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

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One of the fundamental properties of the immune system is its capacity to avoid autoimmune diseases. The mechanism underlying this process, known as self-tolerance, is hitherto unresolved but seems to involve the control of clonal expansion of autoreactive lymphocytes. This article reviews mathematical modeling of self-tolerance, addressing two specific hypotheses. The first hypothesis posits that self-tolerance is mediated by tuning of activation thresholds, which makes autoreactive T lymphocytes reversibly "anergic" and unable to proliferate. The second hypothesis posits that the proliferation of autoreactive T lymphocytes is instead controlled by specific regulatory T lymphocytes. Models representing the population dynamics of autoreactive T lymphocytes according to these two hypotheses were derived. For each model we identified how cell density affects tolerance, and predicted the corresponding phase spaces and bifurcations. We show that the simple induction of proliferative anergy, as modeled here, has a density dependence that is only partially compatible with adoptive transfers of tolerance, and that the models of tolerance mediated by specific regulatory T cells are closer to the observations.

In cell lifespan studies the exponential nature of cell survival curves is often interpreted as showing the rate of death is independent of the age of the cells within the population. Here we present an alternative model where cells that die are replaced and the age and lifespan of the population pool is monitored until a steady state is reached. In our model newly generated individual cells are given a determined lifespan drawn from a number of known distributions including the lognormal, which is frequently found in nature. For lognormal lifespans the analytic steady-state survival curve obtained can be well-fit by a single or double exponential, depending on the mean and standard deviation. Thus, experimental evidence for exponential lifespans of one and/or two populations cannot be taken as definitive evidence for time and age independence of cell survival. A related model for a dividing population in steady state is also developed. We propose that the common adoption of age-independent, constant rates of change in biological modelling may be responsible for significant errors, both of interpretation and of mathematical deduction. We suggest that additional mathematical and experimental methods must be used to resolve the relationship between time and behavioural changes by cells that are predominantly unsynchronized.

The parameters of the immune response dynamics are usually estimated by the use of deterministic ordinary differential equations that relate data trends to parameter values. Since the physical basis of the response is stochastic, we are investigating the intensity of the data fluctuations resulting from the intrinsic response stochasticity, the so-called process noise. Dealing with the CD8+ T-cell responses of virus-infected mice, we find that the process noise influence cannot be neglected and we propose a parameter estimation approach that includes the process noise stochastic fluctuations. We show that the variations in data can be explained completely by the process noise. This explanation is an alternative to the one resulting from standard modeling approaches which say that the difference among individual immune responses is the consequence of the difference in parameter values.