The success of antiviral therapy depends (amongst other things) crucially on whether resistant mutant virus is present before the initiation of therapy [9,10]. If resistant virus is present prior to treatment, then application of the drug leads to declining levels of sensitive wild-type virus and increasing levels of resistant mutant virus. Mutant virus grows either because of an increased supply of target cells [11,12] or a reduction in antiviral immunity. If there is only a small probability that resistant virus is present in a patient before antiviral therapy, then a straightforward calculation shows that the chance that it will arise after the initiation of therapy is even smaller [13,14].
We call a virus mutant ‘resistant’ if it has a positive growth rate in the presence of therapy. For single drug therapy, it is likely that certain one or two point mutations confer resistance. Therefore, it seems likely that virus mutants, resistant to almost any single drug therapy, are present in a patient before treatment. Resistance to two or three drugs may require two or more point mutations.
Mathematical models of antiviral drug treatment have been developed to study the effect of increasing target cell abundance on rebound of wild-type virus , the emergence of resistant mutant virus as a consequence of increasing target cell abundance , the effect of treatment on reducing viral diversity , the effect on virus load of a drug-induced reduction of infectivity , and the emergence of several different mutants of increasing levels of resistance during therapy [18,19]. Mathematical models of antiviral therapy have also provided estimates for kinetic parameters of virus replication in vivo [1,2,4,7,20–23]. De Boer and Boucher developed a mathematical model to study the consequence of preventing CD4 cell increase during therapy on the emergence of resistant virus .
We begin with a simple model that contains wild-type virus and one mutant. It is assumed here that wild-type and mutant differ by only a single point mutation (Fig. 1a). The mutation rate is given by the parameter μ. Furthermore, it is assumed that wild-type virus has a fitness advantage over mutant virus in the absence of drug; the selection coefficient is given by s. The model has three variables: uninfected cells, x; cells productively infected by wild-type virus, y0; and by mutant virus, y1. Uninfected cells are produced at rate λ and die at rate dx. Infected cells die at rates ay0 and ay1. Infected cells give rise to new infected cells at a rate proportional to the abundance of uninfected cells multiplied by infected cells; the rate constant is β. It is assumed that free virus dynamics are fast compared with infected cell turnover . Therefore, we do not have to consider a separate equation for free virions, but simply assume that virion abundance is proportional to infected cell abundance. The model equations are as follows:
At equilibrium, the ratio of mutant to wild-type virus, to first order in μ, is as follow:
For example, if the point mutation rate is µ =3 × 10−5 and the mutant has a selective disadvantage of s = 0.01 , then the relative proportion of mutant to wild-type is 3 × 10−3. In other words, for this choice of parameters, about 1 in 300 cells contains the mutant virus. If the mutant had a large selective disadvantage (s ≊ 1), then the ratio of mutant to wild-type would be approximately 10−5. However, there is some experimental evidence for one-error mutants with frequencies of the order of 10−3  in previously untreated patients.
We now expand the model to consider resistant mutants that differ by two point mutations (Fig. 1b). This model has five variables: uninfected cells, x; cells infected by wild-type virus, y00; cells infected by one-error mutants, y01 and y10; and cells infected by the two-error mutant, y11. Note that we use a binary notation for wild-type virus (00) and the various mutants (01, 10, and 11). Thus, at each of two relevant positions we consider one amino-acid substitution. We assume that these amino-acid substitutions are due to a single base substitution in the viral genome. The replication fidelity of the reverse transcriptase may vary for different positions in the genome and therefore we assume that the mutation rate for the first position is µ1and for the second position is µ2. The selective disadvantage of the mutants y01, y10 and y11 is given by s01, s10 and s11, respectively. The model equations are as follows:
The abbreviations νi = 1 -µi and rij =1 - sij have been used. The exact equilibrium solution of this system is complicated, but we can obtain an elegant approximation by neglecting back-mutations. In this case, we simply solve the following eigenvalue equation:
We obtain the following:
If in addition we assume that the selection coefficients are larger than the mutation rates, then we obtain the following approximate equilibrium ratios:
Again, this approximation is accurate if the selection coefficients are noticeably larger than the mutation rates. For example, if µ1 = µ2 =3 × 10−5 and if s01 = s10= s11 = 0.01 then y11/y00 ≊ 2 × 10−5. In other words, approximately 1 in 50 000 productively infected cells contains resistant mutant virus, which differs from the wild-type by two point mutations. Table 1 shows the effect of different selective disadvantages for the intermediate mutants, s01 and s10, on the relative frequency of y11. For example, if s11 = 0.01, but s01 = s10 = 1 (i.e., the double mutant has a small selective disadvantage, but the intermediate one-error mutants cannot replicate) then the relative abundance of the double mutant is y11/y00 ≊ 9 × 10−8. If, on the other hand, s01 = s10 = 0.001 (i.e., the intermediate mutants have a selective disadvantage of only 0.1% compared with wild-type) then we find y11/y00 ≊ 1.8 × 10−4. The difference in the equilibrium frequency of the double mutant is several orders of magnitude. Therefore, in order to estimate the pretreatment equilibrium frequency of a resistant variant, it is, of course, not sufficient to know the selective disadvantage of this mutant compared with wild-type, but it is also necessary to know the selection coefficients of all intermediate variants.
The above model can be expanded to consider mutants that differ from wild-type virus by more than two point mutations. For simplicity, assume that the average mutation rate for each site is the same, μ, and that all mutants have the same selective disadvantage, s, which is much smaller than 1. In this case we obtain (Fig. 2) the following:
Of course, n!(µ/s)n has to be less than unity otherwise our approximations break down. This is essentially an error threshold condition of quasispecies theory [29,30].
Table 2 shows equilibrium frequencies of mutants that differ from wild-type by up to five point mutations. For s = 0.01, the frequency of a three-error mutant is 1.6 × 10−7, of a four-error mutant is 1.9 × 10−9 and of a five-error mutant is 2.9 × 10−11. Suppose that there are approximately 107-108  productively infected cells in an HIV-1-infected patient. For s = 0.01, a particular four- or five-error mutant is unlikely to be present, while the three-error mutant is likely to exist. For a higher selective disadvantage (s > 0.01) the three-error mutant is also unlikely to exist in a patient prior to treatment. We can also ask what is the maximum selective disadvantage that is still compatible with the survival of a given virus mutant in a patient with N infected cells. If N =108 and µ =3 × 10−5, then a two-error mutant will survive if s < 0.4, a three-error mutant if s < 0.03, and a four-error mutant if s < 0.007. (In this calculation, intermediate mutants have the same selective disadvantage, s. Other assumptions are also possible and can be calculated.)
The model also provides insight into the time evolution of resistant mutant virus during the course of infection. If a patient is initially infected by sensitive wild-type virus, y0, then the frequency of resistant mutant virus, y1, rises as follows:
Note that this calculation assumes that the number of uninfected cells, x, is approximately constant during the rise of resistant mutants. The mutant will arrive at half its equilibrium value after (ln2)/[βx(s-µ)] days. In principle, we can recursively obtain the rate of convergence towards equilibrium for any of the mutant strains, but the analytical expressions are too complex to be shown here. However, it is interesting to compare the times until a two-error (or greater) mutant attains a given frequency relative to the wild-type if (i) the intermediate mutants cannot replicate at all, or (ii) all mutants have the same selective disadvantage. For example, a two-error mutant reaches a frequency of 10−8 about five times faster under scenario (ii) than under scenario (i). Hence, the presence of intermediate mutants with positive growth rate not only strongly affects the equilibrium frequency of a n-error mutant, but also its rate of ascent. Note, however, that the time necessary for a mutant to reach a given percentage of its equilibrium value is larger under scenario (ii), simply because the equilibrium value is much higher in this scenario.
The success of drug therapy depends to a large extent on whether resistant virus is present in patients prior to treatment. Based on a quasispecies model for the balance of selection and mutation between wild-type and mutants, we derived analytical expressions for the pre-treatment frequency of mutant virus. We showed that the mutant frequency depends on the number of point mutations between wild-type and mutant, the selective disadvantage of the mutant and all its intermediates, and the mutation rate. For example, for a mutation rate of 3 × 10−5  and a selective disadvantage for the mutant and all intermediates of 1%, we find that the frequency of a three-error mutant relative to wild-type is approximately 10−7. If there are on average 107–108productively infected cells in a patient , then such a mutant may well be present before the start of therapy.
There are several practical conclusions to be drawn. The probability of treatment failure due to viral resistance can be reduced in three ways: (i) treatment should start early when virus load is still low and the frequency of resistant mutants is small (assuming that most patients are infected by drug-sensitive wild-type virus); (ii) treatment should commence immediately with multiple drugs (three or more) – it is clearly disadvantageous to initiate therapy with one drug and then add other drugs later, giving the virus the possibility to develop resistance to each drug sequentially; and (iii) treatment should combine drugs for which resistance mutations are known to involve a considerable selective disadvantage in the absence of drug. It may be of great importance to identify such combinations, since mutants resistant to these combinations may not exist in patients prior to treatment.
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Keywords:© Lippincott-Raven Publishers.
Antiviral therapy; mathematical model; resistant mutant; viral load