Modeling the Role of TGF-β in Regulation of the Th17 Phenotype in the LPS-Driven Immune System

Abstract

Airway exposure levels of lipopolysaccharide (LPS) are known to determine type I versus type II helper T cell induced experimental asthma. While low doses of LPS derive Th2 inflammatory responses, high (and/or intermediate) LPS levels induce Th1- or Th17-dominant responses. The present paper develops a mathematical model of the phenotypic switches among three Th phenotypes (Th1, Th2, and Th17) in response to various LPS levels. In the present work, we simplify the complex network of the interactions between cells and regulatory molecules. The model describes the nonlinear cross-talks between the IL-4/Th2 activities and a key regulatory molecule, transforming growth factor β (TGF-β), in response to high, intermediate, and low levels of LPS. The model characterizes development of three phenotypes (Th1, Th2, and Th17) and predicts the onset of a new phenotype, Th17, under the tight control of TGF-β. Analysis of the model illustrates the mono-, bi-, and oneway-switches in the key regulatory parameter sets in the absence or presence of time delays. The model also predicts coexistence of those phenotypes and Th1- or Th2-dominant immune responses in a spatial domain under various biochemical and bio-mechanical conditions in the microenvironment.

Appendices

Appendix A: Parameter Estimation and Nondimensionalization

1.1 A.1 Parameter Estimation

Some of the parameters in our model are estimated in the following:

Diffusion coefficient of IL-4/Th2 module (d _H ): From Francis and Palsson (1997), Jansson et al. (2007), we take d _H =0.0036 mm² h⁻¹.

Diffusion coefficient of TGF-β (d _G ): From Brown (1999), Goodhill (1997), we take d _G =0.36 mm² h⁻¹.

LPS injection amount (α): In an experimental study, Kim et al. (2007a) found that a high (100 μg) and low (0.1 μg) dose of LPS induced Th1- and Th2-inflammatory responses, respectively. Therefore, there are 1000-fold differences in LPS amounts and in order to take this into account in our model, we take α=0.1–100 μg.

Decay/death rate of IL-4/ Th2 module (μ _H ): From Borish et al. (1999), Jansson et al. (2007), Murphy (2007), we take μ _H =0.48 h⁻¹.

Decay rate of TGF-β (μ _G ): From Kim and Friedman (2010), Kudlow et al. (1986), we take μ _G =0.4 h⁻¹.

1.2 A.2 Nondimensionalization

Table 1 lists reference values. We take the characteristic length scale L=5.0 mm. We determine the reference values for H, G as follows:

Table 1 Reference variables and estimated values used in the model

IL-4/Th2 module (H ^∗ ): For the reference value of IL-4/Th2 module, we take the concentration of IL-4 in the experiments in Kim et al. (2007a): H ^∗=20 pg cm⁻³.

TGF-β (G ^∗ ): In a study of the suppressive effect of Tregs with IL-2 and TGF-β on a Lupus-like syndrome, Zheng et al. (2004) used Nylon wool nonadherent T-enriched cells (1.5×10⁶ ml) and similar numbers of irradiated adherent, non-T cells with or without TGF-β1 ((0.1–10) ng/ml) and low dose rhuIL-2 (10 U/ml). In a study of conversion of peripheral CD4⁺CD25⁻ Naive T cells to CD4⁺CD25⁺ Treg cells using TGF-β (Chen et al. 2003), CD25⁺ or CD4⁺CD25⁻ cells were stimulated with platebound anti-CD3 (2 μg/ml) and soluble anti-CD28 (2 μg/ml) in the absence or presence of 0.02, 0.2, 2, or 20 ng/ml TGF-β1, 100 U/ml IL-2 or 1 ng/ml IL-10. We take G ^∗=11 ng/cm³.

We nondimensionalize the variables and parameters in the partial differential equations (6)–(7) as follows:

$$\begin{aligned} T =& \mu_G t,\qquad \widetilde{H} = \frac{H}{H^*},\qquad \widetilde{G} = \frac{G}{G^*},\qquad D_H = \frac{d_H}{\mu_G L^2},\qquad D_G = \frac{d_G}{\mu_G L^2}, \\ f(\alpha) =& \frac{r(\alpha)}{\mu_G H^*},\qquad \mu= \frac{\mu_H}{\mu_G}, \qquad\beta= \frac{k_1}{\mu_G H^*},\qquad K_H = \frac{k_2}{H^*}, \\ \gamma =& \frac{k_3 G^*}{(H^*)^n},\qquad \delta= \frac{\lambda_1}{\mu_G G^*},\qquad \zeta= \frac{\lambda_2 H^*}{\mu_G G^*}. \end{aligned}$$

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By dropping tilde and replacing T with t again in the notation, we get the governing equations in dimensionless form

$$\begin{aligned} \frac{\partial {H}}{\partial t} &= D_H \Delta H + f(\alpha) + \frac{\beta H^n}{K_H^n + H^n + \gamma G} - \mu H, \end{aligned}$$

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$$\begin{aligned} \frac{\partial {G}}{\partial t} &= D_G \Delta G + \delta + \frac{\zeta\alpha^m}{K_\alpha^m + \alpha^m} H - G. \end{aligned}$$

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Dimensionless parameters are listed in Table 2.

Table 2 Parameters used in the model

Appendix B: Sensitivity Analysis

In our model, there are a number of parameters for which no experimental data are known. These parameters may significantly affect the computational results and conclusions. In order to see how sensitive the different phenotypes are to all parameters (α, β, K _H , γ, μ, δ, ζ, K _α ) at t=100, we have performed sensitivity analysis. We have chosen a range for each of these parameters and divided each range into 2000 intervals of uniform length. For each of the eight parameters of interest, a partial rank correlation coefficient (PRCC) value with a range between −1 and 1 is calculated. The sign of PRCC determines whether an increase in the parameter value will increase (+) or decrease (−) the concentration of IL-4 (H) and TGF-β (G) at a given time. The sensitivity analysis was carried out using the method from Marino et al. (2008) and MATLAB files available from the website of Denise Kirschner’s Lab: http://malthus.micro.med.umich.edu/lab/usadata/.

Figure 21 shows the sensitivity analysis results for IL-4 (H) and TGF-β (G) concentrations at t=100. We calculate PRCC values and associated p-value (pv) for eight perturbed parameters (α, β, K _H , γ, μ, δ, ζ, K _α ) for both low (0.01<α<1) and high (10<α<100) LPS levels. Tables 3 and 4 summarize the results of the sensitivity analysis in terms of H and G at t=1, 10, 100 for the ODE model (12)–(13). That is, we compute PRCC values for H(t) at time t=1,10,100. The PRCC values for TGF-β (G(t)) were calculated in a similar fashion.

Fig. 21

Sensitivity analysis for local dynamics: general Latin hypercube sampling (LHS) scheme and a partial rank correlation coefficient (PRCC) performed for both low (0.01<α<1) and high (10<α<100) LPS levels on the ODE model (12)–(13). The reference output is the variable H(t), G(t) at time t=100. Eight parameters (α, β, K _H , γ, μ, δ, ζ, K _α ) were taken as the parameters of interest. PRCC values were calculated for a specific sampled values of the eight parameters while other parameters are fixed. The parameter β (α, μ) is shown to be strongly positively (negatively) correlated with the IL-4 level at time t=100 for low LPS level (0.01<α<1), that is, an increase in β (α, μ) results in a significant increase (decrease) in the IL-4 level at t=100. On the other hand, an increase/decrease in the other parameters K _H , K _α will not significantly change IL-4 and TGF-β levels at t=100 for both low and high LPS levels (α). The analysis was carried out using the method of Marino et al. (2008)

Table 3 Partial rank correlation coefficient (PRCC) of the system of ODEs (12)–(13) for dimensionless parameters (α, β, K _H , γ, μ, δ, ζ, K _α ) at t=1,10,100 when LPS levels (α) are low (0.01<α<1)

Table 4 Partial rank correlation coefficient (PRCC) of the system of ODEs (12)–(13) for dimensionless parameters (α, β, K _H , γ, μ, δ, ζ, K _α ) at t=1,10,100 when LPS levels (α) are high (10<α<100). Asterisk (^∗) represents the case with the p-value <0.01

Case 1. Low LPS levels (0.01<α<1) We conclude that the IL-4 level at t=100 is positively (negatively) correlated to the parameter β (α, μ) but is weakly correlated (and thus not sensitive) to K _H , γ, δ, ζ, K _α . On the other hand, TGF-β level is positively correlated to the parameter α, δ but is not sensitive to β, K _H , γ, μ, ζ, K _α .

Case 2. High LPS levels (10<α<100) We conclude that the IL-4 level at t=100 is positively (negatively) correlated to the parameter β (α, γ, μ, ζ) but is weakly correlated (and thus not sensitive) to K _H , δ, K _α . This implies that an increase in parameter β will increase H, leading to higher probability of inducing Th17 and Th2 phenotypes instead of Th1 phenotype (see Fig. 10(A)) while an increase in γ will increase a chance to get the Th1 phenotype instead of Th17 or Th2 phenotypes (see Fig. 10(B)). On the other hand, TGF-β level is positively (negatively) correlated to the parameter β (γ, μ) but is not sensitive to α, K _H , δ, ζ, K _α .