Oscillations by Minimal Bacterial Suicide Circuits Reveal Hidden Facets of Host-Circuit Physiology
(A) Solid lines indicate positive and negative regulation. Dashed lines represent the effect of cell growth on component dilution. a. Increasing cells density causes RNA I degradation (possibly through uncharged tRNAs. See Figure 4 for more details). b. RNA I inhibits plasmid replication (through its interaction with RNA II). c. RNA I is produced from the ePop plasmid; elevated plasmid levels increase RNA I production. d. E protein is produced from ePop plasmid by basal expression from the luxI promoter in the absence of functional LuxR. Elevated plasmid levels increase E protein production. e. E protein decreases cell density by blocking cell-wall synthesis and lysing cells. (B) Dimensionless ODE model of the circuit. Changes in cell density (n) are modeled as logistic growth with an intrinsic growth rate, α. We assume that killing of cells by the E protein is cooperative and describe it using a Hill-type function (Hill coefficient, p). We note that cooperatively of E protein-mediated killing is not required for generating oscillations. The E protein is produced from a plasmid (y) with a rate β1 and degraded with a rate γ1; both processes follow first-order kinetics with regards to the amount of plasmid and E protein, respectively. Plasmid replication is inhibited by RNA I (s), and replication inhibition follows a power of hyperbolic function where r is the effective number of reaction steps in the inhibitory scheme . β2 sets the maximum plasmid replication rate and γ3 the intrinsic decay rate. RNA I is produced from the plasmid with a rate β3 whereas its degradation rate is dependent on the cell density. Degradation of RNA I is described by a Hill-type function (Hill coefficient, v) to account for possible cooperativity. E protein, plasmid and RNA II are subject to dilution with cell growth. (C) The base parameter set that can generate sustained oscillations. Rate coefficients are normalized to a maximum killing rate (i.e. the maximum cell killing rate by E protein is 1). Biologically relevant parameter values have been chosen to illustrate the basic dynamics. E protein production rate β1 is set to be small to reflect leaky expression. Plasmid decay rate γ3 is set small to reflect the stability of plasmid molecules, and under oscillatory conditions plasmid dilution dominates. (D) Bifurcation diagram showing a region of sustained oscillations over varying ‘half-maximal constant for RNAI cleavage’ (δ1). Insets show simulated time courses of cell density for three δ1 values. Damped oscillations can be generated outside the bifurcation region.