Fig. 2
Hysteresis curves as evoked by parameters butyrate in the lumen or LPS in the lumen. a Hysteresis driven by lumen butyrate, as observed for the core model. Abscissa: butyrate in the gut lumen kBo, ordinate: butyrate in the epithelial cell B. The hysteresis curve is a plot of all fixed points, i.e. no further change in concentrations: \(\dot {N}=\dot {R}=\dot {T}=\dot {L}=\dot {B}=0\), of Eqs. 5 to 9, for each value of kBo. Parameters: ka=12,ks=1.0,kai=5,kr=0.5,kbn=4.7,kb=5,kbr=0.5,kn=0.2,kp=7,kt=0.2,ktl=2.8,kbl=1.02,klb=3.4,kl=0.7,kbt=2.1,kd=0.125,kB=2.9. The limit points on the hysteresis of the core model have normal form coefficients a left/right: a=−0.1079160/−0.01114689. b Hysteresis driven by lumen LPS kbl, as observed for the core model. Abscissa: LPS in the gut lumen kbl, ordinate: butyrate in the epithelial cell B. Parameters as for (a), but with fixed kBo=130. c Schematic view of the transformation of the butyrate distribution in the lumen over the hysteresis to the bimodal distribution of within-cell butyrate, as depicted on the y-axis. d Hysteresis observed for the extended model. This model includes a positive feedback loop between butyrate in the epithelial ⇔ mucin production ⇔ SCFA producing microbiota. The area of bi-stability is broadened for this model