Figure 1.
Gene circuit design for group-level transitions.
(A–B) Schematics of two types of group transition in response to changes in cell density. (A) A graded transition is seen when the fraction of cells in the ON state (red) gradually increases with cell density. (B) An all-or-none transition appears when the state change occurs simultaneously across the population. (C–E) Schematics of an autoinducing gene circuit model (C; Eq. 1), dual-positive feedback regulations (D; Eq. 3), and positive-and-negative feedback regulations (E; Eq. 4) in operation.
Figure 2.
Graded and all-or-none transitions in a simple autoinduction circuit.
(A–B) In the population of simple autoinduction circuits (Fig. 1C; Eq. 1), the time average of the synthetase concentration xi within individual cells at the steady state (upper panel) and the fraction of ON cells (xi>0.5; lower panel) are plotted as a function of normalized cell density ρV; ε = 2.5 (A) and 0.25 (B); λ = 100 (A–B). Data points in the lower panel were calculated from multiple randomized sets of threshold ki with identical standard deviations. An analytically derived steady state for the population mean (dashed line; Eq. S2-1 in Text S1) indicates group-level bistability for
= 2∼5 (B). (C–D) Change in the distribution of xi for various cell densities: a bimodal distribution at high ε (
; C) and a unimodal distribution at low ε (D). ε = 2.5 (C) and 0.25 (D). λ = 100 (C–D).
Table 1.
Examples of the simple autoinduction (Eq. 1) and the dual positive-feedback (Eq. 3) circuits in quorum sensing systems.
Table 2.
Examples of the positive-and-negative-feedback circuit (Eq. 4) in dynamical quorum sensing systems.
Figure 3.
Conditions for group-level transitions in quorum sensing.
(A) A phase diagram showing the bistable regions in parameter space (ε, λ) of the simple autoinduction circuit (Eq, 1). The black lines indicate phase boundaries between GB (Group-level Bistability), AB (Autonomous-Bistability), and M (Mono-stability) (Eqs. S2-14 and S2-6 in Text S1). CO (Constitutively ON) means the cells are always in the ON state; i.e., xi∼1, regardless of cell density in the case of M (white region). Log10 CVρ/CVk<0 (blue and cyan region) and log10 CVρ/CVk>0 (red and pink region) indicate regions where cell-cell heterogeneity in the threshold value is reduced or not reduced, respectively. The yellow line indicates log10 CVρ/CVk = 0, determined numerically with spline interpolation. The × and + correspond to Figs. 2A and 2B, respectively. (B) A phase diagram for the simple autoinduction circuit (Eq. 1; Fig. 1C) and the dual positive-feedback circuit (Eq. 3; Fig. 1D). The m and n are cooperativity coefficients (Eq. 3). As in (A), solid and dashed lines denote the phase boundaries between PB/CB and CB/M, respectively (Eqs. S2-15 and S2-23 in Text S1). λ = 20 (grey) and 100 (black). The bottom row shows the parameter ε for the bacterial QS operons, las, rhl, lux, and car estimated from the literature (Table S6). For the las, rhl, and lux systems, estimates for two independent data sets are plotted.
Figure 4.
Graded and all-or-none transitions to collective oscillations.
(A–B) Representative time courses of oscillatory transitions within populations of positive-and-negative feedback circuits (Fig. 1E; Eq. 4) for high ε (ε = 16; A,C) and low ε (ε = 4; B,D). λ = 103, g = 30, and |ηi| = 0. Upper panels: time course of the population mean (red line) and the synthetase concentration of a representative cell (blue line) during an exponential increase in cell density ρ (grey line). Lower panels: pulsatile responses of individual cells. Y-axis indicates the cell index. (C–D) Percentage of oscillatory cells plotted as a function of cell density.
Figure 5.
Conditions for collective decision making in dynamical quorum sensing.
(A–B) CVρ/CVk (see Models) in the absence (A, |ηi| = 0) or the presence (B, |ηi| = 0.1) of noise in a positive-and-negative feedback circuit (Eq. 4). NT: No Transition, i.e., constitutive quiescence or constitutive oscillations irrespective of cell density (white region). The × and + correspond to Figs. 4A and 4B, respectively. (C) Log10 CVρ/CVk = 0 in the absence (green curve) and the presence (yellow curve) of noise. The black solid line indicates the boundary ε∼2 (Eq. S2-27 in Text S1). The region surrounded by the black dotted curves marks the parameter region where the cells are autonomously oscillatory in an isolated condition (ρ = 0). The values of the parameters are the same as those in Fig. 4.