Figure 2. Increasing T-cell avidity (decreasing ) induces bistability.

(A) The phase plane of the scaled version of system (2a)–(2b) (see Supplementary Material S1), displaying the - and -nullclines for (i.e. for a high level of T-cell avidity). The three vertical gray lines are the -nullclines, while the Hill-like black line is the -nullcline. Stable steady states and are shown as black dots, while the unstable steady state is shown as a white dot. The dashed line traces the time-dependent level of T and plasma cells obtained from the scaled version of full one-clone model (1a)–(1f) superimposed on this phase plane (the arrow-heads represent the direction of flow). (B) The time evolution of (solid) and (dashed), approaching the autoimmune (black) and healthy (gray) states, depend on the initial level of T cells (). The bifurcation diagrams of (C) , and (D) with respect to are shown displaying the stable steady states and in solid lines and the unstable steady state in a dashed line. As demonstrated in Supplementary Material S1, bistability is only observed for , while at , and merge together at a saddle node bifurcation point, leaving the healthy state to become a global attractor for (i.e. for a low level of T-cell avidity).