Dynamics and control of the ERK signaling pathway: Sensitivity, bistability, and oscillations

Abstract

Cell signaling is the process by which extracellular information is transmitted into the cell to perform useful biological functions. The ERK (extracellular-signal-regulated kinase) signaling controls several cellular processes such as cell growth, proliferation, differentiation and apoptosis. The ERK signaling pathway considered in this work starts with an extracellular stimulus and ends with activated (double phosphorylated) ERK which gets translocated into the nucleus. We model and analyze this complex pathway by decomposing it into three functional subsystems. The first subsystem spans the initial part of the pathway from the extracellular growth factor to the formation of the SOS complex, ShC-Grb2-SOS. The second subsystem includes the activation of Ras which is mediated by the SOS complex. This is followed by the MAPK subsystem (or the Raf-MEK-ERK pathway) which produces the double phosphorylated ERK upon being activated by Ras. Although separate models exist in the literature at the subsystems level, a comprehensive model for the complete system including the important regulatory feedback loops is missing. Our dynamic model combines the existing subsystem models and studies their steady-state and dynamic interactions under feedback. We establish conditions under which bistability and oscillations exist for this important pathway. In particular, we show how the negative and positive feedback loops affect the dynamic characteristics that determine the cellular outcome.

Fig 1. The ERK signaling pathway.

Different subsystems and feedback loops are hierarchically organized to process the extracellular signal introduced by Growth Factor (GF). Red arrows indicate internal positive feedback loops. Green arrows represent the external feedback loops. Blue dashed lines and black dashed lines indicate the subsystems and the nucleus, respectively.

Fig 2. The closed-loop system.

(A) Controller (reaction network), signaling pathway, measured output (green species) and manipulated input (yellow species). (B) The external feedback controller interacting with the pathway.

Fig 3. Steady-state response curves without the external feedback loops.

GF (growth factor) is the bifurcation parameter that is being changed. Red and blue curves are the stable and unstable branches, respectively. LP: Limit Point bifurcation also called the turning point at which the switch between the low and high stable branches occurs. (A) Since bistability curve is so flat for the SOS_complex, the unstable blue branch is squeezed between the two red stable branches and is not visible. (B) RasGTP response. The switch between the low and high stable branches occurs at the turning points, and it is shown by the arrows. (C) Ultrasensitive MEK^PP response. (D) Bistability is sustained in ERK^PP response.

Fig 4. Steady-state response of ERK^PP to MEK^PP.

Fig 5. Dynamic simulations showing the transition between the low and high states.

(A) Input GF pulse which increases from 0.5 to 2 and returns back to 0.5 nM. (B) Different responses to the input pulse.

Fig 6. Sensitivity of RasGTP and MEK^PP bistability.

(A) kcat5 is the rate constant for hydrolysis. (B) Response of the of MAPK pathway when it is disconnected from the rest of the pathway. Sensitivity with respect to k3 which is the rate constant for phoshorylation of MEK. (C) Sensitivity with respect to phosphatase, P’ase 1. (D) Bistability is lost for high levels of phosphatase.

Fig 7. The effect of negative feedback loop FBL4.

(a) k_nfb = 0.03, Km_nfb = 1e4. (b) k_nfb = 0.1, Km_nfb = 1e4. (c) k_nfb = 0.03, Km_nfb = 1e3. (d) k_nfb = 0.1, Km_nfb = 1e3. HB shows the Hopf Bifurcation. In all plots kcat3 = 1.75.

Fig 8. Oscillatory responses.

(A) Damped RasGTP oscillations caused by combination of negative and positive feedback loops. GF stimulus is increased from 0.1 to 2.4 (nM). As kcat3 decreases, strength of positive feedback decreases and negative feedback results in more oscillations. (B) Damped ERK oscillations caused by combination of negative and positive feedback loops. (C) Hopf bifurcation and periodic oscillations for ERK^PP and (D) for RasGTP. Negative feedback when combined with positive feedback give rise to limit cycles. Green points form the locus of the limit cycles. Four values for GF are considered for dynamic simulations (see Fig 10). k_nfb = 0.01 and kcat3 = 1.75.

Fig 9. Limit cycles.

(A) The limit cycles in phase-plane. (B) The response starting from a lower value of RasGTP converges to the limit cycle indicated by the spiral trajectory. (C) The response starting from a high RasGTP value converges to the stable non-oscillatory high state.

Fig 10. Dynamics of periodic oscillations.

(A) ERK^PP oscillations. (B) RasGTP oscillations.

Fig 11. The impact of the autocrine positive feedback loop EFBL1.

(A) ERK response to GF for different gpT values. (B) RasGTP response to GF for different gpT values. (C) ERK response to GF for different grT values. The feedback strength is represented by gpT. The linear gain in ligand production is denoted by grT. Less GF ligand is required to switch on ERK and Ras, and more ligands need to be removed to switch off the active Ras and ERK. For gpT = 10 or grT = 10, LP appears at negative GF, suggesting that once ERK and Ras are active, they cannot be switched off. knfb = 0.01, kcat3 = 1.75.

Fig 12. The impact of the negative feedback loop EFBL2.

(A) ERK^PP response to GF. Argos introduces two new HB points. knfb = 0.0, kcat3 = 1.75 (B) Oscillations emanate from the HB points and their amplitudes depend on the GF. grA = 1.

Fig 13. Generation of a desired ERK response profile by the external feedback loops.

(A) ERK^PP response. (B) GF pulse. (C) External feedback loops EFBL1 and EBFL2. k_EFBL = 1 indicates that the loop is on and k_EFBL = 0 indicates it is off. kcat3 = 1.75, gpT = 10, knfb = 0.0. grA = 1.