Cluster formation in a symmetrical network: a dynamical system for the description of the suppression among non-immune T lymphocytes and its application to the effects of immunization

Quantitating Effector and Regulatory T Lymphocytes in Immune Responses by Limiting Dilution Analysis Modeling

Although there is currently no doubt that regulatory lymphocytes represent a master player in the immune system, a major unresolved problem is the accurate quantitation of these cells among unfractionated cell populations. This difficulty mainly arises because there are no specific immunophenotypic markers that can reliably discriminate between effector and regulatory lymphocytes. To face this problem, we have developed computational models of limiting dilution analyses addressing the question of the accurate estimation of the frequencies of effector and regulatory cells functionally engaged in an immune response. A set of generic equations were provided to form a framework for modeling limiting dilution data, enabling discrimination between qualitatively different models of suppression. These models include either one or two subpopulations of regulatory cells, featured by either low or potent regulatory activity. The potential of this modeling approach was illustrated by the accurate determination of the frequencies of effector and regulatory T lymphocytes in one real limiting dilution experiment of CD4+ CD25+ T lymphocytes performed in the context of an allogeneic response in the human system. The crucial advantage of the limiting dilution method over the "static, phenotype-based" method is the dynamic evaluation of effector and regulatory T cell biology through their actual functional activity.

Fitting limiting dilution experiments with generalized linear models results in a test of the single-hit Poisson assumption

Limiting dilution analysis is a common technique that is used in immunology to estimate accurately the frequency of cells possessing a wide variety of functional activities such as growth, cytotoxicity and production of lymphokines. The reliability of the estimated frequency is usually checked by a standard chi-square (x2) test validating the goodness-of-fit to the single-hit Poisson model (SHPM). We present evidence that modelling limiting dilution data according to a generalized linear model offers an alternative to the standard x2 test for detecting departures from the SHPM, with a considerable increase in power compared to the x2 test.

The standard chi 2 test used in limiting dilution assays is insufficient for estimating the goodness-of-fit to the single-hit Poisson model

Limiting dilution analysis is a common technique that is used in immunology to estimate accurately the frequency of cells possessing a wide variety of functional activities, such as growth, cytotoxicity and production of lymphokines. In the literature, most experiments are fit well by the single-hit Poisson model (SHPM), which assumes that only one cell of one defined cell subset is necessary for a positive response. This is somewhat surprising since other models such as multi-hit or multi-target models that involve the interaction of one or more cells from one or more cell subpopulations for generating or inhibiting a positive response are conceivable. Since the validity of the SHPM is usually investigated by performing a standard chi 2 test, based on the number of observed and expected positive and negative responses, we questioned here the efficiency of this test in comparison with other validity tests for the SHPM, the log likelihood test derived by Cox, and the modified Weibull plot tests, the principles of which are entirely different from that of the standard chi 2 test. We used the following theoretical approach. First, we generated artificial data corresponding to multi-hit and multi-target models. Second, considering that these data were derived from real experiments, we calculated the frequency of the desired cell subset according to the SHPM using the maximum likelihood method. Then, the goodness-of-fit of these data with the SHPM was evaluated. The log likelihood test and the modified Weibull plot tests rejected the SHPM hypothesis, while the standard chi 2 test did not. Thus, the standard chi 2 test is unable to discriminate sensitively between the SHPM and more complicated (non-single-hit) Poisson models. We concluded that the results of limiting dilution studies published thus far must be evaluated with caution. The statistical tests presented here should be routinely applied for each limiting dilution experiment.

Memory but no suppression in low-dimensional symmetric idiotypic networks

We present a new symmetric model of the idiotypic immune network. The model specifies clones of B-lymphocytes and incorporates: (1) influx and decay of cells; (2) symmetric stimulatory and inhibitory idiotypic interactions; (3) an explicit affinity parameter (matrix); (4) external (i.e. non-idiotypic) antigens. Suppression is the dominant interaction, i.e. strong idiotypic interactions are always suppressive. This precludes reciprocal stimulation of large clones and thus infinite proliferation. Idiotypic interactions first evoke proliferation, this enlarges the clones, and may in turn evoke suppression. We investigate the effect of idiotypic interactions on normal proliferative immune responses to antigens (e.g. viruses). A 2-D, i.e. two clone, network has a maximum of three stable equilibria: the virgin state and two asymmetric immune states. The immune states only exist if the affinity of the idiotypic interaction is high enough. Stimulation with antigen leads to a switch from the virgin state to the corresponding immune state. The network therefore remembers antigens, i.e. it accounts for immunity/memory by switching between multiple stable states. 3-D systems have, depending on the affinities, 9 qualitatively different states. Most of these also account for memory by state switching. Our idiotypic network however fails to account for the control of proliferation, e.g. suppression of excessive proliferation. In symmetric networks, the proliferating clones suppress their anti-idiotypic suppressors long before the latter can suppress the former. The absence of proliferation control violates the general assumption that idiotypic interactions play an important role in immune regulation. We therefore test the robustness of these results by abandoning our assumption that proliferation occurs before suppression. We thus define an "escape from suppression" model, i.e. in the "virgin" state idiotypic interactions are now suppressive. This system erratically accounts for memory and never for suppression. We conclude that our "absence of suppression from idiotypic interactions" does not hinge upon our "proliferation before suppression" assumption.