On the impact of correlation between collaterally consanguineous cells on lymphocyte population dynamics.

During an adaptive immune response, lymphocytes proliferate for five to twenty-five cell divisions, then stop and die over a period of weeks. Based on extensive flow cytometry data, Hawkins et al. (Proc Natl Acad Sci USA 104:5032-5037, 2007) introduced a cell-level stochastic model of lymphocyte population dynamics, called the Cyton Model, that accurately captures mean lymphocyte population size as a function of time. In Subramanian et al. (J Math Biol 56(6):861-892, 2008), we performed a branching process analysis of the Cyton Model and deduced from parameterizations for in vitro and in vivo data that the immune response is predictable despite each cell's fate being highly variable. One drawback of flow cytometry data is that individual cells cannot be tracked, so that it is not possible to investigate dependencies in the fate of cells within family trees. In the absence of this information, while the Cyton Model abandons one of the usual assumptions of branching processes (the independence of lifetime and progeny number), it adopts another of the standard branching processes hypotheses: that the fates of progeny are stochastically independent. However, new experimental observations of lymphocytes show that the fates of cells in the same family tree are not stochastically independent. Hawkins et al. (2008, submitted) report on ciné lapse photography experiments where every founding cell's family tree is recorded for a system of proliferating lymphocytes responding to a mitogenic stimulus. Data from these experiments demonstrate that the death-or-division fates of collaterally consanguineous cells (those in the same generation within a founding cell's family tree) are strongly correlated, while there is little correlation between cells of distinct generations and between cells in distinct family trees. As this finding contrasts with one of the assumptions of the Cyton Model, in this paper we introduce three variants of the Cyton Model with increasing levels of collaterally consanguineous correlation structure to incorporate these new found dependencies. We investigate their impact on the predicted expected variability of cell population size. Mathematically we conclude that while the introduction of correlation structure leaves the mean population size unchanged from the Cyton Model, the variance of the population size distribution is typically larger. Biologically, through comparison of model predictions for Cyton Model parameterizations determined by in vitro and in vivo experiments, we deduce that if collaterally consanguineous correlation extends beyond cousins, then the immune response is less predictable than would be concluded from the original Cyton Model. That is, some of the variability seen in data that we previously attributed to experimental error could be due to intrinsic variability in the cell population size dynamics.

Determining the expected variability of immune responses using the cyton model.

During an adaptive immune response, lymphocytes proliferate for 5-20 cell divisions, then stop and die over a period of weeks. The cyton model for regulation of lymphocyte proliferation and survival was introduced by Hawkins et al. (Proc. Natl. Acad. Sci. USA 104, 5032-5037, 2007) to provide a framework for understanding this response and its regulation. The model assumes stochastic values for division and survival times for each cell in a responding population. Experimental evidence indicates that the choice of times is drawn from a skewed distribution such as the lognormal, with the fate of individual cells being potentially highly variable. For this reason we calculate the higher moments of the model so that the expected variability can be determined. To do this we formulate a new analytic framework for the cyton model by introducing a generalization to the Bellman-Harris branching process. We use this framework to introduce two distinct approaches to predicting variability in the immune response to a mitogenic signal. The first method enables explicit calculations for certain distributions and qualitatively exhibits the full range of observed immune responses. The second approach does not facilitate analytic solutions, but allows simple numerical schemes for distributions for which there is little prospect of analytic formulae. We compare the predictions derived from the second method to experimentally observed lymphocyte population sizes from in vivo and in vitro experiments. The model predictions for both data sets are remarkably accurate. The important biological conclusion is that there is limited variation around the expected value of the population size irrespective of whether the response is mediated by small numbers of cells undergoing many divisions or for many cells pursuing a small number of divisions. Therefore, we conclude the immune response is robust and predictable despite the potential for great variability in the experience of each individual cell.