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 Mitogen-activated protein kinase (MAPK) pathway regulations, which is the phophorylation and dephosphorylation of extracellular signal-regulated kinase (ERK). Firstly, distributive ordered MAPK models (BIOMD0000000026, BIOMD0000000027) were build in order to detect how bistable behavior could arise from two-site phosphorylation and dephosphorylation regardless of the specific monophosphorylation site (Fig. 1).

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 phospho-threonine or phospho-tyrosine 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.

Notably, the author also compared different models based on either Michaelis-Menten kinetics (BIOMD0000000027, BIOMD0000000029) or reaction rates of elementary steps (BIOMD0000000028, BIOMD0000000030). The saddle-node 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 macro-level (e.g. Michaelis-Menten kinectics) and the micro-level descriptions (e.g. reaction rate for elementary steps) should not be neglected [5]. Michaelis-Menten kinetics assumes a well-stirred 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 enzyme-substrate conplex may always be assumed to be constant (quasi-steady 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.