Early low-titer neutralizing antibodies impede HIV-1 replication and select for virus escape

Single genome sequencing of early HIV-1 genomes provides a sensitive, dynamic assessment of virus evolution and insight into the earliest anti-viral immune responses in vivo. By using this approach, together with deep sequencing, site-directed mutagenesis, antibody adsorptions and virus-entry assays, we found evidence in three subjects of neutralizing antibody (Nab) responses as early as 2 weeks post-seroconversion, with Nab titers as low as 1∶20 to 1∶50 (IC(50)) selecting for virus escape. In each of the subjects, Nabs targeted different regions of the HIV-1 envelope (Env) in a strain-specific, conformationally sensitive manner. In subject CH40, virus escape was first mediated by mutations in the V1 region of the Env, followed by V3. HIV-1 specific monoclonal antibodies from this subject mapped to an immunodominant region at the base of V3 and exhibited neutralizing patterns indistinguishable from polyclonal antibody responses, indicating V1-V3 interactions within the Env trimer. In subject CH77, escape mutations mapped to the V2 region of Env, several of which selected for alterations of glycosylation. And in subject CH58, escape mutations mapped to the Env outer domain. In all three subjects, initial Nab recognition was followed by sequential rounds of virus escape and Nab elicitation, with Nab escape variants exhibiting variable costs to replication fitness. Although delayed in comparison with autologous CD8 T-cell responses, our findings show that Nabs appear earlier in HIV-1 infection than previously recognized, target diverse sites on HIV-1 Env, and impede virus replication at surprisingly low titers. The unexpected in vivo sensitivity of early transmitted/founder virus to Nabs raises the possibility that similarly low concentrations of vaccine-induced Nabs could impair virus acquisition in natural HIV-1 transmission, where the risk of infection is low and the number of viruses responsible for transmission and productive clinical infection is typically one.

Distinct kinetics of Gag-specific CD4+ and CD8+ T cell responses during acute HIV-1 infection

HIV infection is characterized by a gradual deterioration of immune function, mainly in the CD4 compartment. To better understand the dynamics of HIV-specific T cells, we analyzed the kinetics and polyfunctional profiles of Gag-specific CD4(+) and CD8(+) T cell responses in 12 subtype C-infected individuals with different disease-progression profiles, ranging from acute to chronic HIV infection. The frequencies of Gag-responsive CD4(+) and CD8(+) T cells showed distinct temporal kinetics. The peak frequency of Gag-responsive IFN-γ(+)CD4(+) T cells was observed at a median of 28 d (interquartile range: 21-81 d) post-Fiebig I/II staging, whereas Gag-specific IFN-γ(+)CD8(+) T cell responses peaked at a median of 253 d (interquartile range: 136-401 d) and showed a significant biphasic expansion. The proportion of TNF-α-expressing cells within the IFN-γ(+)CD4(+) T cell population increased (p = 0.001) over time, whereas TNF-α-expressing cells within IFN-γ(+)CD8(+) T cells declined (p = 0.005). Both Gag-responsive CD4(+) and CD8(+) T cells showed decreased Ki67 expression within the first 120 d post-Fiebig I/II staging. Prior to the disappearance of Gag-responsive Ki67(+)CD4(+) T cells, these cells positively correlated (p = 0.00038) with viremia, indicating that early Gag-responsive CD4 events are shaped by viral burden. No such associations were observed in the Gag-specific CD8(+) T cell compartment. Overall, these observations indicated that circulating Gag-responsive CD4(+) and CD8(+) T cell frequencies and functions are not synchronous, and properties change rapidly at different tempos during early HIV infection.

Killing of targets by CD8 T cells in the mouse spleen follows the law of mass action

It has been difficult to correlate the quality of CD8 T cell responses with protection against viral infections. To investigate the relationship between efficacy and magnitude of T cell responses, we quantify the rate at which individual CD8 effector and memory T cells kill target cells in the mouse spleen. Using mathematical modeling, we analyze recent data on the loss of target cells pulsed with three different peptides from the mouse lymphocytic choriomeningitis virus (LCMV) in mouse spleens with varying numbers of epitope-specific CD8 T cells. We find that the killing of targets follows the law of mass-action, i.e., the death rate of individual target cells remains proportional to the frequency (or the total number) of specific CD8 T cells in the spleen despite the fact that effector cell densities and effector to target ratios vary about a 1000-fold. The killing rate of LCMV-specific CD8 T cells is largely independent of T cell specificity and differentiation stage. Our results thus allow one to calculate the critical T cell concentration at which growth of a virus with a given replication rate can be prevented from the start of infection by memory CD8 T cell response.

Stochastic models of lymphocyte proliferation and death

Quantitative understanding of the kinetics of lymphocyte proliferation and death upon activation with an antigen is crucial for elucidating factors determining the magnitude, duration and efficiency of the immune response. Recent advances in quantitative experimental techniques, in particular intracellular labeling and multi-channel flow cytometry, allow one to measure the population structure of proliferating and dying lymphocytes for several generations with high precision. These new experimental techniques require novel quantitative methods of analysis. We review several recent mathematical approaches used to describe and analyze cell proliferation data. Using a rigorous mathematical framework, we show that two commonly used models that are based on the theories of age-structured cell populations and of branching processes, are mathematically identical. We provide several simple analytical solutions for a model in which the distribution of inter-division times follows a gamma distribution and show that this model can fit both simulated and experimental data. We also show that the estimates of some critical kinetic parameters, such as the average inter-division time, obtained by fitting models to data may depend on the assumed distribution of inter-division times, highlighting the challenges in quantitative understanding of cell kinetics.

Transmission of single HIV-1 genomes and dynamics of early immune escape revealed by ultra-deep sequencing

We used ultra-deep sequencing to obtain tens of thousands of HIV-1 sequences from regions targeted by CD8+ T lymphocytes from longitudinal samples from three acutely infected subjects, and modeled viral evolution during the critical first weeks of infection. Previous studies suggested that a single virus established productive infection, but these conclusions were tempered because of limited sampling; now, we have greatly increased our confidence in this observation through modeling the observed earliest sample diversity based on vastly more extensive sampling. Conventional sequencing of HIV-1 from acute/early infection has shown different patterns of escape at different epitopes; we investigated the earliest escapes in exquisite detail. Over 3–6 weeks, ultradeep sequencing revealed that the virus explored an extraordinary array of potential escape routes in the process of evading the earliest CD8 T-lymphocyte responses – using 454 sequencing, we identified over 50 variant forms of each targeted epitope during early immune escape, while only 2–7 variants were detected in the same samples via conventional sequencing. In contrast to the diversity seen within epitopes, non-epitope regions, including the Envelope V3 region, which was sequenced as a control in each subject, displayed very low levels of variation. In early infection, in the regions sequenced, the consensus forms did not have a fitness advantage large enough to trigger reversion to consensus amino acids in the absence of immune pressure. In one subject, a genetic bottleneck was observed, with extensive diversity at the second time point narrowing to two dominant escape forms by the third time point, all within two months of infection. Traces of immune escape were observed in the earliest samples, suggesting that immune pressure is present and effective earlier than previously reported; quantifying the loss rate of the founder virus suggests a direct role for CD8 T-lymphocyte responses in viral containment after peak viremia. Dramatic shifts in the frequencies of epitope variants during the first weeks of infection revealed a complex interplay between viral fitness and immune escape.

Persistence of viral infection despite similar killing efficacy of antiviral CD8(+) T cells during acute and chronic phases of infection

Why some viruses establish chronic infections while others do not is poorly understood. One possibility is that the host's immune response is impaired during chronic infections and is unable to clear the virus from the host. In this report, we use a recently proposed framework to estimate the per capita killing efficacy of CD8(+) T cells, specific for the polyoma virus (PyV), which establishes a chronic infection in mice. Surprisingly, the estimated per cell killing efficacy of PyV-specific effector CD8(+) T cells during the acute phase of the infection was very similar to the efficacy of effector CD8(+) T cells specific to lymphocytic choriomeningitis virus (LCMV-Armstrong), which is cleared from the host. Our results suggest that persistence of PyV does not result from the generation of an inefficient PyV-specific CD8(+) T cell response, and that other host or viral factors are responsible for the ability of PyV to establish chronic infection.

In mice, tuberculosis progression is associated with intensive inflammatory response and the accumulation of Gr-1 cells in the lungs

Background

Infection with Mycobacterium tuberculosis (Mtb) results in different clinical outcomes ranging from asymptomatic containment to rapidly progressing tuberculosis (TB). The mechanisms controlling TB progression in immunologically-competent hosts remain unclear.

Methodology/Principal Findings

To address these mechanisms, we analyzed TB progression in a panel of genetically heterogeneous (A/SnxI/St) F2 mice, originating from TB-highly-susceptible I/St and more resistant A/Sn mice. In F2 mice the rates of TB progression differed. In mice that did not reach terminal stage of infection, TB progression did not correlate with lung Mtb loads. Nor was TB progression correlated with lung expression of factors involved in antibacterial immunity, such as iNOS, IFN-γ, or IL-12p40. The major characteristics of progressing TB was high lung expression of the inflammation-related factors IL-1β, IL-6, IL-11 (p<0.0003); CCL3, CCL4, CXCL2 (p<0.002); MMP-8 (p<0.0001). The major predictors of TB progression were high expressions of IL-1β and IL-11. TNF-α had both protective and harmful effects. Factors associated with TB progression were expressed mainly by macrophages (F4-80⁺ cells) and granulocytes (Gr-1^hi/Ly-6G^hi cells). Macrophages and granulocytes from I/St and A/Sn parental strains exhibited intrinsic differences in the expression of inflammatory factors, suggesting that genetically determined peculiarities of phagocytes transcriptional response could account for the peculiarities of gene expression in the infected lungs. Another characteristic feature of progressing TB was the accumulation in the infected lungs of Gr-1^dim cells that could contribute to TB progression.

Conclusions/Significance

In a population of immunocompetent hosts, the outcome of TB depends on quantitatively- and genetically-controlled differences in the intensity of inflammatory responses, rather than being a direct consequence of mycobacterial colonization. Local accumulation of Gr-1^dim cells is a newly identified feature of progressing TB. High expression of IL-1β and IL-11 are potential risk factors for TB progression and possible targets for TB immunomodulation.

Explicit kinetic heterogeneity: mathematical models for interpretation of deuterium labeling of heterogeneous cell populations

Estimation of division and death rates of lymphocytes in different conditions is vital for quantitative understanding of the immune system. Deuterium, in the form of deuterated glucose or heavy water, can be used to measure rates of proliferation and death of lymphocytes in vivo. Inferring these rates from labeling and delabeling curves has been subject to considerable debate with different groups suggesting different mathematical models for that purpose. We show that the three most common models, which are based on quite different biological assumptions, actually predict mathematically identical labeling curves with one parameter for the exponential up and down slope, and one parameter defining the maximum labeling level. By extending these previous models, we here propose a novel approach for the analysis of data from deuterium labeling experiments. We construct a model of “kinetic heterogeneity” in which the total cell population consists of many sub-populations with different rates of cell turnover. In this model, for a given distribution of the rates of turnover, the predicted fraction of labeled DNA accumulated and lost can be calculated. Our model reproduces several previously made experimental observations, such as a negative correlation between the length of the labeling period and the rate at which labeled DNA is lost after label cessation. We demonstrate the reliability of the new explicit kinetic heterogeneity model by applying it to artificially generated datasets, and illustrate its usefulness by fitting experimental data. In contrast to previous models, the explicit kinetic heterogeneity model 1) provides a novel way of interpreting labeling data; 2) allows for a non-exponential loss of labeled cells during delabeling, and 3) can be used to describe data with variable labeling length.

Understanding of cellular processes is impossible without quantitative estimates of how quickly cells in an organism divide and die. The most widely used approach to measure rates of cell turnover in humans is by labeling dividing cells with deuterium given in the form of deuterated glucose or heavy water. Surprisingly, quantitative estimates of the rates of cell turnover obtained from accumulation and decay of the labeled nucleotides in the cell population varied between different studies. We demonstrate that these differences were not likely to arise because of different mathematical models used in data fitting, since the previously used models have an identical mathematical structure. We extend these previous models to allow for cell populations with different rates of turnover and show how such a new explicit kinetic heterogeneity model can be applied to simulated and experimental data. The new model opens a new way of interpreting data from deuterium labeling experiments and will likely lead to new insights into how infections and/or treatments affect cell turnover in humans.

The first T cell response to transmitted/founder virus contributes to the control of acute viremia in HIV-1 infection

Identification of the transmitted/founder virus makes possible, for the first time, a genome-wide analysis of host immune responses against the infecting HIV-1 proteome. A complete dissection was made of the primary HIV-1–specific T cell response induced in three acutely infected patients. Cellular assays, together with new algorithms which identify sites of positive selection in the virus genome, showed that primary HIV-1–specific T cells rapidly select escape mutations concurrent with falling virus load in acute infection. Kinetic analysis and mathematical modeling of virus immune escape showed that the contribution of CD8 T cell–mediated killing of productively infected cells was earlier and much greater than previously recognized and that it contributed to the initial decline of plasma virus in acute infection. After virus escape, these first T cell responses often rapidly waned, leaving or being succeeded by T cell responses to epitopes which escaped more slowly or were invariant. These latter responses are likely to be important in maintaining the already established virus set point. In addition to mutations selected by T cells, there were other selected regions that accrued mutations more gradually but were not associated with a T cell response. These included clusters of mutations in envelope that were targeted by NAbs, a few isolated sites that reverted to the consensus sequence, and bystander mutations in linkage with T cell–driven escape.

Lymphocyte kinetics in health and disease

Quantitative understanding of immunology requires the development of experimental and mathematical techniques for estimation of rates of division and death of lymphocytes under different conditions. Here, we review the advantages and limitations of several labelling methods that are currently used to quantify turnover of lymphocytes in vivo. In addition to highlighting insights into lymphocyte kinetics which have recently been gained thanks to the development of novel techniques, we discuss important directions for future experimental and theoretical work in the field of lymphocyte turnover.

Discriminating between different pathways of memory CD8+ T cell differentiation

Despite the rapid accumulation of quantitative data on the dynamics of CD8(+) T cell responses following acute viral or bacterial infections of mice, the pathways of differentiation of naive CD8(+) T cells into memory during an immune response remain controversial. Currently, three models have been proposed. In the "stem cell-associated differentiation" model, following activation, naive T cells differentiate into stem cell-like memory cells, which then convert into terminally differentiated short-lived effector cells. In the "linear differentiation" model, following activation, naive T cells first differentiate into effectors, and after Ag clearance, effectors convert into memory cells. Finally, in the "progressive differentiation" model, naive T cells differentiate into memory or effector cells depending on the amount of specific stimulation received, with weaker stimulation resulting in formation of memory cells. This study investigates whether the mathematical models formulated from these hypotheses are consistent with the data on the dynamics of the CD8(+) T cell response to lymphocytic choriomeningitis virus during acute infection of mice. Findings indicate that two models, the stem cell-associated differentiation model and the progressive differentiation model, in which differentiation of cells is strongly linked to the number of cell divisions, fail to describe the data at biologically reasonable parameter values. This work suggests additional experimental tests that may allow for further discrimination between different models of CD8(+) T cell differentiation in acute infections.

Dynamics of CD8+ T cell responses during acute and chronic lymphocytic choriomeningitis virus infection

Infection of mice with lymphocytic choriomeningitis virus (LCMV) is frequently used to study the underlying principles of viral infections and immune responses. We fit a mathematical model to recently published data characterizing Ag-specific CD8+ T cell responses during acute (Armstrong) and chronic (clone 13) LCMV infection. This allows us to analyze the differences in the dynamics of CD8+ T cell responses against different types of LCMV infections. For the four CD8+ T cell responses studied, we find that, compared with the responses against acute infection, responses against chronic infection are generally characterized by an earlier peak and a faster contraction phase thereafter. Furthermore, the model allows us to give a new interpretation of the effect of thymectomy on the dynamics of CD8+ T cell responses during chronic LCMV infection: a smaller number of naive precursor cells is sufficient to account for the observed differences in the responses in thymectomized mice. Finally, we compare data characterizing LCMV-specific CD8+ T cell responses from different laboratories. Although the data were derived from the same experimental model, we find quantitative differences that can be solved by introducing a scaling factor. Also, we find kinetic differences that are at least partly due to the infrequent measurements of CD8+ T cells in the different laboratories.

Do most lymphocytes in humans really reside in the gut?

It is widely believed that the gut, and particularly the lamina propria (LP) of the gut, contains most of the lymphocytes in humans. The strong depletion of CD4(+) T cells from the gut LP of HIV-infected patients was, therefore, suggested to be such a large, irreversible insult that it could explain HIV disease progression. However, reviewing data from different mammalian species, we found that only 5%-20% of all lymphocytes reside in the gut, and that only 1%-9% of the total lymphocyte number is located in the gut LP. Our findings suggest that spleen and lymph nodes, rather than the gut, are the largest immune compartments in mammals.

IL-2 regulates expansion of CD4+ T cell populations by affecting cell death: insights from modeling CFSE data

It is generally accepted that IL-2 influences the dynamics of populations of T cells in vitro and in vivo. However, which parameters for cell division and/or death are affected by IL-2 is not well understood. To get better insights into the potential ways of how IL-2 may influence the population dynamics of T cells, we analyze data on the dynamics of CFSE-labeled polyclonal CD4(+) T lymphocytes in vitro after anti-CD3 stimulation at different concentrations of exogenous IL-2. Inferring cell division and death rates from CFSE-delabeling experiments is not straightforward and requires the use of mathematical models. We find that to adequately describe the dynamics of T cells at low concentrations of exogenous IL-2, the death rate of divided cells has to increase with the number of divisions cells have undergone. IL-2 hardly affects the average interdivision time. At low IL-2 concentrations 1) fewer cells are recruited into the response and successfully complete their first division; 2) the stochasticity of cell division is increased; and 3) the rate, at which the death rate increases with the division number, increases. Summarizing, our mathematical reinterpretation suggests that the main effect of IL-2 on the in vitro dynamics of naive CD4(+) T cells occurs by affecting the rate of cell death and not by changing the rate of cell division.

Pathology during acute infections: contributions of intracellular pathogens and the CTL response

Previous work has shown how, in the case of cytotoxic T-lymphocyte (CTL) responses to persistent viral infections, pathology may arise as a consequence of cell destruction directly by the virus or indirectly due to the CTL response, leading to maximum pathology at intermediate efficacy of the immune response. We expand these studies to consider pathology arising during acute infections with intracellular pathogens controlled by the CTL response. We show that, in contrast to persistent infections, pathology during acute infections is minimized with increasing efficacy of the immune response. The implications of these results for vaccination are discussed.

How does cross-reactive stimulation affect the longevity of CD8+ T cell memory?

Immunological memory—the ability to “remember” previously encountered pathogens and respond faster upon re-exposure is a central feature of the immune response in vertebrates. The cross-reactive stimulation hypothesis for the maintenance of memory proposes that memory cells specific for a given pathogen are maintained by cross-reactive stimulation following infections with other (unrelated) pathogens. We use mathematical models to examine the cross-reactive stimulation hypothesis. We find that: (i) the direct boosting of cross-reactive lineages only provides a very small increase in the average longevity of immunological memory; (ii) the expansion of cross-reactive lineages can indirectly increase the longevity of memory by reducing the magnitude of expansion of new naive lineages which occupy space in the memory compartment and are responsible for the decline in memory; (iii) cross-reactive stimulation results in variation in the rates of decline of different lineages of memory cells and enrichment of memory cell population for cells that are cross-reactive for the pathogens to which the individual has been exposed.

Immunological memory—the ability to “remember” previously encountered pathogens and respond faster on re-exposure—is a central feature of the immune response of vertebrates. Exposure to a pathogen results in the clonal expansion of a few relatively rare clones of immune cells which are specific for the pathogen to form a population large enough to control the pathogen. Immunological memory arises from the maintenance of an elevated numbers of these pathogen-specific immune cells. There has been much debate on the contribution of different processes such as the persistence of antigen, cross-reactive stimulation, and homeostasis to the maintenance of the elevated number of “memory” cells. Models have been useful in understanding the contributions of these various processes to the maintenance of memory. The models have shown that the decline rate of memory specific for previously encountered pathogens arises due to exposure to new pathogens—this causes the replacement of a fraction of “old” memory cells with memory cells specific for new pathogens. In this paper Ganusov, Antia, and colleagues use mathematical models to explore how the ability of cross-reactive memory cells to respond to the antigens on more than one pathogen can help in the maintenance of immunological memory.

Estimating Lymphocyte Division and Death Rates from CFSE Data

The division tracking dye, carboxyfluorescin diacetate succinimidyl ester (CFSE) is currently the most informative labeling technique for characterizing the division history of cells in the immune system. Gett and Hodgkin [Nat. Immunol. 1:239-244, 2000] have pioneered the quantitative analysis of CFSE data. We confirm and extend their data analysis approach using simple mathematical models. We employ the extended Gett and Hodgkin [Nat. Immunol. 1:239-244, 2000] method to estimate the time to first division, the fraction of cells recruited into division, the cell cycle time, and the average death rate from CFSE data on T cells stimulated under different concentrations of IL-2. The same data is also fitted with a simple mathematical model that we derived by reformulating the numerical model of Deenick et al. [J. Immunol. 170:4963-4972, 2003]. By a non-linear fitting procedure we estimate parameter values and confidence intervals to identify the parameters that are influenced by the IL-2 concentration. We obtain a significantly better fit to the data when we assume that the T cell death rate depends on the number of divisions cells have completed. We provide an outlook on future work that involves extending the Deenick et al. [J. Immunol. 170:4963-4972, 2003] model into the classical smith-martin model, and into a model with arbitrary probability distributions for death and division through subsequent divisions.

Imperfect vaccines and the evolution of pathogens causing acute infections in vertebrates

A study by Gandon et al. (2001) considered the potential ways pathogens may evolve in response to vaccination with imperfect vaccines. In this paper, by focusing on acute infections of vertebrate hosts, we examine whether imperfect vaccines that do not completely block a pathogen's replication (antigrowth) or transmission (antitransmission) may lead to evolution of more or less virulent pathogen strains. To address this question, we use models of the within-host dynamics of the pathogen and the host's immune responses. One advantage of the use of this within-host approach is that vaccination can be easily incorporated in the models and the trade-offs between pathogen transmissibility, host recovery, and virulence that drive evolution of pathogens in these models can be easily estimated. We find that the use of either antigrowth or antitransmission vaccines leads to the evolution of pathogens with an increased within-host growth rate; infection of unvaccinated hosts with such evolved pathogens results in high host mortality and low pathogen transmission. Vaccination of only a fraction of hosts with antigrowth vaccines may prevent pathogens from evolving high virulence due to pathogen adaptation to unvaccinated hosts and thus protection of vaccinated hosts from pathogen-induced disease. In contrast, antitransmission vaccines may be beneficial only if they are effective enough to cause pathogen extinction. Our results suggest that particular mechanisms of action of vaccines and their efficacy are crucial in predicting longterm evolutionary consequences of the use of imperfect vaccines.

Estimating Costs and Benefits of CTL Escape Mutations in SIV/HIV Infection

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.

The role of models in understanding CD8+ T-cell memory

Immunological memory - the ability to 'remember' previously encountered pathogens and respond faster on re-exposure - is a central feature of the immune response of vertebrates. We outline how mathematical models have contributed to our understanding of CD8(+) T-cell memory. Together with experimental data, models have helped to quantitatively describe and to further our understanding of both the generation of memory after infection with a pathogen and the maintenance of this memory throughout the life of an individual.

Quantifying cell turnover using CFSE data

The CFSE dye dilution assay is widely used to determine the number of divisions a given CFSE labelled cell has undergone in vitro and in vivo. In this paper, we consider how the data obtained with the use of CFSE (CFSE data) can be used to estimate the parameters determining cell division and death. For a homogeneous cell population (i.e., a population with the parameters for cell division and death being independent of time and the number of divisions cells have undergone), we consider a specific biologically based "Smith-Martin" model of cell turnover and analyze three different techniques for estimation of its parameters: direct fitting, indirect fitting and rescaling method. We find that using only CFSE data, the duration of the division phase (i.e., approximately the S+G2+M phase of the cell cycle) can be estimated with the use of either technique. In some cases, the average division or cell cycle time can be estimated using the direct fitting of the model solution to the data or by using the Gett-Hodgkin method [Gett A. and Hodgkin, P. 2000. A cellular calculus for signal integration by T cells. Nat. Immunol. 1:239-244]. Estimation of the death rates during commitment to division (i.e., approximately the G1 phase of the cell cycle) and during the division phase may not be feasible with the use of only CFSE data. We propose that measuring an additional parameter, the fraction of cells in division, may allow estimation of all model parameters including the death rates during different stages of the cell cycle.

The rescaling method for quantifying the turnover of cell populations

The dynamic nature of immune responses requires the development of appropriate experimental and theoretical tools to quantitatively estimate the division and death rates which determine the turnover of immune cells. A number of papers have used experimental data from BrdU and D-glucose labels together with a simple random birth-death model to quantify the turnover of immune cells focusing on HIV/SIV infections [Mohri et al. 279 (1998) 1223-1227, Hellerstein et al. 5 (1999) 83-89, Bonhoeffer et al. 164 (2000) 5049-5054, Mohri et al. 87 (2001) 1277-1287]. We show how uncertainties in the assumptions of the random birth-death model may lead to substantial errors in the parameters estimated. We then show how more accurate estimates can be obtained from the more recent CFSE data which allow to track the number of divisions each cell has undergone. Specifically, we: (i) describe a general stage-structured model of cell division where the probabilities of division and death are functions of time since the previous division; (ii) develop a rescaling method to identify invariant parameters (i.e. the ones that are independent of the specific functions describing division and death); (iii) show how these invariant parameters can be estimated, and (iv) illustrate this technique by applying it to CFSE data taken from the literature.

Trade-offs and the evolution of virulence of microparasites: do details matter?

Models of the within-host dynamics of parasites have been used to consider the evolution of microparasites causing acute infections in vertebrate hosts. In this paper, we use these models to examine how the level of virulence to which a parasite evolves, depends on factors such as the relationship between parasite density and its rate of transmission from infected hosts, and the mechanism of parasite-induced pathogenesis. We show that changes in the terms describing transmissibility and pathogenesis may lead to dramatic differences in the level of virulence to which a parasite evolves. This suggests that no single factor is likely to be responsible for the differences in virulence of different parasites, and that understanding of the evolution of virulence of parasites will require a detailed quantitative understanding of the interaction between the parasite and its host.

The role of the cytotoxic T-lymphocyte response and virus cytopathogenicity in the virus decline during antiviral therapy

Although it is clear that HIV can lyse HIV-infected CD4 T cells, it is still controversial whether the depletion of CD4 T cells seen in HIV-infected patients after years of asymptomatic disease is caused by the direct cytopathic effects of the virus or is mediated by the immune response. Assuming the initial decline in viraemia during highly active antiretroviral therapy (HAART) is caused by the death of cells productively infected with HIV, I investigate how the rate of the virus decline is affected by the efficiency of the cytotoxic T-lymphocyte (CTL) response. I find that whether the stronger immune response causes a more rapid virus decline depends critically on how the virus is controlled by the CTL response (lytic versus non-lytic mechanisms). Moreover, variation in the efficiency of the immune response does not always cause variation in the rate of the virus decline (and, therefore, in the death rate of infected cells), implying that the constancy of the virus decline rate measured in different patients does not necessarily indicate that the virus is cytopathic. The potential problems associated with the model and the approach undertaken are also discussed.

Estimating the instability parameters of plasmid-bearing cells. I. Chemostat culture

What determines the stability of plasmid-bearing cells in natural and laboratory conditions? In order to answer this question in a quantitative manner, we need tools allowing the estimation of parameters governing plasmid loss in different environments. In the present work, we have developed two methods for the estimation of the instability parameters of plasmid-bearing cells growing in chemostat. These instability parameters are: (i) selection coefficient (or cost of the plasmid)alpha and (ii) the probability of plasmid loss at cell division tau(0). We have found that generally selection coefficient alpha changes during elimination of plasmid-bearing cells due to changes in substrate concentration; hence, methods which assume constancy of alpha are intrinsically imprecise. Instead, one can estimate selection coefficient at the beginning and the end of cultivation when the substrate concentration is approximately constant. Applying developed techniques to two sets of experimental data, we have found that (i) the cost of the plasmid pBR322 depended on the dilution rate in chemostat and was higher at low dilutions; (ii) high levels of plasmid gene expression led to a high cost of the plasmid pPHL-7; (iii) the probability of plasmid loss was lower at high levels of plasmid gene expression and independent of the dilution rate. We have also discussed the application of our results to understanding the basic biology of bacterial plasmids.

Within-host population dynamics and the evolution of microparasites in a heterogeneous host population

Why do parasites harm their hosts? The general understanding is that if the transmission rate and virulence of a parasite are linked, then the parasite must harm its host to maximize its transmission. The exact nature of such trade-offs remains largely unclear, but for vertebrate hosts it probably involves interactions between a microparasite and the host immune system. Previous results have suggested that in a homogeneous host population in the absence of super- or coinfection, within-host dynamics lead to selection of the parasite with an intermediate growth rate that is just being controlled by the immune system before it kills the host (Antia et al. 1994). In this paper, we examine how this result changes when heterogeneity is introduced to the host population. We incorporate the simplest form of heterogeneity--random heterogeneity in the parameters describing the size of the initial parasite inoculum, the immune response of the host, and the lethal density at which the parasite kills the host. We find that the general conclusion of the previous model holds: parasites evolve some intermediate growth rate. However, in contrast with the generally accepted view, we find that virulence (measured by the case mortality or the rate of parasite-induced host mortality) increases with heterogeneity. Finally, we link the within-host and between-host dynamics of parasites. We show how the parameters for epidemiological spread of the disease can be estimated from the within-host dynamics, and in doing so examine the way in which trade-offs between these epidemiological parameters arise as a consequence of the interaction of the parasite and the immune response of the host.

Computer modeling of the biotic cycle formation in a closed ecological system

The process of biotic turnover in a closed ecological system (CES) with an external energy flow was analyzed by mathematical modeling of the biotic cycle formation. The formation of hierarchical structure in model CESs is governed by energy criteria. Energy flow through the ecosystem increases when a predator is introduced into a "producer-reducer" system at steady state. Analysis of the model shows that under certain conditions the presence of the primary predator with its high mineralization ability accelerates the biotic turnover measured by primary production. We, therefore, conclude that for every system it is possible to find a suitable predator able to provide the system with a higher biotic turnover rate and energy consumption. Grant numbers: 99-04-96017/2000.

[Mathematical modeling of population dynamics of unstable plasmid-containing bacteria during continuous cultivation in a chemostat]

A structural approach to studying the regularities of the population dynamics of unstable recombinant bacterial strains in a chemostat was elaborated. The approach is based on the mathematical modeling of cell distribution in a population with different numbers of plasmid copies. The effect of decreased selective preference of plasmidless variants of the recombinant strain in the chemostat, which is related to a decrease in the number of plasmid copies in cells upon long-term incubation was analyzed. It is shown that the time of half-elimination of plasmids from the bacterial population in the steady state in the chemostat T1/2 does not depend on the maximum number of plasmid copies in cells N but is determined only by the mean time of generation g and the probability of the loss of one plasmid copy tau. The dependence of the preference of bacterial plasmidless variants on the efficiency of expression of genes cloned into plasmids in chemostat was analyzed using the recombinant strain E. coli Z905, whose plasmids pPHL-7 contain cloned genes for the luminescence system of marine luminescing bacteria Photobacterium leiognathi.

Modelling of genetically engineered microorganisms introduction in closed artificial microcosms

The possibility of introducing genetically engineered microorganisms (GEM) into simple biotic cycles of laboratory water microcosms was investigated. The survival of the recombinant strain Escherichia coli Z905 (Apr, Lux+) in microcosms depends on the type of model ecosystems. During the absence of algae blooming in the model ecosystem, the part of plasmid-containing cells E. coli decreased fast, and the structure of the plasmid was also modified. In conditions of algae blooming (Ankistrodesmus sp.) an almost total maintenance of plasmid-containing cells was observed in E. coli population. A mathematics model of GEM's behavior in water ecosystems with different level of complexity has been formulated. Mechanisms causing the difference in luminescent exhibition of different species are discussed, and attempts are made to forecast the GEM's behavior in water ecosystems.