Fig 1.
Monotone-symmetric and nonmonotone-asymmetric toggle designs.
S design (top panel): Activation of the expression of gene x (lacI) occurs by binding of autoinducer G (C14-HSL) to promoter PG (Pcin). Inhibition of the expression of both genes x (lacI) and u (cinI) occurs by binding of the gene product Y (TetR) of gene y (tetR) to a single promoter PY (Ptet). Symmetrically, activation of the expression of gene y (tetR) occurs by binding of autoinducer R (C4-HSL) to promoter PR (Prhl), while inhibition of the transcription of both genes y (tetR) and w (rhlI) occurs by binding of X (LacI) to a single promoter PX (Plac). A design (bottom panel): Activation of the expression of gene x (lacI) occurs by binding of autoinducer R (C4-HSL) to promoter PR (Prhl). Expression of genes y (tetR) and w (rhlI) is driven by a common single promoter PX. Gene products U and W are synthases CinI and Rhil, respectively. Gray horizontal strips correspond to integration plasmids. Genes gfp and rfp correspond to green and red florescent proteins, GFP and RFP, respectively.
Table 1.
A toggle molecular part catalog (explanations of variables are given in S model).
Fig 2.
Homogeneous and heterogeneous (mixed) populations.
An example of a population consisting of 10 cells is shown. The left panel demonstrates a homogeneous G-population. The center panel demonstrates a heterogeneous (1:1)-population, where the homogeneous G- and R-subpopulations have equal number of cells. The right panel demonstrates a heterogeneous (9:1)-population formed of two unequal subpopulations which represent a spontaneous synchronization error, when one or a few toggles spontaneously flip from green (G) to red (R) states.
Fig 3.
Application of Monotone Systems Theory to the S design.
The top panel presents a monotonicity diagram for a single-cell S design, while the bottom panel represents an example of three identical S toggles interacting via common autoinducers, see the main text for details. In all cases, solid arrows and lines highlighted in red color correspond to monotone parameter dependencies. In the bifurcation analysis, the values of all monotone parameters are varied for all cells simultaneously.
Fig 4.
Examples of monotone parametric dependencies.
Panels (A) and (B) correspond to the unsaturated S design, while Panels (C) and (D) correspond to the saturated S design. Panels (A) and (B). The following color coding schema is used: (i) black plots are used for G-homogeneous solutions at d = 0.1; (ii) red plots are used for G-homogeneous solutions at d = 10; (iii) blue plots are used for (1:1)-mixed states at d = 0.1; and (iv) green plots are used for (9:1)-mixed states at d = 0.1. Red filled circles in panel (B), labeled with LP1 and LP2, correspond to Limit Point (LP) (or, equivalently, Saddle-Node) bifurcation points. Here, the blue curve connecting the origin (0, 0) and the LP1-point corresponds to the stable branch of the (1:1)-mixed state. The green curve connecting the origin (0, 0) with the LP2-point corresponds to the stable branch of the (9:1)-mixed state. Because the green curve was plotted after plotting the blue curve, a part of the blue curve is hidden beneath the green curve. Projections of the corresponding plots on 2D-planes often overlap, mixing different colors, which should not lead to any difficulty in recognizing similar monotone (“overlapping”) dependencies. Panels (C) and (D). The following color coding schema is used: (i) red plots correspond to stable homogeneous G-states, (ii) violet plots correspond to stable (1:1)-mixed states, and (iii) green plots correspond to stable (9:1)-mixed states. In all cases, blue plots correspond to unstable states. All red filled circles correspond to the LP bifurcations. In the panel (D), the unstable branches for both (1:1) and (9:1)-mixed states are not shown because they overlap with the stable ones.
Fig 5.
Examples of monotone parametric dependencies for a (1:1)-mixed state.
Panels (A) and (B) correspond to dependencies of LacI and TetR levels on parameter δg, respectively. Panels (C) and (D) correspond to dependencies of LacI and TetR levels on parameter δr, respectively. The dependencies for the G-subpopulation are shown only, within which LacI is activated, while TetR is repressed. Green and red solid curves correspond to stable branches of (1:1)-equilibrium solutions, while all blue curves correspond to unstable branches of the solutions. Red filled circles correspond to an LP-bifurcation point. In panel (A), projections of stable and unstable branches coincide and, so, only the stable branch is shown.
Fig 6.
Bistability regions for S and A toggles (top), and a reduced SR toggle design discovered from the bistability region (bottom).
(Top panel). The region between two blue color coded LP-bifurcation loci corresponds to a bistability region for the A toggle model Eqs (9) and (10) at d = 0. A red filled circle corresponds to a cusp point (CP). For the S toggle model, bistability exists for all parameter values a1 ≥ 0 and a2 ≥ 0 at d = 0. Other fixed parameter values are given in Eqs (11)–(14). (Bottom panel). The reduced SR toggle is obtained from the original S toggle (Fig 1) after removal of genes lacI and tetR from the corresponding plasmids bearing promoters PY and PX, respectively. This reduction procedure corresponds to setting zero values a1 = a2 = 0 as discussed in the main text.
Fig 7.
Symmetry breaking in a (1:1)-mixed population of S toggles.
Panels (A) and (B) show the dependencies of LacI and TetR levels for the G-subpopulation of a (1:1)-population of S toggles, respectively. Blue color-coded plots correspond to all unstable equilibrium solution branches, while green and red color-coded plots correspond to all stable equilibrium solution branches. All blue filled BP-labeled points correspond to d ≈ 1.43. All red filled LP-labeled points correspond to d ≈ 2.07.
Fig 8.
An interpretation of symmetry breaking in a (1:1)-mixed population of S toggles.
A new (1:1)-asymmetric mixed state arises, in which there are two subpopulations, one in which green gene-expression dominates (but with different expression levels of LacI in each of them), and another one which red gene-expression dominates (also with different TetR levels).
Fig 9.
(9:1)-mixed population of S toggles.
Panels (A) and (B) correspond to large and small subpopulations of a (9:1)-population of S toggles. All notations and color-coding schemes are as in Fig 7. Red filled circles correspond to the same LP-bifurcation point. In panel (A), projections of stable and unstable solution branches overlap. Because TetR is totally suppressed in the large (90%) subpopulation, the levels of TetR are not shown. Contrarily to panel (A), both TetR and LacI levels are plotted in panel (B) since LacI is only moderately suppressed in the small (10%) R-subpopulation.
Fig 10.
Self-correction of spontaneous errors by S toggles.
Panel (A) shows the S toggle which cannot self-correct a (9:1)-spontaneous synchronization error for a small value of the diffusion parameter d = 0.01 (a weak coupling between all cells). Panels (B) and (C) show the S toggle which can self-correct a (9:1)-spontaneous synchronization error for a medium (d = 10) and large (d = 100) values of parameter d (a medium and strong coupling between all cells, respectively). For the values of parameter d used in Panels (B) and (C), the mixed states become unstable, see Fig 9.