TY - JOUR
T1 - Oscillations and bistability in a model of ERK regulation
AU - Obatake, Nida
AU - Shiu, Anne
AU - Tang, Xiaoxian
AU - Torres, Angélica
PY - 2019/9/1
Y1 - 2019/9/1
N2 - This work concerns the question of how two important dynamical properties, oscillations and bistability, emerge in an important biological signaling network. Specifically, we consider a model for dual-site phosphorylation and dephosphorylation of extracellular signal-regulated kinase (ERK). We prove that oscillations persist even as the model is greatly simplified (reactions are made irreversible and intermediates are removed). Bistability, however, is much less robust—this property is lost when intermediates are removed or even when all reactions are made irreversible. Moreover, bistability is characterized by the presence of two reversible, catalytic reactions: as other reactions are made irreversible, bistability persists as long as one or both of the specified reactions is preserved. Finally, we investigate the maximum number of steady states, aided by a network’s “mixed volume” (a concept from convex geometry). Taken together, our results shed light on the question of how oscillations and bistability emerge from a limiting network of the ERK network—namely, the fully processive dual-site network—which is known to be globally stable and therefore lack both oscillations and bistability. Our proofs are enabled by a Hopf bifurcation criterion due to Yang, analyses of Newton polytopes arising from Hurwitz determinants, and recent characterizations of multistationarity for networks having a steady-state parametrization.
AB - This work concerns the question of how two important dynamical properties, oscillations and bistability, emerge in an important biological signaling network. Specifically, we consider a model for dual-site phosphorylation and dephosphorylation of extracellular signal-regulated kinase (ERK). We prove that oscillations persist even as the model is greatly simplified (reactions are made irreversible and intermediates are removed). Bistability, however, is much less robust—this property is lost when intermediates are removed or even when all reactions are made irreversible. Moreover, bistability is characterized by the presence of two reversible, catalytic reactions: as other reactions are made irreversible, bistability persists as long as one or both of the specified reactions is preserved. Finally, we investigate the maximum number of steady states, aided by a network’s “mixed volume” (a concept from convex geometry). Taken together, our results shed light on the question of how oscillations and bistability emerge from a limiting network of the ERK network—namely, the fully processive dual-site network—which is known to be globally stable and therefore lack both oscillations and bistability. Our proofs are enabled by a Hopf bifurcation criterion due to Yang, analyses of Newton polytopes arising from Hurwitz determinants, and recent characterizations of multistationarity for networks having a steady-state parametrization.
KW - Bistable
KW - Chemical reaction network
KW - Hopf bifurcation
KW - Mixed volume
KW - Newton polytope
KW - Oscillation
UR - http://www.scopus.com/inward/record.url?scp=85069644400&partnerID=8YFLogxK
U2 - 10.1007/s00285-019-01402-y
DO - 10.1007/s00285-019-01402-y
M3 - Journal article
C2 - 31346693
AN - SCOPUS:85069644400
SN - 0303-6812
VL - 79
SP - 1515
EP - 1549
JO - Journal of Mathematical Biology
JF - Journal of Mathematical Biology
IS - 4
ER -