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. 2014 Jul 24;9(7):e102873.
doi: 10.1371/journal.pone.0102873. eCollection 2014.

Gene Networks of Fully Connected Triads With Complete Auto-Activation Enable Multistability and Stepwise Stochastic Transitions

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Free PMC article

Gene Networks of Fully Connected Triads With Complete Auto-Activation Enable Multistability and Stepwise Stochastic Transitions

Philippe C Faucon et al. PLoS One. .
Free PMC article

Abstract

Fully-connected triads (FCTs), such as the Oct4-Sox2-Nanog triad, have been implicated as recurring transcriptional motifs embedded within the regulatory networks that specify and maintain cellular states. To explore the possible connections between FCT topologies and cell fate determinations, we employed computational network screening to search all possible FCT topologies for multistability, a dynamic property that allows the rise of alternate regulatory states from the same transcriptional network. The search yielded a hierarchy of FCTs with various potentials for multistability, including several topologies capable of reaching eight distinct stable states. Our analyses suggested that complete auto-activation is an effective indicator for multistability, and, when gene expression noise was incorporated into the model, the networks were able to transit multiple states spontaneously. Different levels of stochasticity were found to either induce or disrupt random state transitioning with some transitions requiring layovers at one or more intermediate states. Using this framework we simulated a simplified model of induced pluripotency by including constitutive overexpression terms. The corresponding FCT showed random state transitioning from a terminal state to the pluripotent state, with the temporal distribution of this transition matching published experimental data. This work establishes a potential theoretical framework for understanding cell fate determinations by connecting conserved regulatory modules with network dynamics. Our results could also be employed experimentally, using established developmental transcription factors as seeds, to locate cell lineage specification networks by using auto-activation as a cipher.

Conflict of interest statement

Competing Interests: The authors have declared that no competing interests exist.

Figures

Figure 1
Figure 1. Multistability arises from small gene networks and underlies cell differentiation.
Fully-connected triads (FCTs) are important, recurring transcriptional networks in the development and maintenance of cellular states. Notably, the Oct4-Sox2-Nanog triad has been implicated in inducing and maintaining stem cell pluripotency. In the Waddington model for cell differentiation , the cell's underlying developmental landscape is governed by the dynamical potential of these small gene networks to realize multistability. Transition from one state to another can be guided via altered topology or strength of wiring.
Figure 2
Figure 2. High-throughput computational screening for multistable networks.
(A) FCTs were enumerated and modeled by ODEs (parameters denoted as P). (B) The probability of multistability for each of the 104 FCTs generated by the computational screen is plotted. FCTs with a high probability of multistability are marked with an X with the color representing the multistability group; the two highest probability groups are labeled by their indices. Colored backgrounds serve as visual aids to distinguish the top three groups. (C) Fifteen FCTs with significant multistable potentials are listed with groups ranked according to their probability for multistability. The red arrows indicate activating regulation and blue T-head arrows indicate inhibitory regulation. Network 54 was included because it was the only network with complete auto-activation that did not have a high multistable potential.
Figure 3
Figure 3. Bifurcation and Stochastic Analysis of Network 84.
(A) The topology of Network 84 was predicted to have the highest probability for multistability. (B) The bifurcation diagram of Network 84 plots transcription factor concentrations of X and Y at each SSS against α, the self-activation strength. The mutual inhibition parameters are also set equal and referred to as β. Here β is set to 0.1 and α is used as the bifurcation parameter ranging between .01 and 100. By including both stable (colored) and unstable (grey) steady states we can see that as α increases the SSS move to higher concentrations, approximately at the same rate as the increase in α. SSS are color coded and listed in the legend to distinguish different SSS, where ‘+’ indicates the gene is ‘ON’ and ‘-’ means ‘OFF’. Spheres are also attached to each SSS at α = 1 (red) and α = 5 (gray), sphere size correlates with the size of the spectral radius, a measure of SSS stability. (C) Simulations of noise-induced state transitioning in Network 84 under different levels of noise. Simulations were performed with auto-regulation equal to 1 and mutual inhibition equal to 0.1, with noise levels of 0.5, 0.85, and 1 from left to right. The locations of the deterministically calculated states are indicated with red spheres, with their size correlated to their spectral radius. The blue ribbons indicate the temporal trajectories of the simulations. The black arrows indicate initial direction of state transitions. Potential cloud shape for each SSS is illustrated in Figure S6A. (D) The number of states traveled in the stochastic simulations plotted versus noise strength. The red line represents simulations that were initiated from the all-ON state, and the blue line represents simulations that were initiated from the origin.
Figure 4
Figure 4. Stable Steady State Analysis of Network 84.
(A) A matrix presenting the number of stable steady states generated by combinations of different auto-regulatory strengths (rows) and mutual-regulatory strengths (columns). This provides an overview of network stability at various points in the parameter regulation space; the green and orange regions are visualized as FCT diagrams independently in B. (B) The green-boxed diagram corresponds to the green-boxed parameter combination regime from (A), where genes have regulatory strengths α = 1 and β = 0.1. By contrast, the orange-boxed diagram corresponds to the orange-boxed parameter combination regime from (A), where nodes have regulatory strengths α = 5 and β = 0.1. This parameter combination regime allows the system to be stable in 7 of the 8 SSS, losing only the all-OFF state. (C) Many of the parameter combinations yielding multi-stable systems are not represented by the matrix, and are instead abstracted here. As an example, here we present a parameter combination regime that can support 6 SSS and auto-activation strengths are similar to those in the orange box (thick red arrows) but have one unrestrained pair of mutual inhibition that can exist at a very high strength (blue T-head repression lines).
Figure 5
Figure 5. Bifurcation and Stochastic Analysis of Network 1.
(A) The topology of Network 1 was also predicted to have high probability for multistability. Network 1 shares the topological architecture of the Oct4-Sox2-Nanog triad, and is believed to underlie stem cell differentiation and reprogramming. (B) As in Figure 3B, the X and Y protein concentrations of stable (colored) and unstable (gray) steady states at various self-activation strengths can be seen. A legend is shown as in Figure 3. (C) Simulations of noise-induced state transitioning in Network 1 under different levels of noise. Simulations were performed with auto-regulation equal to 1 and mutual regulation equal to 0.1, with noise levels of 0.5, 0.85, and 1 from left to right. The locations of the deterministically calculated states are indicated with red spheres, with their size being correlated to their spectral radius. The blue ribbons indicate the temporal trajectories of the simulations. The black arrows indicate initial direction of state transitions. Potential cloud shape for each SSS is illustrated in Figure S6B. (D) The number of states traveled in the stochastic simulations plotted versus noise strength.
Figure 6
Figure 6. Stable Steady State Analysis of Network 1.
(A) A matrix defined by examining the number of SSS present at points where all auto-regulatory strengths (row) are equal and mutual-regulatory strengths (column) are also equal. This provides an overview of network stability at various points in the parameter regulation space; the green and brown regions are visualized as FCT diagrams independently in B. (B) The green-boxed diagram corresponds to the green-boxed parameter combination regime from (A), where genes have regulatory strengths α = 1 and β = 0.1. The orange-boxed diagram corresponds to the orange-boxed parameter combination regime from (A), where nodes have regulatory strengths α = 5 and β = 0.1. This parameter combination regime allows the system stability in 5 of the 8 Stable Steady States, losing only the two-ON states. (C) As in Figure 4C, there exist parameter combinations not captured by the matrix representation, where increased strength of any one of the five regulations (represented by thick arrows) will enable 5 SSS.
Figure 7
Figure 7. Bifurcation and Stochastic Analysis of Network 1 with constitutive basal expression.
(A) The topology of Network 1 is used to mimic iPSC experiments with added constitutive expression of all three genes. (B) As in Figure 5B, the X and Y protein concentrations of stable and unstable steady states are plotted against self-activation strengths. With the addition of the constitutive expression we can see a further decrease in stabilities, specifically two-ON states are never stable. Additionally one-ON states lose stability very rapidly, approximately at α = 1, and the all-OFF state destabilizes after α = 9 leaving only the all-ON state. This can also be seen in the sizes of the spectral radii, deterministically these steady states exist but in practice they have very little influence. (C) Simulations of Network 1 modeled in the presence of an additional constitutive overexpression term consistently show a transition from an all-OFF state to an all-ON state after a period of latency.
Figure 8
Figure 8. Stochastic simulations capture latency distributions in induced pluripotency.
(A) Distribution of simulated latency for the modified Network 1 model illustrates a skewed bell shape. The histogram was generated from 2000 simulations. (B) Similar results generated from 2000 simulations with Network 84.

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