Fig 1.
Illustration of multistep cell fate transition.
A. transition from one cellular state to another via two intermediate states. Dashed arrow indicates the limited reversibility of each transition. B. stepwise changes of the levels of two transcription factors during the multistep transitions involving four states. C. metaphoric energy landscape depicting the four-attractor system. Colors for cell states and transition arrows in B and C match those in the illustration in A.
Table 1.
Examples of multistep transitions with restricted reversibility.
Fig 2.
Network motifs governing four-attractor systems.
A. Illustration of the network topology searching. Dashed arrows are regulations sampled. The topologies were screened by the criterion of the four attractors with stepwise changes of TFs. B. Complexity atlas for selected topologies. Closed circles denote minimum motifs. Open circles denote topologies containing more regulations than those in the minimum motifs. Each arrow denotes the difference by one regulation in the network. Examples of minimum motifs are shown at the bottom. Red: Type I motif. Blue: Type II motif. Green: Hybrid motif. C. Overlaid four attractors for each of the 29 minimum topologies. Factor A denotes the TF on the left of the network diagram. Factor B denotes the TF on the right of the network diagram. In some topologies A and B and positively correlated (left panel), whereas they are negatively correlated in other topologies (right panel). Colored dots denote the stable steady states. Colored lines connect states of their corresponding topologies. The colors of the cell states match the illustration in Fig 1. The colors of the lines denote different representative models. The z-score is calculated by shifting the mean of each four attractors to 0 and then normalizing the four data points to unit variance data. D. Example phase planes for two minimum topologies (Type I and Type II respectively). In each case, four out of the seven steady states (intersections denoted by solid dots) are stable. Network structures and phase planes for all 29 minimum motifs are included in S1 and S2 Figs. All models shown in this figure are built with additive form of Hill functions.
Table 2.
Definitions and key features of network motifs that generate systems with four ordered attractors.
Table 3.
Numbers of sampled network structures and discovered motifsa.
Fig 3.
Comparison of Type I and Type II motifs for their performance in producing systems with four ordered attractors.
A. The numbers of parameter sets per 106 samples per topology that generate four ordered attractors from two types of minimum motifs. B. The numbers of parameter sets per 106 samples per topology (limited to 30 per topology) that generate four ordered attractors from three types of motifs. C. The numbers of parameter sets per 106 samples per topology that generate four unordered attractors from two types of minimum motifs. D. Inter-attractor distances for each parameter set that generates four ordered attractors were calculated and summarized. For each set of parameters associated with the four attractors, the minimum, the maximum and the standard deviation of the distances were analyzed. Minimum Type I and Type II motifs were compared using these statistics.
Fig 4.
Four-attractor motifs in the early T cell transcriptional network.
A. Influence diagram for transcriptional regulations among four core factors controlling the early T cell development. Arrows represent activations and short bars represent inhibitions. B. Functional subnetworks of the T cell network were systematically obtained by removing regulations from the network. These subnetworks were screened by the criterion that four attractors with stepwise changes of TFs exist in the absence of Notch signal. C. Complexity atlas showing the relationships of the two four-attractor motifs in the subnetworks of the T cell model. Top callout shows the full network in the absence of Notch. Bottom callouts show examples of the minimum functional subnetworks of the two types with particular numbers of regulations. Red: Type I motif. Blue: Type II motif. Green: Hybrid motif. D. Overlaid four attractors for each of the 66 minimum topologies. Colored dots denote the stable steady states. Colored lines connect states of their corresponding topologies. All models shown in this figure are built with the multiplicative form of Hill functions.
Fig 5.
Enrichment of Type I and Type II motifs in the T cell model.
Top panel: total occurrences of various types of motifs in the T cell network. Middle panel: empirical p-values of the single positive feedback loops and the sum of the two types of motifs. Bottom panel: an illustration of the p-values with the distributions of background population. Random networks were obtained by 1) permuting the regulations in the existing network by randomly assigning their sources and targets (red) and 2) assigning random regulations (positive, negative or none) between each pair of TFs (blue). 105 random networks were generated with each method. Empirical p-values were obtained by counting the number of the random networks with the number of motifs not less than those in the T cell network. See Methods for details of the p-value definition. Distributions of motif frequencies obtained from the random networks using the second method are shown in the bottom panel. The yellow vertical bars represent the number of occurrences in the T cell network. The right-tail areas defined by the vertical bars correspond to the p-values shown in the middle panel (blue bars).
Fig 6.
Comparisons of motifs with different complexity and types.
A. Two specific network topologies were selected for comparing models with different complexity. Network 1 contains multiple Type I motifs, whereas Network 2 is a single Type I motif. The color code of the complexity atlas is the same as that in Fig 2 and Fig 4. Red: Type I motif. Blue: Type II motif. Green: Hybrid motif. B. Performances of the two subnetworks are compared. Performance was quantified with the sum of squared distance (SSD) from a predefined hypothetical continuous production function (gray curve) of PU.1 level that have 7 intersections with the degradation function, which generates four attractors (see details in supplementary text). The purple and red curves represent the optimized functions fitted to the gray curve. The gray curve is closer to the purple curve than to the red curve, suggesting a better fit with Network 1. C. SSD values obtained from 500 optimization runs for each of Network 1 and Network 2. Each value was calculated using the procedure shown in B. D. Numbers of parameter sets that generate the four-attractor systems per 106 random samples per topology. Three types of network motifs with 7 regulations are compared.
Fig 7.
Stability analysis of the T cell model.
The full model shown in Fig 4A is used for all the analysis. A. Bifurcation diagrams for the steady states of the four core factors with respect to the Notch signal. Solid curve: stable steady state. Dashed curve: unstable steady state. B. Bifurcation diagram under Bcl11b knockout condition with respect to Notch signal. Solid curve: stable steady state. Dashed curve: unstable steady state. C. Bifurcation diagram with respect to BCL11B production rate parameter. Solid curve: stable steady state. Dashed curve: unstable steady state. D. Illustration of the observed transitions among the four states. Colors of the stable branches of the bifurcation diagrams and the cell icons are matched to the cellular states shown in Fig 1.
Fig 8.
Multistep lineage transitions under the influence of varying dynamics Notch signals.
A. Strength and duration of the Notch signal were varied in each simulation. 200X200 combinations of different signal strengths and durations were tested, and the final cellular phenotypes were determined using the levels of the four core factors. B. Dynamics of PU.1 in response to increasing Notch with significant fluctuations. The mean of the Notch signal increases linearly in the first phase, then it is attenuated in the second phase. Fluctuations were simulated with additive noise in small time intervals.
Fig 9.
Energy landscape for T cell development.
The landscape and corresponding minimum action paths (MAPs) for the T cell developmental network are shown in 3-dimensional (A) and 4-dimensional (B) figure (the full model shown in Fig 4A is used for all the analysis). A. The blue regions represent higher probability or lower potential and the yellow regions represent lower probability or higher potential. White solid lines represent the MAP from ETP state to DN2a, DN2b, and DN3 states. Magenta solid lines represent the MAP from DN3 to DN2b, DN2a, and to ETP state. Dashed lines represent the direct MAP from ETP to DN3 and from DN3 to ETP states, respectively. Here, TCF-1 and PU1 are selected as the two coordinates for landscape visualization. B. The landscape and paths are displayed in a 4-dimensional figure, where the three axes represent three TFs (TCF-1, BCL11B and PU. 1), respectively, and the fourth dimension (color) represents the energy U. The normalized gene express data (including TCF-1, BCL11B, GATA3, and PU. 1) of four stages for T cell development are mapped to the landscape, where the golden balls represent the four steady states (four stages of T cell development) from the models, the red balls (Data1) represent the data from Zhang et al. [47], and the green balls represent the data from Mingueneau et al. [17]. See S7 Fig and S8 Fig for the landscapes and the gene expression data using other pairs of TFs.
Fig 10.
Discrete kinetic transition paths for T cell model.
Transition paths from ETP state to DN3 state in terms of levels of 4 different TFs. A. The relative TF levels are discretized to 0 or 1. 1 represents that the corresponding TFs are in the on (activated) state and 0 represents that the corresponding TFs are in the off (repressed) state. B. The relative TF levels are discretized to five values from low to high. X axis shows the time along the transition path.
Fig 11.
Global sensitivity analysis for T cell developmental model.
Sensitivity analysis was performed for the 39 parameters in the T cell model. The transition actions between different states (SETP->DN2a and SDN3->DN2b) were calculated to quantify the sensitivity of parameters on the landscape. The Y-Axis represents the 39 parameters. The X-Axis represents the percentage of the transition action (S) changed relative to S without parameter changes. Here, SETP->DN2a represents the transition action from attractor ETP to attractor DN2a (cyan bars), and SDN3->DN2b represents the transition action from attractor DN3 to attractor DN2b (magenta bars). A. Each parameter is increased by 1%, individually. B. Each parameter is decreased by 1%, individually.