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.