Bulletin of Mathematical Biology

, Volume 77, Issue 6, pp 1046–1064

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

Original Article

## 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

11538_2015_76_MOESM1_ESM.docx (131 kb)
Supplementary material 1 (docx 130 KB)

## References

1. Antebi YE et al (2013) Mapping differentiation under mixed culture conditions reveals a tunable continuum of T cell fates. PLoS Biol 11:e1001616. doi:
2. Ball J, Schaeffer D (1983) Bifurcation and stability of homogeneous equilibrium configurations of an elastic body under dead-load tractions. In: Mathematical proceedings of the Cambridge philosophical society. Cambridge Univ Press, Cambridge, pp 315–339Google Scholar
3. Bell ML, Earl JB, Britt SG (2007) Two types of Drosophila R7 photoreceptor cells are arranged randomly: a model for stochastic cell-fate determination. J Comp Neurol 502:75–85. doi:
4. Chang HH, Hemberg M, Barahona M, Ingber DE, Huang S (2008) Transcriptome-wide noise controls lineage choice in mammalian progenitor cells. Nature 453:544–547. doi:
5. Cinquin O, Demongeot J (2002) Positive and negative feedback: striking a balance between necessary antagonists. J Theor Biol 216:229–241. doi:
6. Cinquin O, Demongeot J (2005) High-dimensional switches and the modelling of cellular differentiation. J Theor Biol 233:391–411. doi:
7. Clewley R (2012) Hybrid models and biological model reduction with PyDSTool. PLoS Comput Biol 8:e1002628. doi:
8. Crotty S (2011) Follicular helper CD4 T cells (TFH). Annu Rev Immunol 29:621–663. doi:
9. Fang M, Xie H, Dougan SK, Ploegh H, van Oudenaarden A (2013) Stochastic cytokine expression induces mixed T helper cell states. PLoS Biol 11:e1001618–e1001618. doi:
10. Fontenot JD, Gavin MA, Rudensky AY (2003) Foxp3 programs the development and function of CD4+CD25+ regulatory T cells. Nat Immunol 4:330–336. doi:
11. Gerlach C et al (2013) Heterogeneous differentiation patterns of individual CD8+ T cells. Science 340:635–639. doi:
12. Ghoreschi K et al (2010) Generation of pathogenic T(H)17 cells in the absence of TGF-beta signalling. Nature 467:967–971. doi:
13. Golubitsky M, Stewart I, Schaeffer DG (1988) Singularities and groups in bifurcation theory, vol. II. Applied Mathematical SciencesGoogle Scholar
14. Guantes R, Poyatos JF (2008) Multistable decision switches for flexible control of epigenetic differentiation. PLoS Comput Biol 4:e1000235. doi:
15. Hofer T, Nathansen H, Lohning M, Radbruch A, Heinrich R (2002) GATA-3 transcriptional imprinting in Th2 lymphocytes: a mathematical model. Proc Natl Acad Sci USA 99:9364–9368. doi:
16. Hong T, Xing J, Li L, Tyson JJ (2011) A mathematical model for the reciprocal differentiation of T helper 17 cells and induced regulatory T cells. PLoS Comput Biol 7:e1002122. doi:
17. Hong T, Xing J, Li L, Tyson JJ (2012) A simple theoretical framework for understanding heterogeneous differentiation of CD4+ T cells. BMC Syst Biol 6:66. doi:
18. Hosken NA, Shibuya K, Heath AW, Murphy KM, O’Garra A (1995) The effect of antigen dose on CD4+ T helper cell phenotype development in a T cell receptor-alpha beta-transgenic model. J Exp Med 182:1579–1584
19. Hsieh CS, Macatonia SE, Tripp CS, Wolf SF, O’Garra A, Murphy KM (1993) Development of TH1 CD4+ T cells through IL-12 produced by Listeria-induced macrophages. Science 260:547–549
20. Huang S (2013) Hybrid T-helper cells: stabilizing the moderate center in a polarized system. PLoS Biol 11:e1001632–e1001632. doi:
21. Huang S, Guo YP, May G, Enver T (2007) Bifurcation dynamics in lineage-commitment in bipotent progenitor cells. Dev Biol 305:695–713. doi:
22. Hwang ES, Szabo SJ, Schwartzberg PL, Glimcher LH (2005) T helper cell fate specified by kinase-mediated interaction of T-bet with GATA-3. Science 307:430–433. doi:
23. Ivanov II et al (2006) The orphan nuclear receptor RORgammat directs the differentiation program of proinflammatory IL-17+ T helper cells. Cell 126:1121–1133. doi:
24. Kusam S, Toney LM, Sato H, Dent AL (2003) Inhibition of Th2 differentiation and GATA-3 expression by BCL-6. J Immunol 170:2435–2441
25. Luckheeram RV, Zhou R, Verma AD, Xia B (2012) CD4(+)T cells: differentiation and functions. Clin Dev Immunol 2012:925135. doi:
26. Manu SS et al (2009) Canalization of gene expression and domain shifts in the Drosophila blastoderm by dynamical attractors. PLoS Comput Biol 5:e1000303. doi:
27. Maruyama T et al (2011) Control of the differentiation of regulatory T cells and T(H)17 cells by the DNA-binding inhibitor Id3. Nat Immunol 12:86–95. doi:
28. Mendoza L (2006) A network model for the control of the differentiation process in Th cells. Bio Syst 84:101–114. doi: Google Scholar
29. Mendoza L (2013) A virtual culture of CD4+ T lymphocytes. Bull Math Biol. doi:
30. Messi M, Giacchetto I, Nagata K, Lanzavecchia A, Natoli G, Sallusto F (2003) Memory and flexibility of cytokine gene expression as separable properties of human T(H)1 and T(H)2 lymphocytes. Nat Immunol 4:78–86. doi:
31. Mjolsness E, Sharp DH, Reinitz J (1991) A connectionist model of development. J Theor Biol 152:429–453
32. Mucida D, Park Y, Kim G, Turovskaya O, Scott I, Kronenberg M, Cheroutre H (2007) Reciprocal TH17 and regulatory T cell differentiation mediated by retinoic acid. Science 317:256–260
33. Murphy E, Shibuya K, Hosken N (1996) Reversibility of T helper 1 and 2 populations is lost after long-term stimulation. J Exp Med 183:901–913Google Scholar
34. Murphy KM, Stockinger B (2010) Effector T cell plasticity: flexibility in the face of changing circumstances. Nat Immunol 11:674–680. doi:
35. Naldi A, Carneiro J, Chaouiya C, Thieffry D (2010) Diversity and plasticity of Th cell types predicted from regulatory network modelling. PLoS Comput Biol 6:e1000912. doi:
36. O’Shea JJ, Paul WE (2010) Mechanisms underlying lineage commitment and plasticity of helper CD4+ T cells. Science 327:1098–1102. doi:
37. Oguz C, Laomettachit T, Chen KC, Watson LT, Baumann WT, Tyson JJ (2013) Optimization and model reduction in the high dimensional parameter space of a budding yeast cell cycle model. BMC Syst Biol 7:53. doi:
38. Peine M et al (2013) Stable T-bet(+)GATA-3(+) Th1/Th2 hybrid cells arise in vivo, can develop directly from naive precursors, and limit immunopathologic inflammation. PLoS Biol 11:e1001633–e1001633. doi:
39. Szabo SJ, Kim ST, Costa GL, Zhang X, Fathman CG, Glimcher LH (2000) A novel transcription factor, T-bet, directs Th1 lineage commitment. Cell 100:655–669
40. Tyson JJ, Novak B (2010) Functional motifs in biochemical reaction networks. Annu Rev Phys Chem 61:219–240. doi:
41. Usui T, Preiss JC, Kanno Y, Yao ZJ, Bream JH, O’Shea JJ, Strober W (2006) T-bet regulates Th1 responses through essential effects on GATA-3 function rather than on IFNG gene acetylation and transcription. J Exp Med 203:755–766. doi:
42. van den Ham HJ, de Boer RJ (2008) From the two-dimensional Th1 and Th2 phenotypes to high-dimensional models for gene regulation. Int Immunol 20:1269–1277. doi:
43. Wilson HR, Cowan JD (1972) Excitatory and inhibitory interactions in localized populations of model neurons. Biophys J 12:1–24. doi:
44. Yamashita M, Kimura M, Kubo M, Shimizu C, Tada T, Perlmutter RM, Nakayama T (1999) T cell antigen receptor-mediated activation of the Ras/mitogen-activated protein kinase pathway controls interleukin 4 receptor function and type-2 helper T cell differentiation. Proc Natl Acad Sci USA 96:1024–1029
45. Yates A, Callard R, Stark J (2004) Combining cytokine signalling with T-bet and GATA-3 regulation in Th1 and Th2 differentiation: a model for cellular decision-making. J Theor Biol 231:181–196. doi:
46. Zheng W, Flavell RA (1997) The transcription factor GATA-3 is necessary and sufficient for Th2 cytokine gene expression in CD4 T cells. Cell 89:587–596
47. Zhou L et al (2008) TGF-beta-induced Foxp3 inhibits T(H)17 cell differentiation by antagonizing RORgammat function. Nature 453:236–240. doi:
48. Zhu J, Paul WE (2010) Peripheral CD4(+) T-cell differentiation regulated by networks of cytokines and transcription factors. Immunol Rev 238:247–262. doi:
49. Zhu J, Yamane H, Paul WE (2010) Differentiation of effector CD4 T cell populations. Annu Rev Immunol 28:445–489. doi: