Mean-field versus stochastic models for transcriptional regulation

We introduce a minimal model description for the dynamics of transcriptional regulatory networks. It is studied within a mean-field approximation, i.e., by deterministic ODE’s representing the reaction kinetics, and by stochastic simulations employing the Gillespie algorithm. We elucidate the different results that both approaches can deliver, depending on the network under study, and in particular depending on the level of detail retained in the respective description. Two examples are addressed in detail: The repressilator, a transcriptional clock based on a three-gene network realized experimentally in E. coli, and a bistable two-gene circuit under external driving, a transcriptional network motif recently proposed to play a role in cellular development.

Figure 1

(Color online) The four basic types of gene gates: (1) The null gate (a gate without control input); (2) the neg gate (repression of transcription); (3) the pos gate, activation of expression; (4) the posneg gate, a multi-input gate with one activating and one repressing input.

Figure 2

(Color online) The two main classes of simple circuits: Circular (1) and linear (2). Shown are only the repressive circuits; activatory circuits and mixtures of both types can be built in a similar fashion. Circuits shown in (1): The autoinhibitive circuit, a bistable switch, the repressilator. Circuits shown in (2): A linear array and a linear array with a head feedback, hence a mixture of a circular and a linear circuit.

Figure 3

(Color online) Switching in the stochastic bistable circuit without cooperativity. Simulation parameters are $r=1$, $\epsilon =0.4$, $\eta =0.2$, $\gamma =5\times {10}^{-3}$. The inset indicates output on the $\pi$-calculus channels $\text{!}a$, $\text{!}b$, equivalent to protein numbers.

Figure 4

Top: The repressilator dynamics without gene gates (fixed at the nullclines of the gates) for the parameters $r=1$, $\gamma =0.1$, $\epsilon =0.3$, $\eta =0.9$, $h=3$. The limit cycle is absent, the fixed-point is stable. Bottom: Plot of the repressilator dynamics for the full system with identical parameters. The limit cycle persists in a wider range of parameters.

Figure 5

Parameter regimes for the repressilator dynamics. I, stable fixed point; II, stable fixed point for the reduced system, limit cycle for the full system; III, limit cycle. Parameters are the same as in Fig. 4.

Figure 6

(Color online) Top: The limit cycle of the stochastic repressilator. Simulation parameters are $r=r_p=1$, $\epsilon =0.1$, $\eta ={10}^{-2}$, $\gamma ={10}^{-3}$. Bottom: The deterministic version for comparison (reduced system in region III of Fig. 5, parameters identical to the stochastic version, with $h=3$).

Figure 7

(Color online) Top: The rewired repressilator, a positive loop is added (see arrow), so that one of the genes is doubly regulated. Bottom: The limit cycle of the (reduced) rewired repressilator circuit; the additional activation interaction breaks the symmetry, as discernable in the difference in maximal concentrations. Simulation parameters are $r=1$, $r_p={10}^{-4}$, $\epsilon =0.1$, ${\eta }_1={\eta }_2={10}^{-2}$, $\gamma ={10}^{-3}$, $h=3$.

Figure 8

(Color online) Top: The bistable switch under external control by $a$. Bottom: The system starts at zero concentrations of both proteins and enters the state with higher stability, as given by higher production rates. The signal $c$ dominates widely over $a,b$ which are indistinguishable from the baseline.

Figure 9

(Color online) Top: A change of the activator transcription rate by one order of magnitude makes both proteins compete; note the concentration overshoots of the previously stable protein. Bottom: A further increase of the activator transcription rate makes the system switch between the two states. Simulation parameters are (bottom): $r=1$, ${\epsilon }_a={10}^{-3}$, ${\epsilon }_b=0.1$, ${\epsilon }_c={10}^{-3}$, ${\eta }_{ab}=2\times {10}^{-3}$, ${\eta }_{ac}=2\times {10}^{-2}$, $\eta =2\times {10}^{-1}$, ${\gamma }_a=5\times {10}^{-3}$, $\gamma =2\times {10}^{-4}$.