# A Mathematical Framework for Understanding Four-Dimensional Heterogeneous Differentiation of \(\hbox {CD4}^{+}\) T Cells

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## Abstract

At least four distinct lineages of \(\hbox {CD4}^{+}\) T cells play diverse roles in the immune system. Both in vivo and in vitro, naïve \(\hbox {CD4}^{+}\) T cells often differentiate into a variety of cellular phenotypes. Previously, we developed a mathematical framework to study heterogeneous differentiation of two lineages governed by a mutual-inhibition motif. To understand heterogeneous differentiation of \(\hbox {CD4}^{+}\) T cells involving more than two lineages, we present here a mathematical framework for the analysis of multiple stable steady states in dynamical systems with multiple state variables interacting through multiple mutual-inhibition loops. A mathematical model for \(\hbox {CD4}^{+}\) T cells based on this framework can reproduce known properties of heterogeneous differentiation involving multiple lineages of this cell differentiation system, such as heterogeneous differentiation of \(\hbox {T}_\mathrm{H}1\)–\(\hbox {T}_\mathrm{H}2, \hbox {T}_\mathrm{H}1\)–\(\hbox {T}_\mathrm{H}17\) and \(\hbox {iT}_\mathrm{Reg}\)–\(\hbox {T}_\mathrm{H}17\) under single or mixed types of differentiation stimuli. The model shows that high concentrations of differentiation stimuli favor the formation of phenotypes with co-expression of lineage-specific master regulators.

## Keywords

\(\hbox {CD4}^{+}\) T cells Cell differentiation Mathematical model## Notes

### Acknowledgments

This work was supported by Grant R01GM078989-07 from the National Institutes of Health to JJT. The authors thank the two anonymous reviewers for their insightful and constructive comments, which helped us to improve the manuscript

## Supplementary material

## References

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