistent with the experimental data.

Model

We assume that the virus is propagated under serial transfer, and model the change in mutant concentrations from passage to passage. We assume that the difference between wild type and mutant lies in the capsid proteins, such that the mutation affects the ability of the virus to enter a cell, but not the viability of the virus inside the cell. Under phenotypic mixing and hiding, there are four types of virions that can occur: wild-type genotype with wild-type phenotype, wild-type genotype with mutant phenotype, mutant genotype with wild-type phenotype, and mutant genotype with mutant phenotype. We will call virions whose genotype coincides with their phenotype as pure, and virions for which genotype and phenotype do not coincide as mixed. In the following paragraphs, we outline the derivation of a set of deterministic equations that describe the change in relative concentrations of the different pure and mixed virions in an infinite virus population. More details are given in the Methods section. Our model is similar to models studied by Szathmáry [38, 39].

The probability p ww that an offspring virion is pure wild type is p ww = (1 - r)x w + r https://static-content.springer.com/image/art%3A10.1186%2F1471-2180-3-11/MediaObjects/12866_2003_Article_84_IEq1_HTML.gif [Methods section, Eq. (14)], where x w is the probability with which a particular virion that infects a cell contains the wild-type genome, and r is a parameter related to the multiplicity of infection. Likewise, the probabilities p wm and p mw that an offspring virion is mixed of either type are p wm = p mw = rx w x m [Eq. (15)], where x m = 1 - x w , and the probability p mm that an offspring virion is pure mutant is p mm = (1 - r)x m + r https://static-content.springer.com/image/art%3A10.1186%2F1471-2180-3-11/MediaObjects/12866_2003_Article_84_IEq2_HTML.gif . The formula that relates the parameter r to the multiplicity of infection, Eq. (13), is somewhat unwieldy. For reference, and in order to provide an intuitive meaning for r, we have tabulated r (which we will call in the following also the mixing parameter) versus the m.o.i. in Table 1.
Table 1

Mixing parameter r as a function of the m.o.i.

m.o.i.

r

m.o.i.

r

m.o.i.

r

0.01

0.002

1.0

0.233

20

0.947

0.1

0.025

2.0

0.423

30

0.965

0.2

0.049

3.0

0.567

40

0.974

0.3

0.074

4.0

0.670

50

0.980

0.4

0.098

5.0

0.742

60

0.983

0.5

0.121

6.0

0.792

70

0.986

0.6

0.144

7.0

0.828

80

0.987

0.7

0.167

8.0

0.853

90

0.989

0.8

0.189

9.0

0.872

100

0.990

0.9

0.233

10.0

0.887

200

0.995

The probabilities x w and x m depend on the composition of the types of free virions on infection. Assume that y ww is the concentration of pure wild-type virions at passage t, y wm that of mixed virions with wild-type genotype and mutant phenotype, y mw that of mixed virions with mutant genotype and wild-type phenotype, and y mm that of pure mutant virions. Further, assume that virions with wild-type phenotype infect cells at a rate γ w , while virions with mutant phenotype infect at a rate γ m < γ w . Then, the total rate at which cells are infected with wild-type genomes is γ w y ww + γ m y wm . Likewise, the total rate at which cells are infected with mutant genomes is γ w y mw + γ m y mm . Therefore, we have x w = (γ w y ww + γ m y wm )/[γ m (y wm + y mm )]. The probability x m follows from x m = 1 - x w .

Using the various formulas given in the previous two paragraphs, we find that the virion concentrations at passage t + 1 (primed quantities) are given by

https://static-content.springer.com/image/art%3A10.1186%2F1471-2180-3-11/MediaObjects/12866_2003_Article_84_Equa_HTML.gif

We now introduce mutations into this model. We assume that the wild type mutates into the mutant with probability μ. For simplicity, we assume that back mutations occur at the same rate. (It is safe to make this assumption, because the rate of back mutations from deleterious to advantageous mutants tends to have little effect on mutation-selection balance.) Mutations will typically not affect the phenotype of the offspring virion, which is determined by the distribution of mutant or wild-type capsid proteins in the cell. The extended model with mutations therefore becomes

https://static-content.springer.com/image/art%3A10.1186%2F1471-2180-3-11/MediaObjects/12866_2003_Article_84_Equb_HTML.gif

Results

Extinction of mutant for μ = 0

In order to evaluate whether the mutant can coexist with the wild type or will ultimately go extinct if it is not constantly regenerated through mutations, we have to analyze the fixed points of Eqs. (1a)–(1d). There are two trivial fixed points, corresponding to the cases in which all virions are either pure wild-type or pure mutant. We could not find any non-trivial fixed points. Since the expressions involved are extremely unwieldy, we searched for non-trivial fixed points only numerically (using the function NSolve of Mathematica [54]).

A linear expansion around the fixed points tells us whether the mutant can invade an established wild-type population or vice versa. If we introduce a small amount ε of mutant virions into an established wild-type population, the total amounts of wild-type genomes Y w = y ww + y wm and mutant genomes Y m = y mm + y mw after the next passage become to first order in ε:

https://static-content.springer.com/image/art%3A10.1186%2F1471-2180-3-11/MediaObjects/12866_2003_Article_84_Equc_HTML.gif

As long as γ m w < 1 (as we assume throughout this paper), the amount of mutant genomes in the population decreases in the first passage, which implies that the mutant cannot invade. Likewise, if we introduce a small amount ε of wild-type virions into an established mutant population, the total amounts of mutant and wild-type genomes after the next passage become to first order in ε:

https://static-content.springer.com/image/art%3A10.1186%2F1471-2180-3-11/MediaObjects/12866_2003_Article_84_Equd_HTML.gif
In this case, the amount of the invading wild-type genomes in the population increases, which implies that the wild-type can invade an established mutant population. The invasion of a mutant population by a small amount of wild-type virions is shown in Fig. 1.
https://static-content.springer.com/image/art%3A10.1186%2F1471-2180-3-11/MediaObjects/12866_2003_Article_84_Fig1_HTML.jpg
Figure 1

Invasion of an established mutant population by the wild type (m.o.i. = 5, γ m w = 0.8, μ = 0). Concentrations at passage 0 were y ww = 0.0001, y mm = 0.9999, y wm = y mw = 0.

The preceeding analysis shows that Eqs. (1a)–(1d) have only a single stable fixed point, the one in which the whole population consists of pure wild-type virions. The mutant will therefore always go extinct eventually.

The rate at which the mutant disappears depends on the m.o.i. Figure 2 shows the ratio between mutant and wild-type concentrations as a function of passage, for various m.o.i. Concentrations are measured with respect to the genotype, that is, mutant and wild-type concentrations are Y m and Y w , respectively. In the example given, we have γ m w = 0.3. At low m.o.i., the mutant-to-wild-type ratio drops from 1 at passage zero to values of 10-2 and lower after four passages. At high m.o.i., the reduction in mutant concentration proceeds much slower. For example, at an m.o.i. of three, the mutant-to-wild-type ratio is only 10-1 after four passages. An interesting aspect of Fig. 2 is that the reduction in mutant frequency during the first passage is independent of the m.o.i. This effect is a consequence of our assumption that the mutant has impaired ability of cell entry. In the first round of infection, when all virions are pure wild type or pure mutant, the mutant virions cannot benefit from phenotypic hiding. Therefore, their frequency will decrease corresponding to the true fitness difference to the wild type, independent of the m.o.i. From the second passage onward, the balance between pure and mixed virions is established, and further reduction of the mutant frequency is slowed down at high m.o.i.
https://static-content.springer.com/image/art%3A10.1186%2F1471-2180-3-11/MediaObjects/12866_2003_Article_84_Fig2_HTML.jpg
Figure 2

Ratio Y m /Y w of mutant to wild type as a function of passage for various m.o.i. (γ m w = 0.3, μ = 0). Concentrations at passage 0 were y ww = y mm = 0.5, y mm = y mw = 0.

Figure 3 shows the individual concentrations of the various pure and mixed virions at an m.o.i. of five. We observe that the concentration of pure mutant virions decreases much faster than that of mixed virions with mutant genomes. After ten passages both concentrations are equal, and after fifty passages, there are more than twice as many mixed virions with mutant genomes than there are pure mutant virions. When the mutant is rare, it is mostly hiding behind wild-type capsids, which slows down its extinction. Interestingly, the fraction of mutant genomes encapsidated with wild-type protein is equal to the fraction of wild-type genomes encapsidated with mutant protein at zero mutation rate. This symmetry is a consequence of the symmetry of equations (1b) and (1c), and disappears for μ > 0.
https://static-content.springer.com/image/art%3A10.1186%2F1471-2180-3-11/MediaObjects/12866_2003_Article_84_Fig3_HTML.jpg
Figure 3

Relative concentrations of pure and mixed virions at m.o.i. = 5 (γ m w = 0.8, μ = 0). Concentrations at passage 0 were y ww = y mm = 0.5, y mm = y mw = 0.

Mutation-selection balance for positive μ

We now turn to the general case of a positive mutation rate μ. We first consider the case of vanishingly low m.o.i., r = 0. When we set r = 0, Eq. (2) becomes the discrete-time version of a model of maternal effects in quasispecies evolution [55]. In Ref. [55], the fitness of a virion is assumed to be the product of a contribution from the genotype of the virion's parent and one from the virion's own genotype. The situation here is simpler, because we assume that fitness is only determined by the phenotype (efficiency of cell entry). In the language of Ref. [55], this means we have b i = 1 for all genotypes i. From Eqs. (5) and (6) in Ref. [55], we obtain the steady-state concentrations of pure and mixed virions (neglecting back mutations from mutant to wild type):

https://static-content.springer.com/image/art%3A10.1186%2F1471-2180-3-11/MediaObjects/12866_2003_Article_84_Eque_HTML.gif

We mentioned in the previous section that y wm = y mw for μ = 0. Clearly, this symmetry does not exist for μ > 0. For 0 < μ << 1, the ratio in concentration of the two classes of mixed virions is y mw /y wm = (1 - μ - γ m w )/μ ≈ 1/μ. There will typically be several orders of magnitude more mutant genomes encapsidated in wild-type protein than wild-type genomes encapsidated in mutant protein. Note, however, that this result holds only under the assumption of vanishingly low m.o.i. As the m.o.i. increases, the mixed virions with mutant genomes will still be at higher concentration than the mixed virions with wild-type genomes, but the ratio between the two will be much smaller than 1/μ. The ratio of mixed virions with mutant genomes to pure mutant virions is y mw /y mm ≈ 1 - γ m w < 1 at low m.o.i., that is, there are more pure virions than mixed virions with mutant genomes in equilibrium. We found numerically that at high m.o.i., this relationship is reversed, and there are substantially more mixed than pure mutants.

For the total amount of mutant genomes in the population, we obtain after summing Eqs. (5c) and (5d):

https://static-content.springer.com/image/art%3A10.1186%2F1471-2180-3-11/MediaObjects/12866_2003_Article_84_Equf_HTML.gif

For comparison, the standard quasispecies model (without phenotypic hiding) predicts:

https://static-content.springer.com/image/art%3A10.1186%2F1471-2180-3-11/MediaObjects/12866_2003_Article_84_Equg_HTML.gif

This expression is equivalent to the one given by Ruíz-Jarabo et al. in their analysis of quasispecies memory [44]. Equations (6) and (7) differ only in the factor 2(1 - μ) - γ m w . This factor is larger than one for small mutation rates, that is, as long as

μ < (1 - γ m w )/2.     (8)

If the mutant has a substantial fitness disadvantage in comparison to the wild type (say at least 10%), then μ will always be small enough to satisfy this condition. Therefore, phenotypic hiding will lead to an increase in mutant genomes even at low m.o.i. The increase is at most a factor of two (for μ << 1 and γ m w << 1) in comparison to the prediction of the standard quasispecies model.

For the case of positive mutation rate and high m.o.i., there is no analytic solution available, and we have to resort to numerical simulations. Figure 4 shows the decay of mutant genomes over time, starting from an initial state where mutant and wild type are present in equal concentrations. We have displayed trajectories for a wide range of m.o.i., and we have also displayed the corresponding trajectory generated by the standard quasispecies model. All trajectories show the same qualitative behavior: The mutant frequency decays quickly during the initial passages, and then levels off in mutation-selection balance. As discussed in the previous paragraphs, the level of mutant genomes in mutation-selection balance is always higher in our model than in the standard quasispecies model. Moreover, the level of mutant genomes increases with increasing m.o.i. The initial decay of mutant genomes is also influenced by the m.o.i., and slows down with increasing m.o.i. In general, the initial decay of mutant genomes for μ > 0 is identical to the one for μ = 0, as described in the previous subsection. It is interesting to note that phenotypic hiding influences the equilibrium concentration of mutant genomes at low m.o.i., but not the speed at which equilibrium is attained. The initial decay of mutant genomes is identical for our model at low m.o.i. and for the standard quasispecies model.
https://static-content.springer.com/image/art%3A10.1186%2F1471-2180-3-11/MediaObjects/12866_2003_Article_84_Fig4_HTML.jpg
Figure 4

Concentration of mutant genomes Y m as a function of passage (γ m w = 0.4, μ = 10-4). Concentrations at passage 0 were y ww = y mm = 0.5, y wm = y mw = 0. Trajectories labeled "phenotypic hiding" were generated from equations (2a)–(2d). The trajectory labeled "standard quasispecies model" was generated from iterating https://static-content.springer.com/image/art%3A10.1186%2F1471-2180-3-11/MediaObjects/12866_2003_Article_84_IEq3_HTML.gif = [γ m (1 - μ) Y m + γ w μ(1 - Y m )]/[γ m Y m + γ w (1 - Y m )].

Discussion

Ruíz-Jarabo et al. measured the frequency of the minority FMDV RED mutant up to passage 50, and found that the frequency was still declining at passage 50 [44]. They observed mutant frequencies at passage 50 between 6.6 × 10-4 and 10-3. These results are in agreement with our model of phenotypic hiding if we use a relative fitness of 0.4 for the RED mutant as compared to the wild type [42, 44], and assume that (1) the fitness disadvantage is fully explained by impaired entry, (2) the mutation rate is μ = 10-4, and (3) the m.o.i. is approximately five (Fig. 4). The m.o.i. in Ref. [44] was between one and five. Ruíz-Jarabo et al. noted that mutant concentrations around 10-3 would follow from equilibrium state of the standard quasispecies model if the mutation rate were μ = 10-3 rather than μ = 10-4. However, the standard quasispecies model predicts that the mutation-selection balance should be reached around passage 10 (Fig. 4), which is at odds with the observation that the frequency of RED mutants continued to decline up to passage 50.

It is important to note that capsid proteins can be provided in trans during the production of infectious FMDV particles [51], and phenotypic hiding has been known for many years in a variety of picornaviruses [5658]. Other factors may enhance or reduce the effects of complementation. For instance, compartimentalization or physical localization should prevent to some degree the free exchange of genomes and products.

In order to test fully the predictions of our model, we would need data on the decay of the mutant frequency from replicate populations propagated at different m.o.i. Unfortunately, such data does not exist. Although Ruíz-Jarabo et al. propagated their populations at an m.o.i. of 0.1 in their initial study [42] and at an m.o.i. between one and five in their follow-up study [44], we cannot compare the mutant frequencies reported in these two studies, because the method of screening for RED mutants was changed in the follow-up study ([44], Table 1, Note b). Moreover, a caveat applies to the comparison of the experimental data to our model: Ruíz-Jarabo et al. determined the frequency of RED mutants by first screening for monoclonal antibody (MAb SD6) resistant virions, and then determining the fraction of RED mutants among them from sequence data. While this method successfully prevents against counting MAb-resistant mutants that are not RED, it does not measure the fraction of RED mutants that hide behind RGD capsids. If phenotypic hiding plays an important role in this system, then the fraction of hiding RED mutants could be substantial, and the reported mutant frequencies would likely represent underestimates. In summary, the experimental data are consistent with our model, but do not allow us to either accept or reject it with certainty. We have shown in the present paper that phenotypic hiding can increase equilibrium frequencies by factors of two to five, or even higher, for high m.o.i. It is conceivable that other types of complementation can have similar effects [59].

Phenotypic hiding can bias fitness determinations if the phenotype, rather than the genotype, is quantified. This bias occurs simply because a fraction of mutant genomes will hide behind wild-type envelopes and will be missed, an effect that has been previously reported [29, 60]. As a byproduct of our study, we predict two phases of mutant frequency decay specific to phenotypic hiding. In the first round of infection, all virions are pure wild type or pure mutant, so that the mutant virions cannot benefit from complementation. Therefore, their frequency will decrease corresponding to their true fitness and independently of the m.o.i. From the second passage onward, the balance between pure and mixed virions is established, and further reduction of the mutant frequency proceeds much slower if the m.o.i. is sufficiently high, as illustrated in Figure 2. As a consequence, if fitness competitions are halted after the first passage, measurements of the phenotype will not bias fitness determinations. These effect should not be observed in other forms of complementation, but should be characteristic of any situation in which the phenotype is determined by envelope proteins. Host range will often belong to this category, since receptor recognition is one of its main determinants. Other examples include antibody resistant mutants and escape mutants when the antiviral drug targets entry, such as rimantidine and amantidine-resistant influenza strains or WIN compound-resistant picornaviruses [61]. Thus, phenotypic hiding can keep a relatively large frequency of mutants in the absence of selective pressure that can provide a quick response when such selective pressure is restored.

Complementation can also complicate the interpretation of other experimental results involving competition between two (or more) strains. For instance, Miralles et al. [62, 63] tested the effect of clonal interference in VSV populations by changing the population size in competitions between wild type and a MARM mutant. This MARM mutant carries a point mutation that confers resistance to I1 Mab, but it is neutral otherwise [64]. Clonal interference occurs in asexual organisms when population sizes are large enough to allow the simultaneous generation of more than one advantageous mutation, so that the best mutation has to compete not only with an inferior wild type but also with the other fitter mutants [65]. Thus, while the best possible mutation should always be the one becoming dominant, the rate at which it does so should be slowed down in large populations. The VSV competition experiments were carried out at increasing viral population sizes, but the number of infected cells was held constant, that is, the m.o.i. was increased alongside the population size. Therefore, complementation may also have contributed to the slowdown of the speed of adaptation at large population sizes, and the relative contributions of complementation and clonal interference to the slowdown would be hard to disentangle. The only phenotypic hiding effect that the authors could avoid were related to the determination of changes in MARM C frequencies, and these only correlate to the changes in beneficial mutation frequencies if the beneficial mutation mapped to the G glycoprotein.

Many infections in plants and animals are probably initiated by one or few viral particles, thus corresponding to a situation of low m.o.i. As replication proceeds and titers rise, the likelyhood of coinfection increases. The best evidence for multiple infections in vivo comes from the observation of natural recombinants. In the case of HIV-1, for instance, rampant recombination has been reported, even among different subtypes [66, 67], and the number of proviruses per cell was estimated to be approximately three. Another well known example is influenza virus, for which reassortment is a major mechanism of evolutionary change [68]. As in the case of HIV, reassortment depends of coinfection, and once coinfection occurs, complementation is as likely to ensue as reassortment.

Conclusions

We have shown that phenotypic hiding has substantial influence on the equilibrium frequencies of mutants in selection-mutation balance, and on the speed at which deleterious mutants are lost from the population. Our theory makes five predictions that can be tested experimentally:
  1. 1.

    The rate at which the mutant disappears should decrease and the equilibrium concentration of the mutant should increase with increasing m.o.i.

     
  2. 2.

    A substantial fraction of mutant genomes should be encapsidated with wild-type protein.

     
  3. 3.

    The ratio of mutant genomes encapsidated in wild-type protein to wild-type genomes encapsidated in mutant protein should decrease with increasing m.o.i.

     
  4. 4.

    Even at low m.o.i., the fraction of mutant present in mutation-selection balance should be larger (up to a factor of two) than predicted by the standard quasispecies model if the mutant fitness is low compared to the wild-type fitness.

     
  5. 5.

    The decrease of mutant frequency during the first passage should be independent of the m.o.i., and equal to that measured for later passages at very low m.o.i.

     

Future experiments can be designed to test these predictions, and can thereby clarify whether phenotypic hiding is a major contributor to viral memory and elevated equilibrium frequencies of mutants.

Methods

In this section, we present some calculations that form the basis of the derivation of our model equations.

We first calculate the probability that a virion produced by an n-fold coinfected cell is pure. Assume a cell is infected by k virions with wild-type genotype, and by n - k virions with mutant genotype. Then, the probability that an offspring virion is pure wild-type is the probability that it has wild-type genotype, which is k/n, times the probability that it has wild-type phenotype, which is again k/n. Thus, the total probability is (k/n)2. If we want to know the probability that an offspring virion is pure for a number of coinfected cells with different ratios in their multiplicity of infection of wild-type and mutant genotypes, then we have to sum over the different contributions of all the individual cells. In practice, we will not know exactly which cells are infected with what virions. However, it is reasonable to assume that we know the overall probabilities with which cells are infected with either mutant or wild-type genotype. Assume that x w is the probability that a cell is infected with wild-type genotype, and x m = 1 - x w likewise for the mutant genotype. Then, the number of wild-type genomes k in a cell is binomially distributed, and therefore the overall probability ξ ww (n) that an offspring virion from a cell infected by n virions is pure wild-type becomes [38]

https://static-content.springer.com/image/art%3A10.1186%2F1471-2180-3-11/MediaObjects/12866_2003_Article_84_Equh_HTML.gif

Likewise, we find for the probability ξ mm (n) that an offspring virion is pure mutant ξ mm (n) = x w x m /n + https://static-content.springer.com/image/art%3A10.1186%2F1471-2180-3-11/MediaObjects/12866_2003_Article_84_IEq2_HTML.gif .

Now we consider mixed virions. Assume again that a cell is infected by k virions with wild-type genotype, and by n - k virions with mutant genotype. In analogy to the considerations for a pure virion, we find that the probability of a mixed virion is k(n - k)/n2, regardless of whether the genotype is wild-type and the phenotype is mutant or vice versa. The probabilities ξ wm (n) and ξ mw (n) that an offspring virion is mixed of either type are then

https://static-content.springer.com/image/art%3A10.1186%2F1471-2180-3-11/MediaObjects/12866_2003_Article_84_Equi_HTML.gif

Equations (9) and (10) are valid if all cells have exactly the same multiplicity of infection. However, infection is a Poisson process, and some cells will end up being infected by more virions than other cells. Assume that λ is the multiplicity of infection of the whole population. Then, the probability that a single infected cell is infected by k virions is e λ k /[(1 - e)k!]. The factor 1 - e in the denominator corrects for the e cells that will not be infected at all. We obtain the overall probabilities to find a pure or mixed offspring virion by summing over all possible combinations of infections, with their respective probabilities. For example:

https://static-content.springer.com/image/art%3A10.1186%2F1471-2180-3-11/MediaObjects/12866_2003_Article_84_Equj_HTML.gif

After taking the sum, we find:

p ww = https://static-content.springer.com/image/art%3A10.1186%2F1471-2180-3-11/MediaObjects/12866_2003_Article_84_IEq1_HTML.gif + cx w x m ,     (12)

where the constant c is given by

https://static-content.springer.com/image/art%3A10.1186%2F1471-2180-3-11/MediaObjects/12866_2003_Article_84_Equk_HTML.gif

Here, Ei(z) is the exponential integral, https://static-content.springer.com/image/art%3A10.1186%2F1471-2180-3-11/MediaObjects/12866_2003_Article_84_IEq4_HTML.gif , and γ is the Euler constant, γ ≈ 0.577216. Finally, by making use of x w = 1 - x m ; we arrive at

p mm = cx w + (1 - c) https://static-content.springer.com/image/art%3A10.1186%2F1471-2180-3-11/MediaObjects/12866_2003_Article_84_IEq1_HTML.gif .     (14)

Similarly, for a mixed virion, we find

https://static-content.springer.com/image/art%3A10.1186%2F1471-2180-3-11/MediaObjects/12866_2003_Article_84_Equl_HTML.gif

We obtain p mm and p mw from Eqs. (14) and (15) by interchanging w and m in all subscripts. Equations (14) and (15) show that the mixing parameter r used in the main body of this paper is related to c via r = 1 - c.

Declarations

Acknowledgments

COW was supported by the NSF under Contract No. DEB-9981397; ISN was supported by NIH grant AI45686.

Authors’ Affiliations

(1)
Digital Life Laboratory, California Institute of Technology
(2)
Department of Microbiology and Immunology, Medical College of Ohio

References

  1. Nowak MA: What is a quasispecies?. TREE. 1992, 7: 118-121. 10.1016/0169-5347(92)90145-2.PubMedGoogle Scholar
  2. Domingo E, Holland JJ: RNA virus mutations and fitness for survival. Annu Rev Microbiol. 1997, 51: 151-178. 10.1146/annurev.micro.51.1.151.View ArticlePubMedGoogle Scholar
  3. Domingo E, Biebricher CK, Eigen M, Holland JJ: Quasispecies and RNA Virus Evolution: Principles and Consequences. 2001, Georgetown, TX: Landes BioscienceGoogle Scholar
  4. Nowak MA, May RM: Virus Dynamics. 2000, Oxford: Oxford University PressGoogle Scholar
  5. Eigen M, McCaskill J, Schuster P: Molecular quasi-species. J Phys Chem. 1988, 92: 6881-6891.View ArticleGoogle Scholar
  6. Schuster P, Swetina J: Stationary mutant distributions and evolutionary optimization. Bull Math Biol. 1988, 50: 635-660.View ArticlePubMedGoogle Scholar
  7. Nowak M, Schuster P: Error thresholds of replication in finite populations. Mutation frequencies and the onset of Muller's Ratchet. J theor Biol. 1989, 137: 375-395.View ArticlePubMedGoogle Scholar
  8. Eigen M, McCaskill J, Schuster P: The molecular quasi-species. Adv Chem Phys. 1989, 75: 149-263.Google Scholar
  9. Woodcock G, Higgs PG: Population evolution on a multiplicative single-peak fitness landscape. J theor Biol. 1996, 179: 61-73. 10.1006/jtbi.1996.0049.View ArticlePubMedGoogle Scholar
  10. van Nimwegen E, Crutchfield JP, Huynen M: Neutral evolution of mutational robustness. Proc Natl Acad Sci USA. 1999, 96: 9716-9720. 10.1073/pnas.96.17.9716.PubMed CentralView ArticlePubMedGoogle Scholar
  11. Wilke CO, Ronnewinkel C, Martinetz T: Dynamic fitness landscapes in molecular evolution. Phys Rep. 2001, 349: 395-446. 10.1016/S0370-1573(00)00118-6.View ArticleGoogle Scholar
  12. Wilke CO: Selection for fitness versus selection for robustness in RNA secondary structure folding. Evolution. 2001, 55: 2412-2420.View ArticlePubMedGoogle Scholar
  13. Krakauer DC, Plotkin JB: Redundancy, antiredundancy, and the robustness of genomes. Proc Natl Acad Sci USA. 2002, 99: 1405-1409. 10.1073/pnas.032668599.PubMed CentralView ArticlePubMedGoogle Scholar
  14. Wilke CO: Probability of fixation of an advantageous mutant in a viral quasispecies. Genetics. 2003, 163: 467-474.PubMed CentralPubMedGoogle Scholar
  15. Wilke CO, Adami C: Evolution of mutational robustness. Mutation Research. 2003, 522: 3-11. 10.1016/S0027-5107(02)00307-X.View ArticlePubMedGoogle Scholar
  16. Rouzine IM, Wakeley J, Coffin JM: The solitary wave of asexual evolution. Proc Natl Acad Sci USA. 2003, 100: 587-592. 10.1073/pnas.242719299.PubMed CentralView ArticlePubMedGoogle Scholar
  17. King AMQ, McCahon D, Slade WR, Newman JWI: Recombination in RNA. Cell. 1982, 29: 921-928.View ArticlePubMedGoogle Scholar
  18. Lai MM: Genetic recombination in RNA viruses. Curr Top Microbiol Immunol. 1992, 176: 21-32.PubMedGoogle Scholar
  19. Agol VI: Recombination and other genomic rearrangements in picornaviruses. Seminars in Virology. 1997, 8: 77-84. 10.1006/smvy.1997.0112.View ArticleGoogle Scholar
  20. Hirst GK: Genetic recombination with newcastle disease virus, polioviruses, and influenza. Cold Spring Harbor Symp Quant Biol. 1962, 27: 303-View ArticlePubMedGoogle Scholar
  21. Turner PE, Chao L: Prisoner's dilemma in an RNA virus. Nature. 1999, 398: 441-443. 10.1038/18913.View ArticlePubMedGoogle Scholar
  22. Steinhauer DA, Skehel JJ: Genetics of influenza viruses. Annual Review of Genetics. 2002, 36: 305-332. 10.1146/annurev.genet.36.052402.152757.View ArticlePubMedGoogle Scholar
  23. White PS, Morales F, Roossinck MJ: Interspecific reassortment of genomic segments in the evolution of cucumoviruses. Virology. 1995, 207: 334-337. 10.1006/viro.1995.1088.View ArticlePubMedGoogle Scholar
  24. Rodriguez LL, Owens JH, Peters CJ, Nichol ST: Genetic reassortment among viruses causing hantavirus pulmonary syndrome. Virology. 1998, 242: 99-106. 10.1006/viro.1997.8990.View ArticlePubMedGoogle Scholar
  25. Sall AA, Zanotto PMD, Sene OK, Zeller HG, Digoutte JP, Thiongane Y, Bouloy M: Genetic reassortment of Rift Valley fever virus in nature. J Virol. 1999, 73: 8196-8200.PubMed CentralPubMedGoogle Scholar
  26. Novick A, Szilard L: Virus strains of identical phenotype but different genotype. Science. 1951, 113: 34-35.View ArticlePubMedGoogle Scholar
  27. Brenner S: Genetic control and phenotypic mixing of the adsorption cofactor requirement in bacteriophages T2 and T4. Virology. 1957, 3: 560-574.View ArticlePubMedGoogle Scholar
  28. Huang AS, Palma EL, Hewlett N, Roizman B: Pseudotype formation between enveloped RNA and DNA viruses. Nature. 1974, 252: 743-745.View ArticlePubMedGoogle Scholar
  29. Holland JJ, de la Torre JC, Steinhauer DA, Clarke D, Duarte E, Domingo E: Virus mutation frequencies can be greatly underestimated by monoclonal antibody neutralization of virions. J Virol. 1989, 63: 5030-5036.PubMed CentralPubMedGoogle Scholar
  30. Sevilla N, Domingo E: Evolution of a persistent aphthovirus in cytolytic infections: Partial reversion of phenotypic traits accompanied by genetic diversification. J Virol. 1996, 70: 6617-6624.PubMed CentralPubMedGoogle Scholar
  31. Sevilla N, Ruíz-Jarabo CM, Gómez-Mariano G, Baranowski E, Domingo E: An RNA virus can adapt to the multiplicity of infection. J Gen Virol. 1998, 79: 2971-2980.View ArticlePubMedGoogle Scholar
  32. Miralles R, Ferrer R, Sole RV, Moya A, Elena SF: Multiple infection dynamics has pronounced effects on the fitness of RNA viruses. J Evol Biol. 2001, 14: 654-662. 10.1046/j.1420-9101.2001.00308.x.View ArticleGoogle Scholar
  33. Chao L: Evolution of sex in RNA viruses. J theor Biol. 1988, 133: 99-112.View ArticlePubMedGoogle Scholar
  34. Boerlijst MC, Bonhoeffer S, Nowak MA: Viral quasi-species and recombination. Proc R Soc Lond B. 1996, 263: 1577-1584.View ArticleGoogle Scholar
  35. Brown SP: Collective action in an RNA virus. J Evol Biol. 2001, 14: 821-828. 10.1046/j.1420-9101.2001.00317.x.View ArticleGoogle Scholar
  36. Frank SA: Multiplicity of infection and the evolution of hybrid incompatibility in segmented viruses. Heredity. 2001, 87: 522-529. 10.1046/j.1365-2540.2001.00911.x.View ArticlePubMedGoogle Scholar
  37. Bangham CRM, Kirkwood TBL: Defective interfering particles: effects in modulating virus growth and persistence. Virology. 1990, 179: 821-826.View ArticlePubMedGoogle Scholar
  38. Szathmáry E: Natural selection and dynamical coexistence of defective and complementing virus segments. J theor Biol. 1992, 157: 383-406.View ArticlePubMedGoogle Scholar
  39. Szathmáry E: Co-operation and defection: playing the field in virus dynamics. J theor Biol. 1993, 165: 341-356. 10.1006/jtbi.1993.1193.View ArticlePubMedGoogle Scholar
  40. Kirkwood TBL, Bangham CRM: Cycles, chaos, and evolution in virus cultures: A model of defective interfering particles. Proc Natl Acad Sci USA. 1994, 91: 8685-8689.PubMed CentralView ArticlePubMedGoogle Scholar
  41. Frank SA: Within-host spatial dynamics of viruses and defective interfering particles. J theor Biol. 2000, 206: 279-290. 10.1006/jtbi.2000.2120.View ArticlePubMedGoogle Scholar
  42. Ruíz-Jarabo CM, Arias A, Baranowski E, Escarmís C, Domingo E: Memory in viral quasispecies. J Virol. 2000, 74: 3543-3547. 10.1128/JVI.74.8.3543-3547.2000.PubMed CentralView ArticlePubMedGoogle Scholar
  43. Arias A, Lázaro E, Escarmís C, Domingo E: Molecular intermediates of fitness gain of an RNA virus: characterization of a mutant spectrum by biological and molecular cloning. J Gen Virol. 2001, 82: 1049-1060.View ArticlePubMedGoogle Scholar
  44. Ruíz-Jarabo CM, Arias A, Molina-París C, Briones C, Baranowski E, Escarmís C, Domingo E: Duration and fitness dependence of quasispecies memory. J Mol Biol. 2002, 315: 285-296. 10.1006/jmbi.2001.5232.View ArticlePubMedGoogle Scholar
  45. Domingo E, Ruíz-Jarabo CM, Sierra S, Arias A, Pariente N, Baranowski E, Escarmís C: Emergence and selection of RNA virus variants: memory and extinction. Virus Research. 2002, 82: 39-44. 10.1016/S0168-1702(01)00385-9.View ArticlePubMedGoogle Scholar
  46. Domingo E, Escarmís C, Baranowski E, Ruiz-Jarabo CM, Carrillo E, Núñez JI, Sobrino F: Evolution of foot-and-mouth disease virus. Virus Research. 2003, 91: 47-63. 10.1016/S0168-1702(02)00259-9.View ArticlePubMedGoogle Scholar
  47. Jackson T, King AMQ, Stuart DI, Fry E: Structure and receptor binding. Virus Research. 2003, 91: 33-46. 10.1016/S0168-1702(02)00258-7.View ArticlePubMedGoogle Scholar
  48. Novella IS, Borrego B, Mateu MG, Domingo E, Giralt E, Andreu D: Use of substituted and tandem-repeated peptides to probe the relevance of the highly conserved RGD tripeptide in the immune-response against foot-and-mouth-disease virus. FEBS Lett. 1993, 330: 253-259. 10.1016/0014-5793(93)80883-V.View ArticlePubMedGoogle Scholar
  49. Verdaguer N, Mateu MG, Bravo J, Domingo E, Fita I: Induced pocket to accommodate the cell attachment Arg-Gly-Asp motif in a neutralizing antibody against foot-and-mouth-disease virus. J Mol Biol. 1996, 256: 364-376. 10.1006/jmbi.1996.0092.View ArticlePubMedGoogle Scholar
  50. Mateu MG, Valero ML, Andreu D, Domingo E: Systematic replacement of amino acid residues within an Arg-Gly-Asp-containing loop of foot-and-mouth disease virus and effect on cell recognition. J Biol Chem. 1996, 271: 12814-12819. 10.1074/jbc.271.22.12814.View ArticlePubMedGoogle Scholar
  51. Leippert M, Beck E, Wetland F, Pfaff E: Point mutations within the βG-βH loop of foot-and-mouth disease virus O1K affect virus attachment to target cells. J Virol. 1997, 71: 1046-1051.PubMed CentralPubMedGoogle Scholar
  52. Ruíz-Jarabo CM, Sevilla N, Dávila M, Gómez-Mariano G, Baranowski E, Domingo E: Antigenic properties and population stability of a foot-and-mouth disease virus with an altered Arg-Gly-Asp receptor-recognition motif. J Gen Virol. 1999, 80: 1899-1909.View ArticlePubMedGoogle Scholar
  53. Martinez MA, Verdaguer N, Mateu MG, Domingo E: Evolution subverting essentiality: Dispensability of the cell attachment Arg-Gly-Asp motif in multiply passaged foot-and-mouth disease virus. Proc Natl Acad Sci USA. 1997, 94: 6798-6802. 10.1073/pnas.94.13.6798.PubMed CentralView ArticlePubMedGoogle Scholar
  54. Mathematica 4.0. Wolfram Research, Inchttp://www.wolfram.com
  55. Wilke CO: Maternal effects in molecular evolution. Phys Rev Lett. 2002, 88: 078-101.Google Scholar
  56. Holland JJ, Cords CE: Maturation of poliovirus RNA with capsid protein coded by heterologous enteroviruses. Proc Natl Acad Sci USA. 1964, 51: 1082-1085.PubMed CentralView ArticlePubMedGoogle Scholar
  57. Itoh H, Melnick JL: Double infection of single cells with Echo 7 and Coxsackie A9 viruses. J Exp Med. 1959, 109: 393-406.PubMed CentralView ArticlePubMedGoogle Scholar
  58. Ledinko N, Hirst GK: Mixed infection of HeLa cells with polioviruses types 1 and 2. Virology. 1961, 14: 207-219.View ArticlePubMedGoogle Scholar
  59. Moreno IM, Malpica JM, Rodríguez-Cerezo E, García-Arenal F: A mutation in tomato aspermy cucumovirus that abolishes cell-to-cell movement is maintained to high levels in the viral RNA population by complementation. J Virol. 1997, 71: 9157-9162.PubMed CentralPubMedGoogle Scholar
  60. Valcarcel J, Ortin J: Phenotypic hiding – The carryover of mutations in RNA viruses as shown by detection of MAR mutants in influenza-virus. J Virol. 1989, 63: 4107-4109.PubMed CentralPubMedGoogle Scholar
  61. Young JAA: Virus entry and uncoating. In Field's Virology. Edited by: Knipe DM, Howley PM. 2001, Philadelphia: Lippincott Williams & Wilkins, 87-103.Google Scholar
  62. Miralles R, Gerrish PJ, Moya A, Elena SF: Clonal interference and the evolution of RNA viruses. Science. 1999, 285: 1745-1747. 10.1126/science.285.5434.1745.View ArticlePubMedGoogle Scholar
  63. Miralles R, Moya A, Elena SF: Diminishing returns of population size in the rate of RNA virus adaptation. J Virol. 2000, 74: 3566-3571. 10.1128/JVI.74.8.3566-3571.2000.PubMed CentralView ArticlePubMedGoogle Scholar
  64. Holland JJ, de la Torre JC, Clarke DK, Duarte EA: Quantitation of relative fitness and great adaptability of clonal populations of RNA viruses. J Virol. 1991, 65: 2960-2967.PubMed CentralPubMedGoogle Scholar
  65. Gerrish PJ, Lenski RE: The fate of competing beneficial mutations in an asexual population. Genetica. 1998, 102/103: 127-144. 10.1023/A:1017067816551.View ArticleGoogle Scholar
  66. Jung A, Maier R, Vartanian JP, Bocharov G, Jung V, Fischer U, Meese E, Wain-Hobson S, Meyerhans A: Recombination: Multiply infected spleen cells in hiv patients. Nature. 2002, 418: 144-10.1038/418144a.View ArticlePubMedGoogle Scholar
  67. Wain-Hobson S, Renoux-Elbe C, Vartanian JP, Meyerhans A: Network analysis of human and simian immunodeficiency virus sequence sets reveals massive recombination resulting in shorter pathways. J Gen Virol. 2003, 84: 885-895. 10.1099/vir.0.18894-0.View ArticlePubMedGoogle Scholar
  68. Webster RG: Antigenic variation in influenza viruses. In Origin and Evolution of Viruses. Edited by: Domingo E, Webster RG, Holland JJ. 1999, San Diego: Academic Press, 377-390.View ArticleGoogle Scholar

Copyright

© Wilke and Novella; licensee BioMed Central Ltd. 2003

This article is published under license to BioMed Central Ltd. This is an Open Access article: verbatim copying and redistribution of this article are permitted in all media for any purpose, provided this notice is preserved along with the article's original URL.