Multi-stability in cellular differentiation enabled by a network of three mutually repressing master regulators

Abstract

Identifying the design principles of complex regulatory networks driving cellular decision-making remains essential to decode embryonic development as well as enhance cellular reprogramming. A well-studied network motif involved in cellular decision-making is a toggle switch-a set of two opposing transcription factors A and B, each of which is a master regulator of a specific cell fate and can inhibit the activity of the other. A toggle switch can lead to two possible states-(high A, low B) and (low A, high B)-and drives the 'either-or' choice between these two cell fates for a common progenitor cell. However, the principles of coupled toggle switches remain unclear. Here, we investigate the dynamics of three master regulators A, B and C inhibiting each other, thus forming three-coupled toggle switches to form a toggle triad. Our simulations show that this toggle triad can lead to co-existence of cells into three differentiated 'single positive' phenotypes-(high A, low B, low C), (low A, high B, low C) and (low A, low B, high C). Moreover, the hybrid or 'double positive' phenotypes-(high A, high B, low C), (low A, high B, high C) and (high A, low B, high C)-can coexist together with 'single positive' phenotypes. Including self-activation loops on A, B and C can increase the frequency of 'double positive' states. Finally, we apply our results to understand cellular decision-making in terms of differentiation of naive CD4⁺ T cells into Th1, Th2 and Th17 states, where hybrid Th1/Th2 and hybrid Th1/Th17 cells have been reported in addition to the Th1, Th2 and Th17 ones. Our results offer novel insights into the design principles of a multi-stable network topology and provide a framework for synthetic biology to design tristable systems.

Keywords: T-cell differentiation; multi-stability; phenotypic plasticity; toggle switch; toggle triad.

Figure 1.

Network schematics. (a) Toggle switch. (b) Dynamics of a toggle switch—different initial conditions can lead to two states: A/B ≫ 1 or A/B ≪ 1. (c) Schematic of the bifurcation diagram of a toggle switch. Solid blue curves indicate stable states; red dotted curves indicate unstable states. Bidirectional arrows show transition among different states. Green shaded region shows bistable region; pink shaded regions show two possible monostable regions. (d) Schematic of a repressilator. (e) Schematics of network topologies—repressilator with one, two or three toggle switches ((i) R + 1TS, (ii) R + 2TS, (iii) R + 3TS = toggle triad), toggle triad with one, two or three self-activations ((iv) TT + 1SA, (v) TT + 2SA, (vi) TT + 3SA).

Figure 2.

RACIPE outputs for networks shown in figure 1e. (a,b) Frequency of parameter sets used by RACIPE that enable monostable, bistable and tristable solutions for different networks. N = 3 independent RACIPE replicates were done; error bars denote standard deviation. * denotes p < 0.05 for a Student's t-test.

Figure 3.

Characterization of a toggle triad. (a) Frequency of monostable, bistable and tristable solutions for a toggle triad. (b) (i) Heatmap showing the all solutions for a toggle triad; (ii) the nomenclature shown capitalizes the node whose levels are relatively high. Thus, Abc denotes (A-high, B-low, C-low), aBc denotes (A-low, B-high, C-low), abC denotes (A-low, B-low, C-high) (three ‘single positive' states). ABc denotes (A-high, B-high, C-low), AbC denotes (A-high, b-low, C-high), and aBC denotes (A-low, B-high, C-high) (three ‘double positive' states). ABC denotes (A-high, B-high, C-high) (triple positive), abc denotes (A-low, B-low, C-low) (triple negative) states. (c) Frequency of 8 = (2³) possible monostable solutions. (d,e) Frequency of different bistable and tristable cases; with the most frequent ones being combinations of Abc, aBc and abC = {Abc, aBc}, {aBc, abC}, {abC, Abc} (bistable) and {aBc, Abc, abC} (tristable). Error bars represent the standard deviation over n = 3 independent replicates of RACIPE. *: p < 0.05 for Student's t-test.

Figure 4.

Bifurcation diagrams and dynamics plots for representative cases of bistable phases. (a) Bifurcation diagram of expression level of component B with kb as bifurcation parameter for the bistable phase {aBc, abC}. (b) The same as (a) but for component C. (c) Dynamics plots of expression levels of components A, B and C for the bistable phase {aBc, abC}, showing convergence to two different states with varied levels of B and C (levels of A are low in both cases). (d) Bifurcation diagram of the expression level of component A with kc as bifurcation parameter for the bistable phase {Abc, abC}. (e) Same as (d) but for component C. (f) Dynamics plots of expression levels of components A, B and C for bistable phase {Abc, abC}, showing convergence to two different states with varied levels of A and C (levels of B are low in both cases). (g) Bifurcation diagram of expression level of component A with ka as bifurcation parameter for the bistable phase {aBc, Abc}. (h) Same as (g) but for component B. (i) Dynamics plots of expression levels of components A, B and C for the bistable phase {aBc, Abc}, showing convergence to two different states with varied levels of A and B (levels of C are low in both cases). Parameter values for columns A–C, D–F and G–I are given correspondingly in electronic supplementary material, table S13. The bifurcation diagrams for ‘low' components (component A in (a–c), component B in (d–f), component C in (g–i)) are shown in electronic supplementary material, figure S11.

Figure 5.

Bifurcation diagrams for bistable phase with combination of a ‘single positive' state and a ‘double positive' state. (a) Bifurcation diagram of expression level of component B with kc as bifurcation parameter for the bistable phase {aBC, abC}. (b) Same as (a) but for component C. (c) Bifurcation diagram of the expression level of component A with ka as bifurcation parameter for the bistable phase {Abc, AbC}. (d) Same as (c) but for component C. (e) Bifurcation diagram of the expression level of component A with kb as bifurcation parameter for the bistable phase {aBc, ABc}. (f) Same as (e) but for component B. Corresponding parameter sets for columns a-b, c-d and e-f are given in electronic supplementary material, table S13.

Figure 6.

Characterization of toggle triad with three self-activations (TT + 3SA). (a) (i) Heatmap showing the all solutions for a TT + 3SA; (ii) the nomenclature shown capitalizes the node whose levels are relatively high. Thus, Abc denotes (A-high, B-low, C-low}, aBc denotes (A-low, B-high, C-low), abC denotes (A-low, B-low, C-high) (three ‘single positive’ states). ABc denotes (A-high, B-high, C-low), AbC denotes (A-high, b-low, C-high) and aBC denotes (A-low, B-high, C-high) (three ‘double positive' states). (b) Frequency of monostable, bistable and tristable solutions in a toggle triad (TT) shown as a pie chart. (c) Frequency of monostable, bistable, tristable, tetrastable and pentastable solutions for a TT + 3SA case shown as a pie chart. (d,e) Frequency of different tetrastable and pentastable phases are combinations of Abc, aBc, abC, aBC, AbC, ABc = {Abc, aBc, abC, aBC}, {Abc, aBc, abC, AbC}, {Abc, aBc, abC, ABc} (tetrastable) and {aBc, Abc, abC, aBC, AbC}, {aBc, Abc, abC, ABc, AbC}, {aBc, Abc, abC, aBC, ABc} (pentastable). Error bars denote the standard deviation of n = 3 independent RACIPE simulations. * denotes statistical significance (p < 0.05 for Student's t-test).

Figure 7.

sRACIPE results for one of the replicates of three different parameter sets used. (a) Dynamics plot showing switching between states for parameter set 1. Colour bars on top representatively mark the regions of each of the states—green bar shows (low A, high B, low C), red bar shows (high A, low B, low C) and blue bar shows (low A, low B, high C). (b,c) Same as (a) but for a replicate of parameter set 2 and 3 respectively. The plots for two other replicates for each of the parameter sets are shown in electronic supplementary material, figure S14, S15. Parameter values for sRACIPE for all the three replicates and parameter sets are given in electronic supplementary material, table S13.

Figure 8.

Design principles of toggle triad (TT) and that with self-activation (TT + 3SA). (a) Frequency of dominance of six inhibitory links for the case of monostable (A high, B low, C low) (i.e. {Abc}) in TT. X –|Y denotes the frequency of parameter sets when the inhibition of X on Y was stronger than the that of Y on X; thus adjacent two columns total up to 1. (b) (i) Probability distribution functions of histograms of frequency of values of x(A–|B)/x(B–|A) (blue) and values of x(A–|C)/x(C–|A) (yellow) in TT for monostable case {Abc}. The x-axis is log₁₀ transformed and the dotted line represents the numerical value 1. (ii) Schematic showing that A inhibiting B and C are stronger links than B and C inhibiting A overall for parameter sets corresponding to monostable {A} (c) Same as (a) but for bistable (i.e. {Abc, aBc}) in TT. (d) Same as (a) but for tristable case {Abc, aBc, abC} in TT. (e) Frequency of parameter cases for which self-activation dominates upon incoming inhibitory link, for monostable case {Abc} in TT + 3SA. X2X/X1Y denotes the percentage of parameter sets for which X self-activation is stronger than Y inhibiting X. (f) (i) Probability distribution functions of histogram of frequency of x(A2A)/x(B1A) and x(A2A)/x(C1A) in TT + 3SA. The x-axis is log₁₀ transformed and the dotted line represents the numerical value of 1. (ii) Schematic showing that for most of the parameter sets corresponding to {Abc}, self-activation of A dominates inhibition of A by B or C. (g) Same as (e) but for bistable case {Abc, aBc} in TT + 3SA. (h) Same as (e) but for tristable case {Abc, aBc, abC} in TT + 3SA. * denotes statistical significance (p < 0.01 for Student’s t-test). ‘ns' denotes statistically non-significant cases. Parameters corresponding to figure 8, electronic supplementary material figure S16 are given in tables S14, S15.

Figure 9.

CD4 T-cell differentiation. (a) Network showing proposed interaction among the master regulators of Th1, Th2 and Th17—T-bet, GATA3 and RORγT—respectively. (b) Two-dimensional scatter plots projecting solutions from the heatmap for a toggle triad network (figure 3). (c) Network of T-bet, GATA3 and RORγT including self-activations. (d) Same as (b) but for solutions from the heatmap for a toggle triad with three self-activations (figure 5). Blue coloured dots denote Th1 (high T-bet, low GATA3, low RORγT), orange coloured dots denote Th2 (low T-bet, high GATA3, low RORγT) and green coloured dots denote Th17 (low T-bet, low GATA3, high RORγT) state. Black dots denote the different hybrid states—Th1/Th2, Th2/Th17 and Th1/Th17: (high T-bet, high GATA3, high RORγT), (high T-bet, low GATA3, low RORγT) and (low T-bet, high GATA3, high RORγT), respectively. Data from respective heatmaps were subjected to k-means clustering to identify these six states (three ‘single positive’ and three ‘double positive' ones).

Figure 10.

Waddington landscape for a toggle triad. Modified Waddington's landscape to demonstrate the differentiation of three distinct ‘single positive' states (states A, B and C), and three putative ‘double positive' states (hybrid states A/B, A/C and C/B) from a common progenitor. These six states can be obtained from a toggle triad with/without self-activation.