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Conceived and designed the experiments: CA RDB. Performed the experiments: CA. Analyzed the data: CA. Contributed reagents/materials/analysis tools: CA. Wrote the paper: CA.

The authors have declared that no competing interests exist.

Several studies have shown that cytotoxic T lymphocytes (CTLs) play an important role in controlling HIV/SIV infection. Notably, the observation of escape mutants suggests a selective pressure induced by the CTL response. However, it remains difficult to assess the definite role of the cellular immune response. We devise a computational model of HIV/SIV infection having a broad cellular immune response targeting different viral epitopes. The CTL clones are stimulated by viral antigen and interact with the virus population through cytotoxic killing of infected cells. Consequently, the virus population reacts through the acquisition of CTL escape mutations. Our model provides realistic virus dynamics and describes several experimental observations. We postulate that inter-clonal competition and immunodominance may be critical factors determining the sequential emergence of escapes. We show that even though the total killing induced by the CTL response can be high, escape rates against a single CTL clone are often slow and difficult to estimate from infrequent sequence measurements. Finally, our simulations show that a higher degree of immunodominance leads to more frequent escape with a reduced control of viral replication but a substantially impaired replicative capacity of the virus. This result suggests two strategies for vaccine design: Vaccines inducing a broad CTL response should decrease the viral load, whereas vaccines stimulating a narrow but dominant CTL response are likely to induce escape but may dramatically reduce the replicative capacity of the virus.

As a result of their high mutation rate, HIV and its counterpart SIV in non-human primates can evade recognition by the host immune response through the generation of viral variants, the so-called escape mutants. This avoidance of cytotoxic T lymphocyte (CTL) mediated killing seems to be one of the major reasons why virus replication is not controlled effectively. However, it remains difficult to investigate the critical properties of the dynamics of immune escape. To this end, we developed a new computational model of HIV/SIV infection consisting of several CTL clones that can recognize specific parts of viral proteins, i.e., epitopes. The simulations allow us to follow the dynamics of immune escape in detail and help to interpret longitudinal data of HIV/SIV infections. Interestingly, changing the relative sizes of the CTL clones leads to a different evolution of the virus. Instead of reducing the number of infected cells, an alternative strategy of vaccine design could be to reduce the replicative capacity of the virus that might have implications for disease progression.

HIV infection in humans and SIV infection in non-human primates is not cleared by the host's immune system. However, there is partial control of virus replication that is mainly attributed to the cytotoxic T lymphocyte (CTL) response

It is a challenge to acquire longitudinal data on CTL escape since the virus diversity within a host has to be followed over a long phase of chronic infection. Additionally, little data is available during the early phase of infection, especially for HIV where the infection is difficult to diagnose during the first weeks. Asquith et al.

The interaction of viral replication, mutation and selection by different CTL responses appears to be complex. Studies have observed CTL escape very early but also years after primary infection

We develop a computational model of HIV/SIV virus dynamics including a cellular immune response consisting of several CTL clones (

A number of

We translate those interactions into a set of ordinary differential equations (ODEs) and add stochastic events for viral mutation. Classically, the processes of infecting target cells or the killing of infected cells by CTLs have been described with simple mass-action terms

Non-infected CD4^{+} target cells _{T}_{i}_{i}^{+} cells (non-infected and infected). After infection of a target cell, reverse transcription occurs during which the virus can mutate with a probability of ^{2n}_{i}_{I}_{ij}_{j}_{k}_{i}_{i}_{i}_{V}_{i}_{j}_{ji}_{E}_{i}

For killing following mass-action dynamics, the killing rate is linearly increasing with increasing number of CTLs (straight line, _{k}^{12}). However, if CTL clones compete with each other to kill infected cells, a saturation effect occurs according to Michaelis-Menten kinetics (dashed (_{k}^{9}) and dotted (_{k}^{8}) line). After a virus escapes recognition from a single CTL clone (blue arrow), the killing of infected cells is reduced differently depending on these functions (red arrows).

Parameter | Value | Explanation and reference |

1.5 d^{−1} | Initial viral growth rate of 1.5 d^{−1} | |

1.0 d^{−1} | Maximal CTL proliferation rate | |

2×10^{7} cells d^{−1} | Source of CD4^{+} target cells, tuned to obtain an infected cell count between 10_{7} and 10^{8} during the chronic phase | |

10^{3} cells | Maintains a small number of 10^{3} ‘naive’ cells per CTL clone in absence of infection. | |

_{T} | 0.01 d^{−1} | Natural death rate of CD4^{+} target cells |

_{I} | 0.1 d^{−1} | Virus-induced death rate of infected cells (includes δ_{T} |

_{E} | 0.01 d^{−1} | Death rate of cytotoxic T lymphocytes |

[1.0, 5×10^{8}] cells | Uniform distribution of avidities for the CTL clones. Generates a few dominant (low | |

Assumes a maximal death rate of infected cells of 1.0 d^{−1} | ||

_{k} | 10^{8}–10^{12} cells | Allows killing of infected cells to follow Michaelis-Menten kinetics (small _{k}_{k} |

3×10^{−5} | HIV-1 mutation rate per nucleotide | |

20 | Approximate number of epitopes that are recognized by the CTL responses |

The most surprising phenomenon of CTL escape in HIV/SIV is the time scale at which it occurs. Selection of escape variants has been found to happen very early after acute infection, but also late after years

Our model describes the virus dynamics of an HIV/SIV infection and the subsequent immune escape in a very realistic manner (_{k}^{12}). _{k}^{9}). In both simulations, the total number of infected cells peaks a few weeks after infection and reaches a set-point level of around 10^{7} to 10^{8} cells (black line in top panels)

In the top panels, the total number of infected cells (black line) is shown together with the emerging escape mutants (colored lines). Escape variants expressed as a frequency of the total viral population are given in the middle panels. These variants can fluctuate in frequency (e.g. red line) and, after dominating the viral population, revert back to wild-type (e.g. green line). In the bottom panels, a number of CTL clones proliferate upon infection (full and dashed lines) but can slowly disappear after the virus population escapes recognition (full colored lines). Starting with the same CTL repertoire, more escapes occur when killing of infected cells approaches mass-action dynamics (_{k}^{12}, shown in A) compared to Michaelis-Menten kinetics (_{k}^{9}, shown in B).

The dynamics of the CTL clones _{i}

It has been suggested that many escapes occur early, i.e. during the decline phase in viral load after the peak of infection and that they potentially prevent clearance of HIV/SIV. However, in our simulations immune escape does not occur before the set-point level is reached around two to three months after infection (^{+} target cell depletion

(A) Viral escapes over time given as an expected number of escapes per month (average of 1000 simulations). It can be seen that most escapes occur during the first year after infection (acute phase) and fewer afterwords (chronic phase). The straight line shows the replicative fitness of the viral population as an average over all simulation runs (standard deviation is given by the gray area). The time of escape is measured when an escape variant breaches a frequency of 50% of the total viral population for the first time. (B) Time-plot of average infected-cell death rates during the chronic phase of infection. The graph shows one simulation representing a single patient. (C) Histogram of the death rates that are bound between 0.1 d^{−1} and 1.0 d^{−1}. For (B) and (C), _{k}^{12}, i.e. killing follows mass-action dynamics.

Infected-cell death rates from different patients have been found to be close to a normal distribution with a mean of 0.45 d^{−1}

Escape variants not only appear at different times during infection but also with different rates. The rate at which an escape variant replaces the wild-type, the so-called escape rate, is determined by the balance between the evaded rate of killing and the fitness cost of the escape mutation

^{−1}, see ^{−1}, and most escape rates are below 0.2 d^{−1}. The lower rates of escape are due to the fitness cost of escape mutations that cannot totally be restored by compensatory mutations. Interestingly, the rates of escape are not distributed equally over the time of infection (

(A) The emergence of escape and the corresponding rates over a time course of 5 years of infection. (B) Given those escapes, the distribution of killing rates given as an average frequency per simulation run. (C) The distribution of escape rates given as an average frequency per simulation run. (D) Average killing and escape rates per year after infection. (E) The number of escapes during 5 years of infection (filled circles) and the average rate of killing and escape as a function of _{k}^{−1} and the mean of the true rates is 0.19±0.11 d^{−1}. All graphs represent data from 1000 simulation runs. The rates are measured when the frequency of the escape variant is 50%. Means are given with standard errors since standard deviations are usually very large. _{k}^{12} if not otherwise indicated.

In _{k}^{12}. However, when CTL effector cells form a complex with infected cells before delivering their lethal hit, the killing should follow Michaelis-Menten kinetics _{k}_{k}_{k}

To estimate rates of escape from

Our analysis shows several properties of the process of escape. Even though the total killing induced by all CTL clones is high, escape rates are expected to be slow. First, escape rates slow down due to the acquired fitness cost in viral replication, especially during later phases of infection. Secondly, if killing of infected cells follows Michaelis-Menten kinetics, the rates of escape are decreased. In addition, estimating rates of escape from ratios of escape variants and wild-type virus is likely to lead to an underestimation of the true rate. These findings highlight that slow escape rates do not necessarily infer low killing rates.

CTL responses during HIV infection generally consist of several CTL clones recognizing many different epitopes. The size of those CTL clones can differ substantially resulting in dominant and sub-dominant responses ^{−1}) and is independent of

(A) The distribution of CTL clones plotted as a function of

To investigate the effect of immunodominance we run simulations for different values of

The interaction of the HIV quasispecies with the CTL response appears to be complex which makes the analysis of experimental data difficult. Mathematical and computational models have been helpful to investigate how different processes influence each other. For example, a previous model described how shifting immunodominance and antigenic oscillations can occur during HIV infection

Another explanation for the late appearance of escape variants that has been put forward is the waiting time for compensatory mutations

Our simulations proved to be useful to analyze the process of escape. We conclude that, although killing can be very efficient, escape rates are expected to be low. The association with a fitness cost and the way how CTL clones interact with infected cells critically influence the rates of escape. Hence, it appears to be important to study those interactions to derive realistic killing terms

Immunodominance affects the viral evolution within a host. A higher degree of immunodominance leads to more frequent escape with a reduced control of viral replication but a substantially impaired replicative capacity of the virus. This is interesting as vaccines generally aim to induce a broad CTL response where escape is unlikely to occur. However, a dramatic reduction of the replicative capacity of the virus due to escape could indeed slow down disease progression and/or reduce transmission

The balance between the CTL response and the viral population acquiring escape mutations appears to be a dynamical process over a long period of infection. We have shown that it is important to analyze the kinetics of this process and also the time scale (acute and chronic phase) at which it occurs. More longitudinal data of HIV infections will help to further investigate this process and future research is likely to go into this direction.

The set of ordinary differential equations (ODEs) is extended with stochastic events for viral mutation (similar as in _{ij}_{ij}_{i}_{i}_{i}_{ij}_{j}_{j}_{ij}

Escape mutations in HIV and SIV are likely to confer a fitness cost in viral replication or infectivity _{wt}

The outgrowth of an escape variant and the subsequent replacement of the wild-type can be considered as a competitive growth between two populations:

(A) Escape variants often only transiently replace the wild-type variant and oscillate thereafter. Sequence measurements are taken at arbitrary time points (squares). (B) When a model is fitted to those data points the initial escape rate is likely to be underestimated.

We assume Michaelis-Menten kinetics for the killing of infected cells by CTL effector cells:_{k}^{−1} during the chronic phase of infection. _{k}^{7}, the infection is always cleared since an infected cell can be killed at a rate of ^{7}<_{k}^{7}, a very rapid escape during acute infection can prevent clearance of the infection. 3) For _{k}^{7}, we approach mass-action kinetics since _{k}

We would like to thank Vitaly Ganusov for extensive discussions.