Quantifying T lymphocyte turnover

Abstract

Peripheral T cell populations are maintained by production of naive T cells in the thymus, clonal expansion of activated cells, cellular self-renewal (or homeostatic proliferation), and density dependent cell life spans. A variety of experimental techniques have been employed to quantify the relative contributions of these processes. In modern studies lymphocytes are typically labeled with 5-bromo-2'-deoxyuridine (BrdU), deuterium, or the fluorescent dye carboxy-fluorescein diacetate succinimidyl ester (CFSE), their division history has been studied by monitoring telomere shortening and the dilution of T cell receptor excision circles (TRECs) or the dye CFSE, and clonal expansion has been documented by recording changes in the population densities of antigen specific cells. Proper interpretation of such data in terms of the underlying rates of T cell production, division, and death has proven to be notoriously difficult and involves mathematical modeling. We review the various models that have been developed for each of these techniques, discuss which models seem most appropriate for what type of data, reveal open problems that require better models, and pinpoint how the assumptions underlying a mathematical model may influence the interpretation of data. Elaborating various successful cases where modeling has delivered new insights in T cell population dynamics, this review provides quantitative estimates of several processes involved in the maintenance of naive and memory, CD4(+) and CD8(+) T cell pools in mice and men.

Figure 1

The immune response of one antigen-specific clone modeled by the ODE model given by Eqs. (1-5). The activation function, $\mathcal{F}\left(B\right)$ (heavy black line), is determined by the antigen concentration of Eq. (1) (which is not shown). At time zero one clone of a 100 naive T cells (N: light black line) becomes activated and starts to proliferate (A or P: red). After about one week proliferation stops because the activation function drops to zero when antigen is rejected. Choosing parameter values that are reasonably realistic for humans, we start with one clone of a hundred naive T cells, i.e., σ = r_N = 0, dN = 0.001 day⁻¹, a_N = day⁻¹. Effector cells, E, disappear by rapid cell death, d_E = 1 day⁻¹, and a small fraction become memory cells, m = 0.05 day⁻¹. Memory cells accumulate and have an expected life span of a hundred days (r_M = 0; a_M = 1 and d_M = 0.01 day⁻¹).

Figure 2

Fitting the model of Eqs. (6-7) to the data of Homann et al. [107] using the precursor frequencies estimated by Kotturi et al. [131], and given in the Table above as the A(0) values. The data is comprised of the CD8⁺ T cell responses to four epitopes from LCMV (GP33, NP396, GP118 & NP205), for which we have both time course data and estimates for the initial number of precursor cells. The light solid lines depict the total number of activated cells, A, per spleen, the dashed lines show the corresponding number of memory cells, M, and the heavy solid lines give the total, T = A+M, number of cells that was fitted to the data (symbols) using non-linear least-square regression [153]. Each inset shows the same data over a time span of 921 days. Ranges in the table indicate 95% confidence intervals determined by bootstrapping the residuals one thousand times. Common parameters: d_M = 0, d_A = 0.395 (0.330-0.479) day⁻¹, m = 0.019 (0.015-0.023) day⁻¹, τ = 7.91 (7.71-8.11) days. Allowing d_M > 0, or allowing any of these common parameters to vary between the epitopes hardly improved the quality of the fit. These four fits are similar to those of De Boer et al. [44] and only extend them with the precursor frequencies estimated by Kotturi et al. [131].

Figure 3

The immune response of one antigen-specific clone modeled by the DDE model given by Eqs. (2) & (4) and Eqs. (11) & (12). The activation function switches at predefined time points: in Panels (a & b) $\mathcal{F}\left(t\right)=1$ when t < 5 days and $\mathcal{F}\left(t\right)=0$ otherwise, whereas in (c) we let antigen persist by defining $\mathcal{F}\left(t\right)=1$ $\mathcal{F}\left(t\right)=0.01$ when t > 5 (which falls off the scale of the vertical axis). At time zero one clone of a 100 naive T cells (N: black) becomes activated and starts to proliferate (A or P: red). After about one week the response starts to contract because for most cells the time window of proliferation, τ_N = 7 days, has ended. After day seven effector cells (E: green) and/or memory (M: blue) cells are formed. The parameters are the same as those in Fig. 1 and are reasonably realistic for humans. In Panel (a) we mimic Fig. 1 by setting a_N = 1 day⁻¹ and in Panels (b & c) we increase the activation of naive T cells 10-fold to a_N = 10 day⁻¹. In Panel (c) memory cells are reactivated because $\mathcal{F}\left(t\right)=0.01$ from day five onwards. By repeated rounds of reactivation of memory T cells and proliferation a chronic immune response is maintained.

Figure 4

One week of deuterated glucose labeling of a healthy human volunteer showing the deuterium enrichment in CD4⁺ and CD8⁺ T cells taken from the blood [163]. The data were fitted with Eq. (26) for n = 2 subpopulations, giving average turnover rates d̄ = 0.006 day⁻¹ and d̄ = 0.0044 day⁻¹, for CD4⁺ and CD8⁺ T cells, respectively. We have shifted time by one day to allow for a short delay before labeled cells appear in the blood [163]. Note that fitting the same data with the temporal heterogeneity model of Eq. (29) would have given identical fits and average turnover rates, with a radically different interpretation of the parameters,; d₁ and d₂, of Eq. (26) [53].

Figure 5

Deuterium labeling curves expected for the temporal heterogeneity model of Eq. (29) for c = 2 (a) and c = 32 (b) taken from De Boer et al. [53]. The model is parameterized for the slowly renewing LCMV specific memory CD8⁺ T cells described by Choo et al. [36]. During the labeling phase one writes for the unlabeled fraction of DNA isolated from resting and activated cells, respectively, dU_R=dt = rU_A−(d_R+a)U_R and dU_A=dt = aU_R−(d_A+r)U_A, whereas during the de-labeling phase one writes the same equation for the labeled fraction dL_R=dt = rL_A (d_R +a)L_R and dL_A=dt = aL_R − (d_A + r)L_A [189], with the total fraction labeled defined as L = −L_R + L_A = 1−U_R−L_R. Parameters in Panel (a): c = 2; a = 0.02 and hence f = 0.01865; r = 1.105 day⁻¹, and an average turnover rate of d̄ = 0.01963 day⁻¹ (which is close to fd_A). The fraction of activated cells, f, and the average turnover rate, d̄, are the same in both panels by setting the parameters in Panel (b) as: c = 32; a = 0.02/31 = 0.0006452 and hence r = 0.08658 [53]. Other parameters: d_R = 0.001 day⁻¹ and d_A = 1 day⁻¹. The up and down-slopes are technically biphasic [189], but this is hardly visible when c = 2 (a).

Figure 6

T cell turnover rates estimated by deuterium labeling. Panel (a): Expected life spans of total T cells estimated by one week of deuterated glucose labeling in Mohri et al. [163] versus 9 weeks of deuterated water labeling in Vrisekoop et al. [223]. Each symbol represents a healthy human volunteer and is plotted at the CD4 or CD8 T cell count (per l blood) of that subject. The horizontal lines depict the average expected life spans, i.e., 250 and 484 days for CD4⁺ T cells, and 400 days for CD8⁺ cells. The vertical lines depict the 95% confidence intervals; since the Vrisekoop et al. [223] data was recalculated from the naive and memory T cell enrichment, we have no confidence intervals for those data points. Note that the CD4⁺ T cells of one volunteer in the Vrisekoop et al. [223] data have an expected life span that is about 2-fold larger than the mean, which is largely due to an approximately 2-fold lower deuterium enrichment in this subject’s CD4⁺ memory T cell compartment (and not to a poor fitting of the data). Judging the CD4 data on either the mean, or the median to exclude the outlier, the difference between the two techniques is about 2-fold. Westera et al. [231] perform a similar comparison of these two data sets by re-fitting the data with a two-compartment version of Eq. (26), and find that the difference remains but becomes somewhat smaller. Panel (b): Mohri et al. [163] also measured the fraction of dividing cells by staining with the Ki67 antibody, and we plot for each individual the estimated average turnover rates, i.e., the p values in their Table 1, as a function of the Ki67 measurements in CD4⁺ and CD8⁺ T cells. The line in Panel (b) results from fitting the linear regression line y = ax to the data, and suggest that the daily turnover is approximately one fifth of the fraction of Ki67⁺ cells.

Figure 7

Three weeks of BrdU labeling of a monkey infected with SIV. The left and right panels show the percentage of BrdU⁺ cells in CD4⁺ and CD4⁻ memory T cells from monkey H1348 in the study of Mohri et al. [162]. Fitting Eq. (37) to these data required at least two compartments (i.e., n = 2), and restricting ι_θ = 0.25, i.e., two divisions to become BrdU⁻, we find for CD4⁺ T cells an average turnover rate of 0.025 day⁻¹ (with α₁ = 0.34, α₂ = 1 − α₁; d₁ = 0.072 and d₂ = 0.0016 day⁻¹), and for CD4⁻ T cells 0.019 day⁻¹ (with α₁ = 0.36, α₂= 1−α₁, d₁ = 0.048 and d₂ = 0.0028 day⁻¹). The fitting of the same data with the source model of Eq. (32) is shown in De Boer et al. [46].

Figure 8

Basic features of a CFSE profile of dividing B cells four days after polyclonal stimulation. The dashed line at the right illustrates the CFSE intensity of unstimulated B cells. These cells serve as the control to locate the undivided cell position. The solid line shows the dividing B cell population revealing the typical asynchronous profile. Progressive divisions are apparent by the accurate two-fold dilutions of the fluorescence intensity, which on the log scale appear as even spacings. The dashed line to the left illustrates the position of an equivalent population that was not labelled with CFSE, allowing the position of the autoflourescence intensity to be determined. This is Figure 1a in Hasbold et al. [91]; reprinted with permission from the Nature Publishing Group.

Figure 9

A simple representation of the Smith-Martin model (a) and a cartoon (b) of the mathematical model of Eq. (58). Panel (c) shows the fit of Eq. (67) to CFSE data fitted with various models in De Boer et al. [43]. Here n is the number of divisions CD4⁺ T cells have completed after polyclonal stimulation, and n = 6+ refers to all cells having completed six or more divisions. Panels (a & b) are adapted from Figure 1 in Ganusov et al. [79].

Figure 10

The decline of the naive T cell numbers, the total number of TRECs, and the TREC content after infection with HIV-1. The steady state before time zero represents a healthy individual with N = 500 naive CD4⁺ T cells per μl blood, a death rate of d = 0.0005 day⁻¹, and a scaled TREC content of Ĉ = 0.2 TREC cell⁻¹, suggesting that p = 0.8d, and σ = 0.5 cells μl⁻¹ blood day⁻¹. After time zero we set σ = 0.0152 cells l⁻¹ blood day⁻¹, p = 0.96d, and d = 0.00152 day⁻¹ to represent HIV-1 infection. Note that the TREC dynamics are much faster than the depletion of the naive T cells because naive T cells also maintain themselves by proliferation.