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

- 440 Downloads
- 3 Citations

## 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

- Antebi YE et al (2013) Mapping differentiation under mixed culture conditions reveals a tunable continuum of T cell fates. PLoS Biol 11:e1001616. doi: 10.1371/journal.pbio.1001616 CrossRefGoogle Scholar
- 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
- 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: 10.1002/cne.21298 CrossRefGoogle Scholar
- 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: 10.1038/nature06965 CrossRefGoogle Scholar
- Cinquin O, Demongeot J (2002) Positive and negative feedback: striking a balance between necessary antagonists. J Theor Biol 216:229–241. doi: 10.1006/jtbi.2002.2544 MathSciNetCrossRefGoogle Scholar
- Cinquin O, Demongeot J (2005) High-dimensional switches and the modelling of cellular differentiation. J Theor Biol 233:391–411. doi: 10.1016/j.jtbi.2004.10.027 CrossRefGoogle Scholar
- Clewley R (2012) Hybrid models and biological model reduction with PyDSTool. PLoS Comput Biol 8:e1002628. doi: 10.1371/journal.pcbi.1002628 CrossRefGoogle Scholar
- Crotty S (2011) Follicular helper CD4 T cells (TFH). Annu Rev Immunol 29:621–663. doi: 10.1146/annurev-immunol-031210-101400 CrossRefGoogle Scholar
- 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: 10.1371/journal.pbio.1001618 CrossRefGoogle Scholar
- 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: 10.1038/ni904 CrossRefGoogle Scholar
- Gerlach C et al (2013) Heterogeneous differentiation patterns of individual CD8+ T cells. Science 340:635–639. doi: 10.1126/science.1235487 CrossRefGoogle Scholar
- Ghoreschi K et al (2010) Generation of pathogenic T(H)17 cells in the absence of TGF-beta signalling. Nature 467:967–971. doi: 10.1038/nature09447 CrossRefGoogle Scholar
- Golubitsky M, Stewart I, Schaeffer DG (1988) Singularities and groups in bifurcation theory, vol. II. Applied Mathematical SciencesGoogle Scholar
- Guantes R, Poyatos JF (2008) Multistable decision switches for flexible control of epigenetic differentiation. PLoS Comput Biol 4:e1000235. doi: 10.1371/journal.pcbi.1000235 CrossRefGoogle Scholar
- 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: 10.1073/pnas.142284699 CrossRefGoogle Scholar
- 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: 10.1371/journal.pcbi.1002122 MathSciNetCrossRefGoogle Scholar
- 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: 10.1186/1752-0509-6-66 CrossRefGoogle Scholar
- 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 182:1579–1584CrossRefGoogle Scholar
- 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 260:547–549CrossRefGoogle Scholar
- Huang S (2013) Hybrid T-helper cells: stabilizing the moderate center in a polarized system. PLoS Biol 11:e1001632–e1001632. doi: 10.1371/journal.pbio.1001632 CrossRefGoogle Scholar
- Huang S, Guo YP, May G, Enver T (2007) Bifurcation dynamics in lineage-commitment in bipotent progenitor cells. Dev Biol 305:695–713. doi: 10.1016/j.ydbio.2007.02.036 CrossRefGoogle Scholar
- 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: 10.1126/science.1103336 CrossRefGoogle Scholar
- 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: 10.1016/j.cell.2006.07.035 CrossRefGoogle Scholar
- Kusam S, Toney LM, Sato H, Dent AL (2003) Inhibition of Th2 differentiation and GATA-3 expression by BCL-6. J Immunol 170:2435–2441CrossRefGoogle Scholar
- Luckheeram RV, Zhou R, Verma AD, Xia B (2012) CD4(+)T cells: differentiation and functions. Clin Dev Immunol 2012:925135. doi: 10.1155/2012/925135 CrossRefGoogle Scholar
- 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: 10.1371/journal.pcbi.1000303 MathSciNetCrossRefGoogle Scholar
- 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: 10.1038/ni.1965 MathSciNetCrossRefGoogle Scholar
- Mendoza L (2006) A network model for the control of the differentiation process in Th cells. Bio Syst 84:101–114. doi: 10.1016/j.biosystems.2005.10.004 Google Scholar
- Mendoza L (2013) A virtual culture of CD4+ T lymphocytes. Bull Math Biol. doi: 10.1007/s11538-013-9814-9
- 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: 10.1038/ni872 CrossRefGoogle Scholar
- Mjolsness E, Sharp DH, Reinitz J (1991) A connectionist model of development. J Theor Biol 152:429–453CrossRefGoogle Scholar
- 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 317:256–260CrossRefGoogle Scholar
- 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
- Murphy KM, Stockinger B (2010) Effector T cell plasticity: flexibility in the face of changing circumstances. Nat Immunol 11:674–680. doi: 10.1038/ni.1899 CrossRefGoogle Scholar
- 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: 10.1371/journal.pcbi.1000912 CrossRefGoogle Scholar
- O’Shea JJ, Paul WE (2010) Mechanisms underlying lineage commitment and plasticity of helper CD4+ T cells. Science 327:1098–1102. doi: 10.1126/science.1178334 CrossRefGoogle Scholar
- 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: 10.1186/1752-0509-7-53 CrossRefGoogle Scholar
- 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: 10.1371/journal.pbio.1001633 CrossRefGoogle Scholar
- Szabo SJ, Kim ST, Costa GL, Zhang X, Fathman CG, Glimcher LH (2000) A novel transcription factor, T-bet, directs Th1 lineage commitment. Cell 100:655–669CrossRefGoogle Scholar
- Tyson JJ, Novak B (2010) Functional motifs in biochemical reaction networks. Annu Rev Phys Chem 61:219–240. doi: 10.1146/annurev.physchem.012809.103457 CrossRefGoogle Scholar
- 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: 10.1084/jem.20052165 CrossRefGoogle Scholar
- 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: 10.1093/intimm/dxn093 CrossRefGoogle Scholar
- Wilson HR, Cowan JD (1972) Excitatory and inhibitory interactions in localized populations of model neurons. Biophys J 12:1–24. doi: 10.1016/S0006-3495(72)86068-5 CrossRefGoogle Scholar
- 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 96:1024–1029CrossRefGoogle Scholar
- 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: 10.1016/j.jtbi.2004.06.013 MathSciNetCrossRefGoogle Scholar
- Zheng W, Flavell RA (1997) The transcription factor GATA-3 is necessary and sufficient for Th2 cytokine gene expression in CD4 T cells. Cell 89:587–596CrossRefGoogle Scholar
- Zhou L et al (2008) TGF-beta-induced Foxp3 inhibits T(H)17 cell differentiation by antagonizing RORgammat function. Nature 453:236–240. doi: 10.1038/nature06878 CrossRefGoogle Scholar
- Zhu J, Paul WE (2010) Peripheral CD4(+) T-cell differentiation regulated by networks of cytokines and transcription factors. Immunol Rev 238:247–262. doi: 10.1111/j.1600-065X.2010.00951.x CrossRefGoogle Scholar
- Zhu J, Yamane H, Paul WE (2010) Differentiation of effector CD4 T cell populations. Annu Rev Immunol 28:445–489. doi: 10.1146/annurev-immunol-030409-101212 CrossRefGoogle Scholar