Abstract
Aiming to further our understanding of T cellmediated suppression, we investigate the plausibility of the hypothesis that regulatory T cells suppress other T cells (target cells), while both cells are conjugated with one APC. We use a mathematical model to analyze the proliferation inhibition scored during in vitro suppression assays. This model is a radical simplification of cell culture reality, assuming that thymidine incorporation is proportional to the number of target cells that would instantaneously form conjugates with APCs that are free of regulatory cells. According to this model the inhibition index should be mainly determined by the number of regulatory cells per APC and should be insensitive to the number of target cells. We reanalyzed several published data sets, confirming this expectation. Furthermore, we demonstrate that the instantaneous inhibition index has an absolute limit as a function of the number of regulatory cells per APC. By calculating this limit we find that the model can explain the data under two nonmutually exclusive conditions. First, only ∼15% of APCs used in the suppression assays form conjugates with T cells. Second, the growth of the regulatory cell population depends on the target cells, such that the number of regulatory cells per APC increases when they are cocultured with target cells and overcomes its limit. However, if neither of these testable conditions is fulfilled, then one could conclude that suppression in vitro does not require the formation of multicellular conjugates.
Several mechanisms contribute to natural tolerance to body tissues. Thymic deletion purges the emergent T cell repertoire from reactivities to peptides expressed intrathymically (1, 2). Mechanisms such as T cell ignorance (3), T cell anergy (4), and clonal exhaustion (5, 6) may contribute to the peripheral unresponsiveness to tissuespecific Ags that are not expressed in the thymus. However, these mechanisms do not seem sufficient to explain tolerance to continuous stimulation by selfAgs, as they can easily be overcome during the course of immune responses and inflammation (7). Dealing with this chronic stimulation by peripheral Ags seems to call into action yet another, maybe complementary, mechanism of T cellmediated tolerance.
The involvement of T cells in natural tolerance was demonstrated by partial T cell depletion (8, 9, 10) or reconstitution of immunodeficient animals with fractionated peripheral T cell subpopulations (9, 11, 12, 13, 14, 15). These experiments demonstrated the existence in normal healthy animals of CD4^{+} T cell subpopulations with the capacity to prevent autoimmune responses (12, 16) or uncontrolled inflammation triggered by other cells (7, 15). Despite their fundamental and clinical significance, such regulatory T cells have not yet been isolated or cloned, and their phenotype is still elusive.
Although the inability to isolate regulatory T cell has been an obstacle in the characterization of their mechanism of action, important clues have nevertheless been derived from recent in vivo and in vitro studies. In a system in which T cellmediated tolerance is induced to skin grafts, Waldmann reported that suppression requires that the Ags recognized by tolerant and immune T cell populations are expressed on the same cells (17, 18, 19). This type of linked suppression was investigated by several in vitro systems, from systems using T cell lines (20, 21) to ex vivo systems where suppression in vivo and in vitro is well correlated (22, 23). All these studies, with the exception of a recent controversial result (24), indicate that suppression mediated by regulatory T cells in vitro requires direct celltocell contact between APCs, regulatory T cells, and T cells that are targets of regulation (22, 23, 25). These observations led to the hypothesis that suppression takes place while the regulatory T cell and its target T cell are coconjugated with a single APC, i.e., suppression requires simultaneous interaction among three cell types in multicellular conjugates. Alternative hypotheses can be found in the literature, ranging from simple competition for conjugation with APC (20, 26) to more complex interactions in which the APC itself conveys a signal from the regulatory T cell to its target (27, 28, 29, 30). The inability to experimentally assess the different hypotheses has prevented a definitive resolution of this controversy. They have been tested using complex experimental designs whose qualitative results do not have a single straightforward interpretation.
A way of overcoming this impasse is to test directly the hypotheses by their intrinsic capacity to predict and explain the quantitative details of the in vitro suppression assays. Demonstration that one or more hypotheses can be rejected due to incompatibility with experimental data may contribute to reduce the list of candidates. In this context, suppression dependent on simultaneous conjugation of regulatory and target T cells with the APC is an obvious testable candidate due to inherent probabilistic constraints. Mechanisms of interaction involving the formation of conjugates of more than two cell types have been postulated previously (31), the major example being the classical view of linked recognition of Ag aby CD4 and CD8 T cells and their cooperation. It has been argued (32, 33) that such interaction imposes strong constraints on the efficiency of the process where it is imbedded due to the relatively low frequency of its ratelimiting event, the simultaneous encounter between more than two cells. This straightforward quantitative prediction provides an easy way to test the involvement of multicellular conjugates in suppression.
We report here a mathematical analysis of the hypothesis that in vitro suppression takes place in multicellular conjugates and therefore is limited by their formation. We use a general model (34) describing the distribution of multicellular conjugates in a population of regulatory T cells, target T cells, and APCs. We first identify the generic quantitative predictions of the requirement for multicellular conjugates, and then test these predictions on several results of in vitro suppression assays reported in the literature.
Materials and Methods
In vitro suppression assay and inhibition index I
We model the suppression of the response of target cells by regulatory cells in vitro. This is quantified by the following assay. T_{0} target cells are stimulated with A_{0} APCs in the presence or the absence of R_{0} regulatory cells in two parallel cultures that we call here, respectively, test and reference cultures. After a given amplification period, denoted Δ, the cultures are pulsed with [^{3}H]thymidine for a short period, τ, to assess the number of dividing cells. The amount of [^{3}H]thymidine incorporated in the cell fraction during the pulse is quantified. Typically a quantity that we call here the experimental inhibition index, I, is obtained by dividing the counts in the test culture by the counts in the reference culture: I = cpm[test culture]/cpm[reference culture] (Equation 1). Note that this index is inversely related to the actual suppression or inhibition (which can be estimated as 1 − I).
The results of in vitro suppression assays are typically presented as plots of the experimental inhibition index vs the ratio between the numbers of regulatory and target cells originally put into culture R_{0}/T_{0}.
The data sets used in this article are listed in Table I⇓ and were obtained by the following experimental systems.
System A.
The experiments were reported by Lechler and colleagues (20). Briefly, a standard in vitro suppressor assay was performed using the T cell line HC3 as a source of target T cells. The regulatory cells are HC3 cells previously rendered anergic in vitro. The APCs were irradiated BLCLs cells pulsed with a peptide agonistic for HC3 cells. The cultures were pulsed between 48–66 h of culture. Different clonespecific T cells lines were used (including some alloreactive ones) with equivalent results. The major drawback of this experimental system for understanding T cellmediated suppression is that there is not enough evidence that anergic T cells act as regulatory cells in vivo.
System B.
The experiments were reported by Powrie and colleagues (25). CD4 T cells from a normal healthy mouse were sorted into CD45RB^{low} CD38^{+} and the remaining subpopulation. The first subpopulation was used as regulatory cells, and the second subpopulation was used as target cells in the suppression assay. The cells were stimulated with antiCD3 using irradiated T lymphocytedepleted spleen cells as APCs. The cultures were pulsed between 60–78 h of culture. The main drawbacks of these experiments are that there is no clear relationship between the inhibition index in vitro and the immunosuppression in vivo, and the population of CD4^{+}CD45RB^{low}CD38^{+} T cells is enriched, but not pure, in regulatory T cells.
System C.
The experiments were reported by Sakaguchi et al. (23) and Thornton and Shevach (22). CD4^{+} CD25^{+} T cells were used as regulatory cells, and CD4^{+}CD25^{−} T cells were used as target cells in the suppression assay. The stimulus was again soluble antiCD3 and irradiated T celldepleted spleen cells. The thymidine pulse was given between 66–72 h of culture. The major advantage of this system is that that there is a good correlation between the results of the in vitro suppression assay and the results of adoptive transfers in vivo. Its main drawback is that neither the regulatory nor the target T cell population can be assumed to be pure. Indeed, both authors stressed the fact that the CD4^{+}CD25^{+} population is enriched in regulatory cells, but is not pure. The actual purity of regulatory cells is unknown.
A mathematical model of in vitro proliferative responses dependent on the formation of multicellular conjugates
Modeling the in vitro suppression assay and the inhibition index requires a quantitative description of the proliferative response in the individual reference and test cultures. In this section we describe a model of cell proliferation in a culture containing APCs, regulatory cells, and target cells. In the next section we use this model to derive a quantity corresponding to the inhibition index.
The elementary processes and interactions underlying this model are illustrated in Fig. 1⇓. The main postulate, Postulate 1, is that T cell proliferation requires productive conjugation with the APC, and interactions between regulatory and target cells require that both cells are simultaneously conjugated with the APC, i.e., the formation of multicellular conjugates. Following this postulate the number of cells proliferating in a culture that contains, at a given time, A APCs, R regulatory cells, and T target cells is given by where A_{i,j}(A,T,R) is the number of conjugates containing one APC, i target cells, and j regulatory cells in the culture, α_{i,j} are proliferation coefficients determined by the stoichiometry of each conjugate, and s is the number of conjugation sites per APC. Each coefficient α_{i,j} counts how much proliferation is obtained from a conjugate containing i target cells and j regulatory cells. From Equation 2 it is easy to understand that we need to specify the values for every A_{i,j}(A,T,R) and the proliferation coefficients α_{i,j}.
The conjugates with a given stoichiometry present in a culture can be counted using a formalism we proposed previously (34). This formalism is valid as long as the cell cultures fulfill the following additional postulates.
Postulate 2.
Following previous suggestions (35, 36), we assume that the s conjugation sites on each APC are independent and equivalent with either regulatory or target T cells according to conjugation constants K_{R} and K_{T}, respectively. Postulate 3, the conjugation constants are formally analogous to molecular affinities and describe the equilibrium numbers of conjugated target and regulatory T cells (respectively, T_{c} and R_{c}), the numbers of free target and regulatory T cells (respectively, TT_{c} and RR_{c}), and the number of total free conjugation sites (s.ARcTc). Thus, they are defined as:
Postulate 4.
The processes of formation and dissociation of conjugates are very fast processes compared with activation, proliferation, differentiation, and death of T cells. To a good approximation a mixture of cells at any given time will fulfill the equilibrium conditions of Equation 2.
Following these assumptions we calculate the conjugates in two steps. First, the numbers of conjugated target and regulatory cells, T_{c} and R_{c}, respectively are calculated as a function of A, R, and T by solving Equation 2. A general solution for T_{c} and R_{c} was previously presented (34). Throughout this article we will consider two particular cases: one in which the conjugation constants of regulatory and target cells are identical, K_{R} = K_{T} (Equation 3), and one in which the conjugation constant of regulatory cells is infinite, K_{R} = ∞ (Equation 4). Considering these two cases is appropriate because, intuitively, increasing the conjugation constant of regulatory cells will increase the instantaneous inhibition. Setting K_{R} at infinity is certainly unrealistic, because it means that regulatory cells would be permanently “glued” to the APC and could not be displaced by target cells. However, a consideration of both cases helps us to keep track of the more realistic situation that can be expected to lie in between these two extremes.
Once we obtain the total number of conjugated target and regulatory cells we calculate how these are distributed among the A sets of s equivalent sites. The result is a product of two hypergeometric distributions: where H is defined as We turn now to the problem of specifying the proliferation coefficients α_{i,j}. This requires three additional postulates about the actual interactions taking place in the conjugates.
Postulate 5.
The probability that a target cell proliferates following a productive conjugation with an APC is independent of the number of targets cells i in the same conjugate. In other words, there is no cooperativity among target cells.
Postulate 6.
The regulatory cells do not proliferate following conjugation with the APC.
Postulate 7.
If a target cell participates in a conjugate that contains at least one regulatory cell, then its conjugation is not productive, i.e., a single regulatory cell is able to prevent the proliferation of all the target cells in the same conjugate.
Following these postulates we specify the coefficients as follows: where p is the proliferation per productively conjugated target cell.
Postulates 5–7 allow a major simplification of the intercellular interactions. Postulate 6 is based on the fact that regulatory cell populations do not proliferate when cultured with APCs in the absence of target cells (16, 20, 23, 25). Postulate 7 maximizes the strength/efficiency of the inhibition per regulatory cell, and this is adequate for our purpose of identifying a limit in the efficiency of suppression dependent on the formation of multicellular conjugates.
Instantaneous inhibition of proliferation and its index λ(A,T,R)
To compare the predictions of the models with the experimental inhibition index we devised a theoretical quantity that we call instantaneous inhibition index λ(A,T,R), defined as the ratio between the proliferation in an ideal test culture containing at a given time A APCs, T target cells, and R regulatory cells and the proliferation in an ideal reference culture that contains at the same time only A APCs and T target cells: The instantaneous inhibition index estimates the fraction of target cells that is not prevented from having productive conjugation with the APC by the presence of a given number of regulatory cells in the test culture compared with the reference culture. Substitution of Equation 5 in Equation 8 and rearrangement simplify to the product of two factors: The factor χ is the ratio between the numbers of target cells conjugated in the ideal test and reference cultures: where T_{C}(A,T,R) is generically the number of conjugated target cells in a mixture containing A, R, and T cells calculated according to Equation 3 or 4.
The factor θ is simply the fraction of target cells participating in conjugates free from regulatory cells in the ideal test culture: The factors χ and θ represent, respectively, the suppression by competition for conjugation sites and the suppression by active inhibition in multicellular conjugates.
The instantaneous inhibition index can be calculated at any given time for ideal reference and test cultures that have, by construction, the same numbers of target cells and APCs. The comparison of this quantity to the real experimental inhibition index is not straightforward. Although equal numbers of APCs and target cells are put into the reference and test cultures, at the time of the pulse (i.e., after the amplification period) these numbers may have changed due to the interactions and processes, such as proliferation and death in the culture. Keeping this problem in mind, we will equate the inhibition index to the instantaneous inhibition index estimated from the numbers of APCs, target cells, and regulatory cells put into culture, respectively, A_{0}, T_{0}, and R_{0}. This approximation implies two additional postulates.
Postulate 8.
The duration of the pulse of thymidine incorporation τ is negligible compared with cell population doubling time.
Postulate 9.
The numbers of APCs, target cells and regulatory cells do no change during the amplification period Δ. As a corollary, the duration of the amplification period Δ is negligible.
Note that although this approximation is a strong mathematical simplification, it may nevertheless be biologically reasonable. Thus, it is commonly assumed that following stimulation by an APC a T cell may undergo several rounds of division without further interactions between its progeny and the APC (this assumption is substantiated by Ref. 37). Under these conditions only the initial interaction of the T cells with the APC would be relevant.
More rigorous analysis of the results could be made using ordinary differential equations to follow the actual processes of growth and death of the T cell populations during the amplification period Δ and the total number of cells incorporating thymidine in the period τ. The approximation used here, similar to that followed by Borghans et al. (38), consists of calculating the ratio between the positive component of the derivative of the target cells in the test and reference cultures at the limit when both Δ and τ tend to zero (Postulates 8 and 9). There is more to be said about the accuracy of this approximation that will be taken up in Discussion.
Model solving and fitting to experimental data
All the mathematical analyses, both analytic and numeric, were performed with the software Mathematica 3.0 from Wolfram Research (Champaign, IL). The comparison of the models with the experimental data was performed assuming that the populations that were used as sources of APCs, regulatory cells, or target cells in Table I⇑ were 100% pure.
Results
Competitive and active inhibitions contribute to suppression, but one of these components may predominate
According to our model a regulatory cell prevents, by a direct active inhibitory interaction, the proliferation of all target cells that participate in the same multicellular conjugate with the APC. Because regulatory and target cells share the conjugation sites on the APC, a regulatory cell will also inhibit its targets, in an indirect way, when it occupies a site on the APC. These two modes of inhibition, active and competitive, are inevitably associated, because for a regulatory cell to inhibit actively its targets it also needs to occupy a site. Understanding how much of the suppression can be explained by competition alone and how much can be explained by active inhibition becomes a fundamental issue at this stage. This issue is particularly relevant if we consider that competitive inhibition and active inhibition in multicellular conjugates are discussed in the literature as explanations for linked suppression (20, 39).
The first result of our quantitative analysis is the actual derivation of an expression of the instantaneous inhibition index, λ, as the product of two factors, θ and χ, representing, respectively, the contributions of active and competitive inhibition (Equation 9). The presence of the number of conjugated regulatory cells, R_{c}, in the expressions for both θ and χ reveals the interdependence of the two components of suppression (Equations 10 and 11). Both factors vary between 0 and 1; therefore, as expected, suppression including active inhibition is always more efficient than one based on competitive inhibition alone.
The second result of our analysis is that although both competitive and active inhibitions can potentially contribute to in vitro suppression of target cell by regulatory cells, in practice one of these components may predominate depending on the composition of the culture. The interplay between these two components of suppression is illustrated in Fig. 2⇓ (top) where we represent the total suppression, θ × χ, and the suppression due to competitive inhibition alone, χ. The contribution of competitive inhibition becomes significant only when the number of free sites on the APCs becomes reduced, i.e., at a relatively high number of total T cells (regulatory and target) compared with the total number of conjugation sites (s × A). However, active inhibition is already significant when the number of conjugated regulatory cells is on the same order as the number of APCs. Changing the parameters in the system modulates the interplay between the two components. For example, the higher the number of sites per APC, the higher the proportion of suppression due to active inhibition and the lower the proportion due to competitive inhibition (Fig. 2⇓, bottom).
Competitive inhibition and active inhibition have different sensitivities to changes in the numbers of target and regulatory cells
Having demonstrated that suppression may be explained by competitive and/or active inhibitions and that either of these components may predominate depending on the culture conditions, we tried to devise a strategy to clarify under which regimen the in vitro culture system is operating.
To do this we consider the properties of two ideal extreme cases in which suppression is exclusively explained by either competitive inhibition or active inhibition. For an ideal case of suppression by competitive inhibition alone, we set θ to unity and assumed additionally that target cells in both test and reference cultures were in large excess compared with a limited number of conjugation sites, such that the number of free conjugation sites was negligible. Under such conditions, the expression for instantaneous suppression, λ, simplifies to: This expression is dependent on the ratio between the numbers of regulatory and target cells weighted by their respective conjugation constants. This dependence on both cell populations reflects the intrinsic symmetry in competitive interactions; regulatory cells may compete out their targets, but they can also be competed out. Competitive inhibition is expected to be comparably sensitive to changes in the numbers of regulatory and target cells as long as the conjugation constants are not very different. If suppression is dominated by competitive inhibition, then the inhibition index should have a similar sensitivity pattern, which would be adequately captured by plotting the index as a function of the ratio R/T.
In the other case in which suppression is explained exclusively by active inhibition the expression for λ reduces to: This expression is, by construction, independent of the number of target cells. Hence, in the absence of competition between regulatory and target cells for conjugation sites, there is no way in which target cells can interfere with regulatory cells. Under these conditions the number of conjugated regulatory cells, R_{c}, is determined only by the total number of regulatory cells, R, the number of sites, s × A, and the conjugation constant, K_{R}. Furthermore, a simple numerical analysis shows that expression 14 is just a function of the ratio R/A when the number of APCs is large enough (A > 100), and R_{c} is proportional to R (which is always true in the absence of competition, i.e., R_{c}≪s × A). The insensitivity of active inhibition to changes in the number of target cells is qualitatively easy to understand if we consider that it is the result of an intrinsically asymmetric interaction by which regulatory cells inhibit, but are not affected by, their targets. Therefore, when active inhibition predominates, suppression is expected to be a simple function of the number of regulatory cells per APC (R/A) and to be insensitive to changes in the number of target cells.
These predictions are illustrated in the theoretical curves of Fig. 3⇓ (top panel) using the general expression for the instantaneous inhibition, λ (Equation 9). λ is plotted as a function of the ratio of regulatory cells per APC (R/A) or as a function of the ratio regulatory cells per target cells (R/T) for several numbers of target cells. In the R/A representation the different curves overlap when active inhibition predominates (lines are continuous) and start diverging when competition becomes effective (lines are dashed). In contrast, in the R/T representation the curves diverge when competition is negligible (continuous) and converge when competition is more significant (dashed). The dispersion of different experimental curves may therefore provide clues about the relative contributions of competitive and active inhibitions to suppression.
The inhibition index scored by in vitro suppression assays is relatively insensitive to changes in the number of target cells
Following the theoretical considerations of the previous section we asked which representation, R/A or R/T, reduces the dispersion of the experimental inhibition index. A plot of the experimental data sets (Table I⇑) in the coordinates I vs R/A and in the I vs R/T shows that experimental points are significantly less dispersed in the first coordinate system compared with the second (Fig. 3⇑, bottom panel). This is particularly meaningful considering that the data are obtained with different experimental systems and by different laboratories. Moreover, data points resulting from the same experimental system, such as C1 and C2 (circles and diamonds), which are separate in the I vs R/T coordinate system appear as if they belong to the same curve in the R/A representation.
This graphic analysis indicates that the experimental inhibition index is relatively insensitive to changes in the numbers of target cells. This sensitivity pattern suggests that suppression is based on an active inhibitory effect of regulatory cells on target cells, and that competition for conjugation sites plays a minor role. Alternatively, the insensitivity to the number of target cells may result from competition under the particular condition in which the conjugation constant of the regulatory cells is much higher than the conjugation constant of target cells.
The results of the suppression assay are traditionally represented in the coordinates I vs R/T (20, 23, 25), a procedure that stems historically from the cytotoxic assay. Our results indicate that the coordinates I vs R/A are a better representation of the experimental data than the classical coordinates I vs R/T, because the second representation introduces an artificial effect of the number of target cells. Heretofore we adopt the first representation. In the next section we demonstrate that the instantaneous suppression has an absolute limit that is readily revealed in these coordinates.
The instantaneous inhibition index has an absolute asymptotic limit as a function of the number of regulatory cells per APC
As stated in the introduction, the efficiency of suppression by active inhibition should be limited by the frequency of multicellular conjugates. In this section we show that instantaneous inhibition has such a limit, and we estimate its value as a function of the parameters involved. According to the model the instantaneous inhibition, λ, depends on three parameters: the number of conjugation sites per APC, s, and the conjugation constants K_{R} and K_{T} for regulatory and target cells, respectively. These three parameters are explored aiming to identify those values that minimize λ.
Two prototypic combinations of the values of K_{R} and K_{T} are considered: equal conjugation constants of regulatory and target cells and an infinite conjugation constant for regulatory cells. A consideration of the two cases allows us to keep track of the more realistic situation that can be expected to exist in between these two extremes. The conclusions drawn from these particular cases in this section are general.
The parameter dependence of instantaneous inhibition, λ, is illustrated for these two cases in Fig. 4⇓. In the top panel, the ratio R/A required to reach 50% inhibition (λ = 0.5) is plotted as a function of the ratio T/A for different values of K_{T}. In the case of equal conjugation constants (Fig. 4⇓, left), the bigger the value of K_{T}, the smaller the ratio R/A required to reach 50% inhibition. Moreover, as K_{T} increases, the curves converge to a limit curve corresponding to the situation when K_{T} = infinity. This limit curve defines the minimal value of the ratio R/A that can lead to 50% inhibition for a particular number of conjugation sites per APC, s. The absolute minimum is obtained at T/A = s. The shape of the limit curve gives some information about the processes taking place in the system. For low values of the ratio T/A, the curve is a horizontal line. The number of T cells is so small that there is no competition for the conjugation sites on the APC. To reach 50% inhibition one would need to have enough regulatory cells so that about 50% of APC are conjugated with at least one regulatory cell and about 50% are free from regulatory cells. For a short interval of intermediate T/A values there is a transition phase that takes place while the conjugation sites per APC get saturated. During this phase, increases in the number of target cells means that they are occupying the free sites on an APC already containing regulatory cells; this maximizes the inhibitory effect per regulatory cell. For high values of the ratio T/A the curve becomes linear in a regimen in which competition predominates in the system. Any increase in the number of target cells displaces regulatory cells from the APC. To keep the level of inhibition at 50%, the number of regulatory cells must be increased proportionally to the target cells to keep their balance in conjugates.
In the case of infinite conjugation constant of regulatory cells (Fig. 4⇑, top right) the picture is different. In contrast with the previous case, for low values of T/A, the higher the value of K_{T}, the lower the value R/A required to obtain 50% inhibition. For intermediate values of the ratio T/A, there is a value of K_{T} that gives the minimum ratio of R/A. For very high values of T/A the decrease in K_{T} entails an increase in the ratio R/A required to reach 50% suppression. As K_{T} tends to infinity, a limit curve is obtained. The shape of this limit curve resembles that obtained with equal conjugations constants, but here, because target cells cannot by definition compete out the regulatory cells, the linear phase corresponding to competition has a slope of zero. From examination of Fig. 4⇑ (top right) it is clear that the absolute minimum R/A required for 50% suppression is obtained when K_{T} = infinity, and T/A is greater than s. The same absolute minimum is also obtained when K_{T} tends to zero, and a limit horizontal line is obtained, although this limit K_{T} = 0 is biologically uninteresting because it would imply no proliferation in the cultures.
The consideration of these prototypical cases reveals the existence of an absolute minimal number of regulatory cells per APC that can lead to 50% inhibition. In the case where APCs have five conjugation sites, 50% suppression can only be scored only if the ratio R/A is greater than 0.73 in the case of K_{R} = K_{T} and 0.65 in the case of K_{R} = infinity. Although these absolute minima were illustrated in Fig. 4⇑A for a particular number of conjugation sites per APC, s, and a given inhibition index, λ, their existence is generic. For any given value of s, any chosen value of λ cannot be obtained if the ratio R/A is lower than a corresponding minimum ratio R/A. Therefore, we can define for each value s a corresponding limit curve relating the minimal R/A ratio required for scoring a given instantaneous inhibition index. In Fig. 4⇑ (bottom) these limit curves are plotted for a series of values of s for the two cases under study. In the case of equal conjugation constants (Fig. 4⇑, bottom left) as s increases the curves move down and tend asymptotically to an absolute limit curve that is reached when s = infinity. In the case of infinite conjugation constant (bottom right) the curves move up as s increases, reaching the same limit when s = infinity.
In summary, the results in this section show that there is an asymptotic lower limit to the instantaneous suppression λ as a function of the ratio R/A in the test culture. In other words, independently of parameter settings, any chosen value of λ requires a minimum ratio R/A. In the following section this limit of the instantaneous inhibition index is compared with the experimental data.
The instantaneous inhibition model fails to fit the experimental inhibition index
In this section we ask how the experimental data relate to theoretical limit curve identified in the previous section. To do this we will equate the experimental inhibition index I to the instantaneous inhibition index, λ, according to Equation 12. This is a gross approximation that requires Postulates 8 and 9, which assume that the duration of the period before the thymidine pulse and that of the actual pulse are negligibly. Additionally, we assume that the populations of regulatory cells, target cells, and APCs are 100% pure. Under these assumptions, in Fig. 5⇓ (top) the markers represent the experimental data for the inhibition index I during in vitro suppression assays reported by several groups (Table I⇑), and the lines represent the theoretical limit curves for equal conjugation constants and for infinite conjugation of regulatory cells (i.e., the leftmost curves in the bottom panel of Fig. 4⇑). All the experimental points lay outside the range defined by the limit curve of the instantaneous inhibition index predicted by the model. Therefore, the conclusion is that the instantaneous inhibition index cannot be fitted to the available experimental data. It is worth estimating the gap between the experimental data and the limit curve. This can be quantified by determining a multiplication factor to apply to the R_{0}/A_{0} ratio allowing a corrected experimental curve to overlay the theoretical limit curves. This factor f is ∼6 in the case of an infinite conjugation constant for regulatory cells (Fig. 5⇓, middle) and ∼10 in the case of equal conjugation constants of regulatory and targets cells (Fig. 5⇓, bottom).
There are three nonmutually exclusive interpretations for the failure to explain the experimental data. The numbers of cells may have been incorrectly estimated in the experimental assay, such that the true ratio R/A was, in fact, f × R_{0}/A_{0}. Alternatively, some of the approximations in the model or in the procedure for mapping its predictions to the experimental data may be unrealistic. During the period before the thymidine pulse the ratio R/A may have changed by a factor f, violating Postulate 9. Finally, suppression in vitro may proceed by a mechanism that does not require formation of multicellular conjugates.
Discussion
With the purpose of better understanding linked suppression, we addressed the quantitative predictions of the hypothesis that regulatory cells suppress other T cells while both cells are conjugated with the APC. We analyzed the experimental inhibition index in cocultures of regulatory cells, target cells, and APCs using a simple mathematical model. This model is a radical simplification of the reality of cell cultures, as it assumes that thymidine incorporation is proportional to the number of target cells that would instantaneously form conjugates with APCs that are not simultaneously conjugated with regulatory cells. This simplification proved useful, because it allowed us to predict and to explain an insensitivity of the experimental inhibition index to changes in target cell number. However, it is bluntly unable to explain the actual percentage of suppression observed experimentally.
Our theoretical analysis has shown that both competition and active inhibition will quantitatively contribute to the final suppression. To identify whether one or the other component predominates, we put forward a method based on the sensitivity of inhibition index to changes in the number of regulatory cells per APC and in the number of regulatory cells per target. This strategy is only valid when the conjugation constants of the regulatory and target cells are on the same order of magnitude. Theoretically when the conjugation constant of regulatory cells is several orders of magnitude greater than that of target cells our criterion to distinguish the two components is not effective because competition itself becomes insensitive to changes in the number of target cells. How plausible is this extreme case? The conjugation between cells is determined by coordinated signaling and expression of multiple molecules in both cells (40, 41) Conjugation constants are mapped to the average duration of conjugates between two cells; the longer the conjugation constant, the longer the duration of the conjugate. Orders of magnitude differences in conjugation constants require orders of magnitude differences in conjugation times. Conjugation times of T cells with their APC were estimated to range from a few hours to less than a day (42, 43). Therefore, it difficult to squeeze within this interval two values of conjugation constants whose ratio is practically zero. Moreover, in some of the data sets used here regulatory cells exhibit an activated/memory phenotype (CD45RB^{low} or CD25^{+}), while their targets are essentially naive cells (CD45RB^{high} or CD25^{−}). Following these considerations it is unlikely that the conjugation constants of regulatory and target cells differ by orders of magnitude. Full activation of naive cells requires longer conjugation time with the APC than activation of blast or memory cells (42). Thus, eventual differences between conjugation constants of regulatory and target cells may actually be the opposite of that required for competition to be insensitive to changes in target cell numbers. Therefore, in the context of our model, this insensitivity must be interpreted as a predominance of an active inhibitory signal of regulatory delivered to target cells in multicellular conjugates.
Hitherto, it had gone unnoticed in the literature that the inhibition scored in suppression assays is mainly dependent on the number of regulatory per APC and is insensitive to changes in the number of target cells. This is an important finding by itself regardless of the particular explanation we proposed here. Any candidate mechanism for the interactions between regulatory and target cells must be able to account for this property. Generically, any unidirectional interaction (i.e., from regulatory to target cells) whose ratelimiting step is mediated by the APC can potentially explain insensitivity to the number of target cells. This may be true whether the APC mediates the interaction by bringing the two cells together (as studied here) or by conveying a signal from one cell to the other. In contrast, a mechanism involving symmetric interactions between regulatory and target cells, such a competition, can be expected to have difficulty explaining different sensitivities to the numbers of two cell types. We analyzed here competition for conjugation sites, but similar results can be expected for symmetric competition for survival or growth factors (20). Also, interaction mechanisms in which the ratelimiting step is a direct encounter between regulatory and target cells to form 1:1 conjugates (24) may lead to comparable sensitivities to the two cell populations and therefore be incompatible with the result reported here. Due to its relevance for hypothesis testing the relative sensitivities of in vitro inhibition to the three experimental variables (APCs, target and regulatory cell numbers) deserves more experimental attention.
We now turn to the failure to explain the experimental data by instantaneous inhibition. As stated in Results, this can be interpreted in at least three nonmutually exclusive ways. Each of these possibilities is plausible, nontrivial, and biologically relevant, and therefore deserves detailed discussion. However, before such discussion a methodological point deserves being stressed. We used a model that was radically simple both in structure and in the way it was compared with the experimental data. Nevertheless, in the context of this model the actual failure to explain the data becomes informative, because it suggests straightforward extensions that represent quantitative testable hypothesis about the mechanism of suppression. This illustrates a noteworthy aspect of mechanistic modeling of data: even negative results are often informative.
In all the experimental results listed in Table I⇑ it is quite plausible that there was an overestimation of the numbers of regulatory cells, target cells, or APCs that effectively participate in the suppression. Considering these possibilities it is quite straightforward to realize that assuming that the number of regulatory or target cells is overestimated (i.e., assuming that the number of T cells per APC is lower) does not help to bring the theoretical limit curve closer to the experimental data. Actually, in the case of regulatory cells it will just make things worst. In contrast, assuming that there is an overestimation in the number of APCs that form conjugates with T cells can reduce the gap between the prediction of the models and the experimental data. The correction factor estimated before implies that only ∼15% of the APC used in suppression assays is not forming conjugates with T cells. This interpretation may be quite reasonable in the experimental systems where the APCs were obtained in vivo (e.g., T celldepleted splenocytes) or where the APCs are a cell line loaded with an agonist peptide. In other systems in which the APCs are a cell line expressing a particular allogeneic MHC (21) it is more difficult to envisage that the population of APCs is not homogeneous in their capacity to form conjugates with T cells.
Equating the experimental inhibition index, I, to the instantaneous inhibition index calculated for the numbers of cells originally placed in culture is a radical simplification of reality. In all the systems studied here the suppression assay takes about 3 days of culture. During this period it is possible that regulatory and/or target cells undergo several rounds of division, and/or that inhibitory effects are amplified before the thymidine pulse is performed. Regulatory cells might alter the growth kinetics of target cells in the test culture compared with the reference culture, and these differences could be amplified such that at the time of the pulse the number of target cells in the two cultures is different. Therefore, differential kinetics would violate the calculation of instantaneous suppression, which requires cultures with equal numbers of target cells. Another possibility, perhaps biologically more interesting, is that the regulatory cell population expands when cocultured with their targets, increasing the ratio R/A. Indeed, the data can be easily fitted if the number of regulatory cells increases ∼6 and 10fold, respectively, in the cases of infinite and equal affinities of regulatory cells. Because it is well established that regulatory T cells do not proliferate when stimulated with APCs in the absence of target cells, it is not trivial that the regulatory population will grow when cocultured with target cells. However, there are two nonmutually exclusive mechanisms that could explain this target celldependent growth of the regulatory cell population. The regulatory cells may promote differentiation of the target cells into a regulatory phenotype, such that the regulatory population would increase by autocatalytic recruitment of new members. Regulatory populations in the systems studied here are made of anergic, or naturally anergiclike, cells. The proposition that anergic cells may render other cells anergic is not new and has been discussed in the context of immunosuppression and tolerance (26). Alternatively, the size of the regulatory cell population may increase during coculture with target cells, because the later produce a growth factor. The hallmark of anergiclike regulatory cells is the expression of high levels of IL2R and the lack of secretion of IL2. Therefore, regulatory cells are fully responsive to the IL2 produced, which is responsible for the growth of target cells. Explicit modeling of the growth kinetics of regulatory and target cell populations using a model described previously (34), demonstrated that these two alternative explanations can easily account for the data (results not shown). The fitting is degenerated and dependent on several parameters, such as inhibitory and stimulatory coefficients, growth rates of regulatory and target cells, etc., which cannot be measured independently. Overall it is fair to conclude that the impact of the kinetics in the suppression index deserves a proper theoretical analysis. In a complementary approach, it would be worth developing immunosuppression assays in which the assumptions underlying instantaneous suppression would be controlled and validated. Ideally the thymidine incorporation pulse could be performed within the first hours of culture, before an eventual accumulation of kinetic effects. Alternatively, early effects of regulatory cells on target cells might be measured instead of the rather late inhibition of proliferation. Good candidates are the inhibition of expression of early activation markers such as CD69.
Finally, the instantaneous inhibition index may fail to account for the experimental data because the suppression is not strictly dependent on the formation of multicellular conjugates. At the initial proportions of the three cells in culture the probability of formation of multicellular conjugates becomes much too improbable. Therefore, any mechanism of suppression that requires recognition of the same APC for activation of both regulatory and effector cells, but in which the actual mechanism of suppression is operative even when that conjugation with APC is asynchronous, will be more efficient and would more easily fit the data. Recently, Thornton and Shevach (24) suggested that CD25^{+} T cells following activation by APCs might nonspecifically suppress their targets by a mechanism that is no longer APC dependent. Although the underlying mechanism was not detailed, it may be more efficient than that explored here because it does not require multicellular conjugates, but, instead, proceeds by sequential pairwise conjugations of the regulatory first with the APC and then with target cells. However, this proposal is difficult to reconcile with previous reports demonstrating the requirement for linked recognition of the same APC, and also with the fact that suppression does not occur if cells are activated using platebound antiCD3 (22, 23). In agreement with the linked recognition is the alternative hypothesis that regulatory cells may suppress their targets indirectly by modulating the APCs themselves (28, 29). This mechanism can be expected a priori to be more efficient because it involves sequential interactions of the APC first with regulatory cells and then with target cells. Interestingly, the model as formulated here represents a minimal implementation of this mechanism, assuming that the effect of regulatory cells on the APC is very transient. Much higher efficiencies can be achieved if each regulatory cell is able to serially conjugate with different APC, permanently inactivating their capacity to stimulate target cells. Recent reports have shown that CD25^{+} T cells or anergic cells can modulate their APCs (27, 30); however, this mechanism remains difficult to reconcile with the fact that significant suppression is scored in the presence of fixed or irradiated APCs (44), which cannot be modulated. A comparison of the capacities of these alternative hypotheses to explain the experimental data is being conducted.
The results reported here suggest that the population of target cells might promote growth of the population of regulatory cells in vitro. A generic analysis of the population dynamics of regulatory and target cell populations involved in adoptive transfers of tolerance leads us to a similar conclusion (34). These two complementary lines of evidence indicate that the maintenance and growth of a regulatory population might depend on the target population it regulates. It is tempting to suggest that potentially pathogenic cells, which are the targets of tolerogenic regulatory cells, may actually play an active role in the maintenance of tolerance.
In summary, our quantitative analysis of the mechanism of linked suppression dependent on multicellular conjugates allowed us to predict that it is mainly determined by the ratio of regulatory cells per APC and to accurately estimate a limit in efficiency of the suppression. Analysis of experimental data indicates that experimental suppression is mainly determined by the ratio of regulatory cells per APC, as expected from this model. Although having limited efficiency, an active inhibitory signal delivered by regulatory to target T cells in multicellular conjugates with a single APC can still explain the data if the number of regulatory cells increases in cocultures with target cells or if only a small fraction of APCs form conjugates.
Acknowledgments
We thank Prof. Robert Lechler, Prof. Shimon Sakaguchi, and Dr. António Bandeira for discussions from which this work originated. We also thank João Sousa, Jose Faro, and Jocelyne Demengeot for many useful discussions and for critically reading this manuscript.
Footnotes

↵1 This work was supported by Program Praxis XXI of the Ministério para Ciência e Tecnologia, Portugal (Grant Praxis/P/BIA/10094/1998). J.C. was supported by Fundação para a Ciência e Tecnologia, Program Praxis XXI (Fellowship BPD/11789/97). K.L. was supported in part by the Fundação Calouste Gulbenkian.

↵2 Address correspondence and reprint requests to Dr. Jorge Carneiro, Estudos Avançados, Instituto Gulbenkian de Ciência, Apartado 14, 2781901 Oeiras, Portugal. Email address: jcarneir{at}igc.gulbenkian.pt
 Received December 6, 2000.
 Accepted February 16, 2001.
 Copyright © 2001 by The American Association of Immunologists