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

Abstract

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. (PNAS, 2007, 104, 5032-5037) 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., 2008, 56:6, 861-892), 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 for publication) 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-ordivision 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 Ken Duffy and Vijay Subramanian, Hamilton Institute, National University of Ireland, Maynooth, Ireland E-mail: {ken.duffy, vijay.subramanian}@nuim.ie 2 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.

Why the immune system takes its chances with randomness

“…As Reiner and Adams point out, the number of antigen-specific precursors recruited into the immune response is a crucial variable, and may be as low as 20. However, mathematical models in which randomness drives cell fate selection suggest that a reasonably robust immune response can be achieved even with starting cell numbers of this order 4,9,10,15 . Thus, a role for randomness should not be rejected on this basis alone.…”

A Division-Dependent Compartmental Model for Computing Cell Numbers in CFSE-based Lymphocyte Proliferation Assays

“…In this report, the form of the hypothetical function F is determined from the PDE formulation of the compartmental model (2) and the accompanying assumptions regarding the Malthusian rates of proliferation and death. Alternative, one could consider using (29) or its differential form (i.e., (10)) as a starting point, defining the right side of the equation in accordance with the assumptions of the Smith-Martin or cyton models, or their generalizations [34,41,47,58,73]. While previous authors have derived these models specifically in terms of total cell numbers, (29) could be related back to previous work by simple integration.…”

“…Cells with a common precursor may also share a common division destiny [42], which can be altered by stimulation conditions [70]. While computed cell numbers are relatively unaffected provided correlation is limited to cells having undergone the same number of divisions [34,42,45], correlation between subsequent division of cells can alter the dynamics predicted by a mathematical model [73]. For large populations, this effect seems negligible, but may play an important role in vivo where only a small number of responding cells can trigger an immune response [73].…”

“…between any two divisions, and this must induce some minimum cell cycle time. Tools such as delay differential equations or stochastic processes have been used to mathematically model the cell cycle (see, e.g., [30,31,34,37,41,44,45,47,59,66,75]) and have resulted in several successful models. We find the current model with its Malthusian rates to be simple and intuitive while also fully capable of accurately fitting the data (see Section 4).…”

Mathematical Models of Dividing Cell Populations: Application to CFSE Data

“…Branching process models (which can be thought of as generalizations of the probabilistic cyton model [51,77]) have been used to study the effects of correlations between the division characteristics of closely related cells (siblings, cousins, etc.) [38,51,77,82] and the inheritance of parameters [65], both of which have been observed in experimental settings [49].…”