Agent-based modeling and bifurcation analysis reveal mechanisms of macrophage polarization and phenotype pattern distribution

Abstract

Macrophages play a key role in tissue regeneration by polarizing to different destinies and generating various phenotypes. Recognizing the underlying mechanisms is critical in designing therapeutic procedures targeting macrophage fate determination. Here, to investigate the macrophage polarization, a nonlinear mathematical model is proposed in which the effect of IL4, IFNγ and LPS, as external stimuli, on STAT1, STAT6, and NFκB is studied using bifurcation analysis. The existence of saddle-node bifurcations in these internal key regulators allows different combinations of steady state levels which are attributable to different fates. Therefore, we propose dynamic bifurcation as a crucial built-in mechanism of macrophage polarization. Next, in order to investigate the polarization of a population of macrophages, bifurcation analysis is employed aligned with agent-based approach and a two-layer model is proposed in which the information from single cells is exploited to model the behavior in tissue level. Also, in this model, a partial differential equation describes the diffusion of secreted cytokines in the medium. Finally, the model was validated against a set of experimental data. Taken together, we have here developed a cell and tissue level model of macrophage polarization behavior which can be used for designing therapeutic interventions.

Figure 1

The modeling and analyses work flow.

Figure 2

Molecular interaction graphs illustrating pathways that govern macrophage polarization. Two different types of arcs represent activator and inhibitory reactions. (a) Environmental signals are mediated through Toll-Like Receptor 4 (TLR4), IFNγ-R, ILR4α, and IL10-R and activate NFκB, STAT1, STAT6, and STAT3 pathways, respectively. The macrophage fate spectrum is color-coded with M1 (red) and M2 (blue) at opposite ends. (b) Macrophage molecular key cascades obtained by eliminating mediator proteins. (c) Final molecular interaction graph obtained by replacing two-cascaded double negative feedback loops between STAT1, STAT3 and NFκB by a double positive feedback loop between STAT1 and NFκB.

Figure 3

Bifurcation analyses of equations (7–15) for one parameter. Bifurcation diagrams of mediator proteins NFκB, STAT1 and STAT6 considering (a) LPS, (b) IFNγ, (c) IL4 as bifurcation parameter. Solid lines and dashed lines represent stable and unstable steady states, respectively. Arrow heads point to saddle-node bifurcation points and critical switching borders are specified by dotted lines and underlined values.

Figure 4

Bifurcation analyses of equations (7–15) for two parameters. Simultaneous exposure of macrophage to different concentrations of (a) LPS and IL4, (b) IL4 and IFNγ, (c) LPS and IFNγ results in (d) different steady state profiles for the mediator proteins “NFκB, STAT1, STAT6” which determine macrophage phenotype. Underlined values are critical borders obtained from one-parameter bifurcation analysis.

Figure 5

Bifurcation analyses of equations (7–15) for three parameters. Simultaneous exposure of macrophage to different concentrations of LPS, IL4 and IFNγ result in different steady state profiles for “NFκB, STAT1, STAT6” and accordingly into different phenotypes, compare Fig. 6 for phenotype reachability. The left, up and back panels which depict macrophage states in LPS-IL4, IL4-IFNγ and LPS-IFNγ are illustrated separately, compare Fig. 4.

Figure 6

Reachability for the nine specified phenotypes as determined by bifurcation analysis, compare Figs 3–5. Each arc is labeled with the name of the corresponding cytokine causing the fate change. The arc labeled with IL4* switches the macrophage phenotype only, if the IFNγ concentration is low. The arc labeled with IFNγ* switches the macrophage phenotype only, if the LPS concentration is low.

Figure 7

IL4 and IFNγ production in macrophage.This molecular interaction graph illustrates reactions involved when macrophage responds to the population by producing cytokines. The graph depicts reactions governing IL4 and IFNγ cytokine production by the NFκB, STAT1 and STAT6 pathways. IL10 and IL12 are mediator proteins relaying cytokine production. The graph is composed of two types of inhibitory and activator arcs and arc labels specify reaction rate parameters. Compare Fig. 2 which illustrates macrophage molecular pathways involved when macrophage is stimulated by cytokines from population.

Figure 8

Cyclic two-level model simulation of cytokine-based interactions of macrophage population.

Figure 9

Simulation results of the PDE according to equation (30). Macrophage population dominated by (a) M1, (b) M2 phenotypes are exposed to IL4 and IFNγ cytokine, respectively. As a result, each sub-population skews to the opposite phenotype.