Fig 1.
(A) Regulatory mechanism of the lac operon. Expression of permease increases the intracellular concentration of the inducer TMG (thiomethyl β-D-galactoside), which removes the repressor LacI from the promoter, leading to increased expression of permease. Hence the repressor LacI and permease LacY form a positive feedback loop. (B) Cartoon showing the dynamics of operon states. (C) Diagram of the Markovian jumping process of operon states. The O state denotes the free operon; the O*R state denotes the operon bound to the repressor at the auxiliary lac operator O2 or O3(partial dissociation); the OR state denotes the repressor bound to the operon at both the major and auxiliary lac operators; the O*RIm state denotes the repressor bound to the operon at the auxiliary lac operator O2 or O3 and to the inducer.
Fig 2.
Bistability with and without stochastic operon-state switching.
(A) Deterministic bifurcation diagram for wild-type cells. There are two saddle-node bifurcations occurring around Ie = 10μM and 59μM. (B) Deterministic bifurcation diagram containing both the active transcriptional rate kM and the extracellular inducer concentration Ie. The wild-type cells exhibit deterministic bistability inside the parameter region between the blue and brown lines and exhibit monostability otherwise. (C) Deterministic bifurcation diagram of the mutant cells without positive feedback. (D)(E) Deterministic bifurcation diagrams with different association constants for the repressor bound to the operon in the absence of a DNA loop: 5 molec.−1 (D), 8 molec.−1 (E). (F) Stochastic hysteresis response of the probability of induction for wild-type cells. Initial conditions: uninduced (blue line) or fully induced (red line) cells with a period of T = 2000 min. The extracellular inducer concentration must exceed over 350μM to completely activate initially uninduced cells, whereas it must decrease below 10μM to completely deactivate the initially induced cells. See S1 Text for parameter values.
Fig 3.
Positive feedback stabilizes the induced state.
(A-C) Extremely slow operon-state switching is necessary to induce purely stochastic bistability without positive feedback. (D-F) In the presence of positive feedback, the induced state is stabilized, and a bimodal distribution emerges, even when operon-state switching rates are within the physiological region.
Fig 4.
Major transit pathways and transition rates between fully repressed and fully dissociated operon states.
(A, B) The major transit pathways between fully repressed and fully dissociated operon states in the uninduced and induced phenotypic states. (C-F) Transition rates between fully repressed and fully dissociated operon states in the uninduced and induced phenotypic states with very low and high intracellular inducer concentrations respectively. The transition rates from the fully repressed operon state to the fully dissociated state in the uninduced phenotypic state are the lowest, which stabilizes the uninduced state, even outside of the parameter range of deterministic bistability.
Fig 5.
Probability of induction by a single large burst and quasi-steady state.
(A)Two typical single-cell time traces of permease levels. The first shows induction by a single full dissociation event of the repressor from the operon (left), while the second shows a failure to induce (right). (B) The large burst size in the presence of positive feedback is remarkably prolonged compared with the case without positive feedback. (C) Successful probability of induction by a complete dissociation event as a function of the extracellular inducer concentration. (D-G) Probability of induction within different time windows starting from uninduced cells (blue) or induced cells (red); we determined the stochastic threshold through mathematical fitting in the form of for these curves. The deterministic threshold is approximately 20(molec.), while the stochastic thresholds are larger and decrease when the time window is extended. The extracellular inducer concentration, Ie, is set to 40μM.
Fig 6.
Transition rates between phenotypic states and the phenomenon of resonance.
(A) Phenotypic landscape ϕ(x) in the region of deterministic bistability. (B) Phenotypic landscape ϕ(x) outside the region of deterministic bistability. (C) The rate formula (2) is valid for the parameter region of deterministic bistability with the fitted positive barrier V12 = 0.0550. (D) When the switching rates among different gene states are sufficiently rapid, the phenotype transition from the uninduced state to the induced state must occur through the accumulation of many complete dissociation events, rather than through a single dissociation event in wild-type cells, within the parameter region of deterministic bistability. (E) The transition rate increases and is finally saturated when the operon-state switching rate increases in the region of purely stochastic bistability. (F) The mean phenotype transition time varies with the operon-state switching rates at Ie = 25μM.