Virus Replication Strategies and the Critical CTL Numbers Required for the Control of Infection
[['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 [1]–[9] ) .', 'In the standard model the dynamics of infection in a tissue can be described by the abundance of infected cells .', 'During early stages of an infection when susceptible cells are in abundance , and in the absence of specific immunity , grows exponentially as ( the doubling time is then ) .', 'The parameter is the per capita growth rate of the infected cell population and is the net outcome of a plethora of biological processes; virus replication , shedding from infected cells , clearance of virus from the intercellular space , infection of susceptible cells; and loss of infected cells by natural mortality , virus-induced mortality and innate immune mechanisms .', 'Now suppose in addition that infected cells can be killed by CTL .', 'The standard model assumes CTL are tissue-resident and activated , CTL and infected cells are well-mixed , distributed spatially randomly , and a CTL encounters other cells with a constant probability per unit time .', 'These assumptions yield a mass-action term for the killing of infected cells by CTL , ( 1 ) where and are the densities of infected cells and CTL respectively .', 'We define density to be a dimensionless quantity in the range .', 'It represents cell numbers locally as a fraction of all of the cells being surveyed by CTL , uninfected or infected .', 'We use this definition ( rather than cells per unit volume of tissue , cells per unit volume of blood , or the proportion of CTL that are antigen-specific ) because it relates directly to cell frequencies obtained from tissue samples by flow cytometry , and it yields a simple interpretation of the parameter , which we return to in a moment .', 'In this model , sterilising immunity corresponds to a net growth rate .', 'To achieve this requires specific CTL to be present above a critical density where ( 2 ) With our dimensionless definition of density , the parameter is the rate of surveillance , or the expected number of cells of any kind that a CTL encounters per unit time; is the expected time between contacts; and , where is the range , is the expected number of infected cells that a single CTL encounters per unit time ( in Text S1 we discuss how this interpretation of relates to its interpretation when and are measured as numbers of cells per unit volume ) .', 'So if in the absence of CTL , infected cells double in number every days , a simple estimate of the local density of CTL required to control the infection is ( 3 ) However , this simple model neglects at least two major biological processes that may impact the estimate of the critical density – the dynamics of CTL-infected cell conjugates , and the life history of viruses within infected cells .', 'We investigate these separately , and begin with the influence of CTL-target handling time on the rate at which a population of CTL kills a fixed number of target cells .', '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 et al . [10] showed that CTL engaged in lysis can remain bound for up to half an hour after initial contact .', 'Thus accounting for the temporary sequestration of CTL in conjugation with infected cells may be important for the dynamics of killing and for assessing the validity of the mass-action assumption .', 'A simple way to include the time take for CTL to handle infected cells is to explicitly track CTL-infected cell conjugates [11] , such that free CTL ( ) initially present at density successfully bind to infected cells present at density at rate per CTL , and remain in conjugates for a mean time .', 'This time can be a compound of several distinct processes – delivery of the lethal hit following conjugation , release of the apoptotic cell , and a possible recovery time before the CTL is capable of detecting another infected cell .', 'Here is the rate of surveillance of all cell types , and incorporates any time spent conjugated with uninfected cells .', '( 4 ) ( 5 ) ( 6 ) Since if we assume no loss of CTL the number of bound complexes , can be eliminated to give ( 7 ) ( 8 ) We can now identify the conditions under which mass-action holds .', 'If CTL are in excess ( ) , then and so mass-action holds for all and , irrespective of handling or search times .', 'If infected cells are in excess , three regimes exist: 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 [10] and simple mass-action models of in vivo surveillance and killing by CTL , all of which describe experimental data well using the assumption of instantaneous killing and yield estimates of 1–40 cells surveyed per CTL per minute [8] , [9] , [12] , [13] .', 'Extended encounter times between CTL and infected cells are consistent with these models if CTL are in excess .', 'At high ratios of CTL to infected cells ( usually referred to in the literature as the effector∶target ratio , E∶T ) , few CTL will kill more than once , and so handling time becomes irrelevant for the long-term dynamics of infected cell removal .', 'So if the inoculum size of the virus is low compared to existing effector cell densities ( ) , or if handling times are short compared to the time taken to survey cells , we do not need to consider the dynamics of CTL-infected cell conjugates to calculate the minimum density of specific CTL needed for protection .', '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 [8] , [9] , [12] , [13] use the assumption that the infected cells are expressing peptide-MHC complexes at a constant level over the timecourse of the experiments .', 'However , viral replication is a dynamic process and so the window of expression of a given peptide-MHC complex by a cell infected with a live virus may be limited .', 'Thus CTL have to contend with not only locating cells within a tissue , but also the dynamics of the expression of viral epitopes [14] .', 'In the model that follows we assume we are in a regime where mass-action operates .', 'In Text S2 we describe the more general model in which both handling time and virus epitope dynamics are accounted for .', 'How will the dynamics of virion production and virus epitope expression affect our estimate of the local density of CTL required for sterilising immunity , ?', 'With the mass-action assumption , the population dynamics of infected cells becomes ( 13 ) ( 14 ) where is the density of cells at time that have been infected for a time , is the rate of release of virions at age after infection , is the age-dependent mortality of infected cells not ascribed to CTL activity , and is the ( now assumed time-independent ) free CTL density .', 'This equation describes the dynamics of infected cell numbers in infinitesimal age classes as they are generated through new infection , age and die .', 'Since detection upon encounter is dependent on the expression of the appropriate virus epitopes and hence on time since infection , the rate at which a CTL encounters and recognises infected cells in age class is now , and is the total rate at which a CTL encounters and recognises infected cells .', 'Equation ( 14 ) represents the process of new infections , and assumes', '( i ) susceptible cells are in abundance , and', '( ii ) that infection of new cells or clearance of free virus is more rapid than the dynamics of virus replication and release from infected cells .', 'So the rate of infection of susceptible cells at time is proportional to the total rate of virus production at that time , with constant of proportionality .', 'A useful quantity is the survivorship , the proportion of a cohort of cells that are still alive at a time after they were infected .', 'The survivorship is related to the total mortality through ( 15 ) The steady state age-distribution solution of equation ( 14 ) yields the total cell population growing or declining as , where satisfies the Lotka-Euler equation ( see Text S3 for a derivation ) ( 16 ) This has a unique real solution for the growth rate [15] .', '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 Figure 1 .', '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 .', 'A biological motivation for this is that CTL can recognise as few as 10 pMHC complexes [16] and so we expect the efficiency of recognition to saturate quickly in peptide density .', 'With the forms for the virus-induced mortality described in Figure 1 , and , exact expressions can be derived for the survivorship .', 'For budding viruses , we assume that infected cells become visible to CTL before the onset of virus-induced mortality ( Figure 1 ) , but our conclusions are independent of the ordering of these events .', 'We then have ( 17 ) For lytic viruses , ( 18 ) For the budding virus , solving equation 16 for the steady state growth rate reduces to solving the following implicit equation for , ( 19 ) For lytic viruses it yields the direct solution ( 20 ) Identical expressions are derived if visibility to CTL begins after the onset of virus-induced mortality .', '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 [17] , [18] , followed by rapid exponential growth of virus titers in blood .', 'Eventually a combination of mechanisms ( CTL removal of infected cells , virus cytopathicity , innate immunity , the availability of susceptible cells , and antibody responses ) brings virus growth under control and the infection enters the chronic phase .', 'Here we use data from acute SIV infection to estimate the minimum density of tissue-resident CTL that a vaccine needs to elicit for early control of infected cell growth at the site of infection .', 'We assume that the inoculum size is small , and so E∶T will be large .', 'In this regime we argue that handling time can be neglected , and that the mass-action approximation is valid .', 'We assess the validity of the E∶T assumption following the analysis below .'], ['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 .', 'Text S2 describes such a model , and shows how both the models considered here can be derived from it .', 'Simulations suggest that the assumption of a mass-action killing rate may hold in some spatially structured environments [34] .', 'However , as discussed for SIV , mass-action may not hold if infected cells are clustered , which may be likely particularly if infection takes place through cell-cell transmission; or if there is directed motion of CTL towards infected cells driven by chemotactic or inflammatory cues .', 'Incorporating these factors requires the use of spatially explicit dynamical models such as those presented in [34]–[36] .', 'Other effects may complicate the picture – for example , bystander killing of uninfected cells surrounding infected cells may provide a firebreak which limits further transmission [37] , [38] .', '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 [39] , [40] and this effect needs to be considered in the calculation of numbers of effector cells not bound in conjugates .', '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 [41] .', 'Without going into this level of detail , however , it is very likely advantageous for a vaccine to induce a dominant CTL response to very early epitopes , and we show this reduces the estimate of the critical density .', '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 [42] .', 'The use of high inocula in vaccine trials may mask this potential protective effect .', 'Our calculation of the critical CTL density also assumes that the population of infected cells has reached a stable age distribution , which is a reasonable assumption following a small number of rounds of infected cell growth , a timescale that is likely comparable to that of activation of resident memory T cells .', 'Explicit stochastic simulations may be required to study the dependence of the probability of early extinction on local CTL frequencies .', '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 [43] , although in this system there may be significant spatial constraints on the surveillance of liver tissue by CTL .', 'If this is the case , then according to the canonical model of a finite homeostatic capacity for T cell memory , the use of such vaccines may compromise the maintenance of existing memory T cell populations .', 'However , it may be possible to generate very large numbers of CTL with prime-boost vaccination regimens without substantially ablating immunity to other pathogens [44] .', '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 [45] .']]
['Introduction', 'Results', 'Discussion', 'Methods']
['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 vivo assays using virus-infected cells rather than peptide-pulsed targets .']
['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 .']
['systems biology', 'immune cells', 't cells', 'immunology', 'biology', 'computational biology', 'immune response']