Table 1.
Parameter values.
Fig 1.
Schematic overview of the model.
This diagram depicts the proteins considered in the model and regulatory influences among these proteins, as well as model inputs and outputs. Two inputs are considered: the level of rapamycin in complex with FKBP1A and therefore competent to inhibit MTORC1 (i.e., the level of rapamycin*) and the level of AMPK with activating phosphorylation in its kinase domain (i.e., the level of AMPK*). We note that AMPK activity is limited by the level of AMPK* and also the level of inhibitory phosphorylation of AMPK by ULK1. Thus, AMPK* does not necessarily correspond to the level of active AMPK. AMPK negatively regulates MTORC1 and positively regulates ULK1. A dashed arrow is used to represent the negative influence of AMPK on MTORC1 because this influence is not considered until the “Sensitivity analysis” section. Key regulatory influences considered in the model are mutual inhibition of MTORC1 and ULK1 and (slow) negative feedback from ULK1 to AMPK. Two outputs are considered: the level of phosphorylation of EIF4EBP1, a repressor of translation, and the level of phosphorylation of AMBRA1, which is involved in activating autophagy. We consider phosphorylation of EIF4EBP1 to be indicative of a translation state and phosphorylation of AMBRA1 to be indicative of an autophagy state.
Fig 2.
Illustration of site-specific details captured in the model.
Nested boxes represent functional components of proteins. Black lines attached to small square boxes at one end and text labels at the other end indicate sites of serine/threonine phosphorylation. Blue lines that end with a circle indicate substrates of kinases. Black lines that begin and end with arrowheads represent direct binding interactions. Red lines with flat arrowheads indicate inhibitory effects of either serine/threonine phosphorylation reactions or direct binding interactions. Numbers next to arrows refer to sets of rules in the model (see Materials and Methods). The numbering of amino acid residues is consistent with UniProt entries for human proteins; the site of activating phosphorylation in AMPK is T172 (T183) in the PRKAA2 (PRKAA1) isoform of the AMPK α subunit. The following abbreviations are used for names of protein components: RNC, RAPTOR N-terminal conserved domain; WD40, solenoid protein domain consisting of WD40 repeats; HEAT, solenoid protein domain consisting of HEAT repeats; FRB, FKBP12-rapamycin binding domain; and RCR, RAPTOR crosslinking region. In the model, we consider several undefined regions within proteins; these regions are represented by names having all lowercase letters, which refer to binding partners. For further information, see Materials and Methods.
Fig 3.
Results from bifurcation analysis of the system without negative feedback from ULK1 to AMPK.
Each panel is a one-dimensional bifurcation diagram showing stable steady-state levels of phosphorylated AMBRA1 (red curves, top panels) or phosphorylated EIF4EBP1 (blue curves, bottom panels) as a function of the level of AMPK* (left panels) or the level of rapamycin* (right panels). Thus, we took the inputs of the model as our bifurcation parameters. For the left panels, the abundance of rapamycin* is fixed at zero. For the right panels, the abundance of AMPK* is fixed at 30,000 copies per cell. For all panels, the parameters considered in Table 1 are held fixed at their nominal values. The labels SN1, SN2, and SN3 indicate saddle node bifurcation points. Bifurcation analysis was performed numerically (see Materials and Methods).
Fig 4.
Time-dependent levels of AMBRA1 and EIF4EBP1 phosphorylation after various stimuli.
In all panels, red curves indicate the fraction of total AMBRA1 phosphorylated by ULK1 and blue curves indicate the fraction of total EIF4EBP1 phosphorylated by MTORC1. In each panel, we simulate a cellular response to stress; the simulation begins at an “unstressed” steady state where the level of rapamycin* is zero and the level of AMPK* is 30,000 copies per cell. The stresses introduced at time t = 0 are as follows: (A) no stimulus (i.e., rapamycin* and AMPK* remain at their starting levels), (B) an increase in AMPK* level to 150,000 copies per cell, (C) an increase in rapamycin* level to 9,000 copies per cell, (D) an increase in AMPK* level to 90,000 copies per cell, (E) an increase in rapamycin* level to 6,000 copies per cell, and (F) increases in the rapamycin* level to 3,000 copies per cell and AMPK* level to 60,000 copies per cell.
Fig 5.
Results from bifurcation analysis of the system with negative feedback from ULK1 to AMPK.
The solid blue curves indicate stable steady-state levels of AMBRA1 phosphorylation for (A) different levels of AMPK* and (B) different levels of rapamycin*. The dotted curves indicate lower and upper bounds of stable limit cycles. The red curves indicate periods of oscillation (see right vertical axes). In the left panel, the level of rapamycin* is held fixed at zero. In the right panel, the level of AMPK* is held fixed at 30,000 copies per cell. For both panels, the parameters considered in Table 1 are held fixed at their nominal values. The labels SNIC1 and SNIC2 indicate saddle-node-on-invariant-circle bifurcation points, and the label SuperH indicates a supercritical Hopf bifurcation point. The labels SubH and CF refer to subcritical Hopf and cyclic fold bifurcation points, which are very close to each other (panel A).
Fig 6.
Two-dimensional bifurcation diagrams.
(A) The qualitative behavior of the system without negative feedback from ULK1 to AMPK as a function of AMPK* and rapamycin* levels. The shaded region indicates where the system exhibits bistability. (B) The qualitative behavior of the system with negative feedback from ULK1 to AMPK as a function of AMPK* and rapamycin* levels. The shaded region indicates where the system exhibits oscillations. Labeled points correspond to the stress inputs considered in Fig. 4 (e.g., the point labeled F corresponds to the case of Fig. 4F). For both panels, parameter values considered in Table 1 are held fixed at their nominal values.
Fig 7.
Influence of negative regulation of MTORC1 by AMPK on qualitative system behavior.
The rate constant for inhibitory phosphorylation of RPTOR at S792 by AMPK is p9. The panels shown here are two-dimensional bifurcation diagrams. In the left panels, p9 and AMPK* level are the bifurcation parameters. In the right panels, p9 and rapamycin* level are the bifurcation parameters. In the top panels, we consider the system without negative feedback from ULK1 to AMPK. The shaded regions in these panels indicate where the system exhibits bistability. In the bottom panels, we consider the system with negative feedback from ULK1 to AMPK. The shaded regions in these panels indicate where the system exhibits oscillations. Note that the horizontal axes are logarithmic (base 10).
Fig 8.
Parameter sensitivity analysis.
Each bar corresponds to a rate constant in the model and indicates a range of values for that rate constant over which the following pattern of qualitative behavior is obtained as a stress input (AMPK* or rapamycin* level) is varied: operation in a translation state at low stresses, oscillations between translation and autophagy states at intermediate stresses, and operation in an autophagy state at high stresses. For red bars, the stress input is AMPK* level; we considered levels of AMPK* from 0 to 106 copies per cell. For blue bars, the stress input is rapamycin* level; we considered levels of rapamycin* from 0 to 105 copies per cell. To find the upper and lower bounds of a bar, we varied (in discrete steps) the value of its corresponding parameter individually100-fold above and 100-fold below the parameter’s nominal value, which corresponds to 1 on the vertical axis. For each parameter value tested, scans of AMPK* and rapamycin* levels were performed to determine whether responses to varying levels of stress follow the same pattern as the system with nominal parameter values. The height of a bar serves as a measure of robustness. We considered all rate constants of the model with the exception of p9. (Recall that the influence of p9 on system behavior has already been considered, in Fig. 7.)