
Markevich et al. (2004), MAPK
June 2007, model of the month by Lu Li
Original models: BIOMD0000000026 and BIOMD0000000031
A bistable system is a system which can switch between two distinct states without resting in intermediate states. Bistable behavior usually combines with hysteresis, meaning the system lacks reversibility when the stimulus varies [1]. At least one positive feedback loop has been proposed to be necessary to cause bistability in a system (conjecture de Thomas [2]).This conjecture has been proven to be a theorem recently [3]. However, in this present paper, Markevich et al. showed that the distributive kinetic mechanism of a dual phosphorylation and dephosphorylation cycle has the potential to trigger bistability [4]. In fact, the required positive feedback is raised at a system level as a consequence of enzyme saturation and competitive inhibition.
Markevich et al. reached these conclusions by modelling one level of Mitogenactivated protein kinase (MAPK) pathway regulations, which is the phophorylation and dephosphorylation of extracellular signalregulated kinase (ERK). Firstly, distributive ordered MAPK models (BIOMD0000000026, BIOMD0000000027) were build in order to detect how bistable behavior could arise from twosite phosphorylation and dephosphorylation regardless of the specific monophosphorylation site (Fig. 1).


Figure 1: The distributed ordered dual phosphorylationdephosphorylation cycle for MAPK, where M, Mp, and Mpp stand for the unphosphorylated, monophosphorylated, and bisphosphorylated forms of MAPK. From [4]



Figure 2: The distributive random dual phosphorylationdephosphorylation cycle for MAPK, where MpY and MpT stand for monophosphorylated form of MAPK on tyrosine and threonine. From [4]


If M (unphosphorylated ERK) is at the steady state, the rate of its phosphorylation should be the same as its dephosphorylation (v1=v4). Assuming both the kinase and phosphatase are saturated by their corresponding substrates, which are M and Mpp respectively, the increase of Mpp inhibits the dephosphorylation of Mp because of the competitive occupation of the phosphatase. Thus, the rise in Mpp may cause the decrease of M or the increase of Mp, which are required to reduce the phosphorylation rate of M in order to maintain its steady state. As a consequence, the decreased M or the increased Mp stimulates the phosphorylation rate of Mp (v2). To summarize, this increased Mpp indirectly accelerates its production, thus producing apparent product activation, which triggers bistability.
Within each MAP pathway level, however, the dual phosphorylation and dephosphorylation are not ordered. For example, ERK could be firstly phosphorylated on either threonine residue or tyrosine residue; when it is dephosphorylated, either phosphothreonine or phosphotyrosine could be the intermediate state (Fig. 2). Thus, the author further developed the distributive random MAPK models (BIOMD0000000028, BIOMD0000000029, BIOMD0000000030). Although the system becomes more complicated, it shares the same principle to exhibit bistability, which is caused by the emerging positive feedback from Mpp under the steady state condition of M and the steady state of one monophosphorylated form [4].

Furthermore, the author extended the model to a system where the successive phosphorylations are catalysed by different kinases, but dephosrphorylated by the same phosaphatase (BIOMD0000000031). Interestingly, bistability and hysteresis becomes an inherent property, depending on the relative abundance of kinases (Fig. 3). The author concluded that this switch like behavior, controlled by multiple kinases and the same phosphatase, may have a broad regulation effect on protein activation.


Figure 3: The bistability in a cycle where MAPK (M) is phosphorylated by two different kinases (MAPKK1 and MAPKK2) and dephosphorylated by the same phosphatase. (A) The stationary states of the bisphosphorylated MAPK on [MAPKK1] at different [MAPKK2]; (B) The stationary states of the bisphosphoryalted MAPK on [MAPKK2] at different [MAPKK1]. From [4]



Figure 4: The bifurcation diagram regarding Km1 Km2, based on different modeling approaches. (1) domain 1 is calculated for MechaelisMenten kinetics. (2) Domain 2 is calculated for the elementary step model. (3) Domain 3 corresponds to unsaturated phosphatase. From [4]


Notably, the author also compared different models based on either MichaelisMenten kinetics (BIOMD0000000027, BIOMD0000000029) or reaction rates of elementary steps (BIOMD0000000028, BIOMD0000000030). The saddlenode bifurcation analysis regarding Km1 and Km2 showed distinct bistability domains (Fig. 4) based on these two modelling approaches. This showed that, under certain circumstances, especially when the system contains similarly distributed enzymes and substrates, the differences between the macrolevel (e.g. MichaelisMenten kinectics) and the microlevel descriptions (e.g. reaction rate for elementary steps) should not be neglected [5]. MichaelisMenten kinetics assumes a wellstirred compartment where the substrate concentration greatly exceeds the enzyme concentration. Thus, the enzyme may be characterized as an independent catalyst, and the concentration of the enzymesubstrate conplex may always be assumed to be constant (quasisteady state approximation)[6]. Taking this present model as an example, however, the total concentration of ERK (M) is conserved, and the concentrations of intermediate complex either between ERK and MEK or ERK and MKP should not be eliminated from the total ERK concentration balance. Based on this, the elemental step description is always a more precise way to gain an accurate and quantitative understanding of enzyme control.

Bibliographic References

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