A MATHEMATICAL MODEL FOR THE REGULATION OF T HELPER 1, 2 AND 17 CELLS IN ASTHMA

Allergic asthma is characterized by airway inflammation, which is orchestrated by T cell responses. The “Th2 hypothesis” was first suggested by Mosmann, and it states that type 2 helper T (Th2) cells are considered to play a major role in initiating and driving the airway inflammatory responses ([1,6,7]). Recent studies suggests that noneosinophilic asthma can be also developed by Type 1 helper T (Th1) cell and Type 17 helper T (Th17) cell responses when exposed to lipopolysaccharide (LPS)-containing allergens ([3–5]). High LPS levels derive mixed Th1 and Th17 cell responses with noneosinophilic (or neutrophilic) inflammation, while low LPS levels derive Th2 cell induced asthma ([2,5]). In this study we present a simple mathematical model for asthma development. The model focuses on the balance of Th1, Th2, and Th17, and transforming growth factor beta (TGF-β) which is important in regulation of the immune system and the differentiation of Th17 cells. MATHEMATICAL MODEL The main variables areH , the cell density of Th2-modelues, andG, the concentration of TGF-β, and α represent the amount of LPS injection. The local dynamics of the model is given by dH dt = 1 1 + α + βH 1 + γH2 + δG − μH, (1) dG dt = Λ1 + Λ2H −G. (2) We also consider the case with the delay, dH dt = 1 1 + α + βH 1 + γH2 + δG(t− τ1) − μH, (3) dG dt = Λ1 + Λ2H(t− τ2) −G. (4) RESULTS At the steady states (Hs, Gs), dH dt = dG dt = 0 yields 1 1 + α = μHs − βH s (1 + δΛ1) + δΛ2Hs + γH2 s } {{ } =:f(Hs) . (5) 0 1 2 3 4 5 6 0 0.2 0.4 0.6 0.8 1 α = 100 α = 10 α = 4.6 α = 2.2 α = 1 α = 0.46 α = 0.22 α = 0.1 H f( H ) β = 6.0 β = 8.0 β = 8.5 β = 10.0 10 −2 10 0 10 2 0 1 2 3 4 5 6 α H * (s te a d y s ta te o f H ) β = 6.0 β = 8.0 β = 8.5 β = 10.0 Figure 1. The effect of β, the strength of self-activation. (Left) f(Hs) (RHS of the equation (5)). Horizontal lines are 1/(1 + α) for each α, therefore, points of intersection represent the steady states. (Right) Steady states. There are some qualitatively different behaviors among several values of β’s. For small β (for example, β = 6 in Figure 1), there is no bistability at all and there is only one stable steady state for all α. For intermediate values of β, two saddle-node bifurcation exists and the bifurcation diagram is S-shaped (β = 8 in Figure 1). As α increases (decreases), jump from the upper (lower, respectively) branch to another branch may occur. For large values of β (β = 8.5 or 10 in Figure 1), there is only one bifurcation point and it is impossible to jump from the upper branch to the lower branch. In the delay differential equation, on the other hand, jump from the upper branch to the lower branch may occur when proper delay exists in the system (bottom right figure in Figure 2). REFERENCES 1. Grunig G., Warnock M., Wakil A. E., Venkayya R., Brombacher F., Rennick D. M., Sheppard D., Mohrs M., Donaldson D. D., Locksley R. M., Corry D. B., “Requirement for IL-13 independently of IL-4 in experimental asthma”, Science, Vol. 282(5397), 1998, pp. 2261–2263. 2. Kim Y. K., Oh S. Y., Jeon S. G., Park H. W., Lee S. Y., Chun E. Y., Bang B., Lee H. S., Oh M. H., Kim Y. S., Kim J. H., Gho Y. S., Cho S. H., Min K. U., Kim Y. Y., Zhu Z., “Airway exposure levels of lipopolysaccharide determine type 1 versus type 2 experimental asthma”, J Immunol, Vol. 178(8), 2007, pp. 5375–5382. 3. Kim Y. S., Hong S. W., Choi J. P., Shin T. S., Moon H. G., Choi E. J., Jeon S. G., Oh S. Y., Gho Y. S., Zhu Z., Kim Y. K., “Vascular endothelial growth factor is a key mediator in the development of T cell priming and its polarization to type 1 and type 17 T helper cells in the airways”, J Immunol, Vol. 183(8), 2009, pp. 5113–5120. 0 50