^{1}

^{2}

^{*}

^{1}

VVG and RJDB conceived and designed the models. VVG and RJDB analyzed the data. VVG and RJDB wrote the paper.

The authors have declared that no competing interests exist.

Mutations that allow SIV/HIV to avoid the cytotoxic T lymphocyte (CTL) response are well documented. Recently, there have been a few attempts at estimating the costs of CTL escape mutations in terms of the reduction in viral fitness and the killing rate at which the CTL response specific to one viral epitope clears virus-infected cells. Using a mathematical model we show that estimation of both parameters depends critically on the underlying changes in the replication rate of the virus and the changes in the killing rate over time (which in previous studies were assumed to be constant). We provide a theoretical basis for estimation of these parameters using in vivo data. In particular, we show that 1) by assuming unlimited virus growth one can obtain a minimal estimate of the fitness cost of the escape mutation, and 2) by assuming no virus growth during the escape, one can obtain a minimal estimate of the average killing rate. We also discuss the conditions under which better estimates of the average killing rate can be obtained.

Due to their high mutation rate, RNA viruses—like SIV and HIV—can avoid recognition by the host immune response by evolving new variants (i.e., immune escape mutants). Avoiding the cytotoxic T lymphocyte (CTL) immune responses is one of the major obstacles for the development of vaccines to HIV, and this avoidance seems a major mechanism of HIV disease progression to AIDS. Using a relatively general mathematical model, Ganusov and De Boer suggest a simple technique by which two main parameters determining the likelihood of viral escape can be estimated. First is the “cost” of the escape mutation, which is the relative fitness reduction in the virus replication rate. Second is the rate at which the CTL response specific for one epitope “clears” virus-infected cells. Application of their technique to data on virus escape helps to quantify the costs and benefits of CTL escape mutations in SIV/HIV infection.

Several observations suggest that cytotoxic T lymphocyte (CTL) responses play an important role in controlling virus replication in SIV/HIV infections. First, depletion of CD8^{+} T cells during chronic SIV infection of rhesus macaques leads to a rapid increase in viral loads [^{+} T cells prior to SIV infection results in rapid progression and death of animals following infection [

During “escape” experiments in which a wild-type virus is substituted with a mutant, the average rate,

In the reversion experiments, the dynamics of the CTL escape mutant is observed in a host lacking the MHC class I allele presenting the wild-type epitope. In such a host, the CTL response specific for the wild-type epitope is absent, i.e., _{s}_{e}_{s}_{s}_{e}_{e}_{s}, t_{e}

Note that to estimate the rate ^{−1}), the rate of replication may be lower around the peak of viremia (^{−1}), and is likely to approach its lowest value during the stable phase (^{−1}) [

Animals 1 and 2 were infected with both the mutant and wild-type virus at the same time, animals 3 and 4 were infected only with the mutant, and it took about a month before the wild-type virus appeared. Using Equation 1, we estimated the average rate of replacement ^{−1} for these four cases, respectively (_{s}_{e}_{s}_{e}_{max}^{−1} using Equation 2, we obtain the minimal estimate of the cost of the Gag_{206−215} escape mutation

Nevertheless, even if the changes in virus replication rate over time are not known, one can make a minimal estimate of the fitness cost of the escape mutation. By assuming that during the experiment the virus population expands exponentially at a fixed maximal rate _{max},

The fact that Equation 2 provides an underestimate of the fitness cost is demonstrated in _{min}

(A) Plots the hypothetical changes in the replication rate of the virus during the acute phase of infection. We assume that initially (until time _{r}_{max}_{r} ,_{min}

(B) Plots changes in the ratio _{s}_{e}_{max}^{−1} (shown by dashed lines), i.e., _{min}_{min}^{−1}, _{r}^{−1},

Two additional points needs to be stressed. It is often concluded that the time taken for replacement of the mutant by the wild-type in reversion experiments is related to the cost of the escape mutations, i.e., longer reversion times from the start of experiment to a complete reversion correspond to lower fitness costs (e.g., [

We have shown that the rate of replacement is determined by the fitness cost

During escape experiments, the wild-type virus is subjected to additional killing rate

where _{s}, t_{e}_{s}_{e}_{s}_{e},

Two groups have independently proposed to use escape experiments to estimate the rate at which the CTL response kills cells expressing the wild-type CTL epitope [

Importantly, Equation 4 suggests that in order to determine the average killing rate one not only needs to have an estimate of the cost of the escape mutation _{min},_{min}

Second, one could assume that during both the reversion and escape experiment, the virus replication rate is constant and the same (i.e.,

(A) Plots an example of the replication rate

(B) Plots the changes in the ratio of the wild-type virus to the mutant density as a function of time.

Thick solid lines denote the case where the replication rate _{s}_{e}_{max}^{−1}. Note that this estimate of the average killing rate underestimates the maximum killing rate _{max}_{on}_{of} =_{max}^{−1}, _{min}^{−1}. Other parameters are the same as in ^{2}.

where _{s}_{e}_{s}_{e}

While Equation 3 delivers the minimal estimate of the average CTL killing rate, the rate _{min}_{164−172}) was ^{−1}. During the reversion (which occurred before the peak of viremia), the mutant was substituted by the wild-type at a rate ^{−1}. By assuming that the rate of virus replication is higher before the peak of viremia than that after the peak, we obtain the following minimal and maximal estimates of the average killing rate of the CTL response specific for the KP9 epitope (_{min}, K′^{−1}, 1.09 d^{−1}).

Estimating the Average Killing Rate

If during the reversion experiments the rate of virus replication is lower than that during the escape experiments (_{min}_{max}

In this paper we have provided a theoretical basis for estimating the costs of CTL escape mutations and the average rate at which the CTL response specific for a given epitope clears virus-infected cells. We show that by assuming exponential growth of the virus during the reversion experiments, one can obtain a minimal estimate for the cost of escape mutation (using Equation 2). Similarly, by assuming no virus growth during the escape experiments, one can obtain the minimal estimate of the average rate at which the CTL response specific to one viral epitope clears virus-infected cells (using Equation 3). Since our model is relatively general, our conclusions about estimating the costs and benefits of CTL escape mutations of SIV/HIV are equally applied to acute and chronic phases of SIV/HIV infection. However, while during the acute phase there are likely to be substantial changes in the rate of virus replication

We assume the following scenario for viral escape. The wild-type virus has a higher replication rate, and cells infected with the wild-type virus are killed at a higher rate by the CTL response. The mutant virus has a lower replication rate and cells infected with the mutant virus are killed at a lower rate by the CTL response. We formulate a mathematical model describing the dynamics of the density of cells productively infected with the wild-type

where

It is useful to rewrite

where ^{+} T cell responses reduce the rate of virus replication by noncytolytic mechanisms (see below). In the main text we consider how the cost

We consider the case when the escape mutation renders the mutant less fit due to an increased death rate of virus-infected cells. The dynamics of cells productively infected with the wild-type

where

To estimate the cost of the escape mutation and the average CTL killing rate, one needs to know the changes in the death rate of the cells productively infected with the virus with time. Since these changes will be dependent on the CTL responses specific to both wild-type and mutant viruses, the death rate

The model describing the dynamics of the cells infected with the wild-type

where the assumptions on the virus growth are similar to those in the main model. However, we assume that the CD8^{+} T cell response reduces the rate of virus replication. The reduction in the replication rate is due to the CD8^{+} T cell response specific for the wild-type epitope ^{+} T cell responses to other viral epitopes

During the reversion experiments, the specific CD8^{+} T cell response directed against the wild-type epitope is absent (i.e.,

Assuming that the virus replicates at the maximum rate ^{+} T cell response (

During the escape experiments Equation 14 holds. One can rewrite this equation:

While this expression is somewhat complex, its interpretation is similar to that of Equation 8: the first term on the right hand side corresponds to increase in the frequency of the wild-type due to cost of escape mutation, and the second term corresponds to a decrease in the frequency of the wild-type due to the CD8^{+} T cell response specific for the wild-type epitope.

There are two differences with the equation found in the main model, however. First, we find that the rate of accumulation of the mutant in the population depends on the replication rate of the virus: a higher replication rate ^{+} T cell response controls the virus by killing virus-infected cells, where the opposite trend is observed (i.e., the faster rates of virus replication leads to slower replacement of the wild-type by the mutant virus, see Equation 3).

Second, there is no an easy way of estimating the CD8^{+} T cell suppression efficacy

We thank Becca Asquith and colleagues for sharing their unpublished work with us. We also thank Can Keşmir, José Borghans, and Becca Asquith for their comments on the previous versions of the manuscript.

fitness cost

cytotoxic T lymphocyte

total response

average killing rate

killing rate

mutant

virus replication rate

wild-type