Networks in a large motif family govern systems with four attractors with stepwise transcriptional modulation

To find transcriptional network topologies that can generate multiple intermediate states during cell fate transition, we first performed random parameter sampling with a network family containing up to 3 nodes (Fig 2A). In this framework of network topology, each node represents a transcription factor (TF) that can potentially influence the transcription levels of other two TFs and itself. Topology searching with a 3-node network was used for motif discovery for various performance objectives in previous studies [38,39]. We performed exhaustive search for topologies with up to 6 regulations from a total of 9 regulations of the network family, and constructed a mathematical model for each topology (see Methods for details). For each model, we performed random sampling in the parameter space from uniformly distributed values (S1 Table). We selected topologies containing at least one parameter set that is able to generate four attractors with stepwise changes of transcriptional levels. We define the system with four attractors with the stepwise changes of transcriptional levels as the scenario in which there are four stable steady states and they can be consistently ordered by the concentrations of any pairs of TFs. In other words, one TF always monotonically increases or decreases with another TF in these four states, and we term these states ‘ordered’ attractors in this paper. Among the 2114 network topologies that we searched, we found 216 topologies that can produce such behavior. In addition, we found 417 topologies that can only produce four unordered steady states (TF concentrations are non-monotonically correlated among the states) (S11 Fig, S12 Fig).

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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.

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To visualize the relationships among these topologies, we constructed a complexity atlas (Fig 2B), in which the nodes represent the network structures that gave rise to four attractors, and the edges connect pairs of topologies that differ by a single regulation (addition or removal of a transcriptional interaction) [40]. We define the minimum topologies as those of which the reduction of complexity, or the removal of any regulation from the network, will abolish its capability to generate four attractors (solid nodes in Fig 2B and examples in Fig 2C). We found 29 such minimum topologies which represent the non-redundant structures for producing the four-attractor system.

Interestingly, all of the 216 topologies obtained from our search contain three distinct positive feedback loops (including double-negative feedback loops), and they can be categorized into two types of motifs (Fig 2B, bottom panel). The Type I motif contains three positive feedback loops that are closed at a single TF (red nodes and edges in Fig 2B). The Type II motif contains three connected positive feedback loops, two of which do not share any TF but are connected via the third loop (blue nodes and edges in Fig 2B). There is a remarkable diversity of each of the motif types because the interconnected positive feedback loops can share multiple TFs (S1 and S2 Figs). Based on the complexity atlas (Fig 2B), we found that Type II motifs contain 4–6 regulations, and Type I motifs contain 5–6 regulations. Some of the networks with 6 regulations contain subnetworks of both Type I and Type II motifs (Hybrid type, green nodes). The four attractors in the space of two TFs exhibit a variety of patterns of nonlinear monotonic correlations (Fig 2C, S3 Fig), which are governed by intersections of highly nonlinear nullclines in the state space containing the two TFs (Fig 2D, S1 and S2 Figs). The definitions of various types of motifs are listed in Table 2, and the statistics of the topologies discovered are summarized in Table 3 (also see S11 Fig for an illustration). Overall, this motif family represents a large number of networks that can produce a common type of behaviors: multiple stable intermediate states in terms the transcriptional activity.

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Table 2. Definitions and key features of network motifs that generate systems with four ordered attractors.

https://doi.org/10.1371/journal.pcbi.1006855.t002

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Table 3. Numbers of sampled network structures and discovered motifs^a.

https://doi.org/10.1371/journal.pcbi.1006855.t003

We next asked whether there is a difference between Type I and Type II motifs in terms of their ability to generate systems with four ordered attractors. We found that with the same number of sampled parameter sets, Type II motifs have greater fractions of parameter sets that give rise to four ordered attractors than Type I motifs do (Fig 3A and 3B, p-value < 0.0001, Mann-Whitney U test). This suggests that Type II motifs may be more robust for governing the four ordered attractors. However, among the 15 Type I and 14 Type II minimum motifs, all Type II motifs are able to generate four unordered attractors (Some TFs are not monotonically correlated. See S12 Fig), whereas there is no Type I motif that has any parameter set that gives rise to four unordered attractors (Fig 3C). This suggests that as compared to Type II motifs, Type I motifs has higher specificity in generating four ordered attractors, which is more relevant to the stepwise cell fate transitions than the unordered ones. Moreover, we observed that the inter-attractor distances between neighboring attractors in the gene expression space were generally more variable with Type I motifs than those with Type II motifs (Fig 3D, magenta boxes). In particular, among the three inter-attractor distances for each model, Type I motifs generated smaller minimum distance than Type II motifs did (Fig 3D, orange boxes. p-value < 0.0001, Mann-Whitney U test). We did not observe any significant difference between Type I and Type II motifs in terms of the stabilities of the attractors and the kinetic paths that they generate (S13 Fig See Methods for calculation of quasi-energy landscape and kinetic path). In addition to the effects of motif types, we also asked whether the fractions of positive or negative regulations in the network can influence the function of generating four-attractor systems. We found that the fraction of positive regulations has a weak but significant correlation with the fraction of successful parameter sets generating four-attractor systems (S14A and S14B Fig). Although negative regulations are slightly less favorable, they might be important to ensure that the levels of some TFs are inversely correlated during multistep cell fate transitions, which is necessary for having at least one highly expressed TFs in each of the initial and the final cell states (S14C Fig).

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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.

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In summary, we found two types of network motifs that generate four attractors with stepwise changes of the transcriptional profile. Two of these attractors represent the multiple intermediate states observed in various biological systems. This exploratory analysis elicits several interesting questions: what are the biological examples of such network motifs? Can the conclusions with respect to the two types of motifs be generalized to networks with more than three TFs? Is there any advantage of combining both types of motifs? How are the transitions among these states triggered deterministically and stochastically? To provide insights into these questions in a more biologically meaningful context, we will use a specific biological system to describe more detailed analysis of these motifs and their underlying gene regulatory networks in the following sections.

Type I and Type II network motifs are enriched in a transcriptional network controlling early T cell development

We asked whether the motifs that we discovered can be found in any known transcriptional network that potentially control multistep cell differentiation. We used the early T cell differentiation in the thymus as an example to address this question. The differentiation from multipotent lymphoid progenitor cells to unipotent early T cells involves multiple stages at which the cells have differential potentials to commit to non-T lineages and other cellular properties such as proliferation rates. At the early phase of this process, four stages of development T cells (ETP/DN1, DN2a, DN2b, DN3) were identified experimentally, and the progression through these stages is controlled by a myriad of transcription factors including four core factors, TCF-1, PU.1, GATA3 and BCL11B. These TFs form a complex network among themselves (see Fig 4A and supporting experimental observations in S3 Table), and the stepwise changes in the levels of these TFs were observed in the four developmental stages of T cells [20,28]. The interactions involving these core TFs were shown to be critical for the irreversible commitment to the T cell lineage by forming a bistable switch [41]. Among these factors, PU.1 level decreases as the cells commit to later stages, whereas the levels of other three factors increase in this process. It is unclear, however, whether this transcriptional network can serve as a regulatory unit that governs the multistep nature of the T cell differentiation.

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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.

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We noticed that this T cell transcriptional network contains the motifs that we found in our analysis using the generic form of networks, we therefore hypothesized that the models based on this network can have four attractors with sequential changes of the four TFs. Indeed, using random sampling we were able to find parameter sets that give rise to four-attractor systems similar to what we obtained with the generic 3-node framework. To find the functional components that generate this behavior, we analyzed the subnetworks of the complex T cell regulatory network [42]. We removed the regulations from the network systematically, and we found that out of the non-redundant 1553 topologies (2047 subnetworks), there are 568 topologies (701 subnetworks) that can generate four attractors with stepwise changes of the TFs (Fig 4B). We used a complexity atlas to visualize the relationships among these subnetworks (Fig 4C). We found that the network can be reduced to one of the 66 minimum topologies (97 minimum subnetworks) which retains the four-attractor property (solid nodes in Fig 4C). Notably, these networks can be classified into the two types of motifs described earlier (Fig 2B). Similar to the networks that we obtained through the generic framework, the two types of minimum motif have 4–6 regulations. Subnetworks with both types of motifs (green nodes and edges) start to appear when the number of regulations reaches six. The numbers of motifs and subnetworks obtained for the generic framework and the T cell model are summarized in Table 3.

We next quantified the enrichment of the two motif families in the early T cell transcription network. We first generated random networks by perturbing the existing regulations in the network model and computed the empirical p-values for observing the numbers of different types of network motifs. The T cell network contains a large number of positive feedback loops and the two types of motifs that we described earlier (Fig 5, top panel). As expected, the network is significantly enriched with positive feedback loops in general (Fig 5, middle panel, red bars). However, the enrichments of Type I motifs and the combinations of Type I and Type II motifs are even more significant than that of the single positive feedback loops (Fig 5, middle panel, red bars). To exclude the possibility that this differential significance was observed due to the way we generate random networks which gives low p-values (<10⁻⁴) in general, we used another method to generate random networks with an augmented number of regulations (Fig 5, middle panel, blue bars). Each pair of TFs were assigned with a pair of random regulations (positive, negative or none). Consistent with the previous method, the T cell transcriptional network is enriched with positive feedback loops overall, but the enrichment is more significant for Type I motifs or for the combination of Type I and Type II motifs. Interestingly, motifs that are similar to Type I motif but have higher complexity (more positive feedback loops) do not show more significant enrichment than Type I motif does (S15 Fig). In addition, we found that networks controlling switch-like behaviors, but not multistep cell fate transitions, are not enriched with Type I or Type II motif [43–46]. These results suggest the possibility that the network has been evolved to reach more complex performance objectives than those enabled by simple positive feedback loops alone.

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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).

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Since the minimum motifs alone can generate the four-attractor system, we asked whether the combination of these motifs enhances the ability of the network to produce the system. We therefore compared a subnetwork containing only one minimum Type I motif with another one containing multiple such motifs in terms of the performance to generate a particular four-attractor system (Fig 6A See Methods and S1 Text for details). We found that the subnetwork with multiple Type I motifs (Network 1) outperforms the one with only one motif (Network 2) in that Network 1 can give a better fit to a hypothetical four-attractor system (Fig 6A–6C). In this hypothetical ‘target’ system, the four attractors are assumed to be determined by dynamics of PU.1 with multiple feedback loops. The assumed degradation (Fig 6B, gray curve) and production rate functions of PU.1 (Fig 6B, green curve) are specified. The curves of these two functions have 7 intersections, four of which represent attractors. The optimized production function obtained from Network 1 (Fig 6B, purple curve) has more robust intersections with the degradation curve than one obtained from Network 2 (Fig 6B, red curve). This difference was observed for production functions of these two categories from multiple runs of optimization (Fig 6C). This suggests the advantage of combining multiple motifs with similar functions to enhance its overall performance. We next compared the 66 minimum motifs (Fig 4C bottom solid nodes) and the full topology (Fig 4C top node) in terms of the parameter values that gave rise to the four-attractor systems. We found that the full model contains more parameter sets that have low nonlinearity of the regulations than the minimum motifs do (S16A Fig), and the parameter values are distributed in larger regions with the full model than those with the minimum motifs (S16B Fig).

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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.

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We next asked whether the topologies that contain both Type I and Type II motifs have greater probabilities to generate the four-attractor system than the topologies with one type of motifs do. When we explored the parameter space randomly for each topology with a fixed number of samples, a larger number of parameter sets that can generate the four-attractor system were found with the topologies containing both motifs than with those containing either Type I or Type II motifs only (S5 Fig and Fig 6D). This suggests that the combination of both motifs might be a robust strategy to generate the four-attractor system. This pattern was observed for all the topologies in the complexity atlas (S6 Fig) as well as those with the same degree of complexity (Fig 6D, networks with 7 regulations were chosen because they have comparable fractions of the three types of motifs).

In summary, we found that the core transcriptional network controlling early T cell differentiation are enriched with Type I and Type II network motifs. The network composed of these two types of motifs governs a dynamical system containing four attractors, corresponding to four known stages in the early T cell development. The networks with both types of motifs and greater number of such motifs have more robust capability of generating the four-attractor systems than those networks with fewer types of numbers of motifs do.

Stepwise transitions with restricted reversibility provide robustness to fluctuating differentiation signal to multiple intermediate states

We next characterized the dynamical features of the four-attractor system of the T cell development model in response to differentiation signals. For this and subsequent analysis, we focused on a model describing the network topology shown in Fig 4A (the full model). We first performed bifurcation analysis of the system to the changes of Notch signaling (Fig 7A). With the increasing Notch signal, the system undergoes three saddle-node bifurcations, at which the stability of the proceeding cellular states is lost (Fig 7A, black arrows). These bifurcation points therefore represent the cell state transitions from one stage to the next. The structure of the bifurcation diagram shows a remarkable robust multistep commitment program governed by the T cell transcription network: the commitment to each stage of the program has restricted reversibility in that the attenuation or withdrawal of the Notch signaling does not result in de-differentiation of the developing T cells (i.e. the return of the transcription profile to earlier stages that may have greater multipotency). It was previously shown that the commitment from DN2a to DN2b is an irreversible process with respect to Notch signaling, and this transition eliminates developing T cells’ potential to be diverted to any other lineages when Notch signaling is abolished [20,41]. However, simple toggle-switch models do not explain the observation that the multipotency of the early T cells is lost in a stepwise manner. For example, cells at ETP can be differentiated into B cells, macrophages, dendritic cells (DCs), granulocytes, natural killer (NK) cells and Innate lymphoid cellsubset2 (ILC2), whereas the potentials to commit to many of the lineages are blocked even in the absence of Notch signaling at the DN2a stage, at which the cells can only be differentiated into NK cells and ILC2 [20]. Therefore, the stepwise, irreversible transcriptional transitions revealed by our model is consistent with the experimental observations with respect to the loss of multipotency in the stepwise manner.

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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.

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Although the absence of Notch signal does not allow the reversal of lineage progression, it was previously shown that the absence of BLC11B in lymphoid progenitor cells blocks its ability to progress to DN2b stage, whereas the Cre-controlled knockout of Bcl11b in committed T cells (e.g. DN3 cells) reverts its transcriptional profile to DN2a-like cells [28]. Upon blocking the production of BCL11B in our model, we observed the loss of attractors of DN2b and DN3, and the DN2a state is the only stable stage even in the presence of the strong Notch signaling (Fig 7B). As a result, increasing Notch signaling only triggers one saddle-node bifurcation, representing the transition from ETP to DN2a cell (Fig 7B black arrow), whereas the decrease of the BCL11B production triggers the transition back to DN2a instead of ETP (Fig 7C). These results are in agreement with the previous experimental findings [28], and they further support the importance of the multistep differentiation system revealed by our model.

The bifurcation analysis shows how the lineage progression is influenced by stably increasing or decreasing Notch signal strengths. We next asked how the duration of Notch signal may control the multistep lineage transition. By inducing the differentiation with varying durations of the Notch signaling, we found that cells experiencing transient Notch signals may only commit to intermediate stages of differentiation (Fig 8A). In addition, the system is able to integrate the information of the signal intensity and duration to make decision on the lineage progression. These results suggest that the multistep lineage transition can be triggered by the increasing strength of the signal, the increasing duration of the signal, or the combination of both types of signal dynamics. Earlier experimental studies have shown that transient Notch signaling can irreversibly drive the cells to an intermediate, but committed stage with a definitive T cell identity (DN2b) [28,41,47]. This is in agreement with our results, and our model further suggests that the commitment to other intermediate states is also irreversible with respect to the lineage progression (note that this irreversibility does not refer to the establishment of T cell identity).

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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.

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One possible advantage of the multi-stable system is its robustness of response in facing fluctuating signals. We therefore performed numerical simulations of the dynamical system under increasing Notch signaling with significant fluctuations. Under this condition, transient reduction of Notch signaling halted the progress of the lineage commitment but did not trigger the de-differentiation (Fig 8B). Our model suggests that the design of transcriptional network allows the system to stop at intermediate stages before proceeding to the next ones. This strategy has several potential physiological benefits: 1) it protects the cell lineage progression against sporadic fluctuations of Notch signaling; 2) it facilitates the ‘checkpoints’ before lineage commitment in the middle of the developmental process and 3) it allows the stable storage of differentiation intermediates which can be differentiated into mature T cells rapidly when there is an urgent need of new T cells with a diverse T cell receptor repertoire.

Quantitative analysis of the energy landscapes and minimum action paths delineates the patterns of the multiple-attractor system in T cell differentiation

With the deterministic modeling and bifurcation approaches, we described the local stability for multi-stable T cell model. However, the global stability is less clear from the bifurcation analysis alone. In addition, it is important to consider the stochastic dynamics for T cell development model because the intracellular noise may play crucial roles in cellular behaviors [48,49]. The Waddington landscape has been proposed as a metaphor to explain the development and differentiation of cells [50]. Recently, the Waddington epigenetic landscape for the biological networks has been quantified and employed to investigate the stochastic dynamics of stem cell development and cancer [51–57].

Following a self-consistent approximation approach (see Methods), we calculated the steady state probability distribution and then obtained the energy landscape for the model of the early T cell development (full model in Fig 4A). For visualization, we selected two TFs (PU.1 and TCF-1) as the coordinates and projected the 4-dimensional landscape into a two-dimensional space by integrating the other 2 TF variables (Fig 9A). Here TCF-1 is a representative T cell lineage TF, and PU.1 is a TF for alternative cell fates. We also displayed the landscape in a four-dimensional figure, where the three axes represent three TFs (TCF-1, BCL11B and PU.1), and the color represents the energy U (Fig 9B). Note that our major conclusions do not depend on the specific choice of the coordinate (see S7 and S8 Figs for landscapes with PU.1/BCL11B and PU.1/GATA3 as the coordinates).

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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.

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In the case without Notch signal (N = 0), four stable cell states emerge on the landscape for the T cell developmental system (Fig 9). On the landscape surface, the blue region represents lower potential or higher probability, and the yellow region represents higher potential or lower probability. The four basins of attraction on the landscape represent four different cell states characterized by different TF expression patterns in the 4-dimensional state space (Fig 9A and 9B provide two types of projections of the whole 4-dimensional landscape). These states separately correspond to ETP/DN1 (high PU.1/low TCF-1/low BCL11B/low GATA3 expression), DN3 state (low PU.1/high TCF-1/high BCL11B/high GATA3 expression), and two intermediate states (DN2a and DN2b, intermediate expression for the four TFs). The existence of four stable attractors is consistent with experiments [16–19]. As the Notch signal (N) increases, the landscape changes from a quadristable (four stable states coexist), to a tristable (DN2a, DN2b and DN3), to a bistable (DN2b and DN3) and finally to a monostable DN3 state (S9 Fig). These results provide a straightforward explanation for the irreversibility observed in experiments for the stepwise T cell lineage commitment.

To check whether our modelling results match experimental data quantitatively, we acquired two sets of gene expression data of the four core TFs for T cell development from previous publications [17,47], and mapped the normalized values (see Methods) to the landscape (Fig 9A and 9B, S7 and S8 Figs). Here, the golden balls represent the four steady states (characterizing four stages of T cell development) from the models, the red balls represent the gene expression data points (Data1) from Zhang et al. [47], and the green balls represent the gene expression data points (Data2) from Mingueneau et al. [17]. We found that these gene expression data agree well with our landscape models in the sense that each data point is almost positioned in the corresponding basin (Fig 9A and 9B, S7 and S8 Figs). We found that the landscapes give a better fit to Data2 (green points), since each green data point can be well positioned in one of the four basins, corresponding to four stages of T cell development. In fact, the two sets of data points are not very close to each other or to the steady states (golden points) from the models. This is reasonable because these two sets of experimental data are measured separately, probably in different conditions, and these data usually delineate the average of multiple measurements from different samples. Also, the gene expression fluctuations are common in biological systems. Therefore, our landscape pictures provide a natural way to reconcile the two different experimental data, i.e. the gene expression data do not have to be at the positions of steady states. Instead, the gene expression data for each individual stage could be somewhere around that basin because of the fluctuations. This reflects the original spirit of the classic Waddington developmental landscape.

To examine the transitions among individual cell types, we calculated kinetic transition paths by minimizing the transition actions between attractors [58,59], obtaining minimum action paths (MAPs). The MAPs for different transitions are indicated on the landscape (Fig 9). The white MAPs from the ETP state to the DN3 state, correspond to the T cell developmental process while the magenta MAPs from the DN3 state to the ETP state, correspond to reprogramming process. The lines represent the MAPs, and the arrows denote the directions of the transitions. The MAP for T cell developmental process and the MAP for the backward process are irreversible, since the forward and reverse kinetic paths are not identical. This irreversibility of kinetic transition paths is caused by the non-gradient force, i.e. the curl flux [60,61]. Here, the solid white lines represent three stepwise transitions from ETP to DN2a, DNa2 to DN2b, and DN2b to DN3, whereas the dashed white line represents the direct transition path from ETP to DN3. From the MAPs for T cell development, we found that the direct transition path is very similar to the stepwise transition path (the white solid line is similar to the white dashed line, Fig 9, S7 and S8 Figs), which indicates that the T cell developmental process needs to go through the two intermediate states (DN2a and DN2b). This confirms the critical roles of the intermediate states for the T cell differentiation. It is worth noting that the MAPs here quantify the most probable transition paths, which suggest the optimal path (with least transition action) for cells to switch from one state to another. However, in a realistic gene regulatory system, usually a signal is needed to induce rapid cell state transitions (e.g. the Notch signaling is used here to induce T cell development).

To investigate the dynamical developmental process of T cell for multiple TFs, we visualized the 4-dimensional MAP from the ETP to the DN3 state by discretizing the levels of the four TFs. We found that for T cell development, TCF-1 is upregulated first, followed by the activation of GATA3. This leads to the complete inactivation of the alternative fate TF PU.1 and the activation of BCL11B (Fig 10). Interestingly, this temporal order is in good agreement with experimental observations [62] (Also see the gene expression data of four core TFs for four stages in Fig 9, S7 Fig and S8 Fig). These results suggest that the sequence of switching on or off for different TFs can be critical for the lineage commitment of T cell development. Moreover, under the Bcl11b knockout condition (kB = 0), the landscape changes from a quadristable (four stable states coexist), to a bistable (ETP and DN2a) state (S10 Fig), which is consistent with the bifurcation analysis (Fig 7) and experimental observations [28].

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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.

https://doi.org/10.1371/journal.pcbi.1006855.g010

Global sensitivity analysis based on landscape topography reveals the critical factors for T cell development

To identify the critical factors (regulations and TFs) which determine T cell development, we performed a global sensitivity analysis based on the landscape topography. Specifically, we use the transition action between attractors as a measure to quantify the feasibility of a transition between different attractors. A smaller transition action, corresponding to a smaller energy barrier, means a more feasible transition from one attractor to another. In this way, by changing the parameters each at a time we can identify the critical parameters for T cell development (we use the transition from ETP to DN3 as an example). To do this, we constrict the models within the parameter region corresponding to the four-attractor system, so that we can make comparisons for the changes of transition actions as parameters are varied.

We identified some critical parameters of which the variations caused significant changes of transition actions between ETP and DN3 attractor. These parameters include the effective degradation rate of PU.1, (rdP), the regulated production rate of PU.1 (kP), the basal production rate of PU.1 (kP0), the threshold of the self-activation of PU.1 (KPP), and the threshold for the repression of PU.1 on GATA3 (KGP) (Fig 11). In particular, the increase of the self-activation strength of PU.1 (i.e. decreased KPP) reduces the transition action from DN3 to DN2b (Fig 11B), indicating a less stable DN3 state and a more stable ETP state. This is reasonable because the PU.1 is a major TF for alternative cell fates (B-cell, dendritic-cell, and myeloid cell), and silencing of PU.1 is operationally important for T cell commitment [28]. Additionally, the increase of the repression strength of PU.1 on GATA3 (decreased KGP) raises the transition action from ETP to DN2a (Fig 11B), indicating a more stable ETP state and a less stable DN3 state, which is consistent with the observation that GATA3 is a critical TF promoting T cell development. Overall, these results from sensitivity analysis indicate that the PU. 1 synthesis/degradation related parameters, the GATA3 synthesis related parameters, and the regulations between PU.1 and GATA3 are critical to the dynamics and the cell fate decisions of T cell development. This indicates that the regulatory circuit between PU.1 and GATA3 plays critical roles for the cell fate determinations during T cell development.

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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 (S_ETP->DN2a and S_DN3->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, S_ETP->DN2a represents the transition action from attractor ETP to attractor DN2a (cyan bars), and S_DN3->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.

https://doi.org/10.1371/journal.pcbi.1006855.g011

Framework of mathematical modeling

We used ordinary differential equations (ODEs) to describe the dynamics of the concentrations of transcription factors (TFs). We used Hill function to describe the transcriptional regulation by TFs. Each ODE has the following form: (1)

Here, X_i represents the concentration of a transcription factor (TF). is the basal production rate of the TF in the absence of any regulator. is the maximum production rate under the control of the transcriptional activators and inhibitors of this TF. β_i,j denotes the weight of the influence of the TF j on i. The sum of the Hill functions determines the regulation of the production of this TF by other TFs. In each term of the summation, θ = 1 when the regulating TF (X_j) is an activator. θ = 0 when the regulating TF is an inhibitor. K_i,j is the apparent dissociation constant of the regulating TF binding to its regulatory element of the promoter, and it describes the effectiveness of the regulation in terms of the concentration of the TF. n is the total number of regulating TFs. is the effective degradation rate constant. The production rate of the proteins is assumed to be linearly correlated with mRNA production rate. Similar generalized forms of Hill function were previously used for analysis of a variety of gene regulatory networks [52,88]. One time unit of our model corresponds to 6 hours, and all the parameters are dimensionless.

To exclude the possibility that our conclusions are sensitive to the choice of the form of equations, we used an alternative form of ODE to describe the regulatory networks: (2)

In these ODEs, multiplication of Hill functions was used instead of addition. Similar forms of Hill function were also used for modeling a variety of gene regulatory networks [66,89]. With this form, the two types of network motifs that generated the four-attractor behavior are the same as those discovered with the additive form of Hill functions (S3 Fig). In fact, using both forms of equations gave rise to the same number of network topologies (216 topologies with the steady states shown in both S3 Fig and S4 Fig). Therefore, our conclusions are robust in terms of the choice of equation form.

During topology searching, random parameters values were chosen from defined ranges (S1 Table, see below).

Topology searching for four-attractor systems

Network topology searching was first performed for all possible topologies involving up to 3 nodes (TFs) and 6 regulations that are able to generate four-attractor systems with stepwise changes of TF levels. We sampled n of the 9 arrows in Fig 2A, where 1≤n≤6. The procedure for the exhaustive search is the following: 1) Choose n arrows from the 9 arrows in Fig 2A (9 choose n), which gives rise to 465 topology templates. 2) For each topology template containing n regulations, enumerate all 2ⁿ networks (the number 2 represents positive and negative regulations), which gives 12258 networks in total. 3) Remove redundant networks that are different in terms of labels of nodes but are otherwise identical in terms of topology (isometric topologies), which produces 2114 non-redundant topologies. Three-node networks were previously used to explore several types of functional dynamics of network motifs [38,90]. For each topology, we performed random sampling of 10⁶ parameter sets. For each parameter set, we selected 125 initial conditions in the three-dimensional state space ((0, 3.3) for each variable) using Latin Hypercube sampling, and then solved the ODEs numerically. We stopped the simulations at time point 500 and checked if the 125 ODE systems are stabilized at four or more distinct steady states. We next checked if the changes of the TFs are monotonically coupled. We first ordered the steady states by the levels of one TF, and then we looked for scenarios in which all other TFs monotonically increase or decrease with the ordered TF (i.e. the attractors with stepwise changes of the TFs). We excluded the scenarios in which one TF is not monotonically correlated with others in terms of their levels at the four attractors. Models that generated oscillations at the final time point were also excluded. The parameter sets which produced the stepwise changes of steady state were accepted and their associated network topologies were analyzed. Parameter values for the minimum topologies are listed in S2 Table.

Topology classification

To determine whether a network topology contains Type I and/or Type II motifs, we developed a simple algorithm for motif identification. We first enumerated all positive feedback loops (PFLs) in the network. A PFL is defined as any unbranched closed loop structure that has even number of negative regulations. We then followed the scheme described in Table 2 to check if Type I and/or Type II motifs exist in a specific network: we enumerated all combinations of three PFLs in the network. If any three PFLs share one or more TFs, then the network contains a Type I motif. If two of any three PFLs do not share any TF, but they (i.e. at least one TF from PFL 1 and at least one TF from PFL 2) are connected via the third PFL, then the network contains a Type II motif.

Complexity atlas was plotted for the obtained network topologies as described previously by Jiménez et al [40] (Fig 2B and Fig 4C).

Transcriptional network model for early T cell development

We built a model for early T cell development based on the regulations that were previously shown experimentally [91–104]. Information about experimental evidence is described in S3 Table. The form of equations is similar to Eq (2). We chose this multiplicative form of Hill functions because earlier experimental study suggested that regulations of Bcl11b gene are combined via an ‘and’ logic gate [105], which favors the use of multiplication. Although similar detailed information is not available for other TFs, we have shown that our main conclusions with respect to the multistep transitions controlled by a network motif family do not depend on the choice of the form of equations (Fig 2 and S4 Fig). Full list of equations is included in S1 Text. The parameter values were obtained by random searching described above with the four-attractor property as the selection criterion followed by minor manual adjustments. All parameter values are dimensionless. To our knowledge, there is no published experimental measurement that would allow us to directly constrain the ranges of these values except for the degradation rates, which have a unit of the inverse of time. These degradation rates were estimated from a previous study that measured the half-lives (approximately 4 hours) of the transcription factors [41]. The parameter values are listed in S4 Table. To explore the subnetworks of the T cell development model that are essential for the four-stage transition, we performed similar exhaustive search in a set of 1553 non-redundant topologies (2047 subnetworks) to find functional circuit in the model. We obtained 568 topologies (701 topologies) from the search, and we analyzed them with complexity atlas. Isometric topologies were removed in the simulations, but they are included in the complexity atlas so that we do not mix isometric topologies with possibly differential biological meanings specific to certain genes.

In the bifurcation analysis, the value of the parameter N (Notch signal strength) or k_BCL11B (maximum production rate of BCL11B) is varied and the changes of the steady states of the system were analyzed. We let k_BCL11B = 0 to simulate the Bcl11b knockout condition.

To simulate the system under various scenarios of Notch signaling, we first varied the strength and/or duration of the Notch signal and checked the steady state distribution of the system under the varying strengths and durations. We tested 200X200 combinations of strengths and durations of Notch signals and obtained the phenotypes of the cells at the steady state. To simulate the fluctuating Notch signals, we divided the time window of the simulation into small intervals (0.1 unites of time). For each interval, we used a random number with a specified mean and an additive noise. The mean of the Notch signal first increased overtime and then became attenuated.

Enrichment analysis of the four-attractor motifs in the T cell model

To quantify the enrichment of various types of motifs, we used the generic definition of p-value: the p-value for a particular motif is the probability of obtaining at least n number of motifs from a random network population, where n is the observed number of such motif in the T cell network. To compute the p-values, we first counted the frequencies of the positive feedback loop, Type I motif and Type II motif in the T cell model (i.e. n₁, n₂, n₃, n₄ representing the numbers of positive feedback loops, Type I motifs, Type II motifs, and the sum of the Type I and Type II motifs respectively). Random networks were generated using two methods: 1) for each regulation in the existing T cell model, we randomly reassign its source and target TFs (referred to as ‘permuted regulations’), and 2) for each pair of TFs from the network, we randomly assign a regulation (positive, negative or none) (referred to as ‘permuted regulations’). For each of the two methods, we generated 10⁵ networks, and we calculated the empirical p-values by counting the number of the random networks with the numbers of motifs not less than those of respective motifs in the T cell network. The method with permuted regulations is more biologically relevant because the number of the positive and negative regulations are retained in the random networks. We used the second approach as an alternative to exclude the possibility that the conclusion based on the trend of the p-values is due to the low number of networks containing the extreme amount of the motifs.

Optimization for performance comparison of two subnetworks

Due to the difficulty to compare the performances of regulatory circuits with different complexities in general, we selected two specific instances of Type I network motif for comparison. One of them contains only one Type I motif, whereas the other one contains multiple motifs. For each topology, we reduced the system to one ODE with quasi-steady state assumption and defined a continuous production rate function that can produce four attractors as a surrogate function (see S1 Text). Multiple runs of optimization using differential evolution algorithm was used, and 500 converged parameter sets for each circuit were used for comparison. This optimization method was previously used for finding optimum parameter sets and for comparing the performances of regulatory circuits [64,106,107].

Self-consistent mean field approximation for the quantification of energy landscape

The temporal evolution a dynamical system was determined by a probabilistic diffusion equation (Fokker-Planck equation). Given the system state P(X₁,X₂,…,X_N,t), where X₁,X₂,…,X_N, represent the concentrations of molecules or gene expression levels, we have N-dimensional partial differential equation, which are difficult to solve because the system has a very large state space. Following a self-consistent mean field approach [52,61,108], we split the probability into the products of the individual probabilities: and solve the probability self-consistently. In this way, we effectively reduced the dimensionality of the system from M^N to MN (M is the number of possible states that each gene can have), and thus made the computation of the high-dimensional probability distribution tractable.

Based on the diffusion equations, when the diffusion coefficient D is small, the moment equations can be approximated to [109,110]: (3) (4)

Here, σ(t) and A(t) are vectors and tensors. σ(t) denotes the covariance matrix and A(t) is the jacobian matrix of . A^T(t) is the transpose of A(t). The elements of matrix A are specified as: . By solving these equations, we can acquire and σ(t). Here, we consider only the diagonal elements of σ(t) from the mean field approximation. Then, the evolution of the probability distribution for each variable can be acquired from the Gaussian approximation: (5)

The probability distribution acquired above corresponds to one stable steady state or the basin of attraction. If the system has multiple stable steady states, there should be several probability distributions localized at each basin with different variances. Thus, the total probability is the sum of all these probability distributions with different weights. From the self-consistent approximation, we can extend this formulation to the multi-dimensional case by assuming that the total probability is the product of each individual probability for each variable. Finally, with the total probability, we can construct the potential landscape by: U(x) = −lnP_ss(x). Here, P_ss is the steady state probability distribution, and U is dimensionless potential energy. In this work, we define two quantities based on the landscape theory. One is the energy barrier height, which is defined as the energy difference between the local minimum and the corresponding saddle point. Another quantity is the transition action, which is defined as the minimum action from one attractor to the other. These two quantities both measure the difficulty of the transitions. However, the transition actions are suggested to provide a more accurate description for the barrier crossing between attractors or the transition rate [111]. Therefore, we used the transition actions to quantify the difficulty of the transitions between attractors in this work (see the following section for the approach of calculating minimum action paths).

Minimum action paths from optimization

Following the approaches based on the Freidlin-Wentzell theory [58,112,113], for a dynamical system with multistability the most probable transition path from one attractor i at time 0 to attractor j at time T, , t∈[0,T], can be acquired by minimizing the action functional over all possible paths: (6)

Here F(ϕ_ij) is the driving force. This optimal path is called minimized action path (MAP). We calculated MAPs numerically by applying minimum action methods used in [58,112].

Comparison between simulated attractors and experimental data

Gene expression data for the four core TFs were obtained from two previous studies on T cell development (Zhang et al. and Mingueneau et al. [17,47]). We rescaled these data to the range of the attractors obtained from our models by linearly transforming the expression values so that they match the attractors approximately. This rescaling is necessary because some corresponding expression values from these two datasets differ from each other by more than 10-fold. The corresponding ratios between these two datasets vary significantly as well.