## 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.

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## 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

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## Appendix: Methods

### Appendix: Methods

### Dynamical Models

We build mathematical models of three different signaling motifs (two generic motifs and one motif specific to T cell differentiation). For each case, we use a generic form of ordinary differential equations (ODEs) suitable for describing both gene expression and protein interaction networks (Mjolsness et al. 1991; Tyson and Novak 2010; Wilson and Cowan 1972). Each ODE in the model has the form:

Here, \(X_{i}\) is the activity or concentration of protein \(i\). On a time \(\hbox {scale }= 1/\gamma _{i}, X_{i}(t)\) relaxes toward a value determined by the sigmoidal function, \(F\), which has a steepness set by \(\sigma _{i}\). The basal value of \(F\), in the absence of any influencing factors, is determined by \(\omega _i^o \). The coefficients \(\omega _{j\rightarrow i}\) determine the influence of protein \(j\) on protein \(i\). \(N\) is the total number of proteins in the network.

All variables and parameters are dimensionless. One time unit in the simulations corresponds to approximately 1 day. Basal parameter values of each model are listed in supplementary tables (see ‘Cell-to-Cell Variability’ subsection for details).

All simulations and bifurcation analyses were performed with PyDSTool, a software environment for dynamical systems (Clewley 2012).

### Bifurcation Diagrams and Steady State Radar Plots

One parameter bifurcation diagrams were plotted by following the steady state solution of the ODEs with change in the value of a control parameter.

In order to analyze the behavior of multi-variable systems, we use radar plots to illustrate the steady states for a particular parameter set. A radar plot depicts the expression level of each key state variable (i.e., master regulator) on one sub-plot, and multiple sub-plots describe multiple steady states. In principle, a radar plot can illustrate unstable steady states as well as stable steady states, but we plot only stable steady states, which correspond to observable cell phenotypes.

### Cell-to-Cell Variability

To account for cell-to-cell variability in a population, we made many simulations of the system of ODEs, each time with a slightly different choice of parameter values (to represent slight differences from cell to cell). We assumed that the value of each parameter conforms to a normal distribution with CV = 0.05 (CV = coefficient of variation = standard deviation/mean). We refer to the mean value for each parameter distribution as the ‘basal’ value of that parameter. In the bifurcation analysis of the dynamical system, we consider an imaginary cell that adopts the basal value for each of its parameters, and we define this cell as the ‘average’ cell. However, none of the cells in the simulated population is likely to be this average cell, because every parameter value is likely to deviate a little from the basal value.

### Simulation Procedure

In order to simulate the induced differentiation process, we first solved the ODEs numerically with small initial values of master regulator concentrations in the absence of any exogenous signals. After a short period of time, each simulated cell found its own, stable ‘naïve’ steady state in which all master regulators are expressed at low level. Next, we changed the exogenous signals to the values listed in Supplementary Tables S1, S2 and S3 and continued the numerical simulation. Each cell arrived at its corresponding ‘induced’ phenotype, which might vary from cell to cell because of the parametric variability of the population. The expression level of each protein in the network ranges from 0 to 1 unit, and we made the simple assumption that a protein is ‘expressed’ if its level is greater than 0.5 units. We defined the derived population as ‘heterogeneous’ if it contained cells with more than one phenotype.

### Parameter Optimization

Before starting to optimize the model parameters, we defined a hyperbox in the parameter space that is bounded by biologically plausible parameter ranges. These ranges are listed in Supplementary Table S3. A population of 200 parameter vectors, generated by Latin hypercube sampling (LHS), captured from 3 to 14 of the 22 experimental constraints that are listed in Table 1 (there are 22 independent constraints in 14 different experimental conditions). Starting with this population, we next implemented Differential Evolution (DE). The two-stage optimization approach based on LHS and DE has been presented previously by Oguz et al. (2013). Details of LHS and DE are provided in the Supplementary Text. For the initial round of DE, we used an aggressive mutation operator (\(F= 0.1\) in Eq. (2) of Supplementary Text) and a non-greedy selection condition. After 900 generations of DE, we obtained several parameter vectors that captured 19 of the 21 experimental constraints. We also identified that Constraint 1 (shown in Table 1) was the experimental constraint with the lowest acceptance (0.03 %) among the parameter vectors generated by DE (\(200\times 900=180{,}000\) vectors). In the second round of DE, starting with the 57 parameter vectors that captured Constraint 1, we used a more conservative mutation operator (\(F= 0.01\)) and a non-greedy selection condition in order to maximize the number of total constraints captured. In addition, we enforced Constraint 1 at every step; a mutant vector could only replace a parent if it captured Constraint 1. After \(\sim \)500 generations, we found several feasible parameter vectors that captured 18–22 of the 22 experimental constraints. The optimized parameter values from the most robust feasible vector are given in Supplementary Table S3. The robustness measure that we used in the robustness analysis is described in Sect. 3 of the Supplementary Text (last paragraph).

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Hong, T., Oguz, C. & Tyson, J.J. A Mathematical Framework for Understanding Four-Dimensional Heterogeneous Differentiation of \(\hbox {CD4}^{+}\) T Cells.
*Bull Math Biol* **77, **1046–1064 (2015). https://doi.org/10.1007/s11538-015-0076-6

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### Keywords

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