Stability estimation of autoregulated genes under Michaelis-Menten-type kinetics

Feedback loops are typical motifs appearing in gene regulatory networks. In some well-studied model organisms, including Escherichia coli, autoregulated genes, i.e., genes that activate or repress themselves through their protein products, are the only feedback interactions. For these types of interactions, the Michaelis-Menten (MM) formulation is a suitable and widely used approach, which always leads to stable steady-state solutions representative of homeostatic regulation. However, in many other biological phenomena, such as cell differentiation, cancer progression, and catastrophes in ecosystems, one might expect to observe bistable switchlike dynamics in the case of strong positive autoregulation. To capture this complex behavior we use the generalized family of MM kinetic models. We give a full analysis regarding the stability of autoregulated genes. We show that the autoregulation mechanism has the capability to exhibit diverse cellular dynamics including hysteresis, a typical characteristic of bistable systems, as well as irreversible transitions between bistable states. We also introduce a statistical framework to estimate the kinetics parameters and probability of different stability regimes given observational data. Empirical data for the autoregulated gene SCO3217 in the SOS system in Streptomyces coelicolor are analyzed. The coupling of a statistical framework and the mathematical model can give further insight into understanding the evolutionary mechanisms toward different cell fates in various systems.

Figure 1

Illustration of the transcription-translation cycle of an autoregulated gene.

Figure 2

Illustration of Result 1, the stability of a system with Michaelis-Menten formulation. The horizontal dotted lines represent the equilibrium point of the system. The solutions of the system converge to the equilibrium point for different initial conditions.

Figure 3

Illustration of stability scenarios and hysteresis. (a) Bifurcation diagram of the mean gene expression ($\overline{x}$) as the dissociation parameter varies. It describes how the autoregulated gene drops in expression level as the TF dissociates from the promoter region. (b) The same bifurcation diagram as in (a) for both state variables. (c) ODE solution which is convergent to an equilibrium ($\gamma =2.5$ is in the stable region). (d) The ODE solution can converge to two distinct equilibria (based on the choice of initial values) since the system is bistable ($\gamma =7.5$ is in the bistability region).

Figure 4

(a) Solutions of the steady-state equations in the examples of Secs. 2b and 2c. The MM model shows a positive root whereas the two GMM shows one and three roots, respectively. (b) Bistability at a single cell level induces bimodality at the population level. In this figure we show the density plot of the abundance of a protein in a population of cells following the kinetics parameters detailed in Fig. 3.

Figure 5

Illustration of an irreversible transition in which the autoregulated gene shifts from a low expression to a high one. The system jumps up as the basal transcription rate $\phi$ passes the tipping point $F$, but it cannot jump down once $\phi$ decreases since the other tipping point lies in the negative range of parameter space. It describes how an autoregulated gene may be subject to an irreversible genetic switch which might, for instance, explain cell differentiation.

Figure 6

Illustration of critical transitions undergone by an autoregulated gene. The upper surface [right-hand side of (11)] distinguishes transitions between stable and bistable modes of the gene while the lower surface [right-hand side of (10)] distinguishes transitions between bistable and irreversible modes.

Figure 7

Simulated data and results of the four scenarios described in Table 2. (a)–(d) show the data generated from stable, bistable, and bifurcation scenarios. (e) corresponds to the estimated distributions of $\widehat{\mathrm{\Delta }}$ in the simulation study. Vertical dotted white lines and black circles represent the true value of $\mathrm{\Delta }$ for the different scenarios.

Figure 8

(a) Observed data for the autoregulated gene INO4 in S. serevisiae and the obtained ODE solution after estimating the parameters of the system. The value $\widehat{\mathrm{\Delta }}<0$ suggests that the system is stable. (b) Observed data for the autoregulated gene SCO3217 in Streptomyces coelicolor and obtained ODE solution after estimating the parameters of system (4). The value $\widehat{\mathrm{\Delta }}>0$ suggests that the system is bistable, which may provoke bimodal effects at a population level.