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Detection of Suppressor T Lymphocytes and Estimation of Their Frequency in Limiting Dilution Assays by Generalized Linear Regression Modeling

Thierry Bonnefoix^1,*, Philippe Bonnefoix, Jian-Qing Mi^*,, Jean-Jacques Lawrence, Jean-Jacques Sotto^* and Dominique Leroux^*

^* Groupe de Recherche sur les Lymphomes, Equipe Mixte Institut National de la Santé et de la Recherche Médicale 0353, Institut Albert Bonniot, Université Joseph-Fourier, Rond-Point de la Chantourne, and Centre Hospitalo-Universitaire Michallon, Informatic Division, Hot Corporation, and Institut National de la Santé et de la Recherche Médicale, Unité 309, Institut Albert Bonniot, Rond-Point de la Chantourne, Grenoble, France; and Shanghai Institute of Hematology, Rui Jin Hospital, Shanghai Second Medical University, Shanghai, People’s Republic of China

	Abstract

Top Abstract Introduction Materials and Methods Results Discussion Appendix 1 References

 
The estimate of the frequency of suppressor T lymphocytes in unfractionated cell populations remains challenging, mainly because these regulatory cells do not display specific immunophenotypic markers. In this paper, we describe a novel theoretical approach for quantifying the frequency of suppressor cells. This method is based on limiting dilution data modeling, and allows the simultaneous estimation of the frequencies of both proliferating and suppressor cells. We used previously published biological data, characterizing the inhibiting activity of suppressor T cell clones. Starting from these data, we propose a mathematical model describing the interaction between suppressor and proliferating T cells, and applied to a Poisson process. Limiting dilution data corresponding to this non-single-hit, suppressor two-target Poisson model were artificially generated, then modeled according to a generalized linear regression procedure. Deviation from the single-hit Poisson model was revealed by a statistical slope test, and a stepwise analysis of the regression appeared to be an efficient method that strongly argued in favor of the presence of suppressor cells. By using the frequency of proliferating T cells calculated in the first step of the regression, we demonstrated the possibility to provide a reasonable estimate of the frequency of suppressor T cells. Based on these findings, a practical decision-making procedure is given to perform standard analyses of limiting dilution data.

	Introduction

Top Abstract Introduction Materials and Methods Results Discussion Appendix 1 References

 
After a discrediting in the 1980s and early 1990s, the concept of regulatory cells has now re-emerged in various models of cell-mediated immunity. In 2003, there is little doubt that regulatory T (Tr)² cells suppressing T cell proliferating responses play a critical role in the prevention of autoimmune diseases and transplantation tolerance, based on animal models (1, 2, 3, 4). Moreover, it is highly probable that these suppressor T cells have a much broader role in controlling normal and pathologic immune responses. In animal models, it has been shown that removal of immunoregulatory CD25⁺CD4⁺ T cells can break immunological unresponsiveness to autologous tumors in vitro and in vivo, leading to spontaneous development of tumor-specific effector cells (5, 6), and that anergic T cells can function as suppressor cells by inhibiting Ag presentation by dendritic cells (7). Very recently, it has been reported that CD4⁺CD25⁺ T cells in human tumors directly inhibit autologous T cell proliferation and secretion of IFN- (8, 9). These results may partly explain the poor immune response in vivo against tumor Ags. Thus, the detection of suppressor T cells, strengthened by an estimate of their frequency, have important implications for the study of tolerance, notably in the context of autoimmunity, transplantation, and cancer.

Two major suppressor cell types have been described: cytokine-secreting Tr1 (or Th3) cells, which predominantly release IL-4, IL-10, and transforming growth factor (TGF) , and CD4⁺CD25⁺ anergic T cells, which appear to require cell-to-cell contact to mediate their suppressive function (1, 2, 3, 4, 5). However, although membrane markers are now available that allow enrichment of suppressor T cell populations (CD45RB/C^low, CD62 ligand, CD152, CD25), they are still imperfect because all are expressed on other cell types. For example, CD152 (CTLA4) is expressed on CD4⁺CD25⁺ suppressor cells, but this marker is also consistent with an activated/memory phenotype. Moreover, the engagement of CTLA4 on the CD4⁺CD25⁺ T cells by CD80/CD86 might lead to inhibition of the TCR-derived signals that are required for the induction of suppressor activity (10). Finally, the expression of some of these markers, like CD25, may not be stable on these suppressor cells (4, 11). This apparent lack of specific immunophenotypic markers precludes the direct detection and quantitation of suppressor cells in an unfractionated cell population. In the present paper we demonstrate, by using a theoretical approach, that performing a limiting dilution analysis (LDA) of the frequency of proliferating T cells with appropriate modeling of the data can be a simple and efficient method for detecting suppressor T cells mixed with the proliferating T cells. Moreover, this method allows the estimation of the frequencies of both cell subsets. LDA is a method based on the binary response variable (positive or negative cultures) which has gained widespread acceptance as a tool for quantifying the frequency of cells in the immune system that possess various functional activities (12). In a recent paper (13), we developed a method for analyzing limiting dilution data according to a generalized linear regression model, aimed at estimating the fit of the single-hit Poisson model (SHPM) to the data. This methodresults in a powerful statistical slope test able to account for deviation from single-hit kinetics, indicating the presence of regulatory cells mixed with the proliferating cells. This new statistical method accompanied by its graphical representation was used in this study to detect the presence of suppressor T cells in simulated limiting dilution assays.

	Materials and Methods

Top Abstract Introduction Materials and Methods Results Discussion Appendix 1 References

 
Construction of a mathematical model describing the interaction between responsive and suppressor T cells and applied to LDA according to a Poisson process

An important aspect of this model is that its construction is based upon experimental findings, previously published by Lombardi et al. (14). In these experiments conducted with Ag-specific T cell clones, anergic/suppressor T cells were added to cultures containing potentially reactive T cells, APC, and influenza hemagglutinin (HA) peptides as Ags. Anergic/suppressor T cells with the same specificity as the responsive T cells led to titrable inhibition of their proliferation. A reproductive result was that anergic T cells caused marked inhibition (>80%) of responsive T cells, when anergized T cells and responsive T cells were mixed at a ratio of 3:1 (three anergic/suppressor T cells per one proliferating/responder T cell). This effect was Ag-specific, because addition of an anergic clone with a different specificity caused less inhibition. In the current paper, the number of suppressor T cells (T_s) necessary to inhibit the growth of one responsive/proliferating T cell (T_p) is termed the T_s:T_p efficiency ratio. Although a T_s:T_p efficiency ratio of 3:1 has received confirmation in other systems using Ag-specific T cell clones (15, 16, 17, 18), it is highly plausible that the value of the T_s:T_p efficiency ratio may be subject to variability in other experimental protocols, depending on the functional properties of the suppressor and proliferating cells. This possibility is taken into account in the mathematical model developed in this work.

Now, we suppose that these subsets of proliferating and anergic/suppressor T cells are an integral part of an unfractionated cell population under study (total spleen cells for example), and that we want to evaluate in vitro the frequency of proliferating T cells in response to HA peptides, by a conventional limiting dilution method in 96-well plates. In these conditions, the first possibility that a well will be scored negative for T cell proliferation is that it did not receive any HA-specific responsive T cell. Thus, the fraction, P_i, of negative wells is given by the zero term of the Poisson distribution

(1)

where f_p is the frequency of HA-specific responsive T cells, x is the total number of cells per well (called the cell dose, or cell input, in this paper), and i, the ith group of replicate wells, i = 1, 2, ... k, each group corresponding to a cell dose. Now, the second possibility that a well will be scored negative for T cell proliferation is that it received, at least, n-fold more suppressor T cells than responsive T cells, with , the numerical value of the T_s:T_p efficiency ratio and n, the number of proliferating T cells. Thus, a well will be scored negative for growth if it contains simultaneously one proliferating T cell and or more suppressor T cells, or two proliferating T cells and 2 or more suppressor T cells, or three proliferating T cells and 3 or more suppressor T cells, and so on. The joint probability that a well contains one proliferating T cell and or more suppressor T cells is given by

where

is the probability of one proliferating T cell per well, and

is the probability of or more suppressor T cells per well, with f_s, the frequency of suppressor cells. The joint probability that a well contains two proliferating T cells and 2 or more suppressor T cells is given by

and so on. Thus the general formula for the suppressor term, called S_i, is given by

where j refers to the number of suppressor cells. Now, if we note that

therefore

The suppressor term S_i can now be written as

(2)

and results from the product of two terms,

defining the fraction of wells containing n proliferating T cells per well, and

defining the fraction of wells containing j suppressor T cells, with j n. S_i is the summation of this product over all n with n 1.

Because these two possibilities to obtain a negative well are mutually exclusive, the total fraction of negative wells, F_i, is

equivalent to

(3)

Equation (Eq) (3) defines the function of the suppressor model. To estimate the influence of the suppressor term S_i on the limiting dilution assay, a conventional means is to define S_i as a function of P_i rather than x_i. The suppressor term S_i, defined in Eq (2) can be written as

(4)

with

Because P_i = [exp(-f_px_i], which is equivalent to P_i = [exp(-f_px_i)], substituting P_i in Eq (4) we obtain

(5)

Finally, Eq (3) defining the suppressor model can be written as

(6)

Fitting the SHPM to limiting dilution data by generalized linear regression

This step consists of modeling limiting dilution data according to a generalized linear log-log model fitting the SHPM and checking this model by an appropriate slope test, as described previously (13). Briefly, the zero term of the Poisson distribution given by Eq (1) can be written as

defining the log-log model. This equation is now in the form

with Y_i = ln[-ln(P_i)], = ln(f_p), = 1, X_i = ln (x_i). A test of deviation from the SHPM (model checking) is a test of whether the estimate of the slope is significantly different from 1. The appropriate statistical slope test z uses the normal deviates under the null hypothesis that = 1

where var() is the variance of the slope . That statement can be rewritten as

showing that the value, 1, is included in the 95% confidence interval of the slope if the SHPM hypothesis holds. Maximum likelihood estimates of the parameters , , var(), were obtained with the Fisher’s method of scoring according to an iteratively reweighted least-squares procedure (13, 19).

Goodness-of-fit of the log-log linear regression model to the data

The goodness-of-fit of the generalized linear log-log model to the data was evaluated by two statistical tests: the residual deviance, D (19), and a W Wald statistic (20). D is defined to be twice the difference between the maximum achievable log likelihood and that attained under the fitted model, and can be written as

where N_i is the number of replicate wells per cell dilution, m_i is the experimental fraction of negative wells, m_i = r_i/N_i, with r_i, the experimental number of negative wells, and P_i is given by

D is approximately distributed as _k-2². The p value associated to the ² value is denoted by p(_D²). The W Wald test is a test concerning the hypothesis that the slope = 0. The reason for interest in testing whether or not = 0 is that, when = 0, there is no linear association between X_i and Y_i, and thus, the log-log linear regression model is not acceptable. The test statistic

has a standard normal distribution when = 0. The p value associated to the W value is denoted by p(W).

Computation of the frequency of proliferating cells, f_p, according to the SHPM

The estimates of f_p and var(f_p) were calculated by the maximum likelihood method as previously described (13, 21).

Goodness-of-fit of the suppressor model to the data

An objective indication of fit of the proposed suppressor model to the experimental data can be checked by an ordinary likelihood ratio test statistic (22), comparing the number of experimental and observed, positive and negative wells, and defined as

where O_ih and E_ih are the observed and expected numbers for the ihth class. There are two h class numbers with a positive (h = 1) and negative (h = 2) response class for each of the k cell doses. Thus, G can be written as

where F_i is given by Eq (3). G is approximately distributed as _k-2². The p value associated to the ² value is denoted by p(_G²).

Softwares

All theoretical curves with their respective equations were generated and plotted by using the commercially available software Mathcad 8 (Mathsoft, Cambridge, MA). Calculations of , , var(), can be made with the commercially available software S-Plus 6 (Insightful, Seattle, WA) including a generalized linear regression module with a log-log link function applied to binomial data.

	Results

Top Abstract Introduction Materials and Methods Results Discussion Appendix 1 References

 
Features of the suppressor model applied to limiting dilution experiments

The main features of the suppressor model can be summarized by using three graphical representations: the suppressor term S_i plotted against P_i with linear and log scales, and the fraction of negative wells F_i plotted against P_i. To generate the sets of theoretical curves, the values of , the f_s:f_p ratio and , the T_s:T_p efficiency ratio, were tuned with reasonable limits to obtain a realistic model behavior. Curves were obtained with = 1, = 3, = 5, and for each of these values we consider a possible variability of , ranging from 1 to 5. As can be seen in Fig. 1, A, D, G, the magnitude of the suppressor term is highly variable, depending on both the values of and . The consequence was that the resulting theoretical limiting dilution (LD) titration curves demonstrated very different shapes: V-shaped curves characterized by an initial decrease of F_i at high values of P_i followed by an increase of F_i at low values of P_i, curves that level off, and curves demonstrating minor or moderate deviation from single-hit kinetics (Fig. 1, C, F, and I). One other feature is of particular importance. As presented in Fig. 1, B, E, and H, S_i remains negligible or small (<0.1) provided that P_i is high. The consequence is that the contribution of the suppressor effect to the total fraction of negative wells F_i remains negligible or small as far as the observed fraction of negative wells is high. This is clearly illustrated in Fig. 1, C, F, and I. Provided that F_i is high, F_i remains close to P_i, and therefore the initial segment of the LD titration curve conforms to the SHPM. The threshold value of F_i at which the LD titration curve starts to move away from single-hit kinetics depends on the values of and . Based on the shape of curves, this threshold value is around 0.5 or 0.6 for curves exhibiting minor or moderate deviation from the SHPM, around 0.6 or 0.7 for curves that level off, and can reach 0.8 or 0.9 regarding the V-shaped curves.

View larger version (17K): [in this window] [in a new window]   FIGURE 1. A, D, and G, Graph of the suppressor term, Si, as a function of Pi plotted on a log scale. B, E, and H, Graph of Si, as a function of Pi plotted on a linear scale. C, F, and I, Graph of the total fraction of negative wells, Fi, as a function of Pi. Si and Fi were calculated by using Eq (5) and (6), respectively. The sets of theoretical curves were generated with = 1 (A–C), = 3 (D–F), = 5 (G–I). Color plain lines: red: = 1; green: = 2; blue: = 3; color dashed lines: red: = 4; blue: = 5. , is the fs:fp ratio, and , is the Ts:Tp efficiency ratio. Black plain line: this straight line serves as reference curve, graphically representing the SHPM; Fi is plotted versus Pi, with Fi = Pi (C, F, and I).

Very few data are available concerning the frequency of suppressor T cells that develop in an immune response; routine estimation of this frequency is precisely the goal of the current work. A recent study indicated that approximately the same number of autoreactive and suppressor T cells develop in response to self peptides (23). In an other system, Groux et al. (24) reported that, in the presence of IL-10, the chronic activation in vitro of human or murine CD4⁺ T cells by allogeneic cells or OVA peptide gave rise to as many as half suppressor Tr1 clones among the total number of clones screened. Thus, the present simulation of a limiting dilution experiment was done based on the reasonable assumption that the frequency of suppressor cells, f_s, equals the frequency of responsive cells, f_p, i.e., f_s:f_p = 1:1 ( = 1). This frequency was taken as 1/2,000, a plausible frequency of HA-specific proliferating T lymphocytes that may develop during the course of an immune response against influenza viruses (25). The value of the T_s:T_p efficiency ratio was taken as = 3. Simulation of the limiting dilution experiment was run with 48 wells per dilution (n = 48), a number of replicate wells typically used in limiting dilution assays. Eight T cell dilutions were used (k = 8), with the following cell inputs per well: 250, 500, 1,000, 2,500, 5,000, 7,500, 10,000, 15,000. These values were chosen so that the total fraction of negative wells, F_i, ranged from almost all negative wells to almost all positive wells. Theoretical (F_i) and experimental (m_i) fractions of negative wells are given in Table I. The usual graphical representation of limiting dilution data (semi-log plot) is given in Fig. 2 A. The LD titration curve exhibits a slight deviation from linearity.

View larger version (17K): [in this window] [in a new window]   FIGURE 2. A, Graph of the experimental fraction of negative wells, mi (log scale) as a function of the number of cells per well, xi (semi-log plot). The graph corresponds to the data presented in Table I. B, Graphical representation of the slope test z revealing that the SHPM does not fit to the limiting dilution data. The log-log transform of the experimental fraction of negative wells (mi) is plotted against the logarithm of the number of cells per well (xi). Blue plain line: log-log regression line fitted to the experimental data according to the generalized linear log-log model (Yi = -5.826 + 0.737 Xi); blue dotted lines: regression lines with the same intercept as the fitted log-log regression line, but with slope corresponding to the lower and upper values of the 95% confidence interval for (Yi = -5.826 + 0.604 Xi; Yi = -5.826 + 0.870 Xi); red line: theoretical, so-called SHPM regression line with the same intercept as the fitted log-log regression line, but with slope equal to 1 (Yi = -5.826 + Xi). C, Graph of the negative logarithm of mi as a function of xi (linear plot).

Modeling limiting dilution data according to a linear log-log regression procedure results in a statistical slope test z that can be graphed to visualize whether the SHPM is compatible or not with the data (13). Based on the current simulated data, the graphical representation of the slope test z is given in Fig. 2B. The generalized linear log-log model fit the data very well because the deviance D = 1.68, with p(_D²) = 0.94, and W = 10.83 with p(W) < 10^-5. The quantity of major interest is the estimate of the slope , which is = 0.737. This value of was found not compatible with 1 (95% confidence interval ranging from 0.604 to 0.870), establishing that the data do not conform to single-hit kinetics (SHPM), and highlighting that the titrated cell population may contain a subset of Tr cells modulating the growth of the responsive T cells.

The stepwise regression procedure

From a general point of view, it should be kept in mind that many other reasons than the presence of suppressor cells can be invoked to explain why limiting dilution data deviate from single-hit kinetics: technical problems in cell count, such as cell clumps, generating dilution errors, positive-negative discrimination error between wells, multihit models, multitarget helper models, involving the cooperation of two or more cells of two or more different cell subsets to generate a positive well (12). Thus, in practice, if the limiting dilution data derived from a real experiment do not conform to the SHPM, the problem that arises is to take from the data some relevant information in favor of the presence of suppressor T cells. To this end, a strategy of data analysis must be provided. We now demonstrate that such a strategy can be based on the conclusions drawn from in-depth analysis of the suppressor model (Fig. 1). Under the condition that F_i is high, it has been demonstrated that the contribution of S_i to F_i is small or negligible and, therefore, F_i is primarily influenced by the probability P_i of finding no proliferating T cells in the well. The consequence is that the initial segment of the LD titration curve conforms to single-hit kinetics. In the present simulation, this statement can be graphically visualized on the semi-log plot representation (Fig. 2A). The fraction of negative wells, m_i (plotted on a log scale) appears to be linearly related to the cell dose x_i, as long as m_i is greater than a threshold value (around 0.5 or 0.6). The linear part of the curve includes the three lowest cell doses, 250, 500, and 1000 cells per well. It should be noted that the linear segment is better evidenced on a linear plot, where the negative logarithm of m_i is plotted versus x_i (Fig. 2C). Then, the LD titration curve clearly deviates more and more from linearity (single-hit kinetics) as the number of cells per well increases (Fig. 2A). The existence of these two consecutive parts of the curve features the presence of suppressor cells according to the current suppressor model, and can be checked by the linear log-log regression method, according to a stepwise procedure (Fig. 3, A–F). The log-log regression restarts with the three lowest cell doses, 250, 500, 1000 cells per well (Fig. 3A). Then, a forward stepwise regression adds each subsequent cell dose to the log-log regression (Fig. 3, B–F). At each step, the equation of the regression line is given with the following statistics: the deviance D with p(_D²), W, p(W), z, p(z), SE(slope) and 95% CI(slope). As expected, at the three lowest cell inputs, the SHPM is a very good fit to the observed data (Fig. 3A). Based on these three points, it is possible to estimate f_p, the frequency of HA-specific responsive T cells, by the method of likelihood maximization (13, 21), and the result is presented in Fig. 4 (f_p = 1/1946). Next, at each subsequent cell dose added to the log-log regression, it is observed that the value of the slope progressively decreases (Fig. 3, B–F), until becoming not compatible with 1 (Fig. 3D). This progressive decline of the slope is related to the progressive departure of the LD titration curve from linearity (single-hit kinetics) as the number of cells per well increases (Fig. 2A), and features the presence of suppressor cells mixed with proliferating cells.

View larger version (26K): [in this window] [in a new window]   FIGURE 3. The stepwise regression procedure applied to the data presented in Fig. 2B. Blue plain line: log-log regression line fitted to the experimental data according to the generalized linear log-log model; blue dotted lines: regression lines with the same intercept as the fitted log-log regression line, but with slope corresponding to the lower and upper values of the 95% confidence interval for ; red line: theoretical, so-called SHPM regression line with the same intercept as the fitted log-log regression line, but with slope equal to 1. A, Step 1: the log-log regression includes only the three lowest cell doses: 250, 500, 1000 cells per well. Yi = -7.296 + 0.957 Xi; D = 7.919 x 10-5; p(D2) = 0.993; W = 2.98; p(W) = 1.44 x 10-3; z = -0.133; p(z) = 0.893; SE(slope) = 0.321; 95% CI(slope): 0.327 - 1.587. At each subsequent step, the subsequent cell dose is added to the log-log regression. B–F, Steps 2–6, respectively. B, Yi = -6.958 + 0.903 Xi; D = 3.777 x 10-2; p(D2) = 0.981; W = 5.715; p(W) < 10-5; z = -0.613; p(z) = 0.539; SE(slope) = 0.158; 95% CI(slope): 0.593 - 1.213. C, Yi = -6.436 + 0.824 Xi; D = 0.510; p(D2) = 0.916; W = 7.644; p(W) < 10-5; z = -1.631; p(z) = 0.102; SE(slope) = 1.078 x 10-1; 95% CI(slope): 0.613 - 1.035. D, Yi = -6.130 + 0.779 Xi; D = 1.030; p(D2) = 0.905; W = 8.944; p(W) < 10-5; z = -2.532; p(z) = 0.0113; SE(slope) = 8.712 x 10-2; 95% CI(slope): 0.608 - 0.950. E, Yi = -5.962 + 0.755 Xi; D = 1.359; p(D2) = 0.928; W = 9.96; p(W) < 10-5; z = -3.223; p(z) = 1.269 x 10-3; SE(slope) = 7.583 x 10-2; 95% CI(slope): 0.607 -0.904. F. The set of statistical values are indicated on the graph of Fig. 2B.

View larger version (15K): [in this window] [in a new window]   FIGURE 4. Plot of the fraction of negative wells (mi) as a function of the number of cells per well (xi). Plain line: the prediction equation of the regression line is: -ln(Pi) = 5.139 x 10-4 xi, where 5.139 x 10-4 is the cell frequency fp estimated by the maximum likelihood method according to the SHPM hypothesis. Upper and lower dotted lines are plotted by using upper and lower values of the 95% confidence interval of fp.

By using the estimate of the frequency of proliferating T cells calculated at the first step of the regression, it is possible to provide a reasonable estimate of f_s, the frequency of suppressor T cells. We generated a family of curves, graphing the function of the suppressor model (Fig. 5). The theoretical fraction of negative wells, F_i, was plotted against the cell dose, x_i. These curves were obtained by varying the value of f_s, whereas f_p remained fixed to the value 1/1946, obtained at the first step of the regression (Fig. 4). The values of f_s were arranged to correspond to the following f_s:f_p cell ratios: 0.75:1 (f_s = 1/2595), 1:1 (f_s = 1/1946), 2:1 (f_s = 1/973), 3:1 (f_s = 1/649). The experimental data points (m_i, x_i) were superimposed on the graph. By visual inspection, we can choose the most adequate frequency of suppressor T cells, i.e., the value of f_s generating the curve that best fits the experimental data. Obviously, the suppressor model corresponding to the 1:1 f_s:f_p cell ratio (f_s = 1/1946) fits the data almost perfectly. In a real limiting dilution experiment, a more objective indication of the goodness-of-fit of the suppressor model to the data can be checked by an ordinary G likelihood ratio goodness-of-fit statistic. Under the hypotheses that = 3 and f_s = f_p = 1/1946, we get G = 0.138 with p(_G²) = 0.999. To obtain an accurate estimate of f_s, the optimal fit of the suppressor model to the data can be determined by using the quasi-Newton method to maximize the likelihood of the data (T. Bonnefoix, P. Bonnefoix, and D. Leroux, manuscript in preparation).

View larger version (16K): [in this window] [in a new window]   FIGURE 5. Estimation of the frequency of suppressor cells, fs, by fitting a family of curves to the simulated data presented in Table I. The curves graph the function of the suppressor model given in Eq (3), varying the value of fs while fp remains fixed to the value 1/1946, with = 3. The experimental data points (mi, xi) are superimposed on the graph. (...), fs = 1/649; (----), fs = 1/973; (_), fs = 1/1946; (__), fs = 1/2595; (_··_), This straight line graphically represents the SHPM, according to Eq (1). These candidate values of fs correspond, respectively, to the following fs:fp cell ratios: 3:1, 2:1, 1:1, 0.75:1.

The stepwise regression procedure can be applied with efficiency to any value of

The value of the T_s:T_p efficiency ratio is a crucial parameter of the suppressor model, that may vary substantially among the subtype of the suppressor T cells, Tr1, CD4⁺25⁺, CD4⁺CD45RB/C^low. Because the previous simulation of a limiting dilution experiment was performed at a unique value of = 3, it was desirable to demonstrate that the stepwise regression procedure remains efficient at a range of potential values of . To this end, we simulated a set of six limiting dilution experiments, varying the value of from 1 to 5. The pairs of parameter values (, ) used for the simulations led to the generation of various shapes of the LD titration curves: minor deviation from single-hit kinetics ( = 4, = 1, Fig. 1C), moderate deviation from single-hit kinetics ( = 2, = 1, Fig. 1C; = 4, = 3, Fig. 1F), leveling off ( = 1, = 1, Fig. 1C; = 3, = 3, Fig. 1F; = 5, = 5, Fig. 1I). We used the same set of cell doses as in the previous simulated limiting dilution experiment (Table I), with 48 replicate wells per cell dose, and the stepwise regression procedure was repeated. In Fig. 6, A–F, is given a plot of the value of the slope , obtained at each step of the stepwise regression procedure. At the first step of the regression, the value of the slope was compatible with 1, allowing the calculation of f_p according to the SHPM. Next, at any value of studied, it can be observed that the introduction of each subsequent cell dose to the regression (steps 2 to 6) led to a decrease of the value of the slope , resulting in a systematic deviation of the plotted points from the straight line corresponding to the slope = 1. At each step of the regression of each LD experiment, it was checked by both the deviance and the W Wald test that the log-log regression model fitted correctly the data, with p(_D²) > 0.05 and p(W) > 0.05 (data not shown). Therefore, the stepwise regression method can be applied to any value of , provided that neither the deviance nor the Wald test shows evidence of lack of fit, at any step of the regression.

View larger version (23K): [in this window] [in a new window]   FIGURE 6. A–F, Plot of the value of the slope , obtained at each step of the stepwise regression procedure, in a set of six simulated limiting dilution experiments. These simulations were conducted with the eight cell doses presented in Table I, with 48 replicate wells per cell dose. The value of fp remained fixed to 1/2000, whereas the values of fs and were variable. = 1: fs = 1/2000; = 3:fs = 1/666.66; = 5: fs = 1/400. At each cell dose, calculation of Fi, the theoretical fraction of negative wells, was done with Eq (3). The experimental number of negative wells, ri, was given by: ri = 48 x Fi, and rounded to the nearest integer. The experimental fraction of negative wells, mi, was given by: mi = ri/48. Step 1: 250, 500, 1,000 cells per well; step 2: 250, 500, 1,000, 2,500 cells per well; step 3: 250, 500, 1,000, 2,500, 5,000 cells per well; step 4: 250, 500, 1,000, 2,500, 5,000, 7,500 cells per well; step 5: 250, 500, 1,000, 2,500, 5,000, 7,500, 10,000 cells per well; step 6: 250, 500, 1,000, 2,500, 5,000, 7,500, 10,000, 15,000 cells per well. Black plain line: straight line corresponding to the slope = 1. Dotted vertical bars: 95% confidence interval for the value of the slope .

Proposal of a standardized graphical representation of the data

In Fig. 7, we propose a standard four-graph representation of the data that should be adopted to summarize the limiting dilution data. The graphs correspond to the data of the limiting dilution experiment simulated with = 1 (f_s = f_p = 1/2000) and = 3. The first graph (Fig. 7, A) represents the log-log regression of the data with the graphical representation of the slope test z, with its p value, designed to visualize the significant departure of the data from the SHPM. The second graph (Fig. 7B) represents the forward stepwise regression, evidencing the progressive decrease of the slope at each subsequent step of the regression. The third graph (Fig. 7C) illustrates the calculation of f_p, according to the SHPM, and based on the first step of the stepwise regression. The fourth graph (Fig. 7D) plots the curve of the suppressor model, by using the values of and f_s that best fit the experimental data. On this graph is also plotted the SHPM regression line. This graph is accompanied by the equation of the suppressor model, with the values of , f_p, f_s, and the value of the G likelihood ratio test, with its p value.

View larger version (17K): [in this window] [in a new window]   FIGURE 7. Proposal of a standard four-graph representation summarizing the limiting dilution data. The graphs correspond to the data of the limiting dilution experiment simulated with = 1 (fs = fp = 1/2000) and = 3. A, Graphical representation of the slope test z evaluating the adequacy of the SHPM to the limiting dilution data. See the legend of Fig. 2B for details. B, The stepwise regression procedure. This graph corresponds to the data presented in Fig. 3, A–F, and plots the value of the slope obtained at each step of the regression. Black plain line: straight line corresponding to the slope = 1. Dotted vertical bars: 95% confidence interval for the value of the slope . C, Graph illustrating the calculation of fp, according to the SHPM, and based on the first step of the stepwise regression. See the legend of Fig. 4 for details. D, Graph of the function of the suppressor model with the parameters values, fp, fs, , giving the curve that best fits the experimental data (mi, xi). Dotted line: this straight line represents the SHPM, according to Eq (1).

	Discussion

Top Abstract Introduction Materials and Methods Results Discussion Appendix 1 References

The ratio dominance model (12) is based on a more restrictive hypothesis than the dominance model, with the equation

This model assumes that a fixed number of suppressor cells, a, is able to inhibit the growth of an unlimited number of proliferating cells. Dominance and ratio dominance models provide a unique shape of LD titration curves, represented by V-shaped curves. At the lowest cell doses, the fraction of negative wells progressively decreases and the LD titration curve conforms to single-hit kinetics. Next, as the cell number is increased, this first phase is followed by a second phase, at which the fraction of negative wells increases. The model proposed by Dozmorov and colleagues (26, 27, 28) is an extension of the ratio dominance model. As in the ratio dominance model, their model specifies that a fixed number of suppressor cells is able to inhibit the growth of an unlimited number of proliferating cells. Additionally, the model proposes thatthese suppressor cells proliferate in turn when their number reaches a fixed value. The equation of this model is

where b, is the threshold number per culture of suppressor cells at which these cells can proliferate. This model leads to zigzag experimental LD titration curves. Typically, at the lowest cell doses, the fraction of negative wells progressively decreases according to single-hit kinetics, then there is a kink and the fraction of negative wells increases at intermediate cell doses, and ultimately decreases again at the highest cell doses. The suppressor model that we described in the current work proposes that the number of suppressor T cells required to suppress the growth is proportional to the number of proliferating cells, depending on a fixed value of , the T_s:T_p efficiency ratio. Eq (6) defines this model

and it is proposed to call this model the proportional suppressor model. Although the dominance models and the zigzag model can yield a reasonable fit to a number of limiting dilution data (12, 29, 30), one major difficulty arises in attempts to explain the mechanism by which a single cell, or a given fixed number of cells, might be able to suppress the growth of an unlimited number (or, more realistically, a great number) of proliferating cells. Such a capability of suppressor cells remains a matter of speculation, because it has not yet received confirmation at the level of T cell clones. From a biological viewpoint, the proportional suppressor model is more attractive than these previously published models for the following reasons. First, this model is based on studies performed with T cell clones. This is a relevant methodology for estimating accurately the stoichiometry of suppressor and proliferating cells leading to inhibition of T cell growth. Secondly, the shapes of the LD titration curves resulting from this model are highly variable: V-shaped curves as observed in the dominance models, but also curves that level off, and curves exhibiting moderate or slight deviation from single-hit kinetics. Thus, it is probable that the proportional suppressor model should be appropriately fitted to more limiting dilution data than the dominance and zigzag suppressor models.

Suppressor T cell clones have proven difficult to expand in vitro in standard culture conditions, partially due to the suppressive effect of autocrine IL-10 and TGF- (31). The recent observation that Tr1 clones can be expanded with IL-15 (32) would facilitate further biological characterization of the suppressor cells, and would help to generate and improve mathematical models applied to the detection and quantitation of suppressor cells in limiting dilution assays.

We hope that the statistical method described in this work will encourage investigators to reassess their limiting dilution data in search of suppressor T cells.

	Appendix 1

Top Abstract Introduction Materials and Methods Results Discussion Appendix 1 References

 
Decision-making procedure for analyzing limiting dilution data

1. Perform the generalized log-log linear regression including all cell doses. Go to 2).
   
2. Perform the slope test z to evaluate the compatibility of the slope with 1. If the slope is compatible with 1 at the 95% confidence level, i.e., p(z) 0.05, the single-hit Poisson model is compatible with the data; go to 7). If the slope is not compatible with 1 at the 95% confidence level, i.e., p(z) < 0.05, the single-hit Poisson model is not compatible with the data; go to 3).
   
3. Perform the first step of the forward stepwise regression with the lowest cell doses. This first step must include at least three cell doses. These cell doses must correspond to high fractions of negative wells. If the slope is not compatible with 1 at the 95% confidence level, i.e., p(z) < 0.05, go to 8). If the slope is compatible with 1 at the 95% confidence level, i.e., p(z) 0.05, compute the frequency of proliferating cells, f_p, according to the single-hit Poisson model. Then, go to 4).
   
4. Perform the subsequent steps of the forward stepwise regression. If a progressive decrease of the slope at each subsequent step is observed, the presence of suppressor cells mixed with the proliferating cells is plausible, according to the current suppressor model; go to 5). If a progressive decrease of the slope at each subsequent step is not observed, go to 9).
   
5. Graph the function of the suppressor model, according to Eq (3). Use the value of f_p determined at the first step of the regression. A family of curves are generated, by varying the values of f_s and . The experimental points (m_i, x_i) are superimposed on the graph. Go to 6).
   
6. Select the best values of f_s and , i.e. the values of f_s and giving the curve that best fits the experimental data (estimated by visual inspection, and by the G likelihood ratio test). End of the analysis.
   
7. Compute the frequency of proliferating cells, f_p, according to the single-hit Poisson model. End of the analysis.
   
8. Limiting dilution data not exploitable.
   
9. Limiting dilution data not exploitable, except the calculation of f_p at 3).
   

	Footnotes

 
¹ Address correspondence and reprint requests to Dr. Thierry Bonnefoix, Groupe de Recherche sur les Lymphomes, Institut Albert Bonniot, Rond-Point de la Chantourne, 38706 La Tronche, France. E-mail address: Thierry.Bonnefoix{at}ujf-grenoble.fr

² Abbreviations used in this paper: Tr, T regulatory; TGF, transforming growth factor; LDA, limiting dilution analysis; SHPM, single-hit Poisson model; HA, hemagglutinin; Eq, equation; LD, limiting dilution.

Received for publication May 28, 2002. Accepted for publication January 15, 2003.

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Top Abstract Introduction Materials and Methods Results Discussion Appendix 1 References

 

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