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Conceived and designed the experiments: AJY MvB RA. Performed the experiments: AJY. Analyzed the data: AJY. Contributed reagents/materials/analysis tools: N/A. Wrote the paper: AJY MvB RA.

The authors have declared that no competing interests exist.

Vaccines that elicit protective cytotoxic T lymphocytes (CTL) may improve on or augment those designed primarily to elicit antibody responses. However, we have little basis for estimating the numbers of CTL required for sterilising immunity at an infection site. To address this we begin with a theoretical estimate obtained from measurements of CTL surveillance rates and the growth rate of a virus. We show how this estimate needs to be modified to account for (i) the dynamics of CTL-infected cell conjugates, and (ii) features of the virus lifecycle in infected cells. We show that provided the inoculum size of the virus is low, the dynamics of CTL-infected cell conjugates can be ignored, but knowledge of virus life-histories is required for estimating critical thresholds of CTL densities. We show that accounting for virus replication strategies increases estimates of the minimum density of CTL required for immunity over those obtained with the canonical model of virus dynamics, and demonstrate that this modeling framework allows us to predict and compare the ability of CTL to control viruses with different life history strategies. As an example we predict that lytic viruses are more difficult to control than budding viruses when net reproduction rates and infected cell lifetimes are controlled for. Further, we use data from acute SIV infection in rhesus macaques to calculate a lower bound on the density of CTL that a vaccine must generate to control infection at the entry site. We propose that critical CTL densities can be better estimated either using quantitative models incorporating virus life histories or with

In the search for vaccines that provide reliable protection against major diseases such as HIV-AIDS, TB and Malaria, there is now a focus on generating populations of antigen-specific cytotoxic T lymphocytes (CTL), immune cells that recognise and kill infected cells. However, we have little idea of the number or density of CTL a vaccine would need to elicit to provide sterilizing immunity to an infection in a given tissue. In this study we use mathematical models to understand how a virus's replication strategy influences the minimum density of CTL needed to provide immunity at an infection site. We show that traditional models that neglect the viral lifecycle within infected cells will underestimate this density. To illustrate, we use our modelling framework to estimate the CTL density needed to control the spread of virus at the very earliest stages of primary SIV infection in rhesus macaques.

The majority of vaccine design approaches to date have used neutralizing antibody titers as a correlate of efficacy. However, major infectious diseases such as HIV-AIDS, TB and Malaria have not yet fully yielded to vaccines aimed at eliciting antibodies. There is currently much interest in developing vaccines that also elicit pathogen-specific CD4 T cells or, more commonly, CD8 T cells (also known as cytotoxic T lymphocyte, or CTL). Such vaccines need to generate T cells of sufficient functional quality, appropriate tissue tropism, and in sufficient numbers. Manipulating all three features of the CTL response presents a major challenge that requires understanding of the biology of T cell priming and the cells' interactions with their microenvironment during clonal expansion and contraction. However, assuming the first two features can be optimised, the third raises an important question – how many T cells does a vaccine need to generate in order to protect against infection? This of course might be determined empirically in animal models, but another approach is to search for principles that might guide our intuition for human vaccine design.

A CTL response is a dynamic process whose chance of success may depend on precursor frequency, speed of priming and clonal expansion or reactivation, total cell numbers, access to infected tissues, and the rate and efficiency with which they survey potentially infected cells. Mathematical models can help us develop a quantitative understanding of how these processes influence the potential for protection. In this paper we focus on tissue-resident activated CTL and the challenges they face in eliminating a growing population of virus-infected cells, with an emphasis on how virus replication strategies influence the efficiency of CTL surveillance.

What we present here builds on the standard model of virus growth used extensively in the literature (see, for example, refs

In this model, sterilising immunity corresponds to a net growth rate

The mass-action killing term implies that killing is instantaneous on encounter of a CTL with an infected cell. However, CTL take time to lyse and detach from their targets. Mempel

A simple way to include the time take for CTL to handle infected cells is to explicitly track CTL-infected cell conjugates

When search times are small compared to handling times, or

When handling and search times are comparable (

When handling times are short compared with search times (

Considering the dynamics of surveillance and of CTL-target conjugates may resolve some of the discord between microscopy studies showing extended encounter times between CTL and their targets

So if the inoculum size of the virus is low compared to existing effector cell densities (

CTL are triggered by their recognition of peptides derived from virus proteins generated within the infected cell and presented on MHC class I molecules. Existing estimates of the surveillance rate

In the model that follows we assume we are in a regime where mass-action operates. In

How will the dynamics of virion production and virus epitope expression affect our estimate of the local density of CTL required for sterilising immunity,

A useful quantity is the survivorship

The steady state age-distribution solution of equation (14) yields the total cell population growing or declining as

We consider representations of two replication strategies – a budding virus, which after some delay following infection sheds virions from the host cell at a constant rate, with a possible increased burden of mortality for the host cell; and a lytic virus, which replicates in the host cell without release of virions until it lyses the host cell, releasing all its progeny within a short time interval.

For both strategies we need to know three functions. These describe the visibility to CTL, the virus production rate and the virus-induced mortality as functions of age since infection, and are shown schematically in

On the left,

In our representations of these strategies we have made the simplifying assumption that above a certain threshold of epitope expression, an infected cell is capable of being identified by CTL at constant rate

For lytic viruses,

For lytic viruses it yields the direct solution

Identical expressions are derived if visibility to CTL begins after the onset of virus-induced mortality

We can compare the effectiveness of CTL responses against lytic and budding virus infections. For reference we compare both to the standard model of a budding virus that assumes infected cells have exponentially distributed lifetimes, and immediately following infection become visible to CTL and begin to make virus at a constant rate.

For a meaningful comparison of the three model strategies, we choose parameters such that (i) in the absence of CTL, infected cells have the same expected lifetime and the same net growth rate

In the absence of CTL, the expected lifetime of a cell infected with a budding virus is

We compare the standard model (green) with models of a budding virus (black) and lytic (red) strategies. Parameters are chosen so that in the absence of CTL all models yield the same infected cell growth rate, expected lifetime, and for the lytic and budding strategies have the same window of visibility of infected cells to CTL,

We draw three conclusions here. First, if budding and lytic viruses are visible to CTL from the time of infection, both life-history strategies give identical results to the simple birth-death model. If there is any delay in infected cells becoming visible to CTL, the threshold CTL frequency required for immunity increases.

Second, if we control for growth rate, infected cell lifetime and the CTL window of opportunity

Our third conclusion is that to make a parameter-independent comparison meaningful requires controlling for the growth rate, cell lifetime and CTL visibility window

SIV in rhesus macaques buds from its primary target cell population, CD4 T cells. Infection may begin at a mucosal surface and virus remains localised there for 2–7 days before disseminating to other tissues

We assume that the cytopathic effects of SIV begin upon virus shedding (

In what follows, we assume that both the natural mortality of cells and the contribution of the endogenous CTL response to infected cell death in the first few days of acute infection are negligible. The combined process of presentation of SIV epitopes to naive CTL in local lymph nodes, activation, proliferation and migration of CTL to the infection site is likely to take several days; and similar upslopes of virus load are observed in SIV-infected rhesus macaques in the presence or absence of CD8 responses

Li

Now suppose that a vaccine can generate a local SIV-specific memory CTL density,

We can assess the importance of modeling virus epitope dynamics by comparing these estimates of

The quantity

Our estimate of the critical killing rate does not depend on the time required for reactivation and/or migration of tissue resident of SIV-specific CTL to the site of infection. It is also independent of inoculum size, provided

To put the estimate of

These observations suggest that the challenge CTL face in controlling the spread of virus after systemic dissemination is far more severe than the one they face at the infection site. There are several possible reasons for this. First, infected cells are spatially localised. Second, resting CD4 T cells are likely the primary target cell population early in infection, and these cells produce virus at a substantially lower rate than the activated CD4 T cells that are the major source of virus in the disseminated acute phase

As a step further, knowing the required vaccine-induced cell death rate

We have shown that we expect mass-action kinetics to hold if populations are well-mixed, and either the E∶T ratio is high or handling times are much shorter than cell-cell surveillance times. In early SIV infection, the validity of the well mixed and/or the high E∶T assumptions will depend jointly on the degrees to which infected cells and CTL are clustered or spread diffusely across the tissue. If both populations are well mixed, we can make a rough estimate of E∶T early in SIV infection if CTL are at the predicted critical density. We estimate 3000–10000 cells (of all types) per

We also note that the limits of applicability of mass-action models to killing assays are still ill-defined. Ganusov

If infected cells are tightly clustered and sparsely infiltrated by CTL, one would expect the rate of killing by CTL to be limited by the handling time once a cluster has been located. In the well-mixed deterministic model, the total rate of loss of infected cells will then be linear in CTL densities and independent of infected cell numbers (equation (10)). In this regime, the model predicts CTL will ultimately fail to control the infection, assuming susceptible cells are abundant and accessible. These assumptions may not hold, however. The density of susceptible cells in healthy tissue is an upper limit to infected cell densities in very early infection, before the influx of SIV-specific CD4 cells that provide new targets. Resting CD4 T cells are present at a density of 100–200 mm

All of these uncertainties emphasise that more precise estimates for

There are inevitably many qualifications to this result that are specific to SIV, as well as more general issues that we present in the Discussion. Several will tend to increase the estimate of the critical frequency. First, as noted above, our estimate of the infected cell death rate may be too high, as it is taken from data at the infection site 28 days-post infection when the primary CTL and antibody responses are likely present. We incorporate our uncertainty in this death rate by using a wide spread of plausible values. Second, in the early growth phase of natural primary infection, selective pressure on the mutating virus exerted by the developing CTL response is expected to be low. A vaccine-induced memory CTL population will increase this pressure. To minimise this effect, broad coverage of virus epitopes to both early and conserved proteins is required. Third, we neglect the longer-term effect of the early generation of latently infected CD4 T cells that escape CTL detection, for which there is some evidence in acute infection

Other factors will act to reduce our estimate. For example, in addition to cytolysis, CTL secrete soluble factors that may make a substantial contribution to CTL-mediated protection in both the acute

Finally, it is worth noting the difficulties in connecting rates of surveillance and killing by a single CTL in a given tissue to estimates of the total-body contribution of CTL to infected cell death. The problems stem from the dimensionality of rate constants and the implicit averaging of the effects of CTL in different anatomical locations. For example, Wick

Eliciting strong cellular immune responses has the potential to augment vaccine efficacy. To our knowledge, however, there are currently no estimates of how many CTL any given vaccine needs to generate, or even whether the necessary numbers are physiologically possible. Our approach provides first-order estimates of the required CTL densities that may inform the design of

In a vaccinated individual, the E∶T ratio might be expected to be high at the beginning of an infection. In this case we have shown that handling times can be neglected and only the effective rate of CTL surveillance needs to be estimated to obtain the critical density. The effective surveillance rate combines (i) the rate at which CTL move between (survey) cells, (ii) the timecourse of expression of virus epitopes on infected cells, and (iii) the sensitivity of CTL to different levels of epitope expression. We illustrated this by estimating the critical CTL density required for the early control of SIV infection.

We have shown that considering virus life-histories is important for two reasons. First, using the simplest mass-action models of CTL killing with estimates of surveillance rates underestimates the number of CTL required to provide immunity. Second, intuition might have suggested that CTL are more effective against lytic viruses than budding viruses, as removing a cell infected with a lytic virus prevents all transmission from that cell. We show that the converse is true, after controlling for growth rate and infected cell lifetime. Thus knowing the visibility of infected cells to CTL, as well as the virus production schedule, is important for calculating critical CTL densities.

There other factors and potential refinements that need to be considered:

We have discussed the issues of handling time and virus epitope dynamics separately, and have argued that only the latter needs to be considered at high E∶T ratios. When E∶T is low, a model incorporating both processes may be required.

Simulations suggest that the assumption of a mass-action killing rate may hold in some spatially structured environments

Reasonable estimates of the maximal densities of CTL achievable in a tissue might be the order of a few percent. Even at very high ratios of CTL to infected cells, then, if both populations are randomly distributed the probability of multiple CTL binding to a single infected cell is small. However, if infected cells exhibit clustering and/or if CTL migrate preferentially to infected cells, multiple CTL attachments to one cell may occur frequently

A CTL response typically comprises multiple clones with different functional quality and efficiencies of surveillance, and specific for different virus epitopes, each potentially with different timecourses of expression. Our analysis can be interpreted as either describing what is required of a single epitope-specific response, or the net effect of multiple CTL responses. In principle, the effect of multiple CTL responses can be calculated given a numerical and functional immunodominance hierarchy, and the timecourses of epitope expression

Very early in infection, when infected cell numbers are small, chains of transmission from cell to cell have a non-negligible probability of going extinct. The presence of CTL, even at a density insufficient to provide immunity once an infection reaches a deterministic phase of growth, may increase the probability of stopping virus growth very early in infection

One concern with T cell based vaccines is that protection may only be possible with very high CTL densities, suggested by studies of CD8 T cell protection against the liver stage of Malaria infection

In summary, our studies suggest that while there are many caveats with using models of CTL control of infected cell to understand infection dynamics, knowledge of life-history strategies may be important for refining our quantitative understanding of how CTL can contribute to the control of acute infections.

All analyses were performed in

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The authors thank Rob de Boer, Ulrich Kadolsky and the reviewers for helpful comments.