A Sigmoid Functional Response Emerges When Cytotoxic T Lymphocytes Start Killing Fresh Target Cells.

Cytotoxic T lymphocyte (CTL)-mediated killing involves the formation of a synapse with a target cell, followed by delivery of perforin and granzymes. Previously, we derived a general functional response for CTL killing while considering that CTLs form stable synapses (i.e., single-stage) and that the number of conjugates remains at steady state. However, the killing of target cells sometimes requires multiple engagements (i.e., multistage). To study how multistage killing and a lack of steady state influence the functional response, we here analyze a set of differential equations as well as simulations employing the cellular Potts model, in both cases describing CTLs that kill target cells. We find that at steady state the total killing rate (i.e., the number of target cells killed by all CTLs) is well described by the previously derived double saturation function. Compared to single-stage killing, the total killing rate during multistage killing saturates at higher CTL and target cell densities. Importantly, when the killing is measured before the steady state is approached, a qualitatively different functional response emerges for two reasons: First, the killing signal of each CTL gets diluted over several targets and because this dilution effect is strongest at high target cell densities; this can result in a peak in the dependence of the total killing rate on the target cell density. Second, the total killing rate exhibits a sigmoid dependence on the CTL density when killing is a multistage process, because it takes typically more than one CTL to kill a target. In conclusion, a sigmoid dependence of the killing rate on the CTLs during initial phases of killing may be indicative of a multistage killing process. Observation of a sigmoid functional response may thus arise from a dilution effect and is not necessarily due to cooperative behavior of the CTLs.

Notwithstanding Circumstantial Alibis, Cytotoxic T Cells Can Be Major Killers of HIV-1-Infected Cells.

UNLABELLED: Several experiments suggest that in the chronic phase of human immunodeficiency virus type 1 (HIV-1) infection, CD8(+) cytotoxic T lymphocytes (CTL) contribute very little to the death of productively infected cells. First, the expected life span of productively infected cells is fairly long, i.e., about 1 day. Second, this life span is hardly affected by the depletion of CD8(+) T cells. Third, the rate at which mutants escaping a CTL response take over the viral population tends to be slow. Our main result is that all these observations are perfectly compatible with killing rates that are much faster than one per day once we invoke the fact that infected cells proceed through an eclipse phase of about 1 day before they start producing virus. Assuming that the major protective effect of CTL is cytolytic, we demonstrate that mathematical models with an eclipse phase account for the data when the killing is fast and when it varies over the life cycle of infected cells. Considering the steady state corresponding to the chronic phase of the infection, we find that the rate of immune escape and the rate at which the viral load increases following CD8(+) T cell depletion should reflect the viral replication rate, ρ. A meta-analysis of previous data shows that viral replication rates during chronic infection vary between 0.5 ≤ ρ ≤ 1 day(-1) Balancing such fast viral replication requires killing rates that are several times larger than ρ, implying that most productively infected cells would die by cytolytic effects. IMPORTANCE: Most current data suggest that cytotoxic T cells (CTL) mediate their control of human immunodeficiency virus type 1 (HIV-1) infection by nonlytic mechanisms; i.e., the data suggest that CTL hardly kill. This interpretation of these data has been based upon the general mathematical model for HIV infection. Because this model ignores the eclipse phase between the infection of a target cell and the start of viral production by that cell, we reanalyze the same data sets with novel models that do account for the eclipse phase. We find that the data are perfectly consistent with lytic control by CTL and predict that most productively infected cells are killed by CTL. Because the killing rate should balance the viral replication rate, we estimate both parameters from a large set of published experiments in which CD8(+) T cells were depleted in simian immunodeficiency virus (SIV)-infected monkeys. This confirms that the killing rate can be much faster than is currently appreciated.

A general functional response of cytotoxic T lymphocyte-mediated killing of target cells.

Cytotoxic T lymphocytes (CTLs) kill virus-infected cells and tumor cells, and play a critical role in immune protection. Our knowledge of how the CTL killing efficiency varies with CTL and target cell numbers is limited. Here, we simulate a region of lymphoid tissue using a cellular Potts model to characterize the functional response of CTL killing of target cells, and find that the total killing rate saturates both with the CTL and the target cell densities. The relative saturation in CTL and target cell densities is determined by whether a CTL can kill multiple target cells at the same time, and whether a target cell can be killed by many CTLs together. We find that all the studied regimes can be well described by a double-saturation (DS) function with two different saturation constants. We show that this DS model can be mechanistically derived for the cases where target cells are killed by a single CTL. For the other cases, a biological interpretation of the parameters is still possible. Our results imply that this DS function can be used as a tool to predict the cellular interactions in cytotoxicity data.

Estimating frequencies of minority nevirapine-resistant strains in chronically HIV-1-infected individuals naive to nevirapine by using stochastic simulations and a mathematical model.

Nevirapine forms the mainstay of our efforts to curtail the pediatric AIDS epidemic through prevention of mother-to-child transmission of HIV-1. A key limitation, however, is the rapid selection of HIV-1 strains resistant to nevirapine following the administration of a single dose. This rapid selection of resistance suggests that nevirapine-resistant strains preexist in HIV-1 patients and may adversely affect outcomes of treatment. The frequencies of nevirapine-resistant strains in vivo, however, remain poorly estimated, possibly because they exist as a minority below current assay detection limits. Here, we employ stochastic simulations and a mathematical model to estimate the frequencies of strains carrying different combinations of the common nevirapine resistance mutations K103N, V106A, Y181C, Y188C, and G190A in chronically infected HIV-1 patients naïve to nevirapine. We estimate the relative fitness of mutant strains from an independent analysis of previous competitive growth assays. We predict that single mutants are likely to preexist in patients at frequencies ( approximately 0.01% to 0.001%) near or below current assay detection limits (>0.01%), emphasizing the need for more-sensitive assays. The existence of double mutants is subject to large stochastic variations. Triple and higher mutants are predicted not to exist. Our estimates are robust to variations in the recombination rate, cellular superinfection frequency, and the effective population size. Thus, with 10(7) to 10(8) infected cells in HIV-1 patients, even when undetected, nevirapine-resistant genomes may exist in substantial numbers and compromise efforts to prevent mother-to-child transmission of HIV-1, accelerate the failure of subsequent antiretroviral treatments, and facilitate the transmission of drug resistance.

Tissue Dimensionality Influences the Functional Response of Cytotoxic T Lymphocyte-Mediated Killing of Targets.

Cytotoxic T lymphocyte (CTL)-mediated killing of virus infections and tumors occurs over a wide range of conditions. The spatial environments in which CTLs encounter target cells vary from narrow vessels, to two-dimensional epithelial tissues, to densely populated 3-dimensional (3D) T cell areas within lymphoid tissues. How such spatial environments alter the functional response of CTL-mediated killing, i.e., how the killing efficiency depends on cell densities, is unclear. In this study, we perform cellular Potts model simulations in different spatial configurations to investigate how the dimensionality of the space affects the functional response of CTL-mediated killing. Irrespective of the spatial configuration, the function with separate saturation constants for CTL and for target cell densities that we previously proposed can in all cases describe the response, demonstrating its generality. However, the tissue dimensionality determines at which cell densities the killing rate starts to saturate. We show that saturation in a fully 3D environment is stronger than in a "flat" 3D environment, which is largely due to accompanying differences in the CTL-target encounter rates.

What do mathematical models tell us about killing rates during HIV-1 infection?

Over the past few decades the extent to which cytotoxic T lymphocytes (CTLs) control human immunodeficiency virus (HIV) replication has been studied extensively, yet their role and mode of action remain controversial. In some studies, CTLs were found to kill a large fraction of the productively infected cells relative to the viral cytopathicity, whereas in others CTLs were suggested to kill only a small fraction of infected cells. In this review, we compile published estimates of CTL-mediated death rates, and examine whether these studies permit determining the rate at which CTLs kill HIV-1 infected cells. We highlight potential misinterpretations of the CTL-killing rates from the escape rates of mutants, and from perturbations of the steady state viral load during chronic infection. Our major conclusion is that CTL-mediated killing rates remain unknown. But contrary to current consensus, we argue that killing rates higher than one per day are perfectly consistent with the experimental data, which would imply that the majority of the productively infected cells could still die from CTL-mediated killing rather than from viral cytopathicity.